 2304 CHM312.HLP, Generated by A3UTIL.EXE, 1994  (c) PCSCC Inc., 1993,01&║Æ!j¿Ä┐╘∞-BZsèÑ┐╫≡	 Wy║Ç■kµ°Æ╖∙@ü«Ω                                                                             ┌─────────────────────────┐                                             │╒════════════════════════╧╕                                            ││    Application File     ├─────────────────────────────────────       ││                         └──────────────────────────────────────      ││                                                                      ││            Interactive Problem Solving in                            ││                                                                      ││                 COMPUTATIONAL CHEMISTRY                              ││                                                                      ││                Using the ASTRO 3 Team                                ││                                                                      ││      Application File  Version 3.12    03-01-1994                     │                                                                                        by  P. DuBois Murphy, Ph.D.                                                                                                    (c) Copyright 1981, 1994  PC SuperComputer Capital, Inc.                                                                                                            All rights reserved.                                                                                                     Read and understand  WARRANTY INFORMATION . Type J twice to read.                                                                            Instructions on using help file are at end of file. Type K to read.                                                                                                                                                   ╒══════════╕                                                            │ CONTENTS └──────────────────────────────────────────────────────────                                                                          0.00                  Table of Contents                                                                                                               *    To quickly jump to Sections, position the Section line number      *    on the │JUMP│ line (line 12) and type J or type Lxxx (enter)       *    where xxx is line number. For example, to jump to line 139,        *    type L139 (ent).  Type S or R to skip back/forth to Sections.                                                                         SECTION                   TITLE                              PAGE│LINE                                                                           0.00 ... Table of Contents ................................... 1   29  1.00 ... Introduction ........................................ 3  102  2.00 ... About the ASTRO 3 Programs. ......................... 4  164  3.00 ... An Introduction to Calculations Using ASTRO 3 ....... 7  277           Programs.                                                      4.00 ... Installing and Using ASTRO 3 Programs. .............. 9  394  5.00 ... MOLE FRACTIONS IN SO2 OXIDATION TO SULFURIC ACID ... 10  443  6.00 ... LANGMUIR ADSORPTION ISOTHERM ....................... 11  464  7.00 ... FLORY-HUGGINS FORMULA, VAPOR PRESSURE OF SOLVENT ... 11  488           ABOVE SOLUTION                                                 8.00 ... NUMERICAL INTEGRATION .............................. 12  513  9.00 ... ROOTS OF QUADRATIC EQUATIONS ....................... 12  533  10.0 ... SERIES EXPANSION OF EXP(-X) ........................ 13  553  11.0 ... RADIOACTIVE DECAY EXPERIMENT, POISSON .............. 13  574           DISTRIBUTION                                                   12.0 ... TEMPERATURE DEPENDENCE OF THE VAPOR OF ............. 13  598           O-TOLUIDINE, NEWTON-RAPHSON METHOD                             13.0 ... PROTON CONCENTRATION (PH) IN AQUEOUS SOLUTION OF ... 14  623           WEAK MONOBASIC ACID                                            14.0 ... HEAT CAPACITY, POWER SERIES APPROXIMATION, ......... 15  646           SIMULTANEOUS EQUATIONS                                         15.0 ... ANALYSIS OF MASS SPECTROMETRY DATA: MIXTURES OF .... 15  673           ALCOHOLS, ITERATIVE METHOD                                     16.0 ... MIXTURE COMPOSITION BY ELEMENTAL ANALYSIS: ......... 16  699           SOLVING LINEAR SIMULTANEOUS EQUATIONS                          17.0 ... ELECTROLYTE CONDUCTIVITY DATA: PREDICTOR ........... 16  723           EQUATION, NONLINEAR LEAST SQUARES FIT                          18.0 ... DETERMINATION OF THERMODYNAMIC DATA FOR ............ 17  748           DECOMPOSITION PROCESS, NONLINEAR FIT                           19.0 ... PREDICTION OF ACTIVITY COEFFICIENTS, NONLINEAR ..... 17  773           REGRESSION                                                     20.0 ... Circumference of an Ellipse: Calculation by ........ 18  796           Series and Integration                                         21.0 ... Ideal Gas Law Calculations. ........................ 19  851  22.0 ... Statistical Analysis: Linear Regression and ........ 20  885           Correlation.                                                   23.0 ... Useful Calculations for Problem Solving in ......... 21  950           Applied Physics.                                               24.0 ... How to Differentiate and Integrate An Equation: .... 23 1044           Commands D and G                                               25.0 ... Backsolving for Value of Independent Variable: ..... 25 1148           Commands S and U                                               26.0 ... Solving (If possible) Linear Simultaneous .......... 28 1274           Equations.                                                     27.0 ... Solving (If possible) Nonlinear Simultaneous ....... 31 1383           Equations.                                                     28.0 ... Finding Best Values of Transistor Parameters ....... 33 1506           from Experimental Data by Nonlinear Least                               Squares.                                                       29.0 ... Automatic Least-squares Polynomial Regression ...... 39 1780           (Linear Least Squares)                                         30.0 ... Glossary of Terms .................................. 43 1934  31.0 ... Product License and Warranty. ...................... 43 1971  32.0 ... Product Information and How to Order Complete ...... 45 2037           Package.                                                       33.0 ... Order/Registration Form ............................ 46 2108  34.0 ... Instructions for Manually Installing Package. ...... 48 2173  35.0 ... Interested in Becoming the Author of Your Own ...... 49 2218           Interactive Application File?                                  36.0 ... Help File: Summary of File Viewing Commands ........ 50 2278                                                                        ╒═══════════════╕                                                       │ INTRODUCTION  └──────────────────────────────────────────────────────                                                                         1.00          Introduction                                              ────────────────────────────────────────────────────────────────────────                                                                                 To change the display colors to high-intensity white on blue            type Q when the highlight bar is lighted on the EQU below.                                                                             'EQU                                                                    'COL HWHITE BLUE                                                        'HELP                                                                                                                                                    This document is an online application file which discusses             the theory and presents solved problems in Chemistry and                related subjects.  Excellent references, which can serve both           as supplements to an existing course in Computational Chemistry         or as a stand-alone study or self-help guides, are the books:                                                                                            "Computational Chemistry"                                                A. C. Norris                                                            John Wiley & Sons Ltd., New York, 1981                                                                                                          "Introduction to PASCAL for Scientists",                                James W. Cooper                                                         John Wiley & Sons, New York, 1981                                                                                                               "Problem Solving in Analytical Chemistry"                               T. P. Hadjiioannou, G. D. Christian,                                    C. E. Efstathiou and D. P. Nikolelis                                    Pergamon Press, New York, 1988                                                                                                                                                                                Personal computers (PC's) and their automation and calculation          capabilities were not available before 1981.  Proper use of the         PC can reduce or eliminate much of the tedium associated with           reducing theory into algebraic expressions and then into useful         day-to-day applications.                                                                                                                        This PC application file is intended to provide students of             Chemistry with help in mastering the execution of the necessary         algebra and math calculations upon which many chemical analyses         are based.  The application is driven by three versatile                spreadsheet/calculators call AstroFormulaCalculator 3,                  AstroEquationSolver 3 and AstroDataFitter 3 which exists as             the files ACAL3.EXE, ASOL3.EXE and AFIT3.EXE, respectively.             These programs are specifically tailored for analyses based on          algebraic expressions, formulas or equations.  One of their             main functions is to eliminate the "programming-type overhead"          in working with known formulas and to allow meaningful learning         to proceed rapidly.                                                                                                                             The ASTRO 3 team can be used with any traditional outline               manual or text to develop equation templates that solve                 specific problems.  These problems can be bundled together              as application files, like this one, and distributed over the           digital superhighway through online data services, BBS's, etc.                                                                                  This file is organized by topics with are summarized in the             Table of Contents.  This entire file can be searched using the          auto-search command "A".                                                                                                               ╒════════════════════╕                                                  │ ABOUT ASTRO 3 TEAM └──────────────────────────────────────────────────                                                                        2.00     About the ASTRO 3 Programs.                                    ────────────────────────────────────────────────────────────────────────                                                                                 This interactive manual uses three independent programs to              accomplish the various kinds of numerical analyses discussed            below.  The programs are AstroFormulaCalculator 3 or ACAL3.EXE,         AstroEquationSolver 3 or ASOL3.EXE and AstroDataFitter 3 or             AFIT3.EXE.  A discussion of the capabilities of these programs          follows:                                                                                                                                  ∙ AstroFormulaCalculator 3 Product Description                                                                                                        AstroFormulaCalculator 3 is a unique and innovative mathematics         tool which offers productivity enhancements not found in any            other software package either retail or shareware.  This                hands-on program combines the functions of both a spreadsheet           and calculator; but it is better.  Why?  Because it is right            between the two:  Easier than a spreadsheet and more powerful           than a calculator.  With its special format-free interface,             formula or equation-based iterations are much easier to set up          than in a spreadsheet.  It is designed to be "smarter" which            makes it very user-friendly for Professionals of all levels,            especially for the infrequent or "computer-shy" user.  Broad            applications exist in many disciplines where calculations with          formulas, equations or other math recipes are required such as          in Economics, Genetics, Chemistry, Electrical Engineering,              Engineering Mechanics, Fluid Mechanics & Hydraulics, Finance,           physics, Statistics and even Nursing.  Features include:                                                                                        ∙ DOS or Windows as non-Windows application.  Runs on all PC            compatibles.  DOS 2.1+, Monochrome display, 200K RAM, floppy or         disk.                                                                                                                                           ∙ DOS command line support for any direct calculation such as           addition, multiplication, subtraction and division.                                                                                             ∙ Format-free entry of up to 63 equations with at least 256             independent variables and 256 constants.  Smart and                     "hassle-free" interpreter.                                                                                                                      ∙ Equations and formulas are written according to the BASIC             conventions.  Versatile library of built-in functions.                                                                                          ∙ Advanced calculator functions including many automated                numerical methods such as:                                                                                                                      ∙ Newton-Raphson for solving x in y=f(x)                                ∙ Advanced simplex for solving x at W=Y=Z in W=f(x), Y=g(x),              Z=h(x)                                                                ∙ Numerical integration (two methods)                                   ∙ Numerical differentiation                                             ∙ Summation                                                             ∙ Finding maxima and minima                                             ∙ Plotting equations                                                    ∙ Saving/retrieving data files with full file management                  support                                                               ∙ Screen dumps as DOS text files for inclusion in documents                                                                                     ∙ Highly-structured documentation, online with Table of                 Contents, Glossary and search function.  Interactive with               program.                                                                                                                                        ∙ online helpfile.  Comprehensive summary of                              commands/functions with step-by-step examples on uses of all            advanced functions.                                                   ∙ online user manual.  A "learn by example" document which                illustrates the universal nature of problem-solving with many           step-by-step examples from Science, Engineering and Finance.                                                                                  ∙ Full support for ACTION21 and ACTION11 data sets.                                                                                             ∙ Comments from users:                                                                                                                   ∙ "... calculate circuit board drilling and routing speeds...easy to       use, versatile and powerful with good documentation."                                                                    J. L.  New York     ∙ "...used the integration capability to determine defect density...do     not have any tool (either PC or host based) that does for me what       ...(AstroFormCalc 3)...does."                                                                                            J. R.  Virginia     ∙ "...for Electrical Engineering practice it is better than a              spreadsheet and a calculator..."                                                                                         C. B.  Florida      ∙ "...used to do frequency spectrum work and calculations on circuits,     ...of real value and a productivity tool."                                                                               E. S.  California   ∙ "...one of a handful of scarcely available packages...in the types       of numeric problems they solve."                                                                                         W. C.  Chile                                                                                 ∙ AstroEquationSolver 3 - Solves both linear and nonlinear                    simultaneous equations with up to 30 unknowns.  Characterized           by extremely easy setup.  Uses active/inactive method of                AstroFormulaCalculator 3 to select variables.  Strictly high            level.  No reference to matrices at all!.  Includes program             ASOL3.EXE and extensive online help file ASOL3HLP.HLP that              includes many learn-and-use examples.                                                                                                     ∙ AstroDataFitter 3 - Fit data to any algebraic expression.                   Nonlinear optimization.  Fully-automated polynomial regression          with automatic "equation generator".  Easy to use, even by a            novice.  Will generate interpolated tables from fitted                  equations.  Up to 30 "active" variables.  Up to 500 data                points.  "Smart" algorithm requires minimal user input to solve         problems.  DOS text plots of "fit".  Point-by-point listing of          fitted data.  Residual variance, correlation coefficient,               coefficient of determination and much more!  Includes program           AFIT3.EXE and extensive online help file AFIT3HLP.HLP that              includes many illustrated examples.                                                                                                    ╒═══════════════════╕                                                   │ HOW TO CALCULATE  └───────────────────────────────────────────────────                                                                        3.00     An Introduction to Calculations Using ASTRO 3 Programs.        ────────────────────────────────────────────────────────────────────────                                                                                 This Section provides an introduction to numbers, number                formats, variables, variable names, mathematical operations,            equations and related topics.  