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I 




. -,,, j^. ^-Tf^ 






-\ 



t 

f 

I 



MATHEMATICAL MONOGRAPHS 

EDITED BY 

Mansfield Merriman and Robert S. Woodward. 

Octavo, Cloth. 



No. 1. 
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No. 3. 
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No. 16. 
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History of Modern Mathematics. By 

David Bugenb Smith. |i.oo net. 

Synthetic Projective Geometry. By 

George Bruce Ualsted. Ii.oo net. 

Determinants. By Labnas Gifford Weld. 
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Harmonic Functions. 

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Probability and Theory of Errors. By 

Robert S. Woodward. Ii.oo net. 

Vector Analysis and Quaternions. By 

Alexander Macfarlame. |i.oo net. 

Differential Equations. By William 
WooLSEY Johnson. |i.oo net. 

The Solution of Equations* By Mansfield 
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Functions of a Complex Variable. By 

Thomas S. Piske. $i.oo net. 

The Theory of Relativity. By Robert D. 
Carmichael. 1 1.00 net. 

The Theory of Numbers. By Robbrt D. 
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Algebraic Invariants. By Lbonasd B. 
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Mortality Laws and Statistics. By Robert 
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ander Macfarlanb. $1.35 net. 

Elliptic Integrals. By Harris Hancock. 
I1.25 net. 

Empirical Formulas. By Thkodors R. 
Running. Ix.40 net, > 



PUBLISHED BY 

JOHN WILEY & SONS, Inc., NEW YORK. 

CHAPMAN &. HALL, Limited, LONDON. 



I 



MATHEMATICAL MONOGRAPHS 

EDITED BY 

MANSFIELD MERRIMAN and ROBERT S. WOODWARD 



No. 19 

/ 

EMPIRICAL FORMULAS 



• • •• "* 



BT J^^ - 
THEODORE Ri*' RUNNING 

Associate Professor of Mathematics. University of Michigan 



FIRST EDITION 



NEW YORK 

JOHN WILEY & SONS, Inc. 
London: CHAPMAN & HALL, Limited 

1917 



\ 



Copyright, 1917, 

BY 

THEODORE R. RUNNING 



pnna or . 

BRAUNWORTH A,' CO. 

BOOK MANUrACTURCRB 

■OOOKLVN. N. V. 



7 ^ 

> 
o #• 






^ 



PREFACE 



This book is the result of an attempt to answer a number 
of questions which frequently confront engineers. So far as 
the author is aware no other book in English covers the same 
ground in an elementary manner. 

It is thought that the method of determining the constants 
in formulas by the use of the straight line alone leaves little to 
be desired from the point of view of simplicity. The approxi- 
mation by this method is close enough for most problems arising 
in engineering work. Even when the Method of Least Squares 
must be employed the process gives a convenient way of obtain- 
t ing approximate values. 

For valuable suggestions and criticisms the author here 
expresses his thanks to Professors Alexander Ziwet and Horace 
. W. King. 

p^ X. Jx. Jx. 

o^ UNiVERsmr of Michigan, 191 7. 




CONTENTS 



PAGE 

Introduction 9 

CHAPTER I 

I. y=a'^bx'^cx*-\-dx*'^ . . . -\-qo^ 13 

Values of x form an arithmetical series and A^y constant. 

11. y=a+-^+-^+4+ • • • +Zh 22 

X X^ X^ 7^ 

Values of - form an arithmetical series and A*y constant. 

X 

III. - ^a-\-hX'\-cx'^-\-dx^-\- . . . •\-qo^ 25 

y 

Values of x form an arithmetical series and A**- constant. 

y 

rV. y'^-a-\-hx-^cx'^'\-dx^-^ . . . •\-qo^ 25 

Values of x form an arithmetical series and A**;y* constant. 

CHAPTER II 

V. y-alf • 27 

Values of x form an arithmetical series and the values of y 
form a geometrical series. 

VI. y=-a-\-hc' 28 

Values of x form an arithmetical series and the values of Ay 
form a geometrical series. 

^"-^ VII. logy =aH-&c* 32 

Values of x form an arithmetical series and the values of 
A log y form a geometrical series. 

VIII. y—a'\-hx-\-cd' ^^ 

Values of x form an arithmetical series and the values of 
A*y form a geometrical series. 

^X IX. y = ioa+»*+«*' 37 

Values of x form an arithmetical series and the values of 
A* log y constant. 

5 






6 CONTENTS 



PAGE 




X. y^ks'f 37 

Values of x form an arithmetical series and the values ^"^ 
A* log y form a geometrical series. 

XI. y= -^ 38 

Values of x form an arithmetical series and A*- constant. 

y 

CHAPTER III 

^ XII. y=ax^ 42 

Values of x form a geometrical series and the values of y 
form a geometrical series. 

XIII. y =a+6 log x-\-c log*jc .44 

Values of x form a geometrical series and A^y constant. 

XIV. y=a-\-bxf 45 

Values of x form a geometrical series and the values of Ay 

form a geometrical series.^ 

XV. y^aic?^ 49 

Values of x form a geometrical series and the values of 
A log y form a geometrical series. 

CHAPTER IV 
XVI. {x^a)(y^h) =c 53 

Points represented by (x—xky ^ | lie on a straight Une. 

-^ XVIa. y=aid'+' / 56 

Points represented by ( log — , log — ) lie on a 

\x—xk y ytj 

straight Hne. 

XVII. y=ae^-\-be^ 58 

Values of x form an arithmetical series and the points repre- 
sented by I'-^j ^^^^] He on a straight line whose slope, 
\yk yi J 

My is positive and whose intercept, 5, is negative, and also 
M^+^B is positive. 

XVIII. y =e^(c cos bx-\-d smbx) 61 

Values of x form an arithmetical series and the points repre- 
sented by l^^^j 5!i±-M lie on a straight line whose slope, 
\ yk yk J 

M, and intercept, B, have such values that M^-\-/^ 
is negative. 



CONTENTS 7 

PAGE 

XIX. y=ax^-\-hofi 65 

Values of x form a geometrical series and the points repre- 
sented by (^^S y^±l\ lie on a straight line whose slope, 
\ yt yk I 

My is positive and intercept, 5, negative, and also 
JIf 24.45 positive. 

XlXa. y=aofc' 72 

Values of x form a geometrical series and the points repre- 
sented by ixny log 5!^ii J lie on a straight line. 

CHAPTER V 

XX. y =Oo+fli cos X'{-a2 cos 2x+ai cos3ic+ . . . -h^ cos rx 74 

-\-bismx-\-b2sm2x-\-biSm$x-\' . . . -\-br sin rx 
Values of y periodic. 

CHAPTER VI 

Method of Least Squares 90 

Application to Linear Observation Equations. 
Application to Non-linear Observation Equations. 

CHAPTER VII 

Interpolation 100 

Differentiation of Tabulated Fimctions. 

CHAPTER Vni 

Numerical Integration 114 

Areas. 
Volumes. 
Centroids. 
Moments of Inertia. 

APPENDIX 
Figures I to XX 132-143 

Index 144 



.»* 



EMPIRICAL FORMULAS 



INTRODUCTION 

In the results of most experiments of a quantitative nature, 
two variables occur, such as the relation between the pressure 
and the volume of a certain quantity of gas, or the relation 
between the elongation of a wire and the force producing it. 
On plotting the sets of corresponding values it is found, if they 
really depend on each other, that the points so located lie 
approximately on a smooth curve. 

In obtaining a mathematical expression which shall represent 
the relation between the variables so plotted there may be two 
distinct objects in view, one being to determine the physical 
law underlying the observed quantities, the other to obtain 
a simple formula, which may or may not have a physical basis, 
and by which an approximate value of one variable may be 
computed from a given value of the other variable. 

In the first case correctness of form is a necessary considera- 
tion. In the second correctness of form is generally considered 
subordinate to simplicity and convenience. It is with the 
latter of these (Empirical Formulas) that this volume is mostly 
concerned. 

The problto of determining the equation to be used is really 
an indeterminate one; for it is clear that having given a set of 
corresponding values of two variables a number of equations 
can be foimd which will represent their relation approximately. 

Let the coordinates of the points in Fig. i represent different 
sets of corresponding values of two observed quantities, x and y. 
If the points be joined by segments of straight lines the broken 

9 



10 



EMPIRICAL FORMULAS 



line thus formed will represent to the eye, roughly, the relation 
between the quantities. 

It is reasonable to suppose, however, that the irregular dis- 
tribution of the points is due to errors in the observations, and 
that a smooth curve drawn to conform approximately to the 
distribution of the points will more nearly represent the true 
relation between the variables. But here we are immediately 
confronted with a difficulty. Which curve shall we select? a 



I/l 


1 






















U 






















.^ 


/ 






















^ 


^ 




















> 
























.•^ 


/ 


r 


, 


















^ 


^ 






















/ 


r 






















i 






















a 


/ 


C^ 


















"- 




A 


\ 




















# 



Fig. I. 



or J? or one of a number of other curves which might be drawn 
to conform quite closely to the distribution of the points? 

In determining the form of curve to be used reliance must 
be largely placed upon intuition and upon knowledge of the 
experiments performed. 

The problem of determinmg a simple equation which will 
represent as nearly as possible the curve selected is by far the 
more difficult one. 

Ordinarily the equation to be used will be derived from a 
consideration of the data without the intermediate step of 
drawing the curve. 

Unfortunately, there is no general method which will give 
the best form of equation to be used. There are, however, a 
number of quite simple tests which may be applied to a set of 



INTRODUCTION 11 

data, and which will enable us to make a fairly good choice of 
equation. 

The first five chapters deal with the application of these 
tests and the evaluation of the constants entering into the 
equations. Chapter VI is devoted to the evaluation of the 
constants in empirical formulas by the Method of Least Squares. 
In Chapter VII formulas for interpolation are developed and 
their applications briefly treated. Chapter VIII is devoted to 
approximate formulas for areas, volumes, centroids, moments of 
inertia, and a number of examples are given to illustrate their 
application. 

Figs. I to XX at the end of the book show a few of the forms 
of curves represented by the different formulas. 

A few definitions may be added. 

Arithmetical Series. A series of numbers each of which, 
after the first, is derived from the preceding by the addition 
of a constant number is. called an arithmetical series. The con- 
stant number is called the common difference 

6, 6.3, 6.6, 6.9, 7.2, 7.5 .. . 

and 

18.0, 15.8, 13.6, 11.4, 9.2 .. . 

are arithmetical series. In the first the common difference is 
.3, and in the second the common difference is —2.2. 

Geometrical Series, A series of niunbers each term of which, 
after the first, is derived by multiplying the preceding by some 
constant multiplier is called a geometrical series. The constant, 
multiplier is called the ratio. 

1.3, 2.6, 5.2, 10.4, 20.8, 41.6 . . . 

and 

100, 20, 4, .8, .16, .032 . . . 

are geometrical series. In the first the ratio is 2, and in the 
second it is .2. 

Differences are frequently employed and their meaning can 
best be brought out by an example. 



12 



EMPIRICAL FORMULAS 



X 


y 


Ay 


A*y 


A»y 


A*y 


I 


I0.2 










2 


II. I 


0.9 
I.I 


0.2 


0.0 




3 


12.2 


1.3 


0.2 


1.2 


1.2 


4 

5 
6 


13.5 
16.2 

18.0 


2.7 
1.8 
1.0 


1.4 
-0.9 

-0.8 


-2.3 
0.1 
2.8 


-3.5 
2.4 

2.7 


7 


19.0 




2.0 






8 


22.0 


3.0 









In the table corresponding values of x and y are given in 
the first two columns. In the third column are given the values 
of the first differences. These are designated by Ay. The first 
value in the third column is obtained by subtracting the first 
value of y from the second value. The column of second differ- 
ences, designated by A^y, is obtained from the values of Ay in 
the same way that the column of first differences were obtained 
from the values of y. The method of obtaining the higher differ- 
ences is evident. 



CHAPTER I 

I. y—a+bx+co(^+dx'^+ . . . +qaf^ 

Values of x form an arithmetical series and A'^y constant. 

In a tensile test of a mild steel bar, the foUowing observa- 
tions were made (Low's Applied Mechanics, p. i88): Diameter 
of bar, unloaded, 0.748' inch, IT = load in tons, x= elongation in 
inches, on a length of 8 inches. 



w 


I 


2 


3 


4 


5 


6 


X 

Ax 


0.0014 

0.0013 S 


0.0027 
0.0013 


0.0040 
o.oois 


0.0055 
0.0013 


0.0068 
0.0014 


0.0082 



Plotting W and x, Fig. 2, it is observed that the points lie 
very nearly on a straight line.* Indeed, the fit is so good that 
it may be almost concluded that there exists a linear relation, 
between W and x. From the figure it is found that the slope 
of the line is 0.00137 and that it passes through the origin. The 
relation between W and x is therefore expressed by the equation 

x=o,ooi^'jW, 

The observed values of x and the values computed by the 
above formula are given in the table below. 



w 


Observed x 


Computed x 


I 

2 

3 

4 

5 
6 


0.0014 
0.0027 
0.0040 
O.OOSS 
0.0068 
0.0082 


0.00137 
0.00274 
0.00411 
O.00S48 
0.00685 
0.00822 



* By the use of a fine thread the position of the line can be deter- 
mined quite readily. 

13 



14 



EMPIRICAL FORMULAS 



The agreement between the observed and the computed 
values is seen to be quite good. It is to be noted, however, that 
the formula can not be used for computing values of x outside 



¥0 



.0080 
















































y 


► 


4»T& 

.0072 
.0068 














































y 


/ 














































/ 


r- 














































/ 










0061 








































/ 












.0060 




































/ 


/■ 














.0056 


































y 


/ 
















iOOR? 
































J\ 


r 


















0048 






























/ 


r 




















0014 




























/ 
























.0040 
























A 


y 


























.0036 
.0032 






















y 


/ 














































y 


/ 






























.0028 


















/ 


r 
































.0024 
















/ 


T 


































.0020 
jflOlB 














/ 














































V 


/ 








































.0012 
.0008 








■ 


/ 














































/ 


A 












































.0004 




/ 


r 
















































^ 
























































1 

J 


L 






4 


1 
1 






4 

1 


i 






\ 


\ 






1 


S 






6 MO 



Fig. 2. 

the elastic limit. In the experiment 6 tons was the load at the 
elastic limit. 

It is not necessary to plot the points to determine whether 
they lie approximately on a straight line or not. Consider the 
general equation of the straight line 

Starting from any value of x^ give to x an increment, Aa:, and 
y will have a corresponding increment, Ly, 

y'\-Ly=m{^-\-^oi)'\-'k\ 
y=mx+k; 



Ay=mAx, 



DETERMINATION OF CONSTANTS 15 

From this it is seen that, in the case of a straight line, if the 
increment of one of the variables is constant, the increment of 
the other will also be constant. 

From the table it is observed that the successive values of 
W differ by unity, and that the difference between the successive 
values of x is very nearly constant. Hence the relation between 
the variables is expressed approximately by 

x=inW+k, 

where tn and k have the values determined graphically from 
the figure. 

By the nature of the work it is readily seen that the graphical 
determination of the constants will be only approximate imder 
the most favorable conditions, and should be employed only 
when the degree of approximation required will warrant it. 
Satisfactory results can be obtained only by exercising great 
care. Carelessness in a few details will often render the results 
useless. Understanding how a graphical process is to be carried 
out is essential to good work; but not less important is the 
practice in applying that knowledge. 

In experimental results involving two variables the values of 
the independent variable are generally given in an arithmetical 
series. Indeed, it is seldom that results in any other form 
occur. It will be seen, however, that in many cases where 
the values of the independent variable are given in an arithmeti- 
cal series it will be convenient to select these values in a geomet- 
rical series. 

As a special case consider the equation 

y = 2-'^x+x!^. 

If an increment be assigned to Xy y will have a corresponding 
increment. The values of x and y are represented in the table 
below. Ay stands for the number obtained by subtracting any 
value of y from the succeeding value. A^y stands for the num- 



16 



EMPIRICAL FORMULAS 



ber obtained by subtracting any value of Ay from its succeeding 
value. The values of x have the common difference 0.5. 



X 


o.S 


I.O 


i-S 


2.0 


2-5 


30 


35 


4.0 


y 


0.7s 


0.00 


-0.25 


O.CX) 


0.7S 


2.00 


3-75 


6.00 


Ay 


-0.7s 


—0.25 


0.25 


0.7S 


I-2S 


1.75 


2.25 




A^y 


0.50 


0.50 


0.50 


0.50 


0.50 


0.50 







The values A^y, which we call the second differences, are 
constant. 

These differences could equally well have been computed 
as follows: 

y+Ay = 2 —^(x+Ax) + (x+Axy, 
Ay= — 3 (Ax) + 2x(A:r) + (Ax)^, 
Ay+A^y= -siAx) + 2ix+Ax){Ax) + {Axy, 
A^y = 2 (Axy = 0.5 since Ax = 0.5. 

From this it is seen that whatever the value of Ax (in 
y = 2-'^x+x^) the second differences of the values of y are 
constant. 

Consider now the general case where the nth differences are 
constant. For convenience the values of y and the successive 
differences will be arranged in columns. The notation used is 
self-explanatory. 



Vi 


Ay\ 










y2 




A2yi 










Ay2 




A3yi 




etc.. 


yz 




A2y2 




A^yi 






Ayz 




A3y2 




etc.. 


y^ 




A^ya 




A^y2 






Ay4 




A^ys 




etc., 


y-o 


Ays 


t^y\ 


• • • 


• • • 




ye 




• • • 









^ 



DETERMINATION OF CONSTANTS 17 

From the above it is clear that 
y2=yi+Ayi, 

y3=y2+Ay2, 

=yi+Ayi+A(yi+Ayi). 
^yi+2Ayi+A^yi. 

3^4=^3+ Ays, 

==yi+2Ayi+A^yi+A(yi+2Ayi+A^yi), 

=yi+3Ayi+3A^yi+A3yi. 

y6=y4+Ay4, 

= yi +sAyi +3A^yi +A^yi +A(yi +sAyi +sA^yi +A^yi) 
= yi +4Ayi +6A^yi +4A^yi + A^yi. 

In the above equations the coefficients follow the law of 
the binomial theorem. Assuming that the law holds for yt 
it will be proved that it holds for y^+i. 

By hypothesis 

3^=yi+(ife"i)Ayi+ ^^"'y""'W 



ffcife^^^^A3yx+etc. . . . . (i) 
13 



If this equation is true, then 
yt+i=yi+(^-i)Ayi+ ^ ""V "" A ^yi 

|2 



+ ^^ ^-^ — — ^A^y 1 + etc. 

3 



+A[yi + ()fe-i)Ayi+^^-^^^-^A2y 






= y, + My,+^(^A2y+M*:^^ 



4 



J ^ -J 



18 EMPIRICAL FORMULAS 

This is the same law as expressed in the former equation, and 
therefore, if the law holds for yky it must also hold for yt+i. 
But we have shown that it holds for y^, and therefore, it must 
hold for ys. 

Since it holds for ys it will hold for y^. By this process it 
is proved that the law holds in general. 

If now the first differences are constant the second and 
higher differences will be zero, and from (i) 

yk=yi+ik-i)Ayi. 