In general, in order to avoid            ambiguities, one must always express numbers, mathematical              expression and equations in an acceptable way when entering             them into a computer program.  These requirements lead to the           formulation of a set of rules which are summarized in the               online Help File for the ASTRO 3 program.  This help file can           be reached from the main equation display by typing the (F2)            key.                                                                                                                                            For example:  It is very common to have to evaluate an                  expression which might be written as:                                                                                                                              456.789 x 0.345                                                                                                              As written, one would interpret this expression as:  Find the           product that results from multiplying the decimal 456.789 by            the decimal 0.345.  Because the letter symbols are most often           associated with variable names, the use of "x" as a                     mathematical operator for multiplication and the                        equally-acceptable use of "x" as a variable name in an equation         such as "y=x" would result in confusion for the                         expression-interpreter of the program.  As a result, the                multiplication operator is commonly the "*" symbol and the              "by-the-rules" way to enter the above expression would be               456.789 * 0.345 (enter) In the language of "computerese",               "(enter)" means to type the enter key of the keyboard and               usually results in some sort of computer action.                                                                                                The following examples illustrate some common ways to write             mathematical expressions and the "by-the-rules" way they are            entered into the ASTRO 3, and many other, computer programs.            These examples exist as equations templates that can be                 "called-up" by scrolling the screen until the highlighted bar           appears over the appropriate 'EQU line.                                                                                                         After the templates are loaded, follow the instructions and             always type (F2) from the main equation display to return to            this application helpfile immediately after the 'EQU reference.                                                                                                                                                'EQU      ≡** If current program ->#NAME<- is not ACAL3 then type Q   'CHA C    ≡** to load AstroFormulaCalculator for the next calculation.                                                                                                                                                  Topic # 1: ENTERING NUMBERS and EXPRESSIONS                                                                                                     Discussion: The following examples show common representations for       expressions and the proper way to enter these into this program. These  expressions can be entered at the ENTER variable below.                                                                                            commonly used     ASTRO 3 way     Result          Reason                                                                                         24.1 x 56.7       24.1*56.7      1366.47        * means multipl.                                                                                     5                                                                       -            5/180          2.77E-2 or    / means division             180                          2.77*10^-2                                                                                                        6 ∙ 3 ∙ 4      6*3*4          72           * not ∙ for multipl.                                                                                 2(-7)5         2*-7*5         -70            implied multipl.                                                        not allowed.                                                                                       25÷5           25/5           5            / is division operat.                                                                                                                                                         2      5                                                             14 -  + 8 -  (14+2/7)+(8+5/9)   22.84           express compound           7      9                                     fractions as sums                                                                                                                                                                                                                                                                                                 'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE COMCWRIT      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 2: ENTERING EXPONENTS and SQUARE ROOTS                                                                                                  Discussion: The following examples show common representations for       expressions and the proper way to enter these into this program. These  expressions can be entered at the ENTER variable below.                                                                                            commonly used     ASTRO 3 way     Result          Reason                    ___                                                                    √144         (space) sqr(144)  12         use sqr(..) function                                                    to take square root.           2.3                                                                   10             10^2.3           199.5       ^ means raise to power               4                                                              5.67 x 10      5.67e4            56700      e means base 10                 4     3                                                             3x10 -2x10     3e4-2e3 or        28000      calculation in                             3*10^4-2*10^3                scientific notation.            2 -1                                                               (2x10 )        (2e2)^-1          0.005    no implied exponentiation                                                                              ___                                                                    √ 6                                                                     ------    (space) sqr(6)/(4*sqr(3))   .3535  must match parentheses     4√3                                                                                                                                                                                                                                                                                                                                                                'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE COMCEXPO      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒════════════════╕                                                      │ USING ASTRO 3  └──────────────────────────────────────────────────────                                                                        4.00       Installing and Using ASTRO 3 Programs.                       ────────────────────────────────────────────────────────────────────────                                                                                 The COMCHM.ZIP package should be unzipped into a directory with         a name such as ASTRO3 or COMCHM.  The following statements              should be added to the AUTOEXEC.BAT file located in the root            directory:                                                                                                                                      PATH=C:\ASTRO3;                                                                                                                                 SET ASTRO3=C:\ASTRO3                                                                                                                            The online helpfiles ACAL3HLP.HLP, ASOL3HLP.HLP and                     AFIT3HLP.HLP are accessed from the equations display using the          F1 key.  These files can be found in other shareware packages:          ACAL3HLP.HLP in SCIFIN.ZIP or PHYCAL.ZIP; ASOL3HLP.HLP in               SOL311.ZIP; and AFIT3HLP.HLP in FIT311.ZIP.  These packages can         be found in the SCIENCE or IBMNET software libraries on CIS and         in the IBM software library on GEnie.                                                                                                           All v 3.12 ASTRO 3 programs can pass equations, variables and           other information to each other by overlaying or chaining.  To          chain, enter the following apostrophe commands at the blinking          cursor '>' prompt of the equation display.                                                                                                      To chain to ACAL3.EXE, type 'CHAIN CALC (ent) or 'CHA C (ent)           To chain to ASOL3.EXE, type 'CHAIN SOLVE (ent) or 'CHA S (ent)          To chain to AFIT3.EXE, type 'CHAIN FIT (ent) or 'CHA F (ent)                                                                                    For interactive examples on integration, differentiation,               Newton-Raphson (backsolving), linear and nonlinear simultaneous         equations and fitting equations to data, scroll this text so            that the JUMP is on the appropriate topic listed below and type         Q.                                                                                                                                              See ∙differentiate or integrate∙.                                     See ∙backsolve∙.                                                      See ∙linear equations∙.                                               See ∙nonlinear equations∙.                                            See ∙fitting∙.                                                        See ∙polynomial fit∙.                                                                                                                         The remainder of the online manual studies many numerical               methods which when grouped together are commonly called                 "Computational Chemistry".                                                                                                             ╒════════════════╕                                                      │ MOLE FRACTIONS └──────────────────────────────────────────────────────                                                                        5.00       MOLE FRACTIONS IN SO2 OXIDATION TO SULFURIC ACID             ────────────────────────────────────────────────────────────────────────                                                                                                                                                'EQU      ≡** If current program ->#NAME<- is not ACAL3 then type Q   'CHA C    ≡** to load AstroFormulaCalculator for the next calculation.                                                                                                                                                  Topic # 3: MOLE FRACTIONS IN SO2 OXIDATION TO SULFURIC ACID                                                                                     Discussion: What is the fractional conversion of SO2 to SO3 when the     total pressure P equals the reciprocal of the equilibrium constant Kp   squared? (Moore, W. J., Physical Chemistry, Longmans, London, 1963)                                                                            'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE COMC141       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒═══════════════════╕                                                   │ LANGMUIR ISOTHERM └───────────────────────────────────────────────────                                                                        6.00     LANGMUIR ADSORPTION ISOTHERM                                   ────────────────────────────────────────────────────────────────────────                                                                                                                                                'EQU      ≡** If current program ->#NAME<- is not AFIT3 then type Q   'CHA F    ≡** to load AstroDataFitter for the next calculation.                                                                                                                                                         Topic # 4: LANGMUIR ADSORPTION ISOTHERM, SOLID ADSORBENT and GASEOUS     ADSORBATE, B AND V∞, NONLINEAR REGRESSION                                                                                                      Discussion: Calculate B and V∞ for the adsorption of CO on charcoal      using the Langmuir model. Data in (P,V) pairs are: (100,3.38), (200,6.  15), (300,8.44), (400,10.39), (500, 12.21), (600, 13.77) and (700,15.   25) (from Atkins, P.W., Physical Chemistry, Oxford Press, Oxford,       1978).                                                                                                                                         'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE COMC144       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒════════════════╕                                                      │ FLORY-HUGGINS  └──────────────────────────────────────────────────────                                                                        7.00 FLORY-HUGGINS FORMULA, VAPOR PRESSURE OF SOLVENT ABOVE              SOLUTION                                                               ────────────────────────────────────────────────────────────────────────                                                                                                                                                'EQU      ≡** If current program ->#NAME<- is not AFIT3 then type Q   'CHA F    ≡** to load AstroDataFitter for the next calculation.                                                                                                                                                         Topic # 5: FLORY-HUGGGINS FORMULA, VAPOR PRESSURE OF SOLVENT ABOVE       SOLUTION, (P0, ß), NONLINEAR REGRESSION                                                                                                        Discussion: Calculate P0 and ß for a solution of rubber in benzene       using the Flory-Huggins model. Data in (Φ,P) pairs are: (0.119,121.5),  (0.312,119.2), (0.524,111.6), (0.727,87.6), (0.945,25.8) and (0.969,15. 1) (from Gee, G. and Orr, W. J. C., Part VIII, Trans. Faraday Soc. 42,  507(1946).).                                                                                                                                   'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE COMC145       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒═════════════════════╕                                                 │ NUMERICAL INTEGRAL  └─────────────────────────────────────────────────                                                                        8.00     NUMERICAL INTEGRATION                                          ────────────────────────────────────────────────────────────────────────                                                                                                                                                'EQU      ≡** If current program ->#NAME<- is not ACAL3 then type Q   'CHA C    ≡** to load AstroFormulaCalculator for the next calculation.                                                                                                                                                  Topic # 6: NUMERICAL INTEGRATION                                                                                                                Discussion: Calculate the integral of f(x) = exp(4*(x-1)/3))*x^(r+3) on  x from x=0 to x=1 at r=1, 2 and 3.                                                                                                             'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE COMC146       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒═════════════════════╕                                                 │ QUADRATIC EQUATION  └─────────────────────────────────────────────────                                                                        9.00       ROOTS OF QUADRATIC EQUATIONS                                 ────────────────────────────────────────────────────────────────────────                                                                                                                                                'EQU      ≡** If current program ->#NAME<- is not ACAL3 then type Q   'CHA C    ≡** to load AstroFormulaCalculator for the next calculation.                                                                                                                                                  Topic # 7: QUADRATIC EQUATIONS                                                                                                                  Discussion: Find the two roots of the equation y=x^2+Ka*x-Ka*Ca (Ka=1    and Ca=3) using plotting and numerical techniques.                                                                                             'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE COMC147A      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒══════════════════╕                                                    │ SERIES EXPANSION └────────────────────────────────────────────────────                                                                        10.0    SERIES EXPANSION OF EXP(-X)                                     ────────────────────────────────────────────────────────────────────────                                                                                                                                                'EQU      ≡** If current program ->#NAME<- is not ACAL3 then type Q   'CHA C    ≡** to load AstroFormulaCalculator for the next calculation.                                                                                                                                                  Topic # 8: SERIES EXPANSION                                                                                                                     Discussion: Calculate the value of exp(-x) at x = 2 using 2, 6 and 11    terms of its power series expansion exp(-x) = 1 - x + X^2/2! - x^3/3!.. ..                                                                                                                                             'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE COMC1414      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒═══════════════════╕                                                   │ RADIOACTIVE DECAY └─────────────────────────────────────────────────                                                                          11.0     RADIOACTIVE DECAY EXPERIMENT, POISSON DISTRIBUTION             ────────────────────────────────────────────────────────────────────────                                                                                                                                                'EQU      ≡** If current program ->#NAME<- is not AFIT3 then type Q   'CHA F    ≡** to load AstroDataFitter for the next calculation.                                                                                                                                                         