If the second differences are constant the third and higher 
differences will be zero, and it follows from (i) that 



In general, then, if the nth differences are constant 

yt=yi+{k-i)Ayi+- p -A^yi+- ^-^ — - — —A^yi 

\l 13 

. (^-x)(^~2)(^-3)(^~4) . . . (k-n) .. , . 

+ . . . + n A yi. (2) 



The law requires that the values of x form an arithmetical series, 
and hence 

Xt=xi+{k—i)Ax; 
from which follows 

*-^+' « 

Substituting this value of * in equation (2) it is found that 
the right-hand member becomes a rational integral function of 
Xk of the «th degree. Equation (2) takes the form 

yt=a+bxk+cx.^^+dx,.^+ . . . +qX)c''. 

\/ 

Since Xt, and y* are any two corresponding values of x and y 

the subscripts may be dropped and there results the following 

law: 






taB^l^^i^H 



^mt^^ 



DETERMINATION OF CONSTANTS 



19 



If two variables, x and y, are so related thai when values ofjc 
are taken in an arithmetical series the nth differences of the cor- 
responding valines of y are constant y the law connecting the variables 
is expressed by the equation 



y=a+bx+cx^+dx?+ 



• • • 



+?«*. 



The nth differences of the values of y obtained from observa- 
tions are seldom if ever constant. If, however, the nth differ- 
ences approximate to a constant it may be concluded that the 
relation between the variables is fairly well represented by I. 

As an illustration consider the data given on page 131 of 
Merriman's Method of Least Squares. The table gives the 
velocities of water in the Mississippi River at different depths 
for the point of observation chosen, the total depth being taken 
as imity. 



At surface. 
0.1 depth. 
0.2 



0.3 
0.4 

o.S 
0.6 

0.7 
0.8 
0.9 



3 . 1950 
3 2299 
3.2532 
3.2611 
3.2516 
3.2282 
3.1807 
3.1266 

3.0594 
2.9759 



Av 



+349 
+ 233 

+ 79 

- 95 
-234 

-475 
-541 
—672 

-835 



A*if 



-116 

-154 
-174 

-139 

— 241 

— 66 

-131 

— 163 



A»» 



- 38 

- 20 

+ 35 

- 102 

+ 175 

- 6s 

- 32 



A*v 



+ 18 

+ 55 
-137 
+ 277 

— 240 
+ 33 



Ah 



+ 37 
— 192 

+414 
-517 
+273 



From the above table it is seen that the second differences 
are more nearly constant than any of the other series of differ- 
ences. Of equations of form I, 

y==a+bx+cx^y 

where x stands for depth and y for velocity, will best represent 
the law connecting the two variables. It should be emphasized, 
however, that the fact that the second differences are nearly 
constant does not show that I is the correct form of equation 



20 EMPIRICAL FORMULAS 

to be used. It only shows that the equation selected will 
represent fairly well the relation between the two variables. 

It might be suggested that if an equation of form I with 
ten constants were selected these constants could be so deter- 
mined that the ten sets of values given in the table would satisfy 
the equation. To determine these constants we would sub- 
stitute in turn each set of values in the selected equation and 
from the ten equations thus formed compute the values of the 
constants. But we would have no assurance that the equation 
so formed would better express the law than the equation of 
the second degree. 

For the purpose of determining the approximate values of 
the constants in the equation 

y=a+bx+cx^ (i) 

from the data given proceed in the following way: 

Letx=X+oco, ./:' V /- 
y=Y+yo, ^ 

r 

where xq and yo are any corresponding values of x and y taken 
from the data. The equation becomes 

Y+yo^a+b{X+xo)+c{X'{'Xoy 

= a+hocQ+cX(?+{b+2Coco)X'\-cX^. 

Y = {h+2cxQ)X+cX^) (2) 

since yo^a+hoco+coci?. Dividing (2) by X it becomes 

^4l+2cc^-\-cX (3) 

F 

This represents a straight Kne when X and — are taken as 

Ji. 

coordinates. The slope of the line is the value of c and the 

intercept the value of b+2CXo. The numerical work is shown 



in the table and the points represented by [X, — :] 



are seen 



DETERMINATION OF CX)NSTANTS 



21 



in Fig. 3. The value of c is found to be -\o,^6^ When oco==o, 
the intercept, 0.44 is the value of 6. For x^^X, the value of 



.5 


Y 
X 








































.4 
.3 




^ 


^^^ 


' 










































^ 


^^ 


1 




























.2 
.1 
















"--- 


<. 










































■^^^-^ 





















M 
-.2 
-.3 






- 








































































































'^ 


■^ 




















































































""^ 


. * 













































.1 



.2 



.3 



.4 



\ 



.6 



.7 



.8 



.9 



PlG. 3. 



yo is taken from the table to be 3.1950, therefore each value of 
Y will be the corresponding value of y diminished by 3.1950. 





/ 


-z. 





"? 


j^ 


Cr 




-? 




X 


y 


X 




Y 


Y 
X 


.44X— .76X* 


a=y-.44* 
-.76X* 


Computed 

y 


.0 


3.1950 


0.0000 




0.0000 


3.1950 


3.1948 




I 

2 
.3 


3.2299 

3-2532 
3. 261 I 


I 
2 

3 


0.0349 
0.0582 
0.0661 


0.3490 
0.2910 
0.2203 


0.0364 
0.0576 
0.0636 


3 
3 
3 


1935'^ 
1956 

1975 


3.2312 
3.2524 
3.2584 




.4 

s 

.6 

7 
8 


3.2516 
3.2282 
3.1807 
3.1266 

3.0594 


4 

5 
.6 

•7 
.8 


0.0566 

0.0332 

—0.0143 

— 0.0684 

-0.1356 


O.1415 

0.0664 

—0.0238 

-0.0977 

— 0.1695 


0.0544 

0.0300 

—0.0096 

—0.0644 

-0.1344 


3 
3 
3 
3 
3 


1972 

1982 

1903 

. 1910' 

.1938 


3 • 2492 
3.2248 
3.1852 

3 . 1304 
3.0604 




9 


2.9759 


•9 


-0.2191 


-0.2434 


—0.2196 


3.195s 


2.9752 



10 )31.9476 
fl= 3.1948 

The numbers in column 6 were found after 6 and c were deter- 
mined in Fig. 3. The sum of the numbers in the seventh column 
divided by ten gives the value of a. In the last column are 
written the values of y computed from the formula 



y=3.i948-f-.44a?— ^6x2. 



(4) 



22 



EMPIRICAL FORMULAS 



II. :y = a+-+-+_+ . . . ^. 
Values of - form an arithmetical series and A y are constant. 

Another method of determining the constants is illustrated 
in the following example: Let it be required to find an equation 
which shall express approximately the relation between x and y 
having given the corresponding values in the first two columns 
of the table below. 



I 


2 


3 


4 


5 


6 


7 


8 


9 


X 


y 


I 

X 


X 


y 


Ay 


A*y 


2 


Com- 
puted y 


I.O 


4.000 


1.0 


1. 000 


4.00 


-0.68 


0.04 


2.00 


4.000 


1.2 


2.889 


0.9 


I. Ill 


3.32 


—0.64 


0.03 


1.50 


2.889 


1.4 


2.163 


0.8 


1.250 


2.68 


—0.61 


0.05 


I. 14 


2.163 


1.6 


1.656 


0.7 


1.429 


2.07 


-0.56 


0.05 


0.87 


1.656 


1.8 


1.284 


0.6 


1.667 


i-Si 


-0.51 


0.03 


0.67 


1.284 


2.0 


1. 000 


0.5 


2.000 


1. 00 


-0.48 


0.04 


0.50 


1. 000 


2.2 


0.777 


0.4 


2.500 


0.52 


-0.44 




0.36 


0.777 


2.4 


0.597 


0.3 


3-333 


0.08 






0.25 


0.597 



In column 3 are given values of - in arithmetical series 

X 

and the corresponding values of x and y are written in columns 
4 and 5 of the table. The values of y were read from Fig. 4. 

It is seen that the second differences of the values of y given 
in column 7 are nearly constant, and therefore the relation 
between the variables is represented approximately by the 
equation 

y-'H^+ii) ■ ^^ 

This becomes evident if x be replaced by - in I. The law 

X 

may then be stated: 

// two variables, x and y, are so related that when values of - 

X 

are taken in arithmetical series the wth differences of the corre- 



DETERMINATION OF CONSTANTS 



23 



sponding values of y are constant, the law connecting the variables 
is expressed by the eqimtion 



II 



. b . c . d . q 



A 












• 


8 


. 


1 


• 


5 


Values of -^ 
.6 .7 


• 


8 


• 


9 


1 




V 








































^ 


L 






































3 




K 


































V 


/ 








k 


« 




























.A 


/^ 




;^ 






\ 


S^ 


























A 


^— 






o 
id2 




• 




\ 


kv 




















> 


y 
















s 


N 
















y 


/^ 












? 














^ 


v,^ 










/ 


^ 














J 


















^ 




. > 


[y 






































/ 


>< 


^^ 




• 






/ 


























^ 


/ 










^^^^ 


























_^ 


/ 




















— 


^' 


1 




1. 


2 


1. 


i 


h 


6 


1.1 


3 
Val 


2. 
ues 



of a 


2, 


2 


% 


1 


2. 


6 


2. 


8 


3.0 



«l^ 



1® 



OS 



Fig. 4. 



If in equation (s) - be replaced by X, then 



and 



y=a+ftX+cJt2, 



y+A3;=a+6(X+AZ)+c(Z+AX)2. 



By subtracting (6) from this equation 

and from (7) 

Ay+A2y = 6AZ+2(;(AZ)(Z+AZ)+(;(AZ)2. . 

Subtracting (7) from (8) 

A2y = 2c(AZ)2; 

A^y 
"^ 2(AZ)2' 



(6) 



(7) 



(8) 



24 



EMPIRICAL FORMULAS 



From column 7 it is seen that the average value of A^y is 
0.04, and as AX was taken —.1, 



0.04 



Writing the equation in the form 



-5-+KJ- 



z 


y- 


2 

«2 
















n 


























/ 


























/ 
























yi 


' 




« * 


















/ 








14> 


















t 
























/ 


























/ 


























/ 










t 


























X 








































1 
























y 


























/ 














c 












/ 














•0 






' 






























/ 


























1 
























/^ 


























/ 
















1 





•4 


L . 


2 ^ 


r. 


i . 


ij .1 


3 . 


7 . 


B . 


9 1. 





X 



Fig. 5. 



it is seen that it represents a 
straight line when - and y— 

X 

2 

3 are the coordinates. From 

Fig. 5 6 is foimd to be 3 
and a to be — i. The for- 
mula is 



y=-i+3 



©-&)• 



The last colmnn gives the 
values of y computed from 
this equation. 

The following, taken from 
Saxelby's Practical Mathe- 
matics, page 134, gives the 
relation between the poten- 
tial difference V and the cur- 
rent ^-4 in the electric arc. 
Length of arc =2 mm., A is 
given in amperes, V in volts. 



A 

Observed V. 

z 



A 
Computed V , 



1.96 
50.25 



.5102 
50.52 



2.46 
48.70 

.4065 
48.79 



2.97 
4790 

■ 3367 
47.62 



3.45 
47.50 

.2899 
46.84 



3.96 
46.80 

.2525 
46.22 



4.97 
45.70 

.2012 
45.36 



5.97 
45.00 

.1675 

44.80 



6.97 
44.00 

.1435 
44.40 



7.97 
43.60 

.1255 

44.10 



9.00 
43.50 

.iiii 
43.85 



DETERMINATION OF CONSTANTS 



25 



Fig. 6 shows V plotted to — as abscissa. The slope of this 

line is 12.5 divided by .75 or 16.7. The intercept on the V—ax is 
is 42. This gives for the relation between V and A 



F=42-f 



16.7 



Although the 
points in Fig. 6 do 
not follow the straight 
line very closely the 
agreement between 
the observed and the 
computed values of V 
is fairly good. 



55 

60 
45 
40 
35 



r 

— I — I — I — I — I — I — I — I — i — I— —J — > 



.1 



.2 



A 



Fig. 6. 



III. - = a+bx+cx^+d(x^+ . . . +qx^, 

y 

Values of x form an arithmetical series and A**— constant. 



.6 



1 



3^ 



■Z- 



If two variables, x and y, are so related that when values of 
X are taken in an arithmetical series the »th differences of the cor- 
responding valines of - are constant, the law connecting the variables 
is expressed by the eqimtion 



III 



- = a+bx+cx^+dx^+ . . . +qx'^. 

y 



This becomes evident by replacing y in I by -. The con- 
stants in III may be determined in the same way as they were 
in I. 

IV. y2 = a+bx+cx^+dcfi+ . . . +qx''. 
Values of x form an arithmetical series and A" y* constant. 

// two variables, x and y, are so related that when values of 
x are taken in an arithmetical series the wth differences of the cor- 



26 EMPIRICAL FORMULAS 

responding values of y^ are constant^ the law connecting the variables 
is expressed by the equation 

IV :^=^a-{-bx-{-co(^+d:fi+ . . . +gx". 

This also becomes evident from I by replacing y by y^. 

The method of obtaining the values of the constants in 
formulas III and IV is similar to that employed in formulas I 
and II and needs no particular discussion. 



CHAPTER II 

V. y = ah\ 

Values of x form an arithmetical series and the values of y a geometrical 
series. 

// two variables, x and y, are so related that when values of 
X are taken in an arithmetical series the corresponding valtces of 
y form a geometrical series, the relation between the variables is 
expressed by the equation 

V y = ab^. 

If the equation be written in the form 

logy = loga+(log6):x;, 

it is seen at once that if the values of x form an arithmetical 
series the corresponding values of log y will also form an arith- 
metical series, and, hence, the values of y form a geometrical 
series. 

The law expressed by equation V has been called the com- 
poimd interest law. If a represents the principal invested, b the 
amount of one dollar for one year, y will represent the amount 
at the end of x years. 

The following example is an illustration under formula V. 
.In an experiment to determine the coefficient of friction, /i, 
for a belt passing round a pulley, a load of W lb. was hung 
from one end of the belt, and a pull of P lb. applied to the other 
end in order to raise the weight W, The table below gives cor- 
responding values of a and /x, when a is the angle of contact 
between the belt and pulley measured in radians. 



a 


IT 
2 


2ir 
3 


5^ 
6 


IT 


7^ 
6 


4ir 
3 


3^ 
2 


5![ 
3 


6 


P 


5.62 


6.93 


8.52 


10.50 


12.90 


15.96 


19.67 


24.24 


29.94 



>- 



27 



28 



EMPIRICAL FORMULAS 



The values of a form an arithmetical series and the values 
of P form very nearly a geometrical series, the ratio being 1.23. 
The law connecting the variables is 

The constants are determined gi-aphically by first writing 
the equation in the form 

log P = log a+a log b 

and plotting the values of a and P on semi-logarithmic paper; 
or, using ordinary cross-section paper and plotting the values 
of a as abscissas and the values of log P as ordinates. Fig. 7 
gives the points so located. The straight line which most 
nearly passes through all of the points has the slope .1733 and 
the intercept .4750. The slope is the value of log b and the 
intercept the value of log a. 



log a =0.4750, 
log J =0.1733; 

6 = 1.49. 




Vi' K' %»• »• %'' Yi' %^ ^^ ^ 
Values of oc 

Fig. 7. 



or 



The formula expressing the relation between the variables is 

P= 3(1.49)", 

VI. y-=a+b<f. 

Values of x form an arithmetical series and the values of Ly form a 
geometrical series. 

// two variables, x and y, are so related that when values of x 
are taken in an arithmetical series the first differences of the values 



DETERMINATION OF CONSTANTS 



29 



of y form a geometrical series, the relation between the variables 
is expressed by the equation 

VI y = a+bc'. 

By the conditions stated the »th value of x will be 

Xn=xi+{n-i) Air, 

and the series of first differences of the values of y will be 

Ayi, Ayir, Ayir^, Ayir^, A^^ir* . . . Ayir""^. 

The values of y will form the series 

yu yi+Ayi, yi+Ayi+rAyi, yi+rAyi+r^Ayi . . . 

yi+Ayi+fA3;i+r2Ayi+r3Ayi+ . . . +r""^Ayi. 

The »th value of y will be represented by 



yn=yi+Ayi 



i-r 



,n-l 



I— r 



From the wth value of x 



»— 1 = 



X n — Xl 

Lx 



Substituting this value in the above equation there is ob- 
tained 



yn=yi+^yi 
=a+b(^, 



i—r 



Ax 



I—r 



Avi" Avi "^ 

where a stands for yi-\ — =^, b for ^— r ^ , and c for r^. 

i—r I— r 

Let it be required to find the law connecting x and y having 

given the corresponding values in the first two lines of the 

table. 



X 


o 


.1 


.2 


-3 


.4 


-5 


.6 


-7 


.8 


-9 


I.O 


y 

Ay 

y 


1.300 
0.140 

1.300 


1.440 
0.157 
1-439 


1.597 
0.177 
1-597 


1.774 
0.200 

1.774 


1-974 
0.224 

1-973 


2.198 
0.254 
2.198 


2.452 
0.285 
2.452 


2.737 
0.323 
2.738 


3.060 
0.363 
3-059 


3.423 
0.407 

3.421 


3.830 
3.830 



30 EMPIRICAL FORMULAS 

Since the values of Ay form very nearly a geometrical series 
the relation between the variables is expressed approximately 
by 

y^a+bc". 

The constants in this formula can be determined graphically 
in either of two ways. First determine a and then subtract 
this value from each of the values of y giving a new relation 

y—a = b(f; 

which may be written in the logarithmic form 

log {y-a)=^logb+x log Cy 

and b and c determined as in Fig. 7; or, determine c first and 
plot (f as abscissas to y as ordinate giving the straight line 

y^a+b{c'\ 

whose slope is b and whose intercept is a. 

First Method, The determination of a is very simple. 
Select three points P, Q^ and R on the curve drawn through 
the points represented by the data such that their abscissas 
form an arithmetical series. Fig. 8 shows the construction. 

P^{xo,a+b(f'); 

Q=(xo+Ax, a+b(f'c^); 

R= (:r6+2A:x:, a+b(f'(?^). 
Select also two more points S and T such that 

S={xo-{-Ax,a+b(f')\ 

T= (:xk)+2Aa;, a+b(f'c^). 
The equation of the line passing through Q and R is 

y= ^ Lx ^^ -{ocQ+Ax)+a+b(f^c^. (i; 



DETERMINATION OF CONSTANTS 



31 



The equation of the line through the pomts S and T is 



y= \^ ^ -^ '-(x,^+Ax)+a+b(f\ . (2) 

These lines intersect in a point whose ordinate is a. For, 
multiplying equation (2) by c^ and subtracting the resulting 
equation from (i) gives 

y = a. 




Fig. 8 gives the value of a equal to 0.2. The formula now 

becomes 

log (y — .2) =log b+x\og c. 

In Fig. 9 \og(y — .2) is plotted to x as abscissa. The slope of 
the line is 0.5185 which is the value of log c, hence c is. equal to 
$.;i. The intercept is the ordinate of the first point or 0.0414, 
which is the logarithm of 6, hence b is equal to i.i. 
The formula is 

>'-0.2-f 1.1(3.3)^ 



32 



EMPIRICAL FORMULAS 



The last line in the table gives the values of y computed from 
this formula. 

Second Method, For any point (a;,y) the relation between 
X and y is expressed by 

and for any other point (a;+A:r, y+Ay) by 

y+A3; = a+&(f{;^. 

From these two equations is obtained 

Ay = 6(f(c^-i) 

log Ay = log &(c^* — i) +:x; log c. 

.6 

If now log Ay be 

plotted to X as 

p abscissa a straight 

3 line is obtained 

.4 



or 













• 










/ 






















J 


k 






















J 


Y 






















i 


/ 






















1 


/ 










■ 














/ 
























/ 
























/ 
























/ 
























/ 
























/ 
























A 



























.J 


L .: 


2 .. 


Va 


lues 


of a 


• 


\ A 


J .1 


J i. 








£ whose slope is log c. 
8 The value of c hav- 
'I mgbeendetennined, 
the relation 



.1 



Fig. 9. 



y^a+h{(f) 

will represent a 
straight line pro- 
vided y is plotted 
to (? as abscissa. 



The slope of this line is b and its intercept a. 

VII. \ogy=a+h(f. 

Values of x form an arithmetical series and the values of A log y form a 
geometrical series. 

If two variables, x and y, are so related that when values of x 
are taken in an arithmetical series the first differences of the cor- 



DETERMINATION OF CONSTANTS 33 

responding values of log y form a geometrical series, the relation 
between the variables is expressed by the eqimtion 

VII logy = a+Jc'. 

This is at once evident from VI when y is replaced by log y. 
The only difference in the proof is that instead of the series 
of differences of y the series of differences of log y is taken. 

VIII. y^a+bx+cd". 

Values of x form an arithmetical series and the values of A*y form a 
geometrical series. 

// hvo variables, x and y, are so related that when values of x 
are taken in an arithmetical series the values of the second differ- 
ences of the corresponding values of y form a geometrical series, 
the relation between the variables is expressed by the equation 

VIII y^a+bx^-cd^ 
The »th value of x is represented by 

Xn—Xi'\'{n — l)^X. 

The values of y and the first and second differences may be 
arranged in columns 



yi 


Ayi 




y2 


Ay2 


A2yi 


ya 


Ays 


A2y2 


y4 


Ay4 


A2y3 


ys 


Ays 


A2y4 


ye 


etc. 


etc. 


etc. 







34 EMPIRICAL FORMULAS 

Since the second differences of y are to form a geometrical 
series they may be written 

A^^fi, rA^yi, r^^^yi, r^^^yi . . . r^"^A^yi. 

The series of first differences will then be 
A>^i , Ayi -h A^y 1 , Ay 1 -|- A^y i -|-r A^y i , Ay i -|- A^y i -|-r A^y i -f-r^A^y i 

Ayi+A2yi+rA2yi+r2A2yi+ . . . +r^"^A2yi. 

The »th value of y will be equal to the first value plus all 
the first differences. For convenience the wth value of y is 
written in the table below. 

yn=yi 
+Ayi 

+Ayi+A2yi 
+ Ayi + A^yi +rA2yi 
+ Ayi + A^yi +rA2yi +r^^yi 
+ Ayi + A^yi +rA2yi +r^^^yi +r^A2yi 



+Ayi+A2yi+rA2yi+r2A2yi+r3A2yi+ . . . +r'*"^A2yi. 
Adding gives 

r 



yn=yi + (w-i)Ayi+A2yi ? — ^-+- — - + ^ }-- — ? 

Li— r I— r i—r i — 



+-— ^+ 

I— r 



, i-f"-' 1 
I—r J 



The first two terms on the right-hand side represent the sum 
of all the terms in the first column of the value of yn* The 
remaining terms contain the common factor A^yi. The terms 
inside the bracket are easily obtained when it is remembered 
that each line, omitting the first term, in the value of y form a 
geometrical series. It is easily seen that the value of y» may be 
written 



DETERMINATION OF CONSTANTS 



35 



A2y] 



A2y] 



>»♦. 



JW=yi+(«-i)Ayi+^^-^(«-2)-^^^(r+f2+r3+. . .+r-=') 

I— r i—r 



I— f I—r I— f 

=4+5(«-i)+Cr-^ 

where 

^=^,_^, B=Ay,+^, and C=-^3. 
(i— r)2 ^ I — r (i— ^) 

From the value of Xn is obtained 



w--i = 



Ax 



Substituting this in the value of y« it is found 

'=a+bxn+cd^'*. 

Since :![:» and yn stand for any set of corresponding values 
of X and y the resulting formula is 

Vni y^a+bx+cd". 

In the first two columns of the following table are given 
corresponding values of x and y from which it is required to 
find a formula representing the law connecting them. 



X 


y 


Ay 


A^y 


log A^y 


(2.00)* 


y— 1.01(2.00)* 


Computed y 


.0 


1.500 


.048 


.023 


-1.6383 


1. 000 


.490 


1.492 


.2 


1.548 


.071 


.026 


— I. 


5850 


1. 149 


.388 


1.550 


.4 


1. 619 


.097 


.028 


— I 


5528 


1.320 


.286 


1.620 


.6 


1. 716 


.125 


.034 


^I 


4685 


1. 517 


.184 


I.71S 


.8 


1. 841 


.159 


.039 


— I 


4089 


1.742 


.082 


1. 841 


I.O 


2.000 


.198 


.043 


^I 


■3665 


2.000 


— .020 


1.999 


1.2 


2.198 


.241 


•051 


— I 


.2924 


2.300 


-.125 


2.196 


1.4 


2.439 


.292 


.059 


— I 


.2291 


2.640 


— .227 


2.440 


1.6 


2.731 


.351 


.067 


— I 


1739 


3.032 


-.331 


2.735 


X.8 


3". 082 
3.500 


.418 








3.482 
4.000 


-.435 
-•540 


3085 
3 506 


3.0 


• • • • 


• • • • 













36 



EMPIRICAL FORMULAS 



Since the values of x form an arithmetical series and the 
second differences of the values of y form approximately a 
geometrical series, it is .evident that the relation between the 
variables is fairly well represented by 

y = a+bx+c(P. 
Taking the second difference 



or 



log A2y =log c(J^- i)2+(log d)x. 



Plotting the logarithms of the second differences of y from 
the table to the values of x, Fig. lo, it is foimd that log J = .3000 



-1.0 
-1.1 

I" 

-1.6 



-1.7 

































\ 


^ 
























_> 


^ 






^ 


\ 


















y 


x^ 












X 


\ 










y 


y 




















'^ 


^, 


J 


H 


Y 
























/^ 


M 


^, 






















> 


[y 








H 


\, 




■ 












^ 


/ 












^ 


M 


^, 








1 


/" 




















^> 


\ 


^ 


^ 
































( 


) J 


i J 


I .( 


5 .1 


raluei 


JD 1. 
BOtO 


2 L 

> 


i 1. 


« JL 


B 2. 


i) 





.8 



.4 1 

;^ 

CO 

0-3 
n2 
-i4 
-.6 



Fig. 10. 

or J = 1.995, approximately 2. The intercept of this line, 

— 1:6500, is equal to log c(J^ — i)^. 

Since 

.02239 =c(2** — 1)2, 



C = I.OII. 



DETERMINATION OF CONSTANTS 37 

Plotting y— (1.01)2* to Xy Figi lo, the values of a and b 

are found to be 

a= 0.5, 

6= -0.5x5. 

The formula derived from the data is 

y=o.5-o.5i5a;+(i.oi)2*. 

In the last colmnn of the table the values of y computed 
from the formula are written down. Comparing these values 
with the given values of y it is seen that the formula reproduces 
the values of y to a fair approximation. 

IX. y = io«+^+«^*. 
Values of x form an arithmetical series and A* log y constant. 

If two variables, x and y, are so related that when values of x 
are taken in an arithmetical series the second differences of the 
values of log y are constant, the relation between the variables is 
expressed by the equation 

This becomes evident from I when y is replaced by log y. 

log y = a+bx+c:x^y 

which represents a parabola when logy is plotted to x. The 
constants are determined in the same way as they were in 
formula I. 

X. y=ksr/. 

Values of x form an arithmetical series and values of A' log y form a 
geometrical series. 

If two variables., x and y, are so related that when values of x 
are taken in an arithmetical series the second differences of the 
corresponding values of log y form a geometrical series, the relation 
between the variables is expressed by the equation 

X y = kff. 



38 EMPIRICAL FORMULAS 

This becomes evident by taking the logarithms of both sides 

and comparing the equations thus obtained with VIII. X 

becomes 

log y = log * + (log s)x + (log g)d\ 

This is the same as VIII when y is replaced by log y, a by 
log ky b by log Sy and c by log g* 

XL y= ^ 



a+bx+cx^ 
Values of x form an arithmetical series and A*- are constant. 



If two variableSy x and y, are so related that when values of x 
are taken in an arithmetical series the second differences of the 

corresponding valines of - are constant, the relation between the 
variables is expressed by the equation 

X 



XI y = 



a+bx+cx^ 
Clearing equation XI of fractions and dividing by y 

- = a+bx+cx^. 

y 

X • 
This is of the same form as I, and when - is replaced by y 

the law stated above becomes evident. 

If a is zero XI becomes 

_ I 

^'b+ac' 
which, by clearing of fractions and dividing by y, reduces to 

- = b+cXy 

y 

a special case of III. 

* For an extended discussion of X see Chapter VI of the Institute of 
Actuaries' Text Book by George King. 



\ 



DETERMINATION OF CONSTANTS 



39 



If c is zero XI becomes a special case of XVI, or 



X 



y 



X . 



which is a straight line when - is plotted to x, 

y 

Corresponding values of x and y are given in the table below, 
find a formula which will express approximately the relation 
between them. 



2 

3 

4 

5 
6 

7 
8 

9 
o 

I 

2 

3 
4 
5 



y 


X 


a1 
y 


A2^ X 

y 




Y 


Y 
X 


X 

2.S«a 

y 


o.ooo 




I • • • 


• • ■ • • ■ 4 


■ 









1.333 


0.075 


.100 


.050 — 


■9 


-2.703 


3.003 


.050 


1. 143 


0.175 


.150 


.050 — 


.8 


— 2 . 603 


3.254 


•075 


0.923 


0.325 


.200 


.050 — 


.7 


-2.453 


3- 504 


.100 


0.762 


0.525 


.250 


.050 — 


.6 


-2.253 


3.755 


.125 


0.645 


0.775 


.300 


.051 -. 


•5 


— 2.003 


4.006 


.150 


0.558 


I 075 


.351 


.049 - 


4 


-1.703 


4.257 


.175 


0.491 


1.426 


.400 


.047 - 


3 


-1.352 


4.507 


.201 


0.438 


1.826 


.447 


.058 - 


2 


-0.952 


4.760 


.226 


0.396 


2.273 


.505 


.040 — 


I 


-0.503 


5. 030 


.248 


0.360 


2.778 


■545 


.054 





0.000 






0.331 


3.323 


.599 


.056 


I 


0.545 


5. 450 


.298 


0.306 


3.922 


.655 


.051 


2 


1. 144 


5.720 


.332 


0.284 


4.577 


.706 


.03s 


3 


1.799 


5-997 


.352 


0.265 


5.283 


.741 


• • • • 


4 


2.505 


6.262 


.383 


0.249 


6.024 


• • • 


• • • • 


5 


3.246 


6.492 


.399 



Com- 
puted y 

0.000 
1.329 

I/- 140 
0.929 
0.760 
0.644 

0.558 
0.491 

0.438 

0.395 
0.360 

0.331 
0.305 
0.284 

0.265 
0.249 



The values of x form an arithmetical series and since the 

• X • 

second differences of - are nearly constant the values of y will 

y 



be fairly well represented by 

y= 

or 

X 



X 



a+bx+cx^' 



- = a+bx+cx^, 

y 



This represents a parabola when - is plotted to x. 

Let X=x—ij 

F = --2.778. 

y 



40 



EMPIRICAL FOIOiULAS 



From these equations are obtained 



X 



The formula becomes 



- = 7+2.778. 

y 



Y+2,y7S--a+b{X+i)+c(X+iy 

=a+b+c+{b+2c)X+cX^. 
Since the new origin lies on the curve 

a+6+c = 2.778, 




the equation reduces to 



Y=(b+2c)X+cX^y 



or 



-— = b+2c+cX. 



Y . 



This represents a straight line when — is plotted to X. The 

value obtained for c from Fig. 11 is 2.5. The value of b could 
be obtained from the intercept of this line but the approximation 



DETERMINATION OF CONSTANTS 41 

will be better by plotting — 2.501:2 to a;. In this way is obtained 

y 

the line 

y 

From the lower part of Fig. 11 the values of a and b are 

found to be 

a = .025, 

^ = .2525. 

Substituting the values of the constants in XI the formula 
becomes 

X 
y= r. 

.025+. 25250^+2.5^^2 

In the last column of the table the values of y computed 
from this equation are given and are seen to agree very well 
with the given values. 



CHAPTER m 

XII. y^a^. 

Values of x form a geometrical series aijd the values of y form a 
geometrical series. 

Ij Pwo variables, x and y, are so related thai when the values of 
X are taken in a geometrical series the corresponding values of y 
also form a geometrical series, the relation between the variables is 
expressed by the equation 

XII y = ax\ 

From the conditions stated equations (a) and (6) are obtained. 

Xn^xif-^, (a) 

yn=yiFP-\ {b) 

where r is the ratio of any value of x to the preceding one 
and i? is the ratio of any value of y to the preceding one. 
Taking the logarithm of each member of (a) 

log Xn = \ogxi + {n-i) log r, 

loga;n-loga;i 

n — I = -. . 

logr 

Also by substituting this value of n — i in the value of y» in 
equation (6), 

log Xi — log X\ 

yn=yiR ""' 

log XI / 1 \ log Xn 
= yiR l«8r |^2?*ogrj 



DETERMINATION OF CONSTANTS 



43 



where 



and 



log xn 



a^yiR *°«'- 



The following data (Bach, Elastizitat und Festigkeit) refer 
to a hollow cast-iron tube subject to a tensile stress; x represents 
the stress in kilogrammes per square centimeter of cross-section 
and y the elongation in terms of -^ cm. as unit. 



X 


9-79 


20.02 


40.47 


60.92 


81.37 


lOI . 82 


204.00 


408.57 


y 


0.33 


0.695 


1.530 


2.410 


3 295 


4.185 


8.960 


19.490 


log X. . . 


oipQoS 


I. 3014 


1.6072 


I . 7847 


I . 9104 


2.0078 


2.3096 


2.6II3 


logy... 


—0.4815 


-0.1580 


0.1847 


0.3820 


0.5178 


0.6217 


0.9523 


1.2898 


Comp. 


















y.... 


0.324 


0.714 


1. 541 


2.416 


3.323 


4.252 


9.132 


19.600 



Selecting the values of x which form a geometrical series, 
or nearly so, it is seen that the corresponding values of y form 
approximately a geometrical series, and, therefore, the relation 
between the variables is expressed by the equation 



or 



y = ax , 
log 3^ = log a +6 log X, 



If now logy be plotted to logo; the value of b will be the 
slope of the line and the intercept will be the value log a. Fig. 
12 gives 6 = i.iT In computing the slope it must be remembeted 
that the horizontal unit is twice as long as the vertical unit. 
The intercept is —1.5800 or 8.4200—10, which is equal to 
log 0.0263. The formula is 



.V 



y = .02630:' 



The values of y computed from this equation are written 
in the last line of the table. They agree quite well with the 
observed values. 



44 



EMPIRICAL FORMULAS 



XIII. y=a+b log x+c log% 
Values of log x form an arithmetical series and A^y constant. 

If two variables, x and y, are so related that when values of 
log X are taken in an arithmetical series the second differences of 
the corresponding values of y are constant the relation between the 
variables is expressed by the equation 



xin 



y = a+b log x+c log^jc. 



This becomes evident from I by replacing x by log x. The 
law can also be stated as follows : If the values of x form a geo- 



1.4 
1.2 
1.0 

.8 

^\ 

o 

s.2 

!• 

-.2 
-.4 

-.6 

-.8 








































^ 




































,^ 


,^ 


































^*^ 


^ 


































> 


^ 


x^ 


































^ 


^ 


































/ 


^ 


































,^ 


X 




































/ 


r 
































^rf* 


^ 




































y^ 


x' 


































y 


r^' 




































X 


















































































B ,\ 


) 1 


L 1. 


.1 1 


.2 L 


3 1. 


.4 1. 


5 1. 


6 1. 

VaJ 


7 L 
lues 


8 1. 
oil 


9 5 

Og i 


S 2, 

C 


.1 2. 


2 2. 


.3 2. 


4 2. 


S 2. 


6 2. 


7 2.8 



Fig. 12. 

metrical series and the second differences of the corresponding 
values of y are constant the relation between the variables is 
expressed by the equation 

y =a+6 log x+c \o^x. 
If c is zero the formula becomes 

y = a+b log Xy 
which is V with x and y interchanged. 



t/ 



DETERMINATION OF CONSTANTS 45 

Formula XIII represents a parabola when y is plotted to 
logo;. The constants are determined in the same way as the 
constants in I. 

XIV. y=^a+hx\ 

Values of x form a geometrical series and values of ^y form a geometrical 
series. 

If two variables, x and y, are so related that when the values of 
X are taken in a geometrical series the first diferences of the cor- 
responding values of y form a geometrical series, the relation between 
the variables is expressed by the equation 

XIV y = a+bx\ 

As in XII the «th value of x is 

x„=a;ir"-' {c) 

The series of first differences of y may be written 

Ayi, Ayii?, ^ylB?, ^yiK? . . . Ayii?*-2, 

and the values of y are 

yi, yi+Ayi, yi+Ayi+AyiiJ, yi+Ayi+AyiiJ+Ayii?^ . . . 
yi+Ayi+Ayii2+Ayii22+Ayiie3+ . . . +Ayii?*-2. 

That is the «th value of y will be 

y«=yi+Ayi+Ayii?+Ayii22+Ayiie3+ . . . +Ayii?*-2 

=yi+Ayi(i+i2+i22+J23+ . . . +i?-2) 

=yi+Ayi ^_^ {d) 

Taking the logarithm of each member of (c), 

log Xn=\og xi+{n-i) log r 
log^^-log^ 

n — I ; . 

logr 



\y 



46 



EMPIRICAL FORMULAS 



Substituting this value of »— i in the nth value of y given 
in (J), 



yn=yi+^yi 



log xn —log x\ 



i-R 



log xi / 1 \\ogxn 



. . log XI / 1 \ 

^ i-R i-R \ I 



= a+6(io0^°^^» 
=a+6io*°«^»' 

Let it be required to find the law connecting x and y having 
given the values in the first two lines of the table. 