Topic # 9: RADIOACTIVE DECAY EXPERIMENT, POISSON DISTRIBUTION                                                                                   Discussion: For the radioactive decay of U(238), data in (sampling time, counts) or (x,f) pairs are: (0,2), (1,10), (2,32), (3,68), (4,112), (5, 147), (6,160), (7,145), (8,116), (9,86), (10,53), (11,31), (12,18),     (13,9), (14,4), (15,1), (16,1). Does this data conform to a Poisson     distribution? (from Norris, A. C., Computational Chemistry, Wiley, New  York, 1981).                                                                                                                                   'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE COMC352       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒════════════════╕                                                      │ VAPOR PRESSURE └──────────────────────────────────────────────────────                                                                        12.0     TEMPERATURE DEPENDENCE OF THE VAPOR OF O-TOLUIDINE,                   NEWTON-RAPHSON METHOD                                            ────────────────────────────────────────────────────────────────────────                                                                                                                                                'EQU      ≡** If current program ->#NAME<- is not ACAL3 then type Q   'CHA C    ≡** to load AstroFormulaCalculator for the next calculation.                                                                                                                                                  Topic # 10: TEMPERATURE DEPENDENCE OF THE VAPOR OF O-TOLUIDINE, NEWTON-  RAPHSON METHOD                                                                                                                                 Discussion: Calculate the boiling point (pressure=760 torr) of           O-toluidine if the empirical relation of P vs.T is log10(P)=23.         8296-3480.3/T-5.081*log10(T). P in torr and T in Kelvin. (from          Eberhart, J. G. and Sweet, T. R., The Numerical Solution of Non-linear  Equations in Chemistry, J. Chem. Educ. 37, 422(1960).                                                                                          'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE COMC453       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒═════════════════════╕                                                 │ WEAK MONOBASIC ACID └─────────────────────────────────────────────────                                                                        13.0     PROTON CONCENTRATION (PH) IN AQUEOUS SOLUTION                         OF WEAK MONOBASIC ACID                                           ────────────────────────────────────────────────────────────────────────                                                                                                                                                'EQU      ≡** If current program ->#NAME<- is not ACAL3 then type Q   'CHA C    ≡** to load AstroFormulaCalculator for the next calculation.                                                                                                                                                  Topic # 11: PROTON CONCENTRATION (PH) IN AQUEOUS SOLUTION OF WEAK        MONOBASIC ACID                                                                                                                                 Discussion: Calculate the proton concentrations (pH's) of aqueous        solutions of bromoacetic acids (BrAc), pka=2.901, when the total acid   concentrations are 0.0001, 0.01 and 1.0 mol/dm^3.                                                                                              'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE COMC455       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒════════════════╕                                                      │ HEAT CAPACITY  └──────────────────────────────────────────────────────                                                                        14.0 HEAT CAPACITY, POWER SERIES APPROXIMATION, SIMULTANEOUS             EQUATIONS                                                              ────────────────────────────────────────────────────────────────────────                                                                                                                                                'EQU      ≡** If current program ->#NAME<- is not ASOL3 then type Q   'CHA S    ≡** to load AstroEquationSolver for the next calculation.                                                                                                                                                     Topic # 12: HEAT CAPACITY, POWER SERIES APPROXIMATION, SIMULTANEOUS      EQUATIONS                                                                                                                                      Discussion: Calculate the 4 constants (a, b, c and d) needed to          approximate the heat capacity of methyl chloride gas by a power series  of the form Cp=a+b*t+ c*T^2+d*T^3. Measurements of the heat capacities  (J/K/mole) at 4 different temperatures (in K) yield the following data  in (T,Cp) pairs: (400,48.10), (600,61.25), (800,71.26), (1000,78.90).   (from Warn, J. R. W., Concise Chemical Thermodynamics, van Nostrand     Reinhold, London, 1969)                                                                                                                        'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE COMC571       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒═══════════════════╕                                                   │ MASS SPECTROMETRY └───────────────────────────────────────────────────                                                                        15.0     ANALYSIS OF MASS SPECTROMETRY DATA: MIXTURES OF ALCOHOLS,             ITERATIVE METHOD                                                 ────────────────────────────────────────────────────────────────────────                                                                                                                                                'EQU      ≡** If current program ->#NAME<- is not ASOL3 then type Q   'CHA S    ≡** to load AstroEquationSolver for the next calculation.                                                                                                                                                     Topic # 13: ANALYSIS OF MASS SPECTROMETRY DATA: MIXTURES OF ALCOHOLS,    ITERATIVE METHOD                                                                                                                               Discussion: Determine the mole fractions of the 4 alcohols: methanol,    ethanol, propanol and iso-propanol if the mass spectrum yields 5 peaks  with mass/charge and heights of: (19, 5.10), (29, 29.75), (31, 100.0),  (39,5.08) and (59,6.80). The relative intensities and sensitives are    supplied. (from Dickson, T. R., The Computer and Chemistry, Freeman,    San Francisco, 1968).                                                                                                                          'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE COMC572       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒═════════════════════╕                                                 │ MIXTURE COMPOSITION └─────────────────────────────────────────────────                                                                        16.0     MIXTURE COMPOSITION BY ELEMENTAL ANALYSIS:                            SOLVING LINEAR SIMULTANEOUS EQUATIONS                            ────────────────────────────────────────────────────────────────────────                                                                                                                                                'EQU      ≡** If current program ->#NAME<- is not ASOL3 then type Q   'CHA S    ≡** to load AstroEquationSolver for the next calculation.                                                                                                                                                     Topic # 14: MIXTURE COMPOSITION BY ELEMENTAL ANALYSIS: SOLVING LINEAR    SIMULTANEOUS EQUATIONS                                                                                                                         Discussion: Determine the component compositions in a mixture of         aspartic acid C4H7NO4; histidine, C6H9N3O2; phenylalanine; C9H11NO2     and arginine C6H14N4O2 if the elemental weight percents are C, 47.7%;   H, 6.3%; N, 17.3% and O, 28.7%. (From Norris, A. C., op. cit.)                                                                                 'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE COMC574       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒═══════════════════╕                                                   │ CONDUCTIVITY DATA └───────────────────────────────────────────────────                                                                        17.0    ELECTROLYTE CONDUCTIVITY DATA: PREDICTOR EQUATION,                    NONLINEAR LEAST SQUARES FIT                                       ────────────────────────────────────────────────────────────────────────                                                                                                                                                'EQU      ≡** If current program ->#NAME<- is not AFIT3 then type Q   'CHA F    ≡** to load AstroDataFitter for the next calculation.                                                                                                                                                         Topic # 15: ELECTROLYTE CONDUCTIVITY DATA: PREDICTOR EQUATION,           NONLINEAR LEAST SQUARES FIT                                                                                                                    Discussion: For hydrochloric acid at 298.15K, data in (concentration,    molar conductivity) or (c,Γ) pairs are: (0.0005,422.74), (0.001,421.    36), (0.005,415. 80), (0.01,412.0), (0.02,407.24), (0.05,399.09). What  is the molar conductivity at zero concentration? (from Glasstone, S.,   Textbook of Physical Chemistry, 2nd ed., Macmillan, London, 1948.)                                                                             'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE COMC575       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒═══════════════╕                                                       │ DECOMPOSITION └───────────────────────────────────────────────────────                                                                        18.0 DETERMINATION OF THERMODYNAMIC DATA FOR DECOMPOSITION PROCESS,           NONLINEAR FIT                                                     ────────────────────────────────────────────────────────────────────────                                                                                                                                                'EQU      ≡** If current program ->#NAME<- is not AFIT3 then type Q   'CHA F    ≡** to load AstroDataFitter for the next calculation.                                                                                                                                                         Topic # 16: DETERMINATION OF THERMODYNAMIC DATA FOR DECOMPOSITION        PROCESS, NONLINEAR FIT                                                                                                                         Discussion: Calculate the thermodynamic data for the decomposition       process UIn3 = U(s) + 3In(g) at T=1250K from the temperature (K) and    pressure (atm) data or (T,P): (1126, 0.607e-6), (1156, 1.34e-6), (1230, 7.98e-6), (1256,14.3e-6), (1295,32.7e-6) (from Warn, J. R. W., Concise  Chemical Thermodynamics, Van Nostrand Reinhold, London, 1969)                                                                                  'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE COMC872       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒════════════════════════╕                                              │ ACTIVITY COEFFICIENTS  └──────────────────────────────────────────────                                                                        19.0    PREDICTION OF ACTIVITY COEFFICIENTS, NONLINEAR REGRESSION       ────────────────────────────────────────────────────────────────────────                                                                                                                                                'EQU      ≡** If current program ->#NAME<- is not AFIT3 then type Q   'CHA F    ≡** to load AstroDataFitter for the next calculation.                                                                                                                                                         Topic # 17: PREDICTION OF ACTIVITY COEFFICIENTS, NONLINEAR REGRESSION                                                                           Discussion: Determine the equation that gives the best description of    the activity coefficient data for sodium chloride over the              concentration range from 0.001-0.1 mol/dm3. Data in (concentration,ln   f±) pairs are (1e-3, -0.0357), (5e-3, -0.0753), (1e-2, -0.1027), (5e-2, -0.1978) and (0.10, -0.2468). (from Norris, A. C., op. cit.)                                                                                   'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE COMC874       ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒════════════════════════╕                                              │ SERIES and INTEGRATION └──────────────────────────────────────────────                                                                        20.0       Circumference of an Ellipse: Calculation by Series and                Integration                                                    ────────────────────────────────────────────────────────────────────────                                                                                 The circumference of an ellipse with axes lengths of "a" and            "b" as measured from its origin cannot be expressed in a simple         "closed" form by a formula.  The approximate value is                   calculated as:                                                                                                                                         l ≈ 2 * pi * sqr( (a^2 + b^2)/2 )                                                                                                        Thus if a = 50 and b = 10, the circumference, l ≈ 226.5.  The           circumference can be express "exactly" by use of the elliptic           integral of the second kind E(k,Φ):                                                                                                                    l = 4 * a * E(k,Φ)   k= sqr( abs(a^2-b^2) )/a                                                                                            There are at least two methods to obtain a higher degree of             accuracy:  the first is to use a series expansion for E(k,Φ)            and the second is to numerically integrate E(k,Φ).  The                 following two examples show how AstroFormCalc 3 can be used to          sum up a series or to accurately integrate an equation.  The            methods are very general and are applicable to many problems in         physics as well as virtually all other disciplines.  It should          be noted that the "approximation" formula, given above, is only         accurate to about 8%.                                                                                                                                                                                          'EQU      ≡** If current program ->#NAME<- is not ACAL3 then type Q   'CHA C    ≡** to load AstroFormulaCalculator for the next calculation.                                                                                                                                                  Topic # 18: SERIES, ARC LENGTH (CIRCUMFERENCE) OF ELLIPSE                                                                                       Discussion: Estimate the circumference of an ellipse with axes lengths   of 50 and 10 cm as measured from its origin. Use the series             approximation.                                                                                                                                 'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE COMCELSE      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 19: INTEGRATION, ARC LENGTH (CIRCUMFERENCE) OF ELLIPSE                                                                                  Discussion: Estimate the circumference of an ellipse with axes lengths   of 50 and 10 cm as measured from its origin. Use the elliptic integral  method.                                                                                                                                        'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE COMCELIN      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒═════════════════╕                                                     │ IDEAL GAS LAW   └─────────────────────────────────────────────────────                                                                        21.0         Ideal Gas Law Calculations.                                ────────────────────────────────────────────────────────────────────────                                                                                 The behavior of gases at high temperatures and low pressures is         approximated very well by the ideal gas law which combines              Boyle's pressure-volume behavior and Charles' temperature               behavior of ideal gases into a single expression.  The                  following templates are written in a general form which allows          calculation of the "missing" variable from a properly defined           gas problem.                                                                                                                           Topic # 20: IDEAL GAS, PRESSURE, VOLUME, TEMPERATURE, MOLES, GRAMS,      MOLECULAR WEIGHT, DENSITY                                                                                                                      Discussion: Find the density in grams per liter of water at STP.                                                                                'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE COMCGAS1      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 21: IDEAL GAS, PRESSURE, VOLUME, TEMPERATURE, MOLES, GRAMS,      MOLECULAR WEIGHT, DENSITY                                                                                                                      Problem # 21.1: 3 grams of an unknown gas exert a pressure of 0.2654     atm at T=273 K in a 5.5 liter gas cell.                                 (a) What is the molecular weight of the gas?                                                                                                   'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE COMCGAS2      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒════════════╕                                                          │ STATISTICS └──────────────────────────────────────────────────────────                                                                        22.0    Statistical Analysis:  Linear Regression and Correlation.       ────────────────────────────────────────────────────────────────────────                                                                                 Linear regression, also called linear least-squares, is a very          common method used to characterize a data set of (X,Y) values           in at least two ways.  