X 


2 


3 


4 


5 


6 


7 


8 


y 


4.21 


5.25 


6.40 


7.65 


8.96 


10.36 


II. 81 


log JC 


.3010 


.4771 


.6021 


.6990 


.7782 


.8451 


.9031 


X 


2 


2.5 


3.125 


3.906 


4.883 


6.104 


7.630 


y 


4.210 


4.720 


5.388 


6.290 


7.515 


9. no 


11.275 


logjc 


.3010 


.3979 


.4948 


.5918 


.6887 


.7856 


• • • • 


Ay 


.510 


.668 


.902 


1.225 


1 . 595 


2.165 


• • • • 


log Ay 


- .2924 


-.1752 


— .0448 


.0881 


.2028 


.3358 


• • • • 


y— 2.72 


1.49 


2.53 


3.68 


4.93 


6.24 


7.64 


9.09 


log(y-2.72) 


.1732 


.4031 


.5658 


.6928 


•7952 


.8831 


.9586 


Computed y 


4.21 


5.25 


6.41 


7.65 


8.98 


10.36 


II. 81 



In the fourth line values of x are given in a geometrical 
series with the ratio 1.25. In the fifth line are given the cor- 
responding values of y read from Fig. 13. The first differences 
of the values of y are written in the seventh line. These differ- 
ences form very nearly a geometrical series with the ratio 1.336. 
Since the ratio is nearly constant the law connecting x and y 
is fairly well represented by the equation 

y = a+bxf^. 

There are two methods which may be employed for deter- 
mining the values of the constants, either one of which may 
serve as a check on the other. 



■■ 



i^ 



DETERMINATION OF CONSTANTS 



47 



First Method, Select three points, A, P, and Q on the 
cxirve, Fig. 13, such that their abscissas form a geometrical 
series and two other 
points, R and 5, such 
that R has the same 
ordinate as A and the 
same abscissa as P, S 
the same ordinate as P 
and the same abscissa 
as Q, The points may 
be represented as fol- 5^ 
lows: 



12 



11 



10 



8 



o 

m 
o 



3 



«7 



A = (xo, a+bx(y); 
.. P={xor,a+bxoY); 
. Q^ixor^a+bxo'r^y, 

R={xor, a+bxa"); 

S^(xor^,a+bxoY). 3 

The equation of the 2 
line passing through P 
and Q is 















A 












A 


! 












/ 












/ 


f 










A 


V 


• 








X 


V 

1 






^^ 




/, 


/ 




^^ 


/ 


n^ 


A, 


V 


^ 


p 


y' 






/ 


/ 


y^ 










/ 


• 












2 


! 3 


\ 


s 


1 c 


1 


8 



Values of x 

Fig. 13. 



XQr(r—i) r—i 

The equation of the line passing through the points R and 



X(f(r—i) 



r — i 



These two lines intersect in a point whose ordinate is a. In 
Fig. 13 xo is taken equal to 2 and r equal to 2. The value of 
a is foimd to be 2.72. The formula then becomes 



or 



y—2,T2 = bx^j 

log C}'— 2.72)=log6+^logJC. 



1 



L^ 



48 



EMPIRICAL FORMULAS 



In Fig. 14 log (y— 2.72) is plotted to log x and b and c 
determined as in XII. It is seen that the points lie very nearly 
on a straight line. The values of c and h are read from Fig. 14. 

log 6 = 9.7840—10; 
b= .61. 
The law, connecting x and y then is 



y = 2.72+.6ijc^' 



.4 
.2 

o 

r 

-»6 

























I 




























/ 








/ 










. 










/ 


/ 






J 


/ 


















c 


/ 








/ 




















/ 








/ 




















/ 








/ 


f 


















J 


/ 






J 


/ 


















J 


V 






> 


/ 




















/ 








/ 




















/ 








A 




















/ 


/^ 






/ 


f 


















^ 


/ 






t^ 


/ 




















/ 








/ 




















/ 








/ 




















/ 


/ 






/ 


f 




















/ 






^ 


/ 























.1 .2 .3 .4 ^ .6 .7 
Values of log x 

Fig. 14. 



.8 .9 1.0 



1.0 



.9 

.8 
I 

At 



.6 



.4 



.2 



.1 



09 



The values of y computed from- this formula are written in 
the last line of the table. 

Second Method. From the equation 



y^a-^-iof 



^ 



DETERMINATION OF CONSTANTS 49 

we have 

y+Ay=a+bxl^r^; 

Ay = 6a:*(r*'-i); 
log Ay = log 6(r*-i) +c log op. 

This is the equation of a straight line when log Ay is plotted 
to logic. Fig. 14 shows the points so plotted and from the 
line drawn through them the values of 6 and c are obtained. 

^ = 1.3, 
6 = .6i. 

a is foimd by taking the average of all the values obtained 
from the equation 

a is equal to 2.72. 

XV. y=aio^\ 

Values of X form a geometrical series and A log y form a geometrical 
series. 

If two variables J x afui y, are so related that when values of 
X are taken in a geometrical series the first differences of the cor- 
responding valtces of log y form a geometrical series, the relation 
between the variables is expressed by the eqtiation 



J>x' 



XV y^aid 

This equation written in the logarithmic form is 

logy=loga+bxf^. 

Comparing this with XIV it is evident that if the values of x 
form a geometrical series the first differences of the corre- 
sponding values of log y also form a geometrical series. 

In an experiment to determine the upward pressure of water 
seeping through sand a tank in the form shown in Fig. 15 was 
filled with sand of a given porosity and a constant head of 



EUPIRICAL FORMULAS 



water of four feet maintained.* The water was allowed to flow 

freely from the tank at A . The height of the column of water 

in each glass tube, 



f 
















six inches 


apart. 




was measured. In 


1 


the table below x 


t 


represents the dis- 


tance of the tube 


I 


from the water head 




in feet, and y the 




height of the column 




of water in the tube, 




also in feet. It is 




required to find the 




law connecting x 


Fig. 15. 


and>. 


Tube 


, 


, 


J 


4 


s 


6 


, 


, 


9 


I** 


:%6 


' 


3I1J 


'■M^A 


isoio 


\ 


L 


"i'.. 


ih 


Jiu 


-'S.. 


Jog (y-b^) 
Computed y 


'■'^ 


z 


J617 

"mo 

195 


- isiie 

- .oiiS 

.36J. 
1.195 


:;l 


- 


i 


- .J6I7 


~s 


- .644J 
.3769 


- .5J.8 



In the fifth line values of x are selected in a geometrical 
series and the corresponding values of y written in the next 
line. In Fig. 16 log {—A log y) is plotted to log*. The 
points lie on a straight line. On account of the small number 
of points used in the test we select formula XV on trial. 
From the formula 

y = aio'^ 
it follows that 

logy = loga+Aa;' 

* Coleman's Thesis, University of Michigan. 



DETERMINATION OF CONSTANTS 



51 



logyk = loga+bxt' 
log yt+i =log a+bxt'r^ 

log (A log y) = log b(r^-i) +c log x. 

If A log y is negative b is negative, in which case it is only 
necessary to divide the equation by — i before taking the 
logarithms of the two members of the equation. 





























« 










































y 


A 






























A 


Y 
































y 


f* 






0.5^ 
























y 


y 














■ 
















y 


^ 










5 


















y 


r ' 
















-1 ?? 
















/ 


/^ 


















o 












^ 


y 






















i 










/ 


r- 
























4.5^ 






^ 


y 






























y 


^— 












. 


















—2 


-.i 


I 


I 
^•« 


3 


^1 

. 9i 


I 




L 
aluc 


Fig 


log 


.1 

X 

). 




1 


2 


»\ 


) 


A 


L 



The last equation above represents a straight line when 
log (a log y) is plotted to log x. The slope gives the value of 
c and the intercept gives log6(r*'-i). From Fig. i6 values 
of b and c are readily obtained. 

6= —.02282. 

In the next to the last line the value of a is computed for 
each value of x from the equation 



log a = log y+ .022801:^*. 



52 EMPIRICAL FORMULAS 

The average of these values of a gives 

a =2.314. 



The formula obtained is 

y= (2.3 14) 10 



-.0228x^' 



The values of y computed from this equation are written 
in the last line of the table. The agreement is not a bad one. 



CHAPTER W 

XVI. ix+a)(y+b)=c. 

I 

Points represented by Ix—xtj ) lie on a straight line. 

If two variables, x and y, are so related that the points repre- 
sented by Ix—Xky -* ) lie on a straight line, the relation between 

\ y-yk/ 

the variables is expressed by the equation 

XVI {x+a)(y+b)=c. 

Let x—Xic=X, 

where Xt and y* are any two corresponding values of x and y. 
From the above equations 

y=Y+yic. 

Substituting these values of x and y in equation XVI we 
have 

(X+x,+a){Y+yt+b)=c, 
or 

XY+(y,+b)X+{x,-\-a)Y+(x,+a)(yt+b)^c. 

Since (xt, yt) is a point on the curve 

{xt+a)(yjc+b)=Cy 
and 

XY+(yk+b)X+(x,+a)7==o. 

53 



54 



EMPIRICAL FORMULAS 



Dividing the last equation by Y 



X 



X+{yk+b)—+Xk+a=-o, 



or 



I Y__^ +^ 



( 



X 
This represents a straight line when X is plotted to — . 

The theorem is proved directly as follows: If the points 



x—x , 



X — Xk 



lie on a straight line its equation will be 



Clearing of fractions 



X — Xx 

y-y^ 



=p{x—X))-\-q. 



x-x:,=p{x-x^{y'-yi)-\'q{y-y^. 

This is plainly of the form 

{x-{-a)(y-\-h)^c. 

The following tables of values is taken from Ex. i8, page 138 
of Saxelby's Practical Mathematics. It represents the results 
of experiments to find the relation between the potential differ- 
ence V and the current A in the electric arc. The length of 
the arc was 3 mm. 



A (am- 
peres) 
y (volts 
X 
Y 
X 

Y 


1.96 
67.00 




2.46 

62.75 
0.50 

-4.25 


2.97 
59.75 

I.OI 

-7.2s 


3.45 
58.50 

1-49 
-8.50 


3.96 

56.00 

2.00 

—11.00 


4.97 

53.50 

3.0I 

-13-50 


5.97 

52.00 

4.01 

—15.00 


6.97 

51.40 

5.01 

— 15.60 


7.97 

50.60 

6.01 

—16.40 




- ,1176 


- .1393 


- .1752 


- .1817 


— .2228 


— . 2670 


— .3210 


- .3665 


Com- 
puted F 


66.99 


62.74 


59 80 


57.80 


56.19 


53.94 


52.44 


51.36 


50.55 



Let A be taken as abscissa and F as ordinate and transfer 
the origin to the point (1.96, 67.00) by the substitution 

X=^ — 1.96, 

7 = 7-67.00. 



DETERMINATION OF CONSTANTS 



55 



The values of X and Y are given in the third and fourth 

X 
lines of the table. The values of — are plotted to X in Fig. 17 



OD 
0) 















' 


^^ — ' 


' ^r--- 






> 












' -^ 


.. 














^" 




.^^ 




• 












J 


5 


i 3 4 ( 
Values of X 


^ 6 ? 



Fig. 17. 

and are seen to lie nearly on a straight line. It is therefore 
concluded that the formula is 

{V+b){A+a)=c. 

By the equations of substitution this becomes 

(X+i.96+a)(F+67.oo+6) =c, 

or A 

XF+(67.oo+6)X+(i.96+»0F=o. 

Dividing by 7(67.00+6) 

X _ I y 1.96+g 

Y 67.00+6 67.00+6* 



The slope of this line is 

— L2_JI^. From Fig. 17 
67.00+6 



67.00+6 



and the intercept is 



67.00+6 



= •045; 



Solving these equations 



From formula 



i.o6+a 

(^=0.151, 
&= -44-78, 

{;= 46.89. 



56 EMPIRICAL FORMULAS 

These values give 

(.4+o.i5i)(F-44.78)=46.89. 

In the last line of the table are written the values of V com- 
puted from the above formula 

XVIa. y=aio*+^ 

(I y y \ 
log -^^j log — ) lie on a straight line. 
x-xu yt yt/ 

If two variables, x and y, are so related that the points repre- 
sented by I log — , log 2- ) lie on a straight line, the relation 

\x-xic y* yt/ 

between the variables is expressed by the equation 

b 
XVIa y=aIO*+^ 

By the condition stated 

y I 'v 

log^=w log ^+6, 

yt x-xt yt 

where oc* and y^ represent any two corresponding values of 
X and y. m is the slope of the line and b its intercept. Clear- 
ing the equation of fractions 

(log y-log yi){x''Xi) = w(log y-log y*)+&(ap-a;*), 
or 

log y{x-xt-m) = (6+log yi)x-\og yk{xu+m) -bxu. 

iQg y _ (^+log yQa: -log yu{xu+m) -bxt 

x—xt—nt 

Ax+B 



x+C 
B-AC 



iH 



log a-\ 



x+C 
b 



x+C 



DETERMINATION OF CONSTANTS 67 

Therefore 

6 
ft 

For the purpose of determining the constants the equation 
is written in the form 

logy=loga+^^, 

• 

(logy-loga)(a;+c)=6, . 
Let 

logy=logF+logy*, 
and 

Then follows 

(log F+log y,-log a){X+Xi,+c) =6, 

X log F -f-log Y{xk -\-c) -f-ZOog yu -log a) + (log y* -log a) (a;* H-c) = 6. 

But 

(log yic - log a) (ap*+c) = 6, 

since the point (jc*, )^t) lies on the curve. 

X log F+log F(:r*+(;)+Z(log y^-log a) =o. 

Dividing this equation by X ^ 

log F = - (oct+c) -^|:- +log a-log y*. 
Replacing log F and X by their values 

log ^ = - {xi,+c)—^ log — +log a-log y*. 

From this it is seen that if log — be plotted to log ~ 

y* oc-ocfc ^'y* 

a straight line is obtained whose slope is —{xn+c) and whose 

intercept is log a — log y^. If the slope of the line is represented 

by M and the intercept by B 

c^—M—Xtf 
loga=B+logyt. 



68 EMPIRICAL FORMULAS 

By writing XVIa in the logarithmic form 

a line is obtained whose slope is b. 

XVII. y=a€"+6A 

> 1 

yt yt I 

lie on a straight line whose slope, M^ is positive and intercept, B^ is negative, 
and M^-\-4B positive. 

If two variables J x and y, are so related that when values of x 
are taken in an arithmetical series the points represented by 

y}±2.yk+2\ li^ 0fi a straight line whose slope, M, is positive 

, yt yt / 

and whose intercept, B, is negative and also M^+4B is positive 
tJie relation between the variables is expressed by the equation 

XVII y^ae'^+b^, 

. Let {xk,yk), {x+^x, yk+i), {xic+2^x, yk+2) be three sets of 
corresponding values of x and y where the values of x are taken 
in an arithmetical series. We can then write the three equations, 
provided these values satisfy XVII. 

)^, = ae'^*+6/^*, ....... (i) 

)'*+i=a€''^V^''+&/V^, (2) 

y*+2=a€^'^*e2^^^+6e'^V^^^ (3) 

Multipl3dng (i) by e^^ and subtracting the resulting 
equation from (2) 

yu^i-e'^y.^be^^'Ke'^^-'e'^^) (4) 

Multipl3dng (2) by e^^ and subtracting from (3) 

Multiplying (4) by e^^^ and subtracting from (5) there 
results 



DETERMINATION OF CONSTANTS 



59 



or 



yk+2 _ r^Axi^dAx\ Vk+l ^(c -\-d)Ax^ 

yt yt ' ' 

The values of c and d are fixed for any tabulated function 
which can be represented by XVII, and therefore, the last 

equation represents a straight line when ^^^ is plotted to 2!^:^. 

yt yk 

The slope of the line is 
and the intercept is 

It is seen that M is positive, B negative, and M^+4B posi- 
tive, for 

and 

In the first two lines of the table are given corresponding 
values of x and y. It is desired to find a formula which will 
•e:q)ress tjie relations between them. 



X 


I.O 


1.5 


2.0 


2.5 


3.0 


3.5 


4.0 


4.5 


5.0 


y 


+ .3762 


+ .0906 


— .1826 


- .4463 


- .7039 


- .9582 


— 1.2119 


-1.4677 


—1.7280 


yk+i 
y* 






+ .241 


— 2.0IS 


+2.444 


+1.577 


+1.361 


+1.265 


+1.211 


y*+2 
yt 






-.485 


-4.926 


+3.855 


+2.147 


+1.722 


+1.532 


+1.426 


g-A12x 

y^-.mx 


+ .662 
+ .319 


+ .539 
+ .071 


+ .439 
-.131 


+ .359 
- .295 


+ .290 
- .429 


+ .236 
- .538 


+ .192 
- .626 


+ .157 
- .698 


+ .127 
- .757 


Computed y 


+ .371 


+ .087 


-.18s 


- .447 


- .704 


- -957 


—1. 210 


-1.464 


-1.723 



Plotting the points represented by 1^^^, ^!^\ Fig. 18, 

\ yt yt / 

a straight line is obtained whose equation is 



— = 1.97 .90, 

yk yk 



60 



EMPIRICAL FORMULAS 



if =1.97, 

B= — .96. 

Since M is positive, B negative, and AP+4B positive, it 
follows that the relation between the variables is expressed 
approximately by XVTE. It has been shown that the slope 

.4 















/ 


4 








• 






/ 


8 
2 










/ 


« 










/ 




•• 
+ 

I 
t 


1 


1. 








/ 












/ 






-3 
-4 






/ 










i 


/ 










/ 










1 


V 










-5 


/ 











.2 
.1 



-.1 



«4 

o 

d 
13 
>--.4 



-%5 



-.6 



-7 



-^ 



-2 -i „ 1 

Values of ^^ 

Fig. 18. 















/ 














/ 


- 










/ 














/ 












/ 


• 












/ 












/ 














/ 












/ 














/ 








• 




/ 














/ 












/ 

















I .5 


I .; 


i 


1 u 


s 


5 .7 



Values oir**"* 

Fig. 19. 



of the line is equal to e^^-\-ef^, and the intercept is equal to 
_^(c+d)Aa; Since Ax is .5 



eV^ = i.97, 



= .96. 

From these equations are obtained the values of c and d, 



^=-.247, 



^ 



d = .i65.^ 



N 



DETERMINATION OF CONSTANTS 61 

The formula is now 

Dividing both sides of this equation by e*^^^* gives the 
equation 

which represents a straight line when ye"*^^* is plotted to 
^-.412* 'pjjg values of these quantities taken from the table 
are plotted in Fig. 19 and are seen to lie very nearly on a straight 

line. 

This line has the slope 2.00 and intercept — i.oi. Sub- 
stituting these values of a and b in the formula it becomes 

;y = 2e-2*^*-i.oie-^«*'. 

It is seen that the errors in the values of y computed from 
this formula are in the third decimal place. The values are as 
good as could be expected from a formula in which the con- 
stants are determined graphically. For a better determination 
of the constants the method of Chapter VI must be employed. 

XVIII. y=€^{ccosbx+dsmbx). 
Values of x form an arithmetical series, and the points (^ — , 5!L_ ? ] 

\ yt yu / 

lie on a straight line. Also M^+4B is negative. 



( 



// two variables, x and y, are so related that when values of 
X are taken in an arithmetical series the points represented by 

y*+l 2!!±? ] lie on a straight line whose slope M and intercept 

B have such values that M^+4B is negative, the relation between 
the variables is expressed by the equation 

XVIII y = 6^(c cos bx+d sm bx). 

Let X and yt be any two corresponding values of the variables. 
We have the three equations 

yt=ef^(c cos bx+d sin bx), (i) 

yt+i =^€f^[c cos (J)X+bAx)+d sm (bx+bAx)] 



62 EMPIRICAL FORMULAS 

= d^'e^^^[c{cos bx cos bAx — sin bx sin bAx) 

+d(sin bx cos iAic+cos bx sin bAx)] 

=€^ef^[{c cos bAx+d sin JAit:)cos bx 

+ {d cos 6Ax — c sin JAx)sin Jrrj. . (2) 

The value yit+2 can be written directly from the value of 
y + 1 by replacing A^ by 2Ajc. 

yk->f2=ef^e^'^^[{c cos 2bAx+d sin 26A:c)cos 6jc 

+ {d cos 26A:c— c sin 26Aji[:)sin bx] (3) 

Subtracting (i) multiplied by e^(c cos bAx+d sin bAx) 
froiri (2) multiplied by c we have 

cyk+i — ^"^(^ cos bAx+d sin bAx)yk 

=:C€^ef^{d cos bAx—c sin 6Aa:)sin foe 

—def^ef^^c cos 6A^+d sin 6Arc)sin to 
= -(c2+(P)6^6^sm6Aa:sin6x (4) 

Similarly 

cyk+2-'e^*^(c cos 2bAx+d sin 2bAx)yt 

= -((;2+(PV*e2^sin2JA:rsinto (5) 

Multipl)dng equation (4) by e*^ sin 26Aji[: and subtracting 
it from (s) multiplied by sin bAx 

c sin bAxyk-{-2—e'^'^{c cos 26Aa: sin bAx+d sin 26Aa: sin 6Ajc)yfc 

— ce"^ sin 2bAxyt+i +e^'^{c cos 6Ax sin 26Arc 

+d sin bAx sin 26Aa:)y* = o. 
Simplifying 

c sin bAxyk+2'-c€f^ sin 2bAxyk+i +ce^'^ sin ftAjcy* =0. 

Dividing by c sin bAxyk, 

^-^ = 2 cosbAx^^^-e"^. 

yk yk 



DETERMINATION OF CONSTANTS 



63 



The values of a and b will be fixed for any tabulated function 
which can be represented by XVIII, and therefore, the last 

equation represents a straight line when ^^^^ is plotted to 
2-^. The slope of the line is 

y* 



M = 2€f^ COS h^x, 



and the intercept 



B=-(? 



aAx 



It is evident that AP+4B is negative. 
It is possible that in a special case AP+^B might be zero, 
but then b would be zero and hence 

y=cef^y 
which is formula V. 

Corresponding values of x and y are given in the first two 
columns of the table below. It is required to find a formula 
which will represent approximately the relation between them. 





y 


y*+i 
yt 


yk+2 
yt 


.OSz 


co^hx 


tSinhx 


.O&r 
cosbx 


y 


Com- 


X 


«-^cos bX 


puted y 





+ .300 


• • • • 




I. 0000 


I . 0000 


.0000 


1 . 0000 


+ .300 


+ .308 


z 


+ .011 


• • • • 




I . 0833 


+ .8646 


+ .5812 


+ .9366 


+ .012 


+ .018 


2 


- .332 


+ .04 


— 1. 11 


I. 1735 


+ .4950 


+ 1.7556 


+ .5809 


- .571 


- .327 


3 


- .636 


—30.2 


-57.8 


I. 2712 


— .0087 


-114.59 


— .0111 


+57.3 


- .634 


4 


— .803 


+1.92 


+ 2.42 


I. 3771 


— .5100 


— I . 6864 


- .7023 


+1.143 


— .804 


5 


— .761 


+1.26 


+ 1.20 


I. 4918 


- .8732 


- .5581 


—1.3026 


+ .584 


— .761 


6 


- .48s 


+ .95 


+ .60 


I.6I6I 


- .9998 


+ .0175 


-1.6159 


+ .300 


- .48s 


7 


— .017 


+ .64 


+ .02 


1.7507 


- .8557 


+ .6048 


—1. 4981 


+ .011 


— .012 


8 


+ .537 


+ .04 


— i.ii 


I . 8965 


- .4797 


+ I. 8291 


— .9098 


- .590 


+ .545 


9 


+1.027 


—31.6 


—60.4 


2.0544 


+ .0262 


— 38.1880 


+ .0538 


+19.08 


+ 1.035 


10 


+1.298 


+1.91 


+ 2.42 


2.2255 


+ .5250 


— I. 6212 


+ I. 1684 


+ 1 .III 


+1 . 299 



In Fig. 20 the points represented by ( ^^, ^^ ) are plotted. 

\ yt yt I 

They lie very nearly on the straight line whose equation ist 






64 



EMPIRICAL FORMULAS 



Since (1.878)2 —4(1.18) is negative the relation between the 

variables is expressed 
approximately by the 
equation 



y = ef^{ccosbx+dsinbx). 

It was shown that 
the slope of the line is 
equal to 2 (cos bAx)ef^^ 
and the intercept equal 
to-e^*^. Since Ax is 
equal to unity we have 

2^ cos 6 = 1.875, 

^ = 1.175, 
log cos b = 9.9370— ID, 

6=30° 10' ap- 
proximately, 

a = .08. 

The formula is now 























/ 






















/ 


2JD 


















/ 




IS 

4 ^ 
















/ 


f 




1.4 
L3 














> 


/ 


















/ 




















/ 








« 9k 

^ .8 












/ 




















/ 




















/ 










*4 

8 •« 

9 










y 




















/ 


















> 


/ 










- 


.2 








-A 















-s2 
"A 






A 


/- 


















/ 


















i 


















'A 




/ 




















/ 


















"A 
*L0 


/ 




















/ 




















-L2 


^ 






















I .- 


V .( 


S .i 


i 1 


.0 I. 


2 1. 


4 1. 


6 1. 


8 2. 



Values Of ^^17-^ 



Fig. 20. 



y = e-^^{c COS 30-J:r+(i sin 30^0;), 

where 30^ is expressed in degrees. 

Dividing the equation by e-®"* cos 30^0; 



6'^^ COS 30^0?' 



=c+d tan 30^0:®, 



which is a straight line when -775 =^^^ — r—=, is plotted to tan 

e-^^ cos 30^0: 

30^:*;°. In Fig. 21 these points are plotted and are seen to 

lie nearly on a straight line whose slope is — .496 and intercept 



DETERMINATION OF CONSTANTS 



65 



.308. Two of the points are omitted in the figure on account 
of the magnitude of the coordinates. Substituting the values 
of constants just foimd in the formula the equation expressing 
the relation between x and y is 

y«e*®8*(.3o8 cos 30^01;°— .496 sin 30^°). 



u 



S.1.0 



8 

s 



0.5 



The last column 
in the table gives 
the values of y 
computed from the 
equation. The 
agreement with the 
original values is 
fairly good. 

In case c is zero 
XVIII becomes the 
equation for damped vibrations, y=def^ sin bx. 



^ 
f -0J5 



•^_ 
































^ 


^ 




































V, 


































"^ 


^ 


w. 
































V 


^ 




































^ 


































^ 


^ 




































^ 


■ 
































"^ 


[^ 




-1. 


& 


-L 





-0. 


5 


( 


1 


0.5 


1.0 


1J5 


2JQ 



Values of tan Z0}4x° 

Fig. 21. 



XrX. y = ax'^+b(xf^. 



fyt+i yk+2 



Values of x form a geometrical series, and the points i - 

\ yt yk 



) 



lie on a straight line, whose slope, Jf , is positive, and whose intercept, 
By is negative, and M*+4B positive. 

// two variables, x and y, are so related that when values of 
X are taken in a geometrical series the points represented by 

y]^^]^±l\ iIq Qfi d straight line whose slope, M, is positive, 
, yt yt / 

and intercept, B, negative, and also M^+4B positive, the relation 
between the variables is expressed by the equation 

XIX y = axf+bx^. 

Let X and yt be any two corresponding values of the variables. 
The following equations are evident: 

yu^axf'+bxf^, (i) 

yt-^i^-axfr'+bxfr^, ..... (2) 



66 EMPIRICAL FORMULAS 

yt.H2=a^r2^+6^V^ (3) 

yt+i-f'y* =bx^(r^-r'), (4) 

yt+2-r'yj:+i =bo(f'r^{r^-r') (5) 

Multiplying equation (4) by 7" and subtracting it from 
equation (5) there results 

or . 

y* y* 

It is seen that the slope of this line is positive and the inter- 
cept negative, and M^+4B positive. 

In the table* below, the values of x and y from x^.o$ to 
« = .55 are taken from Peddle's Construction of Graphical 
Charts. 











yt+i 


yk+i 


.55 


.85 


y 


Com- 


X 


y 


X 


y 


yk 


yk 


X 


X 


.55 

X 


puted y 


.05 


.283 


.05 


.283 






.192 


.078 


1.470 


.283 


.10 


.402 


.10 


.402 






.282 


.141 


1.426 


.402 


.15 


.488 


• • • 


• • • • 






•352 


.199 


1.385 


.488 


.20 


.556 


.20 


.556 


1.420 


1.965 


.413 


.255 


1.347 


.556 


.25 


.613 




• • • • 






.466 


.308 


1. 315 


.612 


.30 


.658 




• • • • 






.516 


.359 


1.276 


.658 


.35 


.695 




• • • • 






.561 


.410 


1.238 


.697 


.40 


.730 


.40 


.730 


1.383 


1. 816 


.609 


.459 


1.208 


.730 


.45 


.757 




• • • • 






.645 


.507 


1. 174 


.757 


.50 


.78c 




• • . • 






.683 


.555 


1. 142 


.780 


.55 


.800 




• • • • 






.720 


.602 


1. 114 


.799 


.60 


.814 




• • • • 






.755 


.648 


1.078 


.814 


.65 


.826 




• • • • 






.789 


.693 


1.047 


.826 


.70 


.835 




• • • • 






.822 


.738 


1. 016 


.835 


.75 


.840 




• • • • 






.854 


.783 


0.984 


.840 


.8c 


.845 


.80 


.845 


1. 313 


1.520 


.885 


.829 


0.955 


.846 



In column 3 the values of x are selected in geometrical ratio 
and the corresponding values of y are given in column 4. The 

points (y}LtLyh±l\ are plotted in Fig. 22, and although the 
* See Rateau*s ** Flow of Steam Through Nozzles." 



DETERMINATION OF CONSTANTS 



67 



three points do not lie exactly on a straight line the approxima- 
tion is good. The slope of the line is 4.10 and the intercept 
—3.86 which give the equations 



2^+2'' =4.10, 



|C+d_ 



=3-86. 



Yalaes of x-^ 
M SO JSn .40 .60 .60 .70 JSO .90 




Valuesol^ 

Fig. 22 AND Fig. 23. 

Solving these equations the values of c and d are found to be 



The formula now is 



e = i.40, 
^=.55. 



68 EMPIRICAL FORMULAS 

Dividing both members of this equation by r^* 



^66 



which represents a straight line when -^ is plotted to r^^. 

The slope of the line is equal to a and the intercept equal to h. 

From Fig. 23 
♦ a=— .685, 

6 = 1.522. 

The formula after the constants have been replaced by their 
numerical values is 

y = i.522r^*-.68sx^'*^ 

The last column of the table shows that the fit is quite 
good. 

If the errors of observation are so small that the values 
of the dependent variable can be relied on to the last figure 
derivatives may be made use of to advantage in evaluating the 
constants in empirical formulas. But when the values can not 
be so relied on, or when the data must first be leveled graphically 
or otherwise, the employment of derivatives may lead to very 
erroneous results. This will be illustrated by two examples 
worked out in detail. 

The first step in the process is to write the differential equa- 
tion of the formula used and then from this equation find the 
values of the constants. 

Consider the formula 

y = ef^{c cos bx+d sin hx). 

Looking upon a and b as known constants and c and d 

as constants of integration, the corresponding differential 

equation is 

y' — 2ay+(a2+62)y=o. 

Dividing this equation by y 

y y 



DETERMINATION OF CONSTANTS 



69 



which, if the data can be represented by XVIII, represents a 
straight line whose slope is 2a and whose intercept is — (a^+fr^). 
Corresponding values of x and y are given in the table. 







y 


y' 


y 




t 


tL 


X 


«.06x 


cos .o8x tan .o&r 


y 


X 






062 
















y 


y 


De- 

grees 


Min- 
utes 








g-VM* 


cos .o8x 





+ 


.3000 
















I .0000 


I. 0000 


.0000 


+ 


.3000 


I 


-}- 


.2750 










4 


35.02 
10.04 


I. 0618 


.9968 


.0802 


+ 


.2598 


3 


+ 


.2441 


-.03t2 


-.0068 


— 


.1401 


-.0279 


9 


I. 1275 


.9872 


.1614 


+ 


.2193 


3 


+ 


.2065 


-.0459 


-.0066 


— 


.1976 


-.0319 


13 


45.06 


I. 1972 


.9713 


.2447 


+ 


.1776 


4 


+ 


.1622 


— .0481 


—.0078 


— 


.2965 


— .0481 


18 


20.08 


I. 2712 


.9492 


.3314 
.4228 


+ 


i^st 


5 


+ 


.1102 


-.OSS7 


-.007s 


— 


.5054 


— .0681 


22 


55 10 


1.3499 


.9211 


+ 


6 


+ 


.0506 


-.0635 


-.0086 


—I 


.2549 


— .1700 


27 


30.12 


1-4333 


.8870 


.5206 


.+ 


.0398 


7 


— 


.017s 


— .0721 


—.0080 


+4 


.1200 


+.4571 


32 


05.14 


1.5220 


.8472 


.6270 


— 


.0136 


8 


— 


.0937 


— .0805 


—.0087 


+ 


.8591 


+.0928 


36 


40.16 


1.6161 


.8021 


.7446 





.0723 


9 


— 


.1786 


-.0894 
-.0985 


—.0091 


+ 


• SOU 


+.0510 


41 


15.18 


I. 7160 


.7519 


.8771 


^ 


.1384 


10 


— 


.2726 


—.0091 


+ 


:ilg 


+.0334 


45 


50.20 


I. 8221 


.6967 


1.0296 


— 


.2147 


II 


— 


• 3757 


- . 1078 


-.0085 


+ 


+.0226 


SO 


25.22 


1.9348 


.6372 


1.2097 
1.4284 


— 


.3047 


12 


— 


.4881 


-.1168 


—.0087 


+ 


.2393 


+.0178 


55 


00.24 


2.0544 


.5735 


— 


.4143 
.5518 


13 


— 


.6093 


-.1257 


—.0091 


+ 


.2063 


+.0149 


59 


35.26 


2.1815 


.5062 


1.7036 


— 


14 


— 


.7396 


-.1348 


—.0089 


+ 


.1823 


+.0120 


64 


10.28 


2.3164 


.4357 


2.0659 


— 


.7328 


IS 


— 


.8788 


- . 143s 


—.0084 


+ 


.1633 


+.0096 


68 


45.30 


2.4596 


.3624 


2.5722 


— 


.9859 


16 


— 1 


.0264. 












TK 


20.32 


2.6117 


.2867 


3.3414 


— 1 


.3707 


17 


— 1 


:.i8i4 












77 


55. 34 


2.7732 


L 
.2098 


4.6735 


—a 


1.0306 







The values of y' and y^ are obtained by the formulas 



yn = 



i2h 



{yn-2—iyn-i+ 8y«+i-3/»+2), 



y"n = p iyn-2 - i(>yn-i +2>oyn - i6y„+ 1 +yn+2), 

where A = Ajc = i. These formulas are derived in Chapter VI. 
Plotting the points represented by (—,—), Fig. 24, it is 

seen that they lie nearly on a straight line whose slope is .12 
and intercept —.01. Therefore 

2a = .i2, 
a2+ft2 = .oi, 



a = .06, 
b = .08. 



We have then 



y = e'^^{c COS .Q&x+d sin .o8«). 



70 



EMPIRICAL FORMULAS 



Dividing this equation by e^' cos .o8x 



e^^ cos .68x 



=(;+(/ tan ,o8x. 



-1.2 -1.0 -. 


s -. 


B n4 -. 


2 ■- 


/ . 


2 . 


4 . 


6 . 


B 1.0 
















i 


\ 










/ 


























/ 


/ 


X 








■ 














> 


/ 






















> 


/ 


























/ 


























/ 


























A 


^ 








^ 


L' 














/ 


/ 












V 












A 


Y 
























y 


/ 


























/ 


























/ 




















< 






/ 


r 






















/ 


/ 
























y 


/ 


























/ 






















































•:2 



Fig. 24. 



This represents a straight line when 



e"®* cos .08:*: 



is plotted 



to tan .o8x. The slope is d and intercept c. From Fig. 25 

c= .3. 
'the law connecting the variables is represented by 

3; = eO^(.3 cos .oSic— .5 sin .oSoc). 



DETERMINATION OF CONSTANTS 



71 



The values of y computed from this formula agree with 
those given in the second column of the table. 
Consider formula XIX 

The corresponding differential equation is 



■ 


v~ 


























£Q>ft2 


006 


.4 
.2 

1 


N 




























\ 


\. 






















0^ 

4 

-.2 






\ 


k 


























\ 




















-.4 

-A 








\ 


\ 


























\ 


\ 
















-s8 












* 


\ 


\ 




























\ 


\ 


























\ 


\ 




























\ 








^D 






















\ 






■1^ 






















\ 


\ 




■ILA 
























\ 


\ 









1 




t 


I 




3 




1 











8 

s 



09 



Valued of taa .08x 

Fig. 25. 

where c and d are known constants and a and b constants~of 
integration. The differential equation represents a straight line 

when — ^ is plotted to — . The slope is c+d—i and the inter- 
ne y 

cept is —cd. 

The values of x and y in the table below are the same as 
those given in the discussion of XIX. 



72 



EMPIRICAL FORMXTLAS 







# 


// 


xy' 


«y' 


X 


y 


y 


y 


y 


y 


.05 


.283 












10 


.402 












15 


.488 


1.503 


-6.933 


.462 


— .320 




20 


• 556 


1.240 


-4.133 


.446 


-.297 




25 


.613 


1. 015 


-4.967 


.414 


-.506 




30 


.658 


0.803 


-3.267 


.366 


-.447 




35 


•695 


0.720 


—0.400 


.363 


— .071 




40 


.730 


0.623 


-3.533 


.341 


-.774 




45 


• 757 


0.492 


—1.500 


.292 


— .401 




50 


.780 


0.433 


—1.067 


.277 


-.342 




55 


.800 


0.338 


-2.633 


.232 


-.996 




60 


.814 


0.2SS 


-0.633 


.188 


— .280 




65 


.826 


0.213 


—1.200 


.167 


— .614 




70 


• 835 


0.135 


-1.767 


• 113 


-1.037 




75 


.840 












80 


.845 











The values of y' and y" were computed by means of the 
formulas used in the preceding table. In Fig. 26 the points 

-^, --^ 1 are plotted and, as is seen, the points do not deter- 

<y y ^ ^ 

mine a line. It is clear that the constants can not be determined 
by this method. 

XlXa. y-daf(f. 

Points represented by (x„, log -^ — ) lie on a straight line. 

\ yn / 

If two variables^ x and y, are so related that when volumes of x 
are taken in a geometrical series the corresponding values of y are 

such that the points represented by i Xny log ^^^ j lie on a straight 
line, the relation between the variables is expressed by the equation 



XlXa 



y = aa^c'. 



Using logarithms: 

log yn = log a+b log Xn+Xn log Cy 
log y„+i =log (^b log Xn+rxn log c+b log r. 