For correlation analysis of two data             sets, the values of the first data set are assigned to the X            coordinate and the values of the second data set are assigned           to the Y coordinate.  For perfectly correlated data, the                least-squares slope should be ±1 and the intercept should be            0.0.  The second application involves using the least-squares           line as the best estimate of the trend in data which is assumed         to be linear.  Once the slope and intercept are known, one can          use the line to interpolate the original data or, as is very            common, to project the observed trend to values that are not            measured.  For example, one might measure some property Y in            the range from X= -3 to 3 and use the least-squares line to             project the Y value at X=6 and so on.                                                                                                           The least-squares linear regression for the data discussed in           the next problem is shown below.  This plot was generated with          AstroDataFitter 3.                                                                                                                      .53-                                                                  █    |                                                                       |                                                                       |                                                                 a     |                                                             a     .485-                                                         a             |                                                     a                 |                                                 a                     |                                              a                        |                                          a         █              .434-                                      a                                |                                  a█                                   |                               a                                       |                           a                                           |                       a                                           .383-                 █ a                                                   |               a                                                       |            a                                                          |        a                                                              |    a                                                              .332-█a                                                                     ||--------------|--------------|--------------|--------------|------     1             1.9            2.7            3.6             4.5                                                                                     '█' represent the (X,Y) data and 'a' represents the l-s line            with slope of 0.04678, intercept of 0.285 and correlation               coefficient of 0.9809.                                                                                                                 Topic # 22: LINEAR REGRESSION, SLOPE, INTERCEPT, CORRELATION COEFFICIENT                                                                        Discussion: Calculate the slope, intercept and correlation coefficient   for a gold deposition process for which the time (min) vs. thickness    (mm) data are: (1, .3351), (2, .3837), (3, .4244), (4, .4493) and (5, . 5362).                                                                                                                                         'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE COMCLINR      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒═══════════════╕                                                       │ USEFUL CALCs  └───────────────────────────────────────────────────────                                                                        23.0 Useful Calculations for Problem Solving in Applied Physics.        ────────────────────────────────────────────────────────────────────────                                                                                 ∙ARITHMETIC and GEOMETRIC PROGRESSIONS∙                                                                                                         A progression is a sequence of numbers, called TERMS, in which          any term can be obtained from its predecessor by adding,                subtracting, multiplying by or dividing by a fixed number.              When terms are generated by adding/subtraction a fixed number           called the COMMON DIFFERENCE, the progression is called                 ARITHMETIC.  When terms are generated by multiplying/dividing           by a fixed number called the RATIO, the progression is called           GEOMETRIC.  Many financial calculations are based on the                concept of the progression.                                                                                                                     The following example illustrate an arithmetic progression with         a common difference of 4 between terms.                                                                                                ----------------------------------------------------------------------- Arithmetic progression: 4, 8, 12, 16, ...   (common difference = +4)                                                                            com. diff.       +4 =   +4 =   +4 =   +4 =    +4 =    +4 =    +4 =                                                                              term           4      8     12     16     20      24      28      32 ...                                                                        index          1st    2nd   3rd    4th    5th     6th     7th     8th   -----------------------------------------------------------------------                                                                                  To generate a progression, one need only specify any two terms,         their values and positions (or indices).  (Note, they don't             even have to be adjacent!).                                                                                                            Topic # 23: ARITHMETIC PROGRESSION                                                                                                              Discussion: A sequence of numbers in which any number, or term, after    the first can be obtained by adding a fixed number COMMDIFF to its      predecessor. Example 12, 22, 32, 42... or 27, 25, 23, 21... To solve    for progression enter any two term positions INDEX#A and INDEX#B and    their values TERM#A and TERM#B. TERM#N is the value for any index N     and SUM is the sum of the progression up to and including term N.                                                                              'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE COMCARPR      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                          The following examples illustrate a geometric progression with          a common ratio of 2, respectively, between terms.  Note:  '*"           means to multiply and '/' means to divide.                                                                                             ----------------------------------------------------------------------- Geometric progression: 4, 8, 16, 32, ...   (common ratio = +2)                                                                                  com. ratio       *2 =   *2 =   *2 =   *2 =    *2 =    *2 =    *2 =                                                                              term           4      8     16     32     64     128     256     512 ...                                                                        index          1st    2nd   3rd    4th    5th     6th     7th     8th   -----------------------------------------------------------------------                                                                                  To generate a progression, one need only specify any two terms,         their values and positions (or indices).  (Note, they don't             even have to be adjacent!).                                                                                                            Topic # 24: GEOMETRIC PROGRESSION                                                                                                               Discussion: A sequence of numbers in which any number, or term, after    the first can be obtained by multiplying its predecessor by a fixed     number COMMRATIO. Example 12, -24, 48, -96... or 27, 9, 3, 1... To      solve for progression enter any two term positions INDEX#A and INDEX#B  and their values TERM#A and TERM#B. TERM#N is the value for any index   N and SUM is the sum of the progression up to and including term N.                                                                            'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE COMCGMPR      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 Topic # 25: EXACT TIME, APPROXIMATE TIME                                                                                                        Discussion: Exact time is the exact number of days while approximate     time assumes 30 days per month. ADAY1, AAMONTH1 and AYEAR1 is first     date and BDAY2, BBMONTH2 and BYEAR2 is the second or last date. Enter   month as a number: Jan=1, Feb=2, March=3, April=4, May=5, June=6,       July=7, Aug=8, Sept=9, Oct=10, Nov=11 and Dec=12. EDAYS is exact days   between two dates (but not including last day, add 1 to including last  day). ADAYS is approximate days. For example, April 20, 1993 is         entered as AAMONTH=4, ADAY1=20, AYEAR1=1993.                                                                                                   'EQU               ≡ ** If highlighted, type Q to get equations that ** 'EGE COMCEDAY      ≡ ** solve this problem. Scroll down to highlight.**                                                                                                                                                 ╒═════════════════════════════╕                                         │ DIFFERENTIATION/INTEGRATION └────────────────────────────────────────                                                                         24.0           How to Differentiate and Integrate An Equation:                       Commands D and G                                           ────────────────────────────────────────────────────────────────────────                                                                                                                                                'EQU      ≡** If current program ->#NAME<- is not ACAL3 then type Q   'CHA C    ≡** to load AstroFormulaCalculator for the next calculation.                                                                                                                                                           To ∙differentiate or integrate∙ one or more equations, use the          commands D or G (!G) while the program is displaying equations.         The use of these commands will be illustrated by an Example.            Refer to the Manual for more details.  The following example is         intended to illustrate the general procedures only.  The                ∙first derivative∙ and the integral of the equation                     Y=0.2*T^2*EXP(-0.2*T) will be calculated.  In order to use              these commands, one must first enter an equation into the               program with at least 1 independent variable.                                                                                                   Note:  You will frequently start to see apostrophe commands             like 'EQU, 'VAR and 'COM appearing in the helpfile.  The 'EQU           field provides info for the equations display which is loaded           when the highlight bar lights up on the 'EQU line and the "Type         Q...." message appears at the bottom of the screen.  To learn           more about these commands, read the discussion of apostrophe            commands in this helpfile.                                                                                                                    ║ To automatically load these equations, type Q when the ║              ║ highlight bar appears on the screen.  Scroll down, if  ║              ║ necessary, to light it.                                ║       'EQU                                                                    Y=0.2*T^2*EXP(-0.2*T)                                                   'COM "Type (F2) to return to the helpfile"                                                                                                               If the equation is loaded, then skip to step (4).                                                                                               1) When the Program is displaying the equations screen, type            (F6) or "!/" in order to get the "Equations" prompt.  In                response to the prompt "?  Equation(s)", type:  (end esc)               Y=O.2*T^2*EXP(-0.2*T) (enter)                                                                                                                   2) In response to the prompt for a comment Type (end esc) ""            (enter).  Remember to type (F2) to return to this file.                                                                                         3) If more than one equation were to be entered, inactivate any         equation for which the integral/derivative is not desired by            moving the cursor to that variable and typing I.  The equal             sign will disappear.  The D and G operations are not performed          on inactive equations but these equations can still be used as          intermediate equations in the calculation of the active                 equations.                                                                                                                                      4) Load in the comment and initial value of T.                                                                                                 ║To load the comment and jump to the equations display, type Q ║        ║when the highlight bar is lighted on the next line.           ║'EQU                                                                    'VAR T=4.347                                                            'COM                                                                     To calculate the derivative at T=4.347, move the cursor to variable T   and set T=4.347.  Type D to calculate the derivative (about 0.4121).    Set T=10 and type D.  The derivative is very small and this indicates   that the equation has an extreme at T=10.  Note:  Because of the small  value of the derivative (<1E-7) there is a lot of uncertainty in the    calculated errors which results in SIG FIGS=0.  If necessary, type (F2) to return to helpfile.                                                                                                                                 ║To load the comment and jump to the equations display, type Q ║        ║when the highlight bar is lighted on the next line.           ║'EQU                                                                    'COM                                                                     To calculate the ∙first integral∙ from T=0 to 1000 (For this particular equation, this range is essentially from zero to infinity.), move the   cursor to variable T and type G.  In response to the prompt from the    range of the integral, type (end esc) 0 to 1000 (enter).  The           integration is performed in stages which take longer as the accuracy    improves.  After a few iterations, the integral should approach 50.     Type any key to stop an integration in progress once the desired number of SIG FIGS is reached.  If necessary, type (F2) to return to           helpfile.                                                                                                                                      Summary of steps to differentiate or integrate equation(s):                                                                                              1) Enter the equation(s) and comment.  If there are other               equations which are not part of the calculations, use I to              inactivate excess.  If there are more than one independent              variable, the derivative or integral will be calculated for the         independent variable on which the cursor resides when D or G is         typed.                                                                                                                                          2) To calculate the derivative, set the value of the                    independent variable to where the derivative is to be evaluated         and type D.  SIG FIGS is an estimate of the uncertainty.                                                                                        3) To calculate the integral, move the cursor to the                    independent variable of integration and type G for the Gauss            method or !G for the Bode method.  In response to the prompt,           enter the range of the integration.  SIG FIGS is an estimate of         the uncertainty.  SIG FIGS will increase with time.  Type any           key to halt the integration while it is in progress.                                                                                   ╒═════════════════════╕                                                 │ SOLVING X in Y=f(X) └─────────────────────────────────────────────────                                                                        25.0          Backsolving for Value of Independent Variable:                        Commands S and U                                            ────────────────────────────────────────────────────────────────────────                                                                                                                                                'EQU      ≡** If current program ->#NAME<- is not ACAL3 then type Q   'CHA C    ≡** to load AstroFormulaCalculator for the next calculation.                                                                                                                                                           The program can ∙backsolve∙ an equation for the value of a              single independent variable such that:  1) The specified                equation has the desired value (∙S or !S∙) or 2) All active             equations are set equal to each other ("∙U∙").  The program             employs two methods:  The Newton-Raphson algorithm (command             "!S") and the Accelerated Interpolation algorithm (command              "S").  In general, the Newton-Raphson method is much faster but         the Accelerated Interpolation method is more general.  The use          of these commands will be illustrated by an Example.  Refer to          the Manual for more details.  The following example is intended         to illustrate the general procedures only.  The methods will be         applied to the analyses of three equations, described below,            which are the mathematical representations of a straight line,          half circle and cubic function.  In order to use these                  commands, one must first enter an equation into the program             with at least 1 independent variable.  Note:  Two or more               equations are required for the U command.                                                                                                     ║ To automatically load these equations, type Q when the ║              ║ highlight bar appears on the screen.  Scroll down, if  ║              ║ necessary, to light it.                                ║       'EQU                                                                    Y1=1.735551*X+0.209225  Y2=SQR(ABS(17.54480-X^2))                       Y3=0.4374873*X^3+0.1804274                                              'CALC                                                                   'COM "Type (F2) to return to the helpfile"                                                                                                               If the equations are loaded, then skip to step (3).                                                                                             1) When the Program is displaying the equations screen, type            (F6) or "!/" in order to get the "Equations" prompt.  In                response to the prompt "?  Equation(s)", type:  (end esc)               Y1=1.735551*X+0.209225 Y2=SQR(ABS(17.54480-X^2))                        Y3=0.4374873*X^3+0.1804274 (enter)                                                                                                              Note:  Leave a single space between equations.  To start each           equation on a separate line use (alt 2) as the terminator.              Remember "(enter)" enters the equations into the program!                                                                                       2) In response to the prompt for a comment, type (end esc) ""           (enter).  Remember to type (F2) to return to this file.                                                                                         3) Load in the comment and initial value of X.                                                                                                 ║To load the comment and jump to the equations display, type Q ║        ║when the highlight bar is lighted on the next line.           ║'EQU                                                                    'VAR X=1                                                                'COM                                                                     To find the value of independent variable X which results in Y2=3, move the cursor to variable X and set X=1.  To implement the Newton-Raphson  method, type '!S'.  In response to prompt for the name of the dependent variable, type (end esc) Y2 (enter).  In response to the prompt for the value of Y2, type (end esc) 3 (enter).  After a few iterations the      value of X will converge to X=5.152165.  Type (F2) to return to         helpfile.                                                                                                                                              ║To load the comment and jump to the equations display, type Q ║        ║when the highlight bar is lighted on the next line.           ║'EQU                                                                    'VAR X=1                                                                'COM                                                                     Also, you may wish to try the Accelerated Interpolation method.  Follow the above procedure.  Set X=1 and type S.  If not already these values, enter Y2 for the name of the dependent variable and 3 for the desired   value.  The S command allows a user-specified initial range.  Use the   default '1 to 1' by typing '(enter)' in response to the prompt.  Type   any key to stop the iterations at the desired SIG FIGS.  Note:  This    algorithm has found another solution X=2.923149!  Type (F2) to return   to this file.                                                                                                                                          ║To load the comment and jump to the equations display, type Q ║        ║when the highlight bar is lighted on the next line.           ║'EQU                                                                    'VAR X=1                                                                'COM                                                                     To solve for the simultaneous intersection of the line, circle and      cubic function or Y1=Y2=Y3, use command U which 'unites' all active     equations.  Move the cursor to variable X and set X=1.  Next type U.    In response to the prompt for an approximation range, type (end esc) 0  to 4 (enter).  After SIG FIGS reaches 7, type any key to stop.  The     results are that Y1=Y2=Y3=3.680326 at X=2.  Type (F2) to return to this file.                                                                                                                                                  ║To load the comment and jump to the equations display, type Q ║        ║when the highlight bar is lighted on the next line.           ║'EQU                                                                    'COM                                                                     To plot the three equations from X= 1 to 2.5, type P.  In response to   the range for X, type (end esc) 1 to 2.5 (enter).  Type E to exit the   plots.  Type (F2) to return to this helpfile.                                                                                                           Summary of steps to backsolve an equation or equation(s):                                                                                       1) Enter the equation(s) and comment.  If there are other               equations which are not part of the calculations, use I to              inactivate excess.  For the U command, the program only unites          active equations and there must be at least 2 active equations.                                                                                 2) To calculate an independent variable, move the cursor to             that variable and type !S.  Enter the name of the appropriate           equation if more than one has been entered.  Next, enter the            desired value for this equation.  In the event that a solution          cannot be found, type any key to stop the iterations.  Also,            command S can be used.  A user-defined range for the initial            search may be specified.                                                                                                                        3) To calculate the united solution Y1=Y2=Y3=..., first                 inactivate all equations (dependent variables) which are not to         be united.  Next, move the cursor to the independent variable           to be calculated.  Type U to begin the approximation and enter          the initial range or use the default.  Type any key to stop the         iterations that are in progress.                                                                                                       ╒════════════════════════╕                                              │ LINEAR SIMUL EQUATIONS └──────────────────────────────────────────────                                                                        26.0          Solving (If possible) Linear Simultaneous Equations.      ────────────────────────────────────────────────────────────────────────                                                                                                                                                'EQU      ≡** If current program ->#NAME<- is not ASOL3 then type Q   'CHA S    ≡** to load AstroEquationSolver for the next calculation.                                                                                                                                                              To solve simultaneous ∙linear equations∙ use subcommand "S".            The use of this subcommand will be illustrated by an                    interactive example intended to illustrate the general                  procedures only.  The four coefficients A, B, C and D of a              third-order polynomial that passes through four (x,y) points            will be determined.  The third-order polynomials are not linear         equations with respect to the variable X but they are linear            with respect to the coefficients A, B, C and D.  In order to            use the subcommand "S", one must first enter an equation into           the program with at least 1 independent variable.                                                                                               Note:  You will frequently start to see apostrophe commands             like 'EQU, 'VAR and 'COM appearing in the helpfile.  The 'EQU           field provides info for the equations display which is loaded           when the highlight bar lights up on the 'EQU line and the "Type         Q...." message appears at the bottom of the screen.  To learn           more about these commands, read the discussion of apostrophe            commands in this helpfile.                                                                                                                    ║ To automatically load these equations, type Q when the ║              ║ highlight bar appears on the screen.  Scroll down, if  ║              ║ necessary, to light it.                                ║       'EQU                                                                     Y1=A+B*X1+C*X1^2+D*X1^3                                                 Y2=A+B*X2+C*X2^2+D*X2^3                                                 Y3=A+B*X3+C*X3^2+D*X3^3                                                 Y4=A+B*X4+C*X4^2+D*X4^3                                                'INA X1 X2 X3 X4                                                        'COM "Type (F2) to return to the helpfile"                                                                                                               If the equations are loaded, then skip to step (4).                                                                                             1) When the Program is displaying the equations screen, type            (F6) or "!/" in order to get the "Equations" prompt.  In                response to the prompt "?  Equation(s)", type:  (end esc)               Y1=A+B*X1+C*X1^2+D*X1^3 Y2=A+B*X2+C*X2^2+D*X2^3                         Y3=A+B*X3+C*X3^2+D*X3^3 Y4=A+B*X4+C*X4^2+D*X4^3 (enter)                                                                                         2) In response to the prompt for a comment Type (end esc) ""            (enter).  Remember to type (F2) to return to this file.                                                                                         3) If necessary, inactivate variables X1, X2, X3 and X4.  Move          the cursor to each variable and type "I".  The equal sign will          disappear.  Note there are 4 equations and there must be 4 and          only 4 unknowns:  A, B, C and D.  If more than 4 equations were         to be entered, inactivate any equation which is supplemental to         the other equations and not actually involved in the problem.           To inactivate an equation move the cursor to that variable and          typing I.  The equal sign will disappear.  The S operations are         not performed on inactive equations but these equations can             still be used as intermediate equations in the calculation of           the active equations.                                                                                                                           4) Set X1, X2, X3 and X4 to desired values and load in the              desired solutions for Y1, Y2, Y3 and Y4.                                                                                                      ║ To automatically load these values, type Q when the    ║              ║ highlight bar appears on the screen.  Scroll down, if  ║              ║ necessary, to light it.                                ║       'EQU                                                                    'VAR X1=1 X2=2 X3=3 X4=4                                                'VAR Y1=-2 Y2=-23 Y3=-86 Y4=-215                                        'COM                                                                     The four (x,y) data points are (1,-2), (2,-23), (3,-86), (4,-215).  If  necessary, set X1=1, X2=2, X3=3 and X4=4.  Next, set Y1=-2, Y2=-23,     Y3=-86, Y4=-215.  Always set the dependent variables last.  Type S to   solve as a linear system.  The result should be A=1, B=-2, C=3 and      D=-4.  Type (F2) to return to the helpfile                                                                                                              To plot the equations, load the following comment and follow            the directions.                                                                                                                               ║ To automatically load this comment, type Q when the ║                 ║ highlight bar appears on the screen.  Scroll down, if  ║              ║ necessary, to light it.                                ║       'EQU                                                                    'COM                                                                     To plot the polynomial, move the cursor to X1 and type P.  In response  to the prompt for the X1 range, type (esc) 1 to 4 (enter).  The Program will plot all 4 equations.  Equation 'a' will pass through all four     (x,y) points.  Note Equations 'b', 'c' and 'd' are constants with       respect to variable X1 and they intersect the 'a' plot at (2,-23),      (3,-86) and (4,-215), respectively.  Type E to exit the plot.  Type     (F2) to return to the helpfile                                                                                                                          Summary of steps to solve linear simultaneous equations:                                                                                        1) Enter the equation(s) and message.  If there are more                independent variables than equations, use "I" to inactivate             excess.  There may be more independent variables than equations         but the number of active independent variables must always              equal the number of active equations for a linear solution.                                                                                     2) Set the active dependent variables to the desired values and         type "S" to attempt a linear solution for the active                    independent variables.  If a linear solution is not found, the          Program will prompt with a warning message.                                                                                            ╒═══════════════════════════╕                                           │ NONLINEAR SIMUL EQUATIONS └───────────────────────────────────────────                                                                        27.0       Solving (If possible) Nonlinear Simultaneous Equations.      ────────────────────────────────────────────────────────────────────────                                                                                                                                                'EQU      ≡** If current program ->#NAME<- is not ASOL3 then type Q   'CHA S    ≡** to load AstroEquationSolver for the next calculation.                                                                                                                                                              To solve simultaneous ∙nonlinear equations∙ either the                  subcommand "W" or "U" can be used.  The "W" method is the               general nonlinear optimization procedure in which 1 or more             active independent variables is solved subject to the                   constraints of the values of the active dependent equation.             The "U" method is a special case of the more general "W"                method.  In the "U" method, the equations are "united" or set           equal to each other and the problem is solved for the best              values of the active independent variables.                                                                                                     The use of the "W" and "U" nonlinear methods will be                    illustrated with an examples which is intended to illustrate            the general procedures only.  First, three nonlinear equations          Y1, Y2 and Y3 will be solved for three independent variables A,         B and C.  Second, the values of variables A, B and C will be            determined such that Y1=Y2=Y3.  In order to use the commands,           one must enter at least one equation into the Program with at           least one independent variable.  Any equation will do.  Note:                                                                                 ║ To automatically load these equations, type Q when the ║              ║ highlight bar appears on the screen.  Scroll down, if  ║              ║ necessary, to light it.                                ║       'EQU                                                                    Y1=A^2+B+C                                                              Y2=A-SQR(ABS(B))+C                                                      Y3=A+B-C^3                                                              'VAR #ALL=0                                                             'COM "Type (F2) to return to the helpfile"                                                                                                               If the equations are loaded, then skip to step (4).                                                                                             1) When the Program is displaying the equations screen, type            (F6) or "!/" in order to get the "Equations" prompt.  In                response to the prompt "?  Equation(s)", type:  (end esc)               Y1=A^2+B+C Y2=A-SQR(ABS(B))+C Y3=A+B-C^3 (enter)                                                                                                2) In response to the prompt for a comment Type (end esc) ""            (enter).  Remember to type (F2) to return to this file.                                                                                         3) Note:  there are 3 equations and there are 3 unknowns:  A,           B, C.  In general, the number of unknowns does not have to              equal the number of equations.  If necessary, inactivate any            independent variables as well as any equations which are                supplemental to the other equations and not actually involved           in the problem.  To inactivate an equation move the cursor to           that variable and typing I.  The equal sign will disappear.             The W operations are not performed on inactive equations but            these equations can still be used as intermediate equations in          the calculation of the active equations.                                                                                                        4) Set the variables A, B and C to reasonable initial                   "guesses".  Enter A=1.5, B=3.5 and C=3.5.  Set the dependent            variables Y1, Y2 and Y3 to the desired solution.  Enter Y1=8,           Y2=2 and Y3=-22.                                                                                                                              ║ To automatically load this comment, type Q when the    ║              ║ highlight bar appears on the screen.  Scroll down, if  ║              ║ necessary, to light it.                                ║       'EQU                                                                    'VAR A=1.5 B=3.5 C=3.5 Y1=8 Y2=2 Y3=-22                                 'COM                                                                     If necessary, set the variables A, B and C to reasonable initial        'guesses'.  Enter A=1.5, B=3.5 and C=3.5.  Set the dependent variables  Y1, Y2 and Y3 to the desired solution.  Enter Y1=8, Y2=2 and Y3=-22.    To attempt a nonlinear solution, type W.  In response to the prompt for a tolerance, type (end esc) .1 (enter).  After about 100 iterations,    the program will calculate A as 1 and B as 4 and C as 3.  Type any key  to stop iterations after # 100 or so.  