DETERMINATION OF CONSTANTS 



73 



Subtracting the first equation from the second 

log Mi = (r''i)xn log c+b log r. 

By plotting log Mi to acn a line is obtained whose slope is 
(r — i)log c, and since r is known c can be determined. 









• 


o 




-.1 








* 




".2 
-.4 









o 


o 


Q 

o 














O 

1-6 
-?8 














o 




o 


• 


-.9 
-1.0 




o 


— n 







.1 .2 .3 

Values of -y- 

Fig. 26. 



.4 



J5 



From the first equation 

log yn-Xnlogc=b log Xn+log a. 

If then log yn—Xn log c be plotted to log Xn a line is obtained 
whose slope is b and whose intercept is log a. 



CHAPTER V 

XX. y=ao+(iiCOSx-\-(hcos2x-\-aiCOSsx+ . . . -f-OrCosfX 

4-6isinx+ftisin 2x4-6jsin3x+ . . . 4-ftr_iSin (r— i)x. 

Values of y periodic. 

The right-hand member of XX is called a Fourier Series 
when the nxmiber of terms is infinite. In the application of the 
formula the practical problem is to obtain a Fourier Series, 
of a limited nimiber of terms, which will represent to a sufficiently 
close approximation a given set of data. The values of y are 
given as the ordinates on a curve or the ordinates of isolated 
points. 

In what follows it is assumed that the values of y are periodic 
and that the period is known. 

We will determine the constants in the equation 

/ I . ' . 

y =00+^1 cos x+a2 cos 2x+a3 cos 3X,+bi sin x+b2 sin 2x, 

so that the curve represented by it passes through the points 
given by the values in the table. 



X 


o° 


60° 


120° 


180° 


240° 


300° 


360° 


y 


I.O 


1-7 


2.0 


1.8 


i-S 


0.9 


1.0 



Substituting these values in the equation we have the fol- 
lowing six linear relations from which the values of the six 
constants can be determined: 

i.o = ao+ ^1+ (i2-\-ci3j 

i'7 = ao+^ai''la2-as-{ — ^bi-{ — ^62, , 

2 2 

2.0=00-^^1— ha2+as-\ — ^bi ^62, 

2 2 

74 



DETERMINATION OF CONSTANTS 76 

1.8=00— ^1+ ^2 — 03, 

1 .5 = flo - 5^1 - ha2 +a3 Hi H — ^62, 

2 2 

0.9 = 00+1^1 — 2^2—^^3 -bl ^62. 

2. 2 



' J 



Multipl)^g each of the above equations by the coefficient 
of ao, (in this case unity) in that equation and adding the result- 
ing equations we obtain (i) below. Multiplying each equation 
by the coefficient of ai in that equation and adding we obtain 
(2). Proceeding in this manner with each of the constants 

a new set of six equations is obtained. 

I 

f ,600 = 8.9. • (i) 

301 = -1.25 (2) 

3^2= -.25 (3) 

603 = . 10 (4) 

3h = .6s^3' ....... (s) 

3*2 = . 15^3- • (6) 

oo=tI, oi=--A, <3t2=— a, az^-h, 
&i=MV3, 62=1^^3. 

The equation sought is 

y =fj— A cos x—^ cos 2x+^ cos zx-\-\^y/T, sin x+^y/j, sin 2x, 

, It reproduces exactly each one of the six given values. 

The solution of a large number of equations becomes tedious 
and the probability of error is great. It is, therefore, very 
desirable to have a short and convenient method for com- 
puting the numerical values of the coefficients.* 

*The scheme here used is based upon the 12-ordinate scheme of 
Rimge. For a fuller discussion see "A Course in Fourier *s Analysis and 
Periodogram Analysis " by Carse and Shearer. 



76 



EMPIRICAL FORMULAS 



t 
f • 



S. • 



Take the table of six sets of values 



X 


o° 


60° 


120° 


180° 


240** 


300" 


y 


yo 


yi 


y2 


ya 


y* 


ys 



where the period is 2v. 

For the determination of the coeflBcients the following six 
equations are obtained: 

2 2 

2 2 



y3 = ^ — ^1- + ^2 — ^3, 



VT, . V^, 



y4 = ao-§fl^l- 1^2+^3 -hx-\ ^62, 



3^6 = 00+1^1-1^2 — ^3 ^61 ^62. 



2 
2 



Proceeding in the same way as was done with numerical 
equations the following relations are obtained: 

6ao=yo+ yi+ ^2+^3+ 3^4+ ys, 
3^1=^0+ \y\- hy^-yz- -?y4+^y6, 
3^2 =yo- \y\- 1^2+^3- ^y4-^y6, 

6a3=j'o- )'i+ )'2-)'3+ ^4- ys, 



r . ^^3 . ^3 ^^3 "^ 

2 2 2 2 



\/^ V^ 



V^ V3 



3*2 = +— ^y 1 ^3^2 + — ^3'4 -yh 



W 



J 



22 22 

For convenience in computation the values of y are arranged 
according to the following scheme: 

yo yi y2 

y3 y4 ys 

Sum z;o z;i z>2 

Difference w^o 2e;i tt;2 



' 



DETERMINATION OF CONSTANTS 



77 



vo 



V2 



Wo 



Wi 
W2 



Sum po 

Difference 



Pi 
?i 

6ao 
3^1 

3^2 

6a3 



Sum 
Difference 



ro 



n 

Sl 



po+Ph 
Po-^piy 

To— Sly 



3^2 = ^1. 



(6) 



J 



It is evident that the equations in set (b) are the same as 
those in set (a). 

For the numerical example the arrangement would be as 
follows: 











l.O 




1.7 . 


2.0 








. 




1.8 




1-5 


0.9 










Vo 


2.8 




3-2 


2.9 










Wo 


-.8 




.2 


I.I 









2.8 




3-2 




t 


-.8 


.2 




Po 


2.8 




2-9 

6.1 




ro 




I.I 




-.8 


1-3 


» 


91 




I f 


•3 

6ao = 
3^1 = 

3^2 = 
6^3 = 

3*1 = 

3*2 = 


= +8 


Sl 

i.90, 
•25, 

.25, 
.10, 


* 


-•9 



78 EMPIRICAL FORMULAS 

It IS seen that the computation is made comparatively simple. 
The values of the v's are indicated by vo, the first one. The 
values of the p% etc., are indicated in the same way. 

8-ORDiNATE SCHEME. The formula for eight ordinates which 
lends itself to easy computation is 

y = ao+ai cos 6+a2 cos 26+a3 cos ^d+a^ cos 46 
+bi sin d+b2 sin 2^+63 sin 3^. 

For determining the values of the constants eight equations 
are written from the table: 









45° 


90° 


135° 


180° 


225° 


270° 


315° 


y 


yo 


yi 


yi 


ys 


y* 


ys 


y6 


3-7 



yo=(io+ (ii+a2+ ^3+04, 

v2 V^2 V^2 v2 

yi =floH di a3—a4-\ 6i+&2H &3, 

2 2 2 2 

y2=flo —^2 +^4+ fti — fts, 

\/2 V2 V^2 V^ 

y3=flo fli H ^3— ^4H &1— &2H 2J3, 

2 2 2 2 

V^2 V^2 v2 V2 

y6=flo di -\ as—CA bi+b2 63, 

2222 

y6 = flo —(i2 +a4—bi + bzy 

, V2 V2 V2, , V2, 

y7=floH ^1 ^3— ^4 01—02 0^. 

2222 

From which are obtained the following eight equations: 

8ao=yo+ yi+y2+ y3+y4+ y5+y6+ y?, 

\^2 y/2 y/2 y/~2 

4^1 = yo -\ yi ys - y4 ys ^ yr, 

2222 

4^2 =yo ~y2 +y4 -ye, 

V2 . V2 . V2 V2 

4^3 =yo yi -\ y3-y4+: — ys y?, 

2222 



DETERMINATION OF CONSTANTS 



79 



8a4=yo- yi+y2- yz+y^- ys+ye- yr, 



V2 , , V2 

-— yi+y2+-— ys 
2 2 

yi - ys 



461 = 
462= 

\/2 V^2 

463 = — y 1 - y2 +-^-y3 



V2 V2 

— —ys-y^ — — yr, 
2 2 

+ ys - yr, 

V2 , VI 

— —ys+yQ — — yr. 
2 2 



2 ' 2 

For. the purpose of computation the values of y will be 
arranged as follows : 

yo yi y2 ya 

y4 3'5 yc yr 

Sum t;o 



Difference w/o 



V2 



Wi 

V3 



V2 
W2 



vz 



Wo 



Sum 
Difference 



po 



Sum 
Difference 



Pi 
&aQ=po+ • ph 



To 






4^2 = qo, 



4^3 =ro 



V2 



-su 



Sa^^po—pu 

4*1 = ^2 H ri, 

2 

4^2 =gi, 

/^bz=-'r2^ ri. 

2 

The process will be made clear by an example: 



Wl 
Wz 



n 



e 

y 



W2 



r2 






45° 


90° 


135° 


180° 


225° 


270° 


315° 


360° 


4 


— 2 


— I 


2 


3 


3 


—I 


2 


4 



I 



80 



EMPIRICAL FORMULAS 

For computation the arrangement is as follows: 

' 4 —2 —I 2 

3 3-12 





vo 


7 


I 




Wo 


I 


-s 




7 


I 






— 2 


4 




Po 


S 


S 


ro 


qo 


9 


-3 


^1 

Soq = lO, 

4ai = I -f V2, 

4^2 = 9) 

4^3 = I+fV2, 

8^4 = 0, 

4*1= -fv^, 
4*2= -3, 

4*3=-fV2. 



— 2 
O 



4 
o 

-5 
o 



-5 
-5 



The formula becomes 

y = 1. 25 — .634 cos ^+2.25 cos 2^+1.134 cos 3^ 
— .884 sin 6— .75 sin 2d— .884 sin 3^. 

io-Ordinate Scheme 





yo 


yi 


y2 


yz 


^4 










y9 


ys 


yi 


ye 


ys 




Sum 


Vo 


Vl 


V2 


vz 


V4. 


V5 




Difference 




Wi 


W2 


Wz 


Wa 






■ 


Vo 


Vl 


V2 






Wi 


W2 




V5 

po 


va 

pi 


V3 
p2 


Simi 




W4 


Ws 


Sum 


h 


12 


Difference 


qo 


qi 


92 


Difference 


nil 


W2 



DETERMINATION OF CONSTANTS 



81 



1000 = 

5^1 = 

5^2 = 

5^3 = 

5^4 = 

1006 = 

In the above equations 

Ci = cos36°, 
C2 = cos 72®, 



po+pi+p2, 
'■qo+Ciqi+C2q2, 
'P0+C2P1—C1P2, 
'■qo—C2qi—Ciq2, 

'■pO''Clpl+C2p2, 

qo-qi+q2, 

'•Slll-\-S2l2, 
= 52^1 +51^2, 

■S2h—Sil2y 
'Siini—S2fft'2' 




Si = sin 36°, 
52 = sin 72°. 



In the schemes that follow, as in the lo-ordinate one, only 
the results will be given. 



Sum 
Difference 



12-Ordinate Scheme 



yo 



vo 



yi 
yn 



Wi 



y2 
yio 



yz 

y^ 



y^ 



V2 
W2 






V4 



ys 

7i 



"Oh 



ye 



z'e 



Sum 
Difference 



Sum 



t'o ^1 ^2 'Oz 

2^6 H ^4 

pQ p\ p2 pz 

?o ?1 ?2 



po 

p2 



pi 

pz 



lo 



u 



Wi 
W5 



W2 

W4. 



Sum n 72 

Difference ^i ^2 



Wz 



n 



rz 



Difference 



?2 



fe 



82 



EMPIRICAL FORMULAS 



i2ao=/o+/i, 
6ai=goH — -qi+iq2y 

6a4=^po+ps''^(Pi+p2), 

6^6 =?o -qi+hq2, 

i2ae=lo—li, 



66i = 



ri+— ^r2+r3, 

2 

662 =-^(^1+^2), 
6^3= /i, 

664=— ^(51-^2), 
2 



16-ORDINATE Scheme 



Sum 
Difference 



yo yi ^2 ^3 ^4 

yi5 yi4 yi3 712 



t'O 



Wi 



V2 

W2 



V3 



yn 



V4 



W5 



^6 

yio 



1'6 



Wq 



yr 
y9 



V7 



ys 



Vs 



Sum 
Difference 



Vg 



po 

qo 



Vl 

V7 



Pl 

qi 



V2 



vz 

V5 



p2 
?2 



p3 
?3 



V4: 



P4. 



DETERMINATION OF CONSTANTS 83 

Wl W2 W3 W4 

W7 We W5 



Sum ri r2 rs U 

Difference si S2 ss 



) 



pO pi p2 ^^ 

p4: pZ ^2 



Sum /o h h Sum k h 

Difference wo wi Difference xq 

8ai =goH 5'2+Cig'i+C2g3, 

8^2= Wo -I Wl, 

2 

8^3 =?o ?2 — Cig'3+C2?l, 

2 

8tl4 = ^, 

8^5 = ?o ?2 + Cig'3 — C2gi , 

806 = Wo Wl, 

2 

807 = ^0 H ?2 — Cigi — 02^3 , 

2 

86i = r4 H r2 +Cir3 +C2f i, 

2 

862 = ^2 H (^1+^3), 

2 

863= -^4 H — ^r2+Clfl-C2r3, 
864 = ^1— -^3, 



84 



SJs 



EMPIRICAL FORMULAS 



V2 

=^4 r2+Ciri—C2r3y 

2 



866 = 



■S2-\ (^1+53), 

2 



867= -r4 



r2+Cir3+C2ri. 

2 



Ci=cos 22^® = sin 67!°, 
C2 = sin 22|® = cos67^°. 

20-Ordinate Scheme 

yo yi y2 ya y^ ys ye yi y% y^ yic 
yi9 yi8 yi7 yi6 yi5 yi4 yi3 yi2 yw 



Sum 


vo 


1^1 


V2 V3 


Va 


vs 


vq 


V7 Vs 


V9 Z'lC 


Difference 




W\ 


W2 Wz 


; W4: 


W5 


Wq 


Wi Ws Wg 






2^0 


Vl 


V2 


V3 


va 


V5 








t'lO 


V3 


vs 


V7 


. ^6 






Sum 


/>0 


Pl 


p2 


p3 


P^ 


ps 




Difference 




?o 


?1 


92 


?3 


?4 










Wi 


W2 


W3 




W4 


W5 








w^ 


Ws 


W7 




Wq 






Sum 




n 


r2 


ra 




r4 


rs 




Difference 




S\ 


S2 


^3 




S4 








/>o 


Pi 


p2 






90 


91 


92 




^ 


P^ 


p3 






94 


9s 




Sum 


/o 


h 


I2 


Smn 


ko 


ki 


k2 


Difference 


Wo 

/o 

/l 

/2 


mi 


m2 


Wo 
1fl2 


Ifl] 


I 




fl f3 


Sxmi 


/o 




Simi 


no 


ni 


Sum 


Ol O3 



DETERMINATION OF CONSTANTS 85 



Sl 
Si 


S2 
S3 


il 

hi 


g2 
h2 



Sum 
Difference 

2000 = k) 

ioai=qo+qi sin 72°+g2 sin S4°+g3 sin 36°+g4 sm I8^ 
ioa2=wo+wi sin 54^+^2 sin 18°, 
ioa3=5'o— ?3 sin 72°— 54 sin $4^+qi sin 36°— 92 sin 18°, 
ioa4=/o— fc sin S4°+/i sin 18°, 

10^5 = ^0"" ^2, 

1006=^0— W2 sin 54°— wi sin 18°, 

ioa7=5'o+?3 sin 72°— 54 sin S4°— gi sin 36°— 52 sin 18°, 

ioas=lo—h sin S4°+/2 sin 18°, 

ioa9=qo—qi sin 72°+g2 sin S4°— ^3 sin 36°+g4 sin 18°, 

2oaio=wo~"Wi, 

lofti =r6+r4 sin 72°+r3 sin S4°+r2 sin 36°+ri sin 18°, 

10*2 =g2 sm 72°+gi sin 36"*, 

1063= — r5+r2 sin 72°+ri sin 54°— r4 sin 36°+r3 sin 18°, 

10^4=^1 sin 72°+A2 sin 36°, 

10^5= ^1—^3, 

ioft6=gi sin 72°-g2 sin 36°, 

io&7= — rs— r2 sin 72°+ri sin S4°+r4 sin 36°+r3 sin i8^ 

10^8= — fe sin 72°+Aisin 36°, 

ioft9=r6— r4 sin 72°+r3 sin 54°— r2 sin 36°+ri sin 18°. 

24-ORDiNATE Scheme 
yo yi y2 yz y^ ys y^ y? ys yg yio yn yi2 

^23 ^22 ^^21 y20 yi9 ^18 ^17 ^16 yi5 ^14 ^13 

Sima vo vi V2 vz V4: vs vq V7 vs vq z;io vn z;i2 

Difference wi W2 w^3 ^^4 tcs ^e W w'8 ^9 w'lo w'n 



86 



EMPIRICAL FORMULAS 



Sum 
Difference 



Vo Vi V2 V3 V4 Vs Ve 

^'12 ^11 Z^IO V:) Vg V7 

Po pi p2 p3 p4: p5 p6 

yo qi q2 qs q^ qb 

Wi W2 Ws W^ Ws Wq 

Wll Wio Wo Ws W7 



Sum 
Difference 



^1 



r2 

S2 



^3 



^4 
^4 



Tb 
S5 



re 



Sum 
Difference 



po pi p2 p3 

p6 pb p4: 

h h h h 

mo mi 1712 



si 

Sb 



S2 
S4. 



Sum 
Difference 



ki 
ni 



h 



h 



mo 
m2 



k2 
«2 

fill 



Sum 
Difference 

2400 = 
i2ai = 

i2a2 = 
I2a3 = 

12^4 = 

I2a5 = 
I2a6 = 
I2a7 = 
I2a8 = 

I2ag = 

i2aio = 
i2aii= 

24ai2 = 



go 
ho 



gi 
hi 



Sum 
Difference 



Co 
fo 



ei 



=go+gu 

=qo+k^+W2q3+h^q2+Ciqi+C2q6y 
=fno+ifn2+^V^mi, 

^qo-q^+W^iqi-qz-qb), 

= ho+^hiy 

-- qo+C2qi - hVJq2 - W^qz +^^4 +Ci js, 
=/o, 

=?o-C2gi-|V3g2+^V2?3+^g4-Cig6, 

■■qo-q^+W^i-qi+qz+qb), 
■-nio+^fn2-^Vsmi, 

=go-Cigi+iV3g2-^Vjg3+ig4-C2g6, 



S3 



k3 



DETERMINATION OF CONSTANTS 



87 



i2fti = C2ri+§r2+^V2r3+^V3r4+Cir6+r6, 

12b2 = iki+^V^k2+k3, 

1 2*3 = r2 - re H-^v^ (ri +r3 — rs) , 

i2J5 = Ciri+Jr2-^V2r3-|V3r4+C2r5+r6, 

I2J6 = ^l~"^3, 

i2&7 = Ciri-^r2-|V2r3+^V3r4+C2r5-r6, 

I2&8 = iV3(wi-W2), 

i2ft9=r6-r2+jV2(ri+r3-r5), 

I2ftl0=53+i(^l+^5)-|V3(52+54), 

i2Jii = C2ri-ir2+§V2r3-^V3r4+Cir5-r6, 

Ci^ ^ = .96593, 

2V2 

C2 = ^~^ =.25882. 
2V2 

As an illustration let it be required to find a Fourier series 
of 24 terms to fit the data given in the table below. 



x"" 


y 


«« 


y 


x"" 


y 


:r° 


y 


00 


149 


90 


159 


180 


178 


270 


179 


15 


137 


105 


178 


195 


170 


285 


185 


30 


128 


120 


189 


210 


177 


300 


182 


45 


126 


135 


191 


225 


183 


315 


176 


60 


128 


150 


189 


240 


181 


330 


166 


75 


135 


165 


187 


255 


179 


345 


160 



149 137 

160 



128 126 128 13s 159 178 189 191 189 187 178 
166 176 182 185 179 179 181 183 177 170 



vo 149 297 

wi -23 



294 302 310 320 338 357370374366357178 

-38 -50 -54 -50 -20 -I 8 8 12 17 



88 EMPIRICAL FORMULAS 





149 


297 


294 


302 


310 


320 


33» 




178 


357 


366 


374 


370 


357 




Po 


327 


654 


660 


676 


680 


677 


338 


qo 


-29 


-60 


-72 


-72 


-60 


-37 






-23 


-38 


-50 


-54 


-50 


— 20 






17 


12 


8 


8 


— I 






n 


- 6 


-26 


-42 


-46 


-51 


— 20 




Sl 


-40 


-50 


-58 


-62 


-49 








327 


654 660 676 






-40 - 


50 -58 




338 677 680 

665 133 I 1340 676 


ki 




-49 - 


62 


/o 


-89 - 


112 -58 


flto 


— II 


-23 - 


20 


»i 




9 


12 






66s 


133 1 






— II 


-23 


■ 




676 
1341 


1340 
2671 


Co 


• 


— 20 




go 


-31 


-23 


h 




— II 


-9 


/o 




9 





The formula becomes 

y = 167.167 — 19.983 cos X--3.410 cos 2X+5.470 cos 3X 

— 1.292 COS4X+.249 cos Sx+,T$ cos 6X+.310 cos 7^ 
+.458 cos 8x— .304 cos 9X — .090 cos lox— .243 cos iix 

— .083 cos I2X— 12.779 sin X— 16.624 sin 2x— .323 sin3x 
+ i.5i6sin4a:+i.46i sin sx— 2.583 sin6a!:+.32i sin ^x 

— .216 sin 8X+.676 sin 90;- .459 sin loic- .639 sin iix. 

In what precedes the period was taken as 27r. This is not 
necessary; it may be any multiple of 2x. The process of finding 
a Fourier series of a limited number of terms which represent 
data whose period is not 2x will be best set forth by an example. 
In the table below the period is x/3 and the values of y are 
given at intervals of -k/i^^. The 12-ordinate scheme can be 
used by first making the substitution 

x=\0 or ^=3rr. 



DETERMINATION OF CONSTANTS 



89 



a;° 


0° 


y 


a:° 


0** 


y 


x"" 


e° 


y 


GO 


GO 


+27.2 


40 


120 


+9.8 


80 


240 


-II-5 


ID 


30 


+34.5 


50 


150 


+8.5 


90 


270 


-17. s 


20 


60 


+21.5 


60 


180 


+0.2 


100 


300 


—17.2 


30 


90 


+ io.i 


70 


210 


-7.1 


no 


330 


+ i.S 





27.2 


34-5 


21-5 


10 


.1 


5 


(.8 


8-5 


0. 


2 






I.S - 


-17.2 — 


17 


•5 


— I] 


t-5 ■ 


-7.1 






^0 


27.2 


36.0 


4.3 ■ 


-7-4 


— ] 


[-7 


1-4 


0. 


2 


W\ 


po 


33.0 

27.2 
0.2 


38.7 

36.0 

1-4 


27 


.6 


21 

4-3 
1-7 


[-3 


15.6 
7-4 








27.4 


37.4 






2.6 




7-4 






33 
15.6 


27.0 

38.7 
21.3 


34.6 
27.6 






6.0 


27.4 
2.6 


37-4 
-7-4 






fl 


48.6 


60.0 


27.6 




k 




30.0 


30.0 






Si 


17.4 


17.4 










48.6 
27.6 


27.0 
6.0 




• 



21.0 



21.0 



The formula is 



y = 5+9.994 cos ^+8.7 cos 2^+3.5 cos 3^+.oo6 cos 5^ 
+ 17.31 sin ^+5.023 sin 2^+3.5 sin 3^— .01 sin 5^. 

Replacing by its value 3X, 

y ==5+9-994 cos 3:^+8.7 cos 6a!:+3.5 cos 9a!:+.oo6 cos i$x 
+17.31 sin 3:^+5.023 sin 6X+3.5 sin 90;— .01 sin 1501;. 



CHAPTER VI 

EMPIRICAL FORMULAS DEDUCED BY THE METHOD 

OF LEAST SQUARES 

In the preceding chapters we computed approximately the 
values of constants in empirical formulas. The methods em- 
ployed were almost wholly graphical, and although the results 
so obtained are satisfactory for most observational data, other 
methods must be employed when dealing with data of greater 
precision. 

It is not the purpose of this chapter to develop the method 
of least squares, but only to show how to apply the method to 
observation equations so as to obtain the best values of the 
constants. For a discussion of the subject recourse must be 
had to one of the nmnerous books dealing with the method of 
least squares.* 

It was found in Chapter I that the equation 

y=a+bx+ca^ (i) 

represents to a close approximation the relation between the 
values of x and y given by the data 



X 


y 


X 


y 


o 


3-1950 


■5 


3.2282 


.1 


3.2299 


.6 


3.1807 


.2 


3-2532 


-7 


3.1266 


•3 


3.261 1 


.8 


3-0594 


•4 


3-2516 


-9 


2-9759 



* Three well-known books are: Merriman, Method of Least Squares; 
Johnson, Theory of Errors and Method of Least Squares: Comstock, 
Method of Least Squares. 

90 



DEDUCED BY THE METHOD OF LEAST SQUARES 91 

where x represents distance below the surface and y represents 
velocity in feet per second. 

Substituting the above values of x and y in (i), the following 
ten linear observation equations are found: 

a+oft+ 0^=3.1950, 
a+.ib+.oic =3.2299, 
a+.2ft+.o4c=3.2S32, 
^+-3ft+-09C=3.26ii, 
a+.4ft+. 16^=3.2516, 
^+-Sft+-2Sc =3.2282, 
a+.6ft+.36(; = 3.1807, 
^+.76+49^=3.1266, 
a+.8&+.64(; =3.0594, 
a+.9ft+.8ic = 2.9759. 

Here is presented the problem of the solution of a set of 
simultaneous equations in which the niunber of equations is 
greater than the ntmiber of imknown quantities. Any set of 
three equations selected from the ten will suffice for finding 
values of the unknown quantities. But the values so found 
will not satisfy any of the remaining seven equations. Since all 
of the equations are entitled to an equal amount of confidence 
it would manifestly be wrong to disregard or throw out any one 
of the equations. Any solution of the above set must include 
each one of the equations. 

The problem is to combine the ten equations so as to obtain 
three equations which will yield the most probable values of 
the three imknown quantities a, ft, and c. It is shown in works 
on the method of least squares that the first of such a set of 
equations is obtained by multiplying each one of the ten equa- 
tions by the coefficient of a in that equation and adding the result- 
ing equations. The second is obtained by multiplying each one 
of the ten equations by the coefficient of ft in that equation 
and adding the equations so obtained. The third is obtained 
by multiplying each of the ten equations by the coefficient of 



92 EMPIRICAL FORMULAS 

c in that equation and adding the equation so obtained. The 
process of computing the coefficients in the three equations is 
shown in the table. The coefficients of a, ft, and c are represented 
hy A, B, and C respectively, and the right-hand members are 
designated by N. The number 5, which stands for the numeri- 
cal sum of -4 , 5, C and iV, is introduced as a check on the work. 
It must be remembered that this method of finding the values 
of the constants holds only for linear equations. 

The sum of the numbers in the column headed A A = 2-4-4 
= io. The sum of the numbers in the column headed AB = 
2^45=4.5. The sum of the numbers in the column headed 
-4C = 2-4C = 2.85. Also the sum of the numbers in the column 
headed -4iV = 2-4iV=3i. 7616. These sums give the coefficients 
in the first equation.* The second and third equations are 
obtained in the same way. 

The three equations from which we obtain the most probable 
values of the constants are: 

10 a + 4.5ft +2.8sc =31.7616; 
4.5a +2.8sft +2.02SC =14.08957; 
2.8sa +2.o2sft+i.S333c= 8.828813. 

These are called normal equations. From them are obtained 

^=+3-19513; 
. ft = + .44254; 

c=- .76531- 
The check for the first equation is 

2i4^ + 2^5+2^C+2^iV' = 2^5=49.1116; 

for the second equation 

2^5+255+25C+25iV = 255 = 23.46457; 

for the third equation 

2^C+25C+2CC+2CiV = 2C5 = 15.2371x3. 

* Cf. Wright and Hayford, Adjustment of Observations. 



i^mui— i^^^^— Ml 



DEDUCED BY THE METHOD OF LEAST SQUARES 



93 



AA 



lO 



AB 



o 

I 

2 

3 

4 

5 
6 

7 
8 

9 



45 



AC 



o 

.OI 

.04 

.09 
.16 

.25 

.36 

.49 

.64 

.81 



2.85 



AN 



3 1950 
3.2299 

3 2532 
3.2611 
3- 2516 
3.2282 
3.1807 
3.1266 

3 0594 
2.9759 



31.7616 



AS 



4.1950 
4.3399 
4.4932 
4.6511 
4.8116 
4.9782 

5 . 1407 
5.3166 

5 • 4994 
5.6859 



49. II 16 



AB 


BB 


BC 


BN 


BS 


















.01 


.001 


.32299 


•43399 




.04 


.008 


.65064 


.89864 




.09 


.027 


.97833 


I 39533 




.16 


.064 


1.30064 


1.92464 




.25 


•125 


1.61410 


2 . 48910 




.36 


.216 


1.90842 


3.08442 




.49 


.343 


2.18862 


3.72162 




.64 


.512 


2.44752 


4.39952 




.81 


.729 


2.67831 


5.11731 


4.5 


2.85 


2.025 


14.08957 


23.46457 



AC 


BC 


CC 


CN 


CS 



















.0001 


.032299 


•043399 






.0016 


.130128 


.179728 






.0081 


. 293499 


.418599 






.0256 


.520256 


. 769856 






.0625 


. 807050 


I . 244550 






:i296 


I. 145052 


1.850652 






.2401 


1.532034 


2.605134 






.4096 


I. 958016 


3.519616 






.6561 


2.410479 


4.605579 


2.85 


2.025 


I . 5333 


8.828813 


15.237113 



94 



EMPIRICAL TORMULAS 



The formula is 

3^ =3.19513 +.442S4X-. 76531^2, 

For the purpose of comparison the observed values and the 
computed values are written in the table, v (called residual) 
stands for the observed value minus the value computed from 
the formula. 







Observed 


Compute 1 




A 




X 


y 


y 


V 


V^ 







3 1950 


3 1951 


— .0001 


.00000001 




.1 


3 2299 


32317 


— .0018 


.00000324 




.2 


3-2532 


3 2530 


+ .0002 


.00000004 




.3 


3.261I 


3 2590 


+ .0031 


.00000441 




■4 


3- 2516 


3 2497 


+ .0019 


.00000361 




.5 


3.2282 


3-2251 


+ .0031 


.00000961 




6 


3.1807 


3.1851 


-.0044 


.00001936 




7 


3.1266 


3.1299 


-.0033 


.00001089 




8 


3 0594 


3.0594 


.OCXX5 


.00000000 




9 


2-9759 


2.9735 


+ .0024 


.00000576 



+ 0001 .00005493 

This method derives its name from the fact that the siun of 
the squares of the residuals is a minimum. A discussion of 
this will be found in the books referred to above. 

In case the formula selected to express the relation between 
the variables is not linear the method of least squares cannot 
be applied directly. In order to apply the method the formula 
must be expanded by means of Taylor's Theorem. Even when 
the formula is linear in the constants it may be advantageous 
to make use of Taylor's Theorem. In order to make this trans- 
formation clear we will apply it to the formula just considered. 

Suppose that there have been found approximate values of 

a, by and c, oo, bo and cq, say, then it is evident that corrections 

must be added in order to obtain the most probable values of 

the constants. Let the corrections be represented by Aa, AJ, 

and Ac, And let 

(i=ao+Aa, 

b = bo+Aby 

c=Co+Ac, 



DEDUCED BY THE METHOD OF LEAST SQUARES 95 

The formula was 

y=a+bx+cx^. 

This may be written 

y=f{ay by c) =/(ao+Aa, bo+Ab, co+Ac). 

Expanding the right-hand member 

/(oo+Aa, J0+A6, co+Ac)=f{ao,bo,co)+^Aa+^Ab+^Ai 

9^0 oOo qco 



+-^(AaAc)+-^(AjAc)l+ . . . 
9ao9co 9^o9^o J 

where -^ stands for the value of the partial derivative of 
900 

/(a, b, c) with respect to a and ao substituted for a, — ^ stands 

for the value of the second partial derivative of f(a, b, c) with 
respect to a and oo substituted for a, etc. If oq, Jo, and co have 
been found to a sufficiently close approximation the second and 
higher powers of the corrections may be neglected. 

9^0 

dbo 

dco 
The formula becomes 

y-/(ao, bo, Co) =-^Aa+-^Ab+^Ac, 

9^ doo dco 

or 

y— (ao+box+cox^) =Aa+xAb+x^Ac, 



96 CMPIKICAL rOSMULAS 

Selecting for the values oi oo. bo. and co those found in 
Ch^ter I, the new set ot observation equations are 

Aa+ oAb+ oAc= .0002, 

Afl+.iAJ+x>iAc= —.0013, 

Afl+.2Aft+.04Air= joooS^ 

Aa+.3Ab+X)g^=^ 0027, 

Aa+.4Aft+.i6A£:= .0024, 

Aa+.s^+.25Ac= .0034, 

Aa+.6A6+.36Ac= —.0045, 

Aa+.7A6+49Ac= — .0038, 

Aa+.8Ab+,64Ac= —.0010, 

Aa + .9A6 + .8 1 Ac = .0007 . 

From these are obtained the three normal equations 

ioAa+4.5 A6+2.85 Ac=— .0004, 

4.5Aa+2.85 AJ+2.025 Ac=— .00203, 

2.85Aa+2.02sA6+i.5333Ac = - .002059. 
Solving 

Aa= +.00033, 

AJ = +.00254, 

Ac= -.00531, 

which added to the values of oo, Jo, and cq, give 

J= .44254, 

^=-•76531- 
the same as just found. 

The above process may be applied to linear equations con- 
taining more than three constants. But as the method of pro- 
cedure is quite evident from the above the general statement 
of the process will be made with reference to equations con- 
taining only three constants. 



2SZ£kia^daMl 



DEDUCED BY THE METHOD OF LEAST SQUARES 97 

Let the observation equations be represented by 

aix+biy+ciz=ni pi, 
a2X+b2y+C2Z=n2 p2, 
azx+bzy+czz==m ps. 



amX+bmy+CtnZ = nm Pm- 



The normal equations will then be 

2pa^ • X + l^pab • y + ^pac • z = S/^a«, 
S/>aft 'X+7:pP -y+Xpbc • z = Xpbn, 
l^pac ' x+2pbc ' y+Zpc^ • z = Xpcn, 

where a, b, c, and n are observed quantities, and x, y, and z 
are to be determined, />i, />2, />3 • • . /^m are the weights assigned 
to the observation equations. In the problem treated at the 
beginning of the chapter the weight of each equation was taken 
as xmity. 

It was stated on a preceding page that when a formula to be 
fitted to a set of observations is not linear in the constants it 
must be expanded by Taylor's Theorem. 

Take as an illustration a problem considered in Chapter IV. 

The formula considered was 

y=f(A, B, m, n) =Ax"'+Bx'', 

w _ymo 



dAo 

^- 
dBo 



=x^. 



^=Aoxrlogx, 

■^'=Box'^ log*; 
9«o 



98 



EMPIRICAL FORMULAS 



y=f{A,B,m,n) = 



/(^o, 5o, mo, no) +^A^ +^^ 



f ^A«,- 9-^ 



9/wo 



9wo 



Aw; 



y-/(i4o,5o, wo,«o) = 



3/.A^ +4-A5+^Am+^A/^. 



9^0 9^0 



9^0 



9wo 



The observation equations will be of the form 



a/ 



AA 



dAo dBo 



^'^ ■AB+^Afn+-^An-=y-yo. 



dmo 



dno 



Assume the approximate values found in Chapter IV. 

A= 1.522, 
5= -.685, 
w= .55, 
n= 1.4. 



.14 



XT' 

Ao 

Bo 

logjc 

5ox''« log X. 



.05 


.10 


15 


.20 


.25 


.19 


.28 


.35 


.41 


.47 


.02 


.04 


.07 


.10 


.14 


1.522 










- .685 










— 2 . 996 . 


— 2.303 


-1.897 


— 1.609 


-1.386 


- .88 


- -99 


— 1.02 


— 1. 01 


- .98 


.03 


.06 


•09 


.12 


.14 



.30 
.52 
.19 



1.204 
•94 

.15 



.55 
.14 



X 

logic 

i4o«"*" logic.. .. 
fiox**" log X. . . 




The new observation equations become 

.i9Ai4 + .02A5— .88Aw+.03A» = 
.28Ai4 + .04A5— .99Aw+.o6Aw = 
.35Ai4 + .07A5 — 1 .02AW + .09AW = 



.0004, 

.0002, 

— .0001, 



DEDUCED BY THE METHOD OF LEAST SQUARES 99 

.4iAi4+.ioAJ5— i.oiAw+.i2A«= .0000, 
.47Ai4+.i4AJ5— .gSAfn+,i4An^ .0013, 
.S^AA+.igAB— .94Aw+;isAw= — .0001, 
.S6A^+.23A5— .9oAw+.i6Aw= — .0019, 
.6oAi4+.28A5— .84Am+.i7A»= — .0016, 
.64A4+.33A5— .78Aw+.i8A»= — .0001, 
.68Ai4+.38A5— .72Aw+.i8A»= — .0001. 
.72Au4+.43A5— .66Afn+.iSAn= .0011. 

From these the four normal equations are obtained 

2.96oAi4 + i.32iAjB--4.637Aw+ .8o6Aw= —.00071, 

i.32iAi4+ .642 A5 — 1. 802 Aw + .359A»= —.00031, 

— 4.637Ai4 — 1.802A5+8.737AW— i.253Aw = +.ooo85, 

.8o6Ai4+ .359AJB — 1.253AW+ .22iA»=— .00023. 

From which 

AA = — .0068, 

A5=+.0II2, 

Aw =—.0022, 
Af^=— .0070. 