Note the magnitudes of the ERROR and SIG FIGS.  As the solution 'improves', the ERROR will tend toward a minimum and SIG FIGS will grow toward 16.  Type (F2) to return to the   helpfile                                                                                                                                                The W method is the general method of solving for the values of         independent variables.  A variation of this method is the U or          "unite" method.  To solve for Y1=Y2=Y3 follow the same                  procedure as described above.                                                                                                                 ║ To automatically load this comment, type Q when the    ║              ║ highlight bar appears on the screen.  Scroll down, if  ║              ║ necessary, to light it.                                ║       'EQU                                                                    'VAR A=1 B=1 C=1                                                        'COM                                                                     If necessary, set A, B and C to reasonable initial guesses.  Enter A=1, B=1 and C=1.  Now type U to unite the active equations.  Enter a        tolerance of 0.1.  After about 100 iterations or so, Y1=Y2=Y3=.824,     A=.849, B=-2.27E-2 and C=.126. Type (F2) to return to the helpfile                                                                                    Summary of steps to solve nonlinear simultaneous equations:                                                                                       1) Enter the equation(s) and message.  Use "I" to inactivate            equations and/or independent variables that are not involved            with the solution.  The number of active independent variables          does not have to equal the number of active equations!                                                                                          2) Set the active independent variables to reasonable initial           "guesses"                                                                                                                                       3) Set the active dependent variables to the desired values and         type "W" or just designate the active equations and type "U".           "W" attempts to optimize the active independent variables for           the specified values of the active equations at the moment "W"          is typed.  "U" attempts the solution where all active equations         are set equal to each other.  Enter a tolerance (0.1 is                 default).                                                                                                                                       4) After ERROR or SIG FIGS is reached, type any key to stop.            Nonlinear equations may not have 0 or many solutions!  You              should have a good idea of what values you are expecting to             calculate, beforehand, if you attempt to solve nonlinear                equations!                                                                                                                             ╒═══════════════════════════════╕                                       │ EXAMPLE: FIT EQUATION TO DATA └───────────────────────────────────────                                                                        28.0          Finding Best Values of Transistor Parameters                          from Experimental Data by Nonlinear Least Squares.          ────────────────────────────────────────────────────────────────────────                                                                                                                                                'EQU      ≡** If current program ->#NAME<- is not AFIT3 then type Q   'CHA F    ≡** to load AstroDataFitter for the next calculation.                                                                                                                                                                  A very common problem in Engineering and Science is to estimate         the best values of the coefficients of a mathematical model             given a set of experimental data.  In the example discussed             here, the area of application is Electrical Engineering and the         current problem is to extract the best values of parameters             which characterize the behavior of a transistor.  By ∙fitting∙          the theoretical equation, discussed below, to the experimental          data, one attempts to determine the DC current gain, BETA0, the         critical frequency, FC, and the transition frequency, FT.                                                                                       In this example, the values of the DC current gain, BETA0, and          the critical frequency, FC, for a transistor will be estimated          by fitting the theoretical equation of BETA as a function of            frequency F to an experimental data set in which the current            gain BETA is measured at 5 different frequencies.  The                  transition frequency, FT, is derived from the DC current gain           BETA0 and FC and is not directly extracted.                                                                                                     The current gain, BETA, and transition frequency, FT, of a              small signal rf transistor can be approximated as a function of         the measurement frequency, F, by the equations:                                                                                                      BETA=BETA0/SQR(1+F^2/FC^2)                                                                                                                      FT=BETA0*FC                                                                                                                                where:                                                                                                                                          BETA = current gain at frequency F                                                                                                              BETA0 = low frequency current gain, CE mode, zero AC                    potential between collector and emitter                                                                                                         FC = transistor cut-off frequency.  Current gain drops to               0.707 (-3dB) times the low frequency value                                                                                                      FT = transistor unity-gain frequency                                                                                                            The equipment used in these measurements operates to 10 MHz;            therefore, it is usually not possible to measure the unity-gain         frequency, FT, directly.  The data set has been generated by            measuring the current gain, BETA, vs.  frequency, F, from 1 to          10 MHz.  The experimentally measured data points are:                                                                                                F (MHz)         BETA                                                                                                                              1             105                                                       2.5           90                                                        5             80                                                        7.5           60                                                        10            50                                                                                                                         First enter the equations into the program.  Move the text down         until the highlight bar appears on the EQU line below.                                                                                        ║ To automatically load these equations, type Q when the ║              ║ highlight bar appears on the screen.  Scroll down, if  ║              ║ necessary, to light it.                                ║       'EQU                                                                    BETA=BETA0/SQR(1+F^2/FC^2)                                              FT=BETA0*FC                                                             'INA FT                                                                 'COO F                                                                  'COM                                                                     Calculate BETA0 (DC current gain) and FC (critical frequency) for a     transistor by fitting BETA vs.  F (MHz) to theoretical equation.        Type (F2) to return to the helpfile.                                                                                                                    After the equations, a comment and data are entered into AFIT3,         the equation display should appear similar to what is shown             below:                                                                                                                                 ......................................................................  --- Comment -->  Calculate BETA0 (DC current gain) and FC (critical     frequency) for a transistor by fitting BETA vs. F (MHz) to theoretical  equation. Type (F2) to return to the helpfile.                                                                                                               BETA=BETA0/SQR(1+F^2/FC^2)                   BETA =  0                                                                                                  FT=BETA0*FC                            FT    0                                                                                                                                                     11:14:05         *** Accuracy ?  Warning # 8 ***          10-20-1980                                                                                                                                                       BETA0 => 0                    F :  0                   FC =  0       ......................................................................                                                                                   The "Comment", if any, is displayed at the top.  The equations          BETA and FT, in the order in which they are entered, are                displayed from top down.  The time and date are displayed in            the middle along with the "Accuracy ?  Warning #8" Comment.             This frequently-seen warning Comment is used to inform the user         that the displayed values of the dependent variables BETA and           FT may not be related to the values based on the equations and          the independent variables.  To actually calculate the dependent         variables, one would enter a value for an independent variable          or type C.  Finally the 3 independent variables BETA0, F an FC          are listed at the bottom in alphabetical order.  The blinking           cursor ">" is positioned on BETA0.                                                                                                              Data points consist of two numbers:  a data coordinate and a            data value.  In the present problem, data are to be entered in          (F,BETA) pairs.  If necessary, to designate F as the variable           associated with the data coordinate, move the cursor to F and           type O.  The equal sign "=" will be replaced by a colon ":".  A         colon is used to signify that variable F is now a data                  coordinate.  Only one of the two equations, BETA, is to be              associated with the data value.  The other equation must be             made "invisible" or inactive.  If necessary, move the cursor to         FT and type I.  The equal sign after FT will disappear.  At             this time, data points are defined as (F,BETA) pairs.                                                                                           Next data are entered in (F,BETA) pairs, type D.  The screen            will appear as shown below and enter the data.                                                                                                ║ To automatically load these equations, type Q when the ║              ║ highlight bar appears on the screen.  Scroll down, if  ║              ║ necessary, to light it.                                ║       'EQU                                                                    'PT0                                                                    'DATA                                                                     1    105                                                                2.5  90                                                                 5    80                                                                 7.5  60                                                                 10   50                                                               'COM                                                                     If desired, enter the five data points as (F,BETA) pairs:  (1,105),     (2.5,90), (0.5,80), (7.5,60) and (10,50).  To verify the points or      enter them one by one , Type D.  For F(1), type 1 (enter).  For         BETA(1), type 105 (enter).  Now enter F(2) then BETA(2) and so on.      After entering BETA(5) or 50, the Program will advance to point #6.     Type R to remove this unwanted point.  Type E to exit data command and  return to the equation display.  Type (F2) to return to the helpfile.                                                                                   Finally initial "guesses" for the two active independent                variables BETA0 and FC are set and the problem is solved.               The Program will optimize the values of the active independent          variables BETA0 and FC, whose names are followed by the "=", in         order to attempt to minimize the sum of squared errors between          the calculated value of the single active dependent variable            (with the data coordinates of each data point substituted into          the coordinate variable) and the data value of each data point.         In this example, each of the five data coordinates will be              substituted into F, BETA will be calculated and then the                squared differences between BETA and the data value of each of          the five points will be summed and minimized.                                                                                                 ║ To automatically load these equations, type Q when the ║              ║ highlight bar appears on the screen.  Scroll down, if  ║              ║ necessary, to light it.                                ║       'EQU                                                                    'VAR BETA0=1   FC=1                                                     'COM                                                                     If necessary, set the initial values of BETA0=1 and FC=1.  Type S to    start the fit.  Use the default tolerance of 0.1.  Just type (enter).   The program will optimized BETA0 and FC to minimize the ERROR and the   iteration counter will 'roll off' as the fit proceeds.  Note that the   value of the coordinate variable F stays constant.  After about 64      iterations, the fit will converge as the estimated significant figures  approaches 16.  In this case SIG FIGS reaches 14.  Type any key to stop the fit.  Type (F2) to return to the helpfile.                                                                                                          As the fit advances and the number of iterations increases,             the ERROR will tend to minimize but it will not approach 0.0            unless the fit is "perfect".  The screen should appear similar          to what is shown below, although the comment will be different.                                                                        ......................................................................  --- Comment -->  Calculate BETA0 (DC current gain) and FC (critical     frequency) for a transistor by fitting BETA vs. F (MHz) to theoretical  equation.                                                                                                                                                    BETA=BETA0/SQR(1+F^2/FC^2)               BETA =  104.1315                                                                                               FT=BETA0*FC                        FT    568.0856                                                                                                                                                  11:24:15  Iteration # 64   ERROR = 6.3589147  SIG FIGS = 14 10-20-1980                                                                                                                                                     BETA0 =  104.1315             F :  0                 FC =  5.455464  ......................................................................                                                                                   Obtaining a summary of results.  Following optimization, one            may want to list a summary of the results.  Type command R from         the equation display. The screen should appear as below.                                                                               ..................................................................                                                                               #      F    BETA    Calculated     % Error      Relative % Error                                                                                1    1      105      102.425      -2.452405          -----                                                                                      2    2.5    90       94.66504      5.183378              +++++++++                                                                              3    5      80       76.76686     -4.041424        -------                                                                                      4    7.5    60       61.25394      2.089902              ++++                                                                                   5    10     50       49.87006     -.2598841              0                                                                                      #      F    BETA    Calculated     % Error      Relative % Error                                                                               ...................................................................                                                                                      The "results" screen list the data point number (#), the value          of the data coordinate (F), the value of the data value (BETA),         the calculated value of BETA based on the equation and the              optimized values of BETA0 and FC, the % ERROR which is the              actual value of BETA less its calculated value expressed as a           percent of the actual value.  Finally the SYMBOLIC ERROR is             tabulated.  It is a visual representation of the uncertainty in         the fit.  For each point, the actual value of BETA can be               thought of as being in the middle under the C of SYMBOLIC and           would correspond to a 0 if the fit were more-or-less perfect.           "+" mean the fitted curve exceeds the data value at this point          and the number of "+" is a relative measure of how much.  "-"           means the fitted curve underestimates the data value at this            point and the number of "-" is a relative measure of how much.          Note, the number of decimal places for the table entries is             determined by the current setting as determined by the 'DEC             apostrophe command.                                                                                                                             Plotting the fitted equation and experimental data.  The                equation and data can be plotted to verify the "fit" visually.          Type command P from the equation display..  Type Y (enter) in           response to the prompt to include data on plot.  The screen             should appear similar to below:                                                                                                        .......................................................................                                                                         105.-█                                                                      |a a                                                                    |      a                                                                |         a                                                             |             a                                                      91.-            █  a                                                       |                   a                                                   |                     a                                                 |                         a                                             |                           a   █                                    77.-                               a                                       |                                 a                                     |                                     a                                 |                                        a                              |                                           a                        63.-                                               a                       |                                                  █a                   |                                                       a               |                                                           a  a        |                                                                a   49.-                                                                       ||--------------|--------------|--------------|--------------|-----      1             2.96           4.91           6.87           8.83                                    BETA[a]  vs. F                          .......................................................................                                                                                  The first equation BETA is represented by "a".  The 5 data              points as (F,BETA) points are plotted as "█".                                                                                          ╒══════════════════════════╕                                            │ POLYNOMIAL LEAST SQUARES └────────────────────────────────────────────                                                                        29.0          Automatic Least-squares Polynomial Regression                         (Linear Least Squares)                                      ────────────────────────────────────────────────────────────────────────                                                                                                                                                'EQU      ≡** If current program ->#NAME<- is not AFIT3 then type Q   'CHA F    ≡** to load AstroDataFitter for the next calculation.                                                                                                                                                                  The ∙polynomial fit∙ is subcommand "P" of the data entry                subcommand "D".  Data must be entered into the Program first.           The use of the polynomial fit (regression) will be illustrated          with an example.  Polynomials of orders 1 and 3 will be fitted          to four data points.  In order to use the subcommand "P", one           must first enter an equation into the Program with at least 2           independent variables.  Any equation will do.                                                                                                 ║ To automatically load these equations, type Q when the ║              ║ highlight bar appears on the screen.  Scroll down, if  ║              ║ necessary, to light it.                                ║       'EQU                                                                    Y=A*X                                                                   'COO X                                                                  'COM "Type (F2) to return to the helpfile"                                                                                                               If the equation is loaded, then skip to step (4).                                                                                               1) When the Program is displaying the equations screen, type            (F6) or "!/" in order to get the "Equations" prompt.  In                response to the prompt "?  Equation(s)", type:  (end esc)               Y=A*X (enter)                                                                                                                                   2) In response to the prompt for a comment Type (end esc) ""            (enter).  Remember to type (F2) to return to this file.                                                                                         3) Next move the blinking cursor to variable X by typing "L".           Designate X as a coordinate by typing "O".  A colon ":" will            appear after its name.                                                                                                                          4) Enter the 4 (X,Y) data points.                                                                                                              ║To load the comment and jump to the equations display, type Q ║        ║when the highlight bar is lighted on the next line.           ║'EQU                                                                    'PT0                                                                    'DATA 1,-3 2,-57 3,-219 4,-549                                          'COM                                                                     If necessary, enter the four data points as (X,Y) pairs:  (1,-3),       (2,-57), (3,-219) and (4,-549).  Type D.  For X(1), type 1 (enter).     For Y(1), type -3 (enter).  Now enter X(2) then Y(2) and so on.  After  entering Y(4) as -549, the program will advance to point #5.  Type R to remove this unwanted point.  Type E to exit data command and return to  the equation display.  Type (F2) to return to the helpfile.                                                                                                                                                                    ║To load the comment and jump to the equations display, type Q ║        ║when the highlight bar is lighted on the next line.           ║'EQU                                                                    'COM                                                                     The polynomial regression is part of the data entry command.  Type      command D first.  To perform a first order fit, type P.  In response to the prompt for an order, type (end esc) 1 (enter).  A positive order    selects a polynomial of the general form Y=A0+A1*X^1+...AN*X^N and a    negative order, a polynomial of the general form                        Y=A0+A1*X^-1+...AN*X^-N.  Type N (enter) in response to the prompt to   'REPLACE EQUATIONS'.  Type E to exit data command and return to the     equation display.  Type (F2) to return to the helpfile.                                                                                                 The Program will calculate A0 as 243 and A1 as -180.  The               Program will display the polynomial equation and a message              which summarizes the first-order fit.  Note the magnitudes of           the residual variance and the coefficient of determination.  As         the order increases, the fit will improve and the residual              variance will decrease toward 0.0 and the coefficient of                determination will increase to 1.0.                                                                                                            ║To load the comment and jump to the equations display, type Q ║        ║when the highlight bar is lighted on the next line.           ║'EQU                                                                    'COM                                                                     The polynomial regression is part of the data entry command.  Type      command D first.  To perform a third order fit, type P.  In response to the prompt for an order, type (end esc) 3 (enter).  The Program will    now calculate A0 as 3, A1 as -2, A2 as 6 and A3 as -10.  Notice that    the fit is 'perfect'.  The residual variance is 0.0 and the coefficient of determination is 1.0.  Type Y (enter) in response to the prompt to   'REPLACE EQUATIONS'.  Type E to exit data command and return to the     equation display.  Type (F2) to return to the helpfile.                                                                                                 The Program will replace the main equation(s) with the                  polynomial Y=A0+A1*X^1+A2*X^2.  If the comment is empty, it             will replace it with fit information.  The coefficients A0 to           A2 are set equal to their least-squares values and variable X           is designated as the coordinate.  The immediate calc is                 replaced by the polynomial with explicit number coefficients.           Should this form of the equation be needed it can be viewed             with the immediate calc command "|".                                                                                                            An error during the calculation will terminate the fit and              return the Program to the equations display.  Warning # 451             will be generated.  The most likely cause of errors are                 overflows from either coordinate values with large magnitudes           or higher order terms.                                                                                                                 Some important points are:                                                                                                                               1) There must be 2 or more data points.  If only two points,            then the order is 1 by default and need not be specified.                                                                                       2) The maximum order is one less than the number of points or           15, whichever is smaller.  Thus if there are 10 data points,            then a polynomial of order from 1 to 9 can be chosen.                                                                                           3) Polynomial coefficients are completely determined if the             order of the polynomial is one less than the number of points.          Thus if there are 4 points, then a third order fit will be              "perfect".  As the number of points becomes much large than the         order, the polynomial represents a "smoothed" approximation.                                                                                    4) The Program calculates the residual variance RESID VAR and           the coefficient of determination COEFF OF DETERM for each fit.                                                                                  5) The polynomial equation can be used to replace the current           equation(s).  This is useful for further analysis including the         tabulation of results (command "R" in equation display), the            plotting of the equation and data (command "P" in equation              display) and the saving of the equation and data (subcommand            "]" in equation display).                                                                                                                       6) The immediate calculation (command "|") is always                    replaced by the polynomial with number coefficients and without         a dependent variable.  The polynomial can be added to the main          equations by exiting to the equation display, typing "/",               moving the cursor to the end of the equations and typing "(alt)         1" .  Remember to insert a dependent variable with a unique             name, the equal sign and at least one space!                                                                                                    7) An interesting relationship exists between poly                      A0+A1*X^1+...AN*X^N and poly A0+A1*X^-1+...AN*X-N.  These two           polynomials when generated from the same data set will always           intersect at (N+1) points.                                                                                                                      8) Data points can be defined with more than one coordinate.            For example (X:W,Y) where X and W are data coordinates and Y is         the data value.  Polynomial fits of Y vs.  X and Y vs.  W are           possible.  Position the cursor on the desired coordinate before         typing "P".  Note:  Zero-valued terms like "+W-W" will be added         to the equations to represent extra coordinates and variables           such as "W" will be designated as coordinates if the polynomial         equation replaces the current equation(s).                                                                                             ╒═══════════╕                                                           │ GLOSSARY  └───────────────────────────────────────────────────────────                                                                        30.0     Glossary of Terms                                              ────────────────────────────────────────────────────────────────────────                                                                               ║ Note, Use the Jump command to locate a glossary reference.    ║       ║ Scroll the screen until the reference is in the middle at line║       ║ 12.  The Jump box will light up.  Type J to locate.  Repeat   ║       ║ typing A, as needed, to continue the search.                  ║                                                                               See ACTIVITY COEFFICIENTS                                             See ∙ARITHMETIC and GEOMETRIC PROGRESSIONS∙                           See CONDUCTIVITY DATA                                                 See DECOMPOSITION                                                     See ∙differentiate or integrate∙                                      See FIT EQUATION TO DATA                                              See FLORY-HUGGINS FORMULA                                             See HEAT CAPACITY                                                     See IDEAL GAS LAW                                                     See LANGMUIR ADSORPTION ISOTHERM                                      See LINEAR SIMUL EQUATIONS                                            See MASS SPECTROMETRY                                                 See MIXTURE COMPOSITION                                               See MOLE FRACTIONS                                                    See NONLINEAR SIMUL EQUATIONS                                         See NUMERICAL INTEGRATION                                             See POISSON DISTRIBUTION                                              See POLYNOMIAL LEAST SQUARES                                          See ROOTS OF QUADRATIC EQUATIONS                                      See SERIES and INTEGRATION                                            See SERIES EXPANSION                                                  See SOLVING X in Y=f(X)                                               See STATISTICS                                                        See VAPOR PRESSURE                                                    See WEAK MONOBASIC ACID                                                                                                                ╒═════════╕                                                             │ LICENSE └─────────────────────────────────────────────────────────────                                                                        31.0             Product License and Warranty.                          ────────────────────────────────────────────────────────────────────────                                                                                       Software License Agreement  (Registered Users)                                                                                            If you are a registered user, PCSCC grants you the right to use         this Software for a single-user computer or a workstation not           attached to a network.  You may not rent or lease this Software         without the expressed written permission of PCSCC.  You may not         decompile, disassemble, reverse engineer, copy, create a                derivative work, or otherwise use this Software except as               stated in this agreement.  The license authorizes you to create         and use an extra copy of the Software on a home or laptop               computer, as long as the extra copy is never loaded at the same         time this Software is loaded on the primary computer on which           you use this Software.                                                                                                                          If any provision of this agreement is found to be unlawful,             void, or unenforceable, then that provision shall be severed            from this Agreement and will not affect the validity and                enforceability of any of the remaining provisions.  This                agreement shall be governed by the laws of the State of                 Delaware.                                                                                                                                       The Software and accompanying materials are provided with               Restricted Rights.  Use, duplication, or disclosure by the              Government is subject to restrictions as set forth in                   subparagraph (c) (1) (ii) of The Rights in Technical Data and           Computer Software clause at DFARS 252.227-7013, or                      subparagraphs (c) (1) and (2) of the Commercial Computer                Software--Restricted Rights at 48 CFR 52.227-19, as applicable.         Contractor/manufacturer is PC SuperComputer Capital,                    Incorporated, at the address listed below.                                                                                                      Non registered users must comply with the Shareware policy              described elsewhere in this software Package.                                                                                                   Type T to return to the Table of Contents.                                                                                                                       Warranty Information                                                                                                           PC SuperComputer Capital, Inc.  (PCSCC) hereby disclaims all            warranties relating to this Software, whether express or                implied, including without limitation any implied warranties of         merchantability or fitness for a particular purpose.  PCSCC             will not be liable for any special, incidental, consequential,          indirect or similar damages due to loss of data, loss of                profits, lost savings or any other reason, even if PCSCC or an          agent of PCSCC has been advised of the possibility of such              damages.  In no event shall PCSCC's liability for any damages           ever exceed the price paid for the license to use Software,             regardless of the form of the claim.  The person using the              Software bears all risk as to the quality and performance of            the Software.  This Software is licensed AS IS.                                                                                                 Any action or proceeding brought by either party against the            other arising out of or related to this agreement shall be              brought in a STATE or FEDERAL COURT of competent jurisdiction           located in Fairfield County, Connecticut.  The parties hereby           consent to in personam jurisdiction of said courts.                                                                                             Type T to return to the Table of Contents.                                                                                             ╒═══════════════╕                                                       │ PRODUCT  INFO └───────────────────────────────────────────────────────                                                                        32.0     Product Information and How to Order Complete Package.         ────────────────────────────────────────────────────────────────────────                                                                                 Register for AstroFormulaCalculator 3 and receive, in addition          to the most recent version of ACAL3.EXE and ACAL3HLP.HLP, these         3 powerful math analysis programs:  AstroEquationSolver 3,              AstroDataFitter 3 and AstroShadowBox 3.  Each has its own               extensive and interactive online helpfile written just like             the one your reading now.  A print program is included which            prints theses helpfiles to the standard printer or to a DOS             text file.  All ASTRO 3 programs are written with the same              high-level user-friendly front-end requiring minimal effort to          setup and execute relatively elaborate calculations.  All data          sets and helpfiles are compatible among members of the family.          ACAL3.EXE, ASOL3.EXE and AFIT3.EXE can be overlayed or chained          so that all functions can be easily accessed for any problem.           Registration also includes AstroShadowBox 3 and all online              interactive helpfiles.                                                                                                                    ∙ AstroShadowBox 3 - The pop-up resident member (TSR) of the                  ASTRO 3 team.  Provides math support for other DOS programs.            Pop-up via hot key, surround calculation in "shadow box", and           type "(enter)" to pass result down through keyboard buffer or           pop up, enter calculation and pass result down.  Great for              adding powerful and interactive math capabilities to DOS text           editors such as Brief (tm, Borland), PE2 (tm, IBM) or                   WordPerfect (tm, WordPerfect).  Extensive features also include         screen dumps with appends as DOS text files or as parsed                columns in 123 (tm, Lotus) .WK1 files.  Requires as little as           15K of main memory.  Includes program ABOX3.EXE and online              help file ABOX3HLP.HLP                                                                                                                          If you like Engineering, Science or Finance and you have ideas          that you could transform into an interactive helpfile like              SCIFIN.HLP (The ASTRO 3 Science Manual) or PHYCAL.HLP ( The             ASTRO 3 physics Manual) or FINCAL.HLP (The ASTRO 3 Finance              Manual), then you should write one and market it!  If you write         it in a commercial spreadsheet format, you will not likely be           able to distribute a fully functional package which is the mark         of a true Shareware program and the policy of the Association           of Shareware Professionals!  However, the AstroUtility 3                program provides the tools needed to generate an ASTRO 3                interactive application file, like this one, with a minimum of          effort AND REGISTERED USERS OF THE AstroUtility 3 PROGRAM       RECEIVE A LIMITED DISTRIBUTION LICENSE WHICH ALLOWS INCLUSION OF                ANY OR ALL OF THE ASTRO 3 PROGRAMS IN YOUR OWN MANUAL/PACKAGE!                                                                            ∙  AstroUtility 3  -   Support for authors to construct their own             online application help files.  Develop equation files in your          favorite editor and translate them into .EQU format for final           "test and touch-up".  Likewise, develop an application file in          your favorite editor, automatically include information from            previously developed .EQU files, and then compile a                     full-feature .HLP with automatic Table of Contents, line                numbers and so on.  .HLP files support all apostrophe commands          and many other useful features such as CalcMerge with lets you          dynamically stuff values calculated in the main equation                display into the application file.  As with the other ASTRO 3           programs, AstroUtility 3 comes with its own interactive and             extensive online help file which can be printed, if needed.             Application files can contain up to 32,000 lines!                                                                                               These productivity-enhancing tools has been developed by the            team at PC SuperComputer Capital, Inc.  Order the complete              AstroFormulaCalculator 3 package today!                                                                                                         See below for instructions on ∙Registration∙.                                                                                          ╒══════════════╕                                                        │ REGISTRATION └───────────────────────────────────────────────────────                                                                         33.0                 Order/Registration Form                            ────────────────────────────────────────────────────────────────────────                                                                                 If a printer is available, position the order form (part 1) on          the screen and type the "(print screen)" key to make a copy.            Then, advance the text and position the order form (part 2) on          the screen and type "(print screen)" again to finish.                                                                                                  ASTRO 3 Products Order/Registration Form  (part 1)                                                                                  Name:________________________________  Title:________________                                                                                   Company:_____________________________________________________                                                                                   Address:____________________________ City:___________________                                                                                   State:___________   Zip:_____________  Country:______________                                                                                   Date:_____/_____/_____           (Fill out form completely)                                                                                     ---------------------------------------------------------------         Product                     #     Price/ea     S&H       Total          ---------------------------------------------------------------         AstroFormulaCalculator 3   ___    _$29.95_    _$5.00_   _______         Complete Package includes                                               ACAL3, ASOL3, AFIT3 + ABOX3                                                                                                                     AstroUtility 3 + ASTRO 3    ___    _$99.95_    _$5.00_   _______        + distribution license                                              ------------------------------------------------------------------------------------------------------------------------------------------------                ASTRO 3 Products Order/Registration Form  (part 2)                                                                                  * Canadian Orders, Add $2.00 per copy                   _______                                                                                 * International Orders, Add $6.00 per copy              _______                                                                                 * CT Residents add appropriate sales tax                _______                                                                                                                  TOTAL ENCLOSED      __________                                                                                 Disk Size: [ ] 5-1/4"   [ ] 3-1/2"   [ ] Check  [ ] Visa  [ ] MC                                                                                Card# __________________________________     Exp. Date ________                                                                                 Signature ______________________________                                                                                                        How did you hear about our Products?    [ ] BBS   [ ] Friend            [ ] Other _____________________________________________________                                                                                               Mail Completed form with Payment to:                                                                                                                  PCSCC, Inc.   Dept. COMC                                                   P. O. Box 173                                                        Botsford,  CT   06404  USA                                                                                                  Support ONLY,  1-203-270-9079     E-mail to CIS, 72263,1657                                                                                          [Mar 1994]  Prices subject to change without notice.                                                                                                  Type T to return to the Table of Contents.                                                                                       ╒═════════════════════╕                                                 │ MANUAL INSTALLATION └─────────────────────────────────────────────────                                                                        34.0           Instructions for Manually Installing Package.            ────────────────────────────────────────────────────────────────────────                                                                             The following are minimal instructions for manually installing                                 ASTRO 3 programs.                                                                                                   ∙ Minimum distribution files:  README.1ST, ACAL3.ICO, ACAL3.EXE,          ASOL3.ICO, ASOL3.EXE, AFIT3.ICO, AFIT3.EXE, COMCHM.HLP and *.EQU.                                                                             ∙ These files will probably be bundled in a compressed file such as       COMCHM.ZIP, COMCHM.SDN, or something similar.  Please unload.                                                                                 ∙ Create a DOS directory named astro3 and change to it:                   cd\ (enter)    md\astro3 (enter)  cd \astro3 (enter)                                                                                          ∙ Copy all files into the directory ASTRO3                                                                                                      ∙ Add these statements to AUTOEXEC.BAT in the base directory:             SET astro3=C:\astro3                                                    append to existing PATH statement  ;C:\astro3;                                                                                                ∙ Restart the PC. Type (ctrl) (alt) (del).                                                                                                      ∙ Running Under Windows as a Non-Windows Application                                                                                             Follow the installation instructions and install on the hard drive C.   Start WINDOWS (tm, MicroSoft).  Click on FILE then click on PROPERTIES. Enter the following information for each of the fields.  Description:   AstroFormCalc 3, Command:  ACAL3.EXE, Working Directory:  \astro3.      Click on CHANGE ICON.  Enter Filename:  C:\astro3\acal3.ico.  Click on  OK to exit and click on OK to exit FILE.  The AstroFormCalc 3 icon      should appear in the accessory window.  Click on it to run ACAL3.                                                                              ∙ To start from DOS:  cd\astro3 (enter)  then  COMCHM (enter)             or                  DEMO (enter)                                        or                  acal3 \astro3\COMCHM.HLP (enter)                                                                                          ∙ To start with the online help file (if available), type                                     cd\astro3 (enter) then acal3 ? (enter)              or                  cd\astro3 (enter) then asol3 ? (enter)              or                  cd\astro3 (enter) then afit3 ? (enter)                                                                                    ╒═════════════╕                                                         │ AUTHOR ???? └─────────────────────────────────────────────────────────                                                                        35.0     Interested in Becoming the Author of Your Own                        Interactive Application File?                                     ────────────────────────────────────────────────────────────────────────                                                                                 Many technologies are based on math/algebra.  The ASTRO 3               programs and their interactive, problem-solving application             files offer distinct advantages to the Author.  The *.EQU               templates are easy to develop and PCSCC offers the AstroUtility         3 program for compiling these from low-level text files into            sophisticated and interactive application files such as the             help file ACAL3HLP.HLP or COMCHM.HLP (The ASTRO 3 Chemistry             Manual) or SCIFIN.HLP (The ASTRO 3 Science Manual) or                   PHYCAL.HLP (The ASTRO 3 Physics Manual) or FINCAL.HLP (The              ASTRO 3 Finance Manual).  If you develop a manual or book with          calculations, then you want the ASTRO 3 team.  Why?                                                                                             1) Technically, they are among the HIGHEST LEVEL of                     calculation/spreadsheet programs available anywhere.  Minimal           user "brain power" is required to perform some very powerful            math analyses.                                                                                                                                  2) You can include fully functional calculations, written as            application files like this one or as *.EQU templates, within           your manual/book and label your book with "ALL REQUIRED                 SOFTWARE INCLUDED".  This will most certainly boost sales of            your manual/book.                                                                                                                               3) ASTRO 3 programs are FULLY UPWARDS COMPATIBLE (DOS 2.1+) and         will run on virtually any PC (even a barebones one with just a          monochrome display, 256K main memory and a single floppy!).             It is very likely that DOS will be around for some time so              long-term future support for the current ASTRO 3 programs is            very likely.                                                                                                                                    4) A DISTRIBUTION LICENSE is include with AstroUtility 3                registration which allows the author to include all the ASTRO 3         programs within the manual/book.  If you develop your                   calculations with a commercial package, it will be extremely            unlikely that you can include the host programs.  The purchaser         of your manual/book must already own the host programs or buy           them.  This will certainly discourage sales of your manual/book         and might make it obsolete.                                                                                                                     5) Your calculation/analyses WON'T BECOME ORPHANS.  If you use          a commercial package, will it be around in 2 or 5 years?  If it         still is, will it support your calculations?  Will it still do          things the way it did when you developed your calculations and          gave instructions in your manual/book?  With ASTRO 3, you can           include all software. People will always be able to perform             your calculations/analyses.                                                                                                                     6) If you develop an ASTRO 3 application file along with your           hard-copy manual/book, you can send the "DEMO" application file         off along the DIGITAL SUPERHIGHWAY to advertise your product            and increase sales with very little marketing expense!  This            file in its ZIP package is an one such example.                                                                                        ╒═══════════╕                                                           │ HELP FILE └───────────────────────────────────────────────────────────                                                                        36.0           Help File: Summary of File Viewing Commands              ────────────────────────────────────────────────────────────────────────                                                                            A - Auto-search of entire text for user-supplied string. Search is          in uppercase and is case insensitive. If string found,                  type A to continue or to enter new string, type (space) A.          B,N - Move back or ahead 18 lines.  Also use (backspace) for "B".       E, (esc) - Exit from help file to equations display.                    F,T,K - Jump to the first (beginning), T.O.C or last (end) line.        G - Skip to Glossary of terms or summary of functions, if present.      J,(ent) - Search line on │JUMP│, Reference is after . Jump to                    line number or perform autosearch ahead if text reference.    L,P - Jump to line/page. Type Lxxx(ent) or Pyy(ent) to jump to                line #xxx or page #yy. With + or -, ahead or back 56 lines.       S,R - Skip ahead (S) or back (R) in help file to major headings         W - Skip to Summary of useful calculations or warnings, if present      1 to 9 - Typing 1 to 9 advances the text by 1 to 9 lines.               (ctrl Q) - Exit from help file.  Same as command "E".                   Edit keys - Use edit keys: (Home) (up arrow) (PgUp) (left arrow)                    (right arrow) (End) (down arrow) (PgDn).                                                                                             Entry into the application file is at/near the last position.           Type: T to return to Table of Contents or E to exit the file.      