These corrections being applied the final formula becomes 



/■ 



CHAPTER Vn 

INTERPOLATION.— DIFFERENTIATION OF TABULATED 

FUNCTIONS 

Interpolation 
In Chapter 11 we found that the formula 

XI. y ^ 



.02S+.2S2sa;+2.sx2 



represents to a fair degree of approxunation the values of y 
given by the data. Any other value of y, within the range of 
values given, can be obtained in the same way. This rests on 
the assxmiption that the formula derived expresses the law con- 
necting X and y. For example, the value of y corresponding 
toa: = i.os will be 

When a formula is used for the purpose of obtaining values 
of ^ y, within the range of the data given it is called an inter- 
>o(^2^on formula. Interpolation denotes the process of calcu- 
\ l|Lting under some assumed law, any term of a series from values 
of any other terms supposed given.* It is evident that empirical 
formulas cannot safely be used for obtaining values outside 
of the range of the data from which they were derived. 

* For a more extended discussion of the subject the reader is referred 
to Text-book of the Institute of Actuaries, part II (ist ed. 1887, 2nd ed. 
1902), p. 434; Encyklopadie der Mathematischen Wissenchaften, Vol. I, 
pp. 799-820; Encyclopedia Britannica; T. N. Thiele, Interpolationsrechnung. 

As to relative accuracy of different formulas, see Proceedings London 
Mathematical Society (2) Vol. IV., p. 320. 

100 



INTERPOLATION 101 

There are two convenient formulas for interpolation which 
will be developed.* 

The first one of these requires the expression for yx+n in 
terms of yx and its successive differences, yx represents the 
value of a fimction of x for any ctosen value of x, and yx+n 
represents the value of that fimction when x+n has been sub- 
stituted for X. 

yx+i=yx+Ayx; 
yx+2=yx+Ayx+A(yx+Ayx) 

=yx+2Ayx+A^yx; 
yx+3 =yx+2Ayx+A^yx+A(yx+2Ayx+A^yx) 

=yx+3^yx+3^^yx+^^yx; 

yx+^=yx+3^yx+3^^yx+^^yx+A(yx+3Ayx+3A^yx+A^yx) 
= yx+4^yx+6A^yx +4A^yx+A^yx. 

These results suggest, by their resemblance to the binomial 
expression, the general formula 

, . .n(n—i).o , «(w— i)(w— 2) ., , . 
yx+n=yx+nAyx+ ^ ' A^yx-\ — ^ p -A^yx+etc. 



If we suppose this theorem true for a particular value of n, 
then for the next greater value we have 

yx+n+1 =yx+nAyx+^ — -A^yx+— f^ ^A^yx+etc, 

^ 13 



+Ayx+nA^yx+ , ^^ A^y^+etc, 



=yx+(n+i)Ayx+ I A^yx+- V^ ^A^y^+eta 

" 13 



The form of the last result shows that the theorem remains 
true for the next greater value of n, and therefore for the next 

* See Chapter III, Boole's Finite Differences. 



102 



EMPIRICAL FORMULAS 



greater value. But it is true when «=4, therefore it is true 
when n = S' Since it is true ioi n = s it is true when « = 6, etc. 
If now o is substituted for x and x for w, it follows that 

yz=yo+xAyo+-^ — -A^yo+— p ^A^yo+etc. 

If A'*y.=o, the right-hand member of the above equation 
is a rational integral function of x of degree n—i. The formula 
becomes 

, . ,x{x—l).o , x(x — l)(x — 2) .^ , 

yx=yo+xAyo+^ — -A?yo+-^ p -A?yo+ . . . 

^ x{x-l)(x-2) . . . ^^''^'^^\ n^l . . . (l) 



Formula (i) will now be applied to problems. It must not 
be forgotten that in applying this formula x is taken to represent 
the distance of the term required from the first term in the series, 
the common distance of the terms given being taken as unity. 

I. Required to find the value of y corresponding to x = ./^2$ 
having given the values under XIX. In the interpolation 
formula x = .$, 

yo yi y2 y^ 



Ayo 

A^yo. . . . 

A^yo. . . . 

y=yo+2^yo-{A^yo+TEA^yo 

= .730+.oi3S+.ooo5+.oooi 

= .744. 

This is the same as given by XIX. 

2. Find the value of y corresponding to x- 
formula will have the value f if we take yo 

X = 2. 



• 730 


.757 


.780 


.800 


.027 


.023 


.020 




— .004 


-.003 






.001 









2.3. X in the 
• —.1826 when 



INTERPOLATION 



103 



yo 


yi 


y2 


yz 


yi 


ys 


-.1826 


-•4463 


-•7039 


- .9582 


— 1.2119 


-1.4677 


-.2637 


-.2576 


-•2543 


-•2537 


- -2558 




.0061 


•0033 


.0006 


—.0021 






—.0028 


— .0027 


—.0027 








.0001 


.0000 










—.0001 













Ayo 

yx=yo+xLy(i-\ 



--, '-A^yo + -^ p^ ^ A^yo 

2 3 



■1 — ^^ ^-^ — — ^ A^yo + etc. 



(-f) 



l(-f)(~i), 



= -.i826+f(-.2637)+^^^-^(.op6i) + ^' \''^' (-.oo28) 

2 6 

j^iAzMzilizii) {,0001) 
24 

= -•3417- 

3. The following example is taken from Boole's Finite Differ- 
ences. Given log 3.14 = .4969296, log 3.15 = .4983106, log 3.16 = 
.4996871, log 3.17 = .5010593; required an approximate value of 
log 3.14159- 

yo y\ y2 yz 



A>'o. 
A2yo 
A^yo 



.4969296 .4983106 .4996871 .5010593 
.0013810 .0013765 .0013722 
-.0000045 —.0000043 
.0000002 
Here the value of x in the formula is equal to 0.159. 

y:c = 4969296+(.i59)Cooi38io)+ ^^^^^^^^~^ (-. 0000045) 



.i59(.i59~i)(.i59-2) ^ orwv.^o>i 

1 ^.0000002) 

o 



=.4971495. 



104 EMPIRICAL FORMULAS 

This is correct to the last decimal place. If only two terms 
had been used in the right-hand member of the formula, which 
is equivalent to the rule of proportional parts, there would 
have been an error of 3 in the last decimal place. The rapid 
decrease in the value of the differences enables us to judge 
quite well of the acau'acy of the results. The above formula 
<:an be applied only when the values of x form an arithmetical 
series. 

In case the series of values given are not equidistant, that is, 
the values of the independent variable do not form an arithmetical 
series, it becomes necessary to apply another formula. 

Let ya, y*, yc, y^, . . . y* be the given values corresponding 
to a,b,Cyd,,,.k respectively as values of x. It is required 
to find an approximate expression for y^, an unknown term 
corresponding to a value of x between x^a and x=k. 

Since there are n conditions to be satisfied the expression 
which is to represent all of the values must contain n constants. 
Assume as the general expression 



y^=A+Bx+Cx^+Dofi+ . . . +Nxf' 



-1 



Geometrically this is equivalent to drawing through the n 
points represented by the n sets of corresponding values a 
parabola of degree «— i. 

Substituting the sets of values given by the data in the 
equation above n equations are obtained from which to determine 
the values of -4, B, C, etc., 

ya=A+Ba+Ca?+Da?+ . . . Na""-^-, 
yi,=A+Bb+Cb^+Dl^+ . . . iVi'*"^; 



yu=A+Bk+Ck^-\rD]^+ . . . NV"-^. 

But the solution of these equations would require a great 
deal of work which can be avoided by using another but equiva- 
lent form of equation. 



INTERPOLATION 




"Let yx=A{x'-b){x—c){x'-d) . , 


. . {x—k) 


+B(x—a){x—c)(x—d) . 


. . {x—k) 


+C(x-a){x-b)(x-d) . . 


. {x-k) 


+D{x'-a){x—b){x—c) . , 


. {x-k) 


+ etc. to n terms. 





105 



Each one of the n terms on the right-hand side of the equation 
lacks one of the factors x—a^ x—b^ x—c, x—d, . . . x—k, 
and each is affected with an arbitrary constant. The expression 
on the right-hand side of the equation is a rational integral 
function of x. 

Letting x = a gives 



and 



ya=A{a—b){a—c)(a—d) . . . a—k. 



A = 



Ja 



(a—b){a—c){a—d) . . . a—k' 



Letting x = b gives 
B = 



yb 



{b-a){b-c){b-d) . . . (b-k)' 

Proceeding in the same way we obtain values for all of the 
constants and, finally, 

{x--b)(x—c)(x—d) . . . (x — k) 



yx=ya 



+yi 



+y* 



(a--b){a—c){a—d) . 
{x—a){x—c){x--d) . 



{b-a){b-c){b-d) . 
{x—a)(x—b)(x--d) . 



+yd 



{c—a){c—b){c—d) . 

{x—a){x—b){x—c) . 
{d^a){d-b){d-c) : 



+y\ 



{x'-a){x—b){x'-c) 



(k-a){k-b){k-c) 



. (a — k) 
. (x — k) 



. (b-k) 
. (x—k) 



. (x — k) 



. (d-k) 



(2) 



106 EMPIRICAL FORMULAS 

This is called Lagrange's theorem for interpolation. 

1. Apply formula (2) to the data given imder formula XIX 
for finding the value of y corresponding to ii;= 0.425. Select 
two values on either side of the value required, 

(^'^''Z^^ ya = .69S, 
J = 40, y& = .730, 

^ = 45, yc=.757» 
^=.50, yd =.780. 

X in the formula must be taken as 0.5. 

,_/.,,>, K-§)(-f) ,/ ..^x f(-^)(-l) 

y-(-695)(_,)(_,)(_3)+(.73o)(^)(_^)(_^) 

=.744. • 

2. Required an approximate value of log 212 from the fol- 
lowing data: 

log 210 = 2.3222193, 

log 211 = 2.3242825, 
log 213 = 2.3283796, 
log 214 = 2.3304138. 

log 212 = {2 ^22210^) (^)(-^)(-^) +(2 .242820 (^)(~^)(-^) 

= 2.326359. 

This is correct to the last figure. 

In case the values given are periodic it is better to use a 
formula involving circular functions. In Chapter V the approxi- 
mate values of the constants in formula XX were derived. This 
formula could be used as an interpolation formula. But on 
account of the work involved in determining the constants it is 



INTERPOLATION 



107 



much more convenient to use an equivalent one which does not 
necessitate the determination of constants.* The equivalent 
formula given by Gauss is 

_ sm^(x — b) sin^ix—c) . . . sinJC^— ^) 



+yi 



+yc 



sin ^(a — 6) sin f (a — c) . 



sin ^(6— a) sin ^{b—c) . 
sin^ix—a) sm^(x—b) . 



sin ^{c—a) sin ^{c—b) . 
+ etc. . 



. sinj(a— ^) 
. sin^(a::— ^) 



. sin ^(6—^) 
. sm^ix—k) 



sin^(c— ^) 



(3) 



It is evident that the value of ya is obtained from this formula 
by putting x=a. The value of y& is obtained by putting x = by 
and yc by putting x=c. 

The proof that (3) is equivalent to XX need not be given 
here. 

Let it be required to find an approximate value of y cor- 
responding to ic=42*^ from the values given. 



X 


y 


30° 
40° 
50° 


10. 1 
9.8 

8.5 



From (3) 

• o • / o\ 

. >, sini sm (-4) _L 
sm y—5) sm(— 10 ; 



, . sin 6^ sin (-4^) 
sm 5 sm (—5 ) 



+(8.S) n 



• ^O • I 

sm 6 sm i 



sm 10° sin 5° 



- ('Tr^ ^\ (-°^75)(-o698) I /„ o>i (-1 045) (-0698) 

I /-O -X (-1045) (-0175) 

■^^"•^^ (.i736)(.o872) 

= 9.618. 



* Trigometrische Interpolation, Encyklopadie der Mathematischen 
Wissenchaften, Vol. II, pt. I, pp. 642-693. 



108 EMPIRICAL FORMULAS 

A better result would have been obtained by using four sets 
of values. 

Differentiation of Tabulated Functions 

It is frequently desirable to obtain the first and second 
derivatives of a tabulated function to a closer approximation 
than graphical methods will yield. For that piupose we will 
derive diflferentiation formulas from (i) and (2). From 



|2 ^ Is 



. . ,XVX— IKo . XyX—l)\X—2) .^ 



By differentiating it follows that 

2 13 

+ 4^-"^+^^^-V o+ (4) 



Differentiating again 

yJ'=A^yo+{x-i)A?yo+{hx^'-x+H)A'yo+ (s) 

As an illustration let it be required to find the first and 
second derivatives of the fimction given in the table below and 
determine whether the series of observations is periodic* 

The consecutive daily observations of a function being 
0.099833, 0.208460, 0.314566, 0.416871, 0.514136, 0.605186, 
0.688921, 0.764329, show that the function is periodic and deter- 
mine its period. 

* Interpolation and Numerical Integration, by David Gibb. 



INTERPOLATION 



109 



From the given observations the following table may be 



wntten: 

X y =f{x) 

1 0.099833 

2 0.208460 

3 0.314566 

4 0.416871 

5 0.514136 

6 0.605186 

7 0.688921 

8 0.764329 



(- 
(- 
(- 
(- 



0.108627 
0.106106 
0.102305 
)o.097265 
)o.09io5o 
)o.o83735 
)o.o754o8 



From (4) 
y'l^ .108627 



— .000427 
.001260 —.000010 



. 109887 
-.000437 

. 109450 

/a = . 102305 
.002520 



-.000437 



000392 
000019 



104825 —.000411 
000411 



. 104414 



A* 

.002521 
.003801 
.005040 
.006215 
.007315 
.008327 



(+) 
(+) 
(+) 
(+) 



.001280 
.001239 
.001175 
.001100 
.001012 



A« 



.000041 
.000064 
.000075 
.000088 



/2 



/4 = 



. IO6IO6 
.001900 

.108006 
— .000429 

.107577 

.097265 
.003108 

.100373 
.000389 

.099984 



— .000413 
— .000016 

— .000429 



— .000367 
— .000022 

— .000389 



For the remaining first derivatives the order must be reversed 
and the resulting sign changed. 

y'5= — .097265 .002520 y'6= — .091050 .003108 

— .000010 .000413 —.000016 .000392 



097275 
002933 



.002933 



.091066 
003500 



.003500 



.094342 



.087566 



110 



EMPIRICAL FORMULAS 



y7=- 



08373s 
.000019 

•083754 
.004025 

.079729 



003658 /8= -.075408 
000367 —.000022 



004025 



From (5) 



•075430 
.004501 

.070929 



.004164 
000337 

004501 



y"i = — .002521 


.001280 y'2= 


— .003801 


.001239 


.001318 


.000038 
.001318 


.001298 
-.002503 


.000059 


— .001203 


.001298 


y"a=-. 005040 


.001175 /'4 = 


— .006215 


.001100 


.001244 


.000069 
.001244 


.001181 


.000081 


-.003796 


-.005034 


.001181 


y"6=-. 005040 


y\= 


— .006215 




— .001239 




-.001175 




— .006279 


-.007390 




.000038 




.000059 




— .006241 


-.007331 




y'7=-. 007315 


A= 


— .008327 




— .001100 




— .001012 




— .008415 


-.009339 




.000069 




.000081 




— .008346 


— .009258 




X y 


y 


r 


y 


I •099833 


. 109450 


— .001203 


— .0121 


2 . 208460 


.107577 


-.002503 


— .0120 


3 .314566 


. 104414 


-.003796 


— .0121 


4 .416871 


.099984 


-.005034 


— .0121 



5 
6 

7 
8 





INTERPOLATION 




111 


y 


y' 


y" 


y" 
y 


.514136 


.094342' - 


.006241 


— .0121 


.605186 


.087566 - 


.007331 


— .0121 


.688921 


.079729 


.008346 


— .0121 


.764329 


.070929 — 


.009258 


— .0121 



Jl 



Since — is very nearly constant and equal to —.0121, the 
corresponding differential equation is 

y+.oi2iy=o, 
whose solution is 

y^A cos o.iioc+B sin o.wx. 
This shows that y is a period function of oc, and its period is 

27r J 

, or 57.12 days. 

O.II 

Convenient formulas for the first and second derivatives may 
also be obtained by differentiating Lagrange's formula for inter- 
polation. Using five points the formula is 

_ {x—})){x-'C){x—S){x—e) (x—a){x—c){x'--d)(x—e) 
y^-y^ (a-bXa-cXa-dXa-ey^' {b-a){b-c){b-d){b-e) 

, {x—a){x-'b)(x—d){x—e). (x-'a)(x'-b){x'-c)(x—e) 
'^^' {c-a){c-b){c-d){c-e) ^^"^ {d-aXd-bXd-cXd-e) 

{x-a)(x-b)(x-c){x-d) , v 

■^^^ (e-a){e-b){e-c){e-d) ^ ^ 

Selecting the points at equal intervals and letting 

e—d=d-'C=c—b = b—a=hf 
and differentiation 

ya=-^[-25ya+48y6-36yc+i6yd-3yJ, 
/&=— t[- sya-ioyt+iSyc- 6yd+ yd 



112 



EMPIRICAL FORMULAS 



/d = — 7[- ya+ 6>-i8yc+ioyd+3yJ, 

y'e = -^-^[ 3> - le^ft+sfij^c - 48ytf + 2syJ. 



Differentiating again 



y'« = ^[3Sya-io43;6+ii4yc- S6yd+iiyj, 



y"6=m5[iiya- 203^6+ 6yc+ 4yd->'J, 



12^2 



y'c=^^[-ya+ i6y6- 3oyc+ i6yd-yJ, 

y'e = ^[i lya - s6>+ 1 H^c - io4ytf +3SyJ- 



The results of applying these formulas to the function given 
are expressed in the table below. 



X 


y 


y 


y 


I 


.099833 


.109451 


— .001203 


2 


. 208460 


. 107583 


— .002524 


3 


.314566 


.104415 


— .003804 


4 


.416871 


.099986 


-.005045 


S 


.514136 


•094347 


— .006221 


6 


.605186 


.087568 


— .007322 


7 


.688921 


.079733 


-.008334 


8 


.764329 


.070929 


— .009258 



These results agree fairly well with those previously obtained. 
It is probable that the formulas derived from the interpolation 
formula give the most satisfactory results. 



INTERPOLATION 



113 



As another application let us find the maximum or minimum 
value of a function having given three values near the critical 
point. 

Let ya, ybj and yc be three values of a function of x near 
its maximum or minimimi corresponding to the values of x, 
a, b, and c respectively. 
From (2) 

^ (x-b)(x-c) (x-aXx-c) (x-a)(x-b) 
^' ^"^ {a-bXa-cy^'ib-^aXb-cy^' (c-a)(c-6)' 

Equating to zero the first derivative with respect to x 



y. 



=ya 



2X'-b'-C 



4-^6 



2X—a 



x = 



{a-bXa-c) ' '^ (b-aXb-c) 

ya(b^ - c") +y,{c^ - a^) +yc{a' - ^ ) 
2\ya{b - c) +yj,{c - a) +yc{a - b)] 



f-yc 



2x—a-'b 
{c—aXc—b) 



=0; 



(6) 



This is equivalent to drawing the parabola 

y=-A+Bx+Cx? 

through the three points and determining its maximiun or 
minimum. 

From the table of values 



6.0 

6.5 
7.0 



10.05 
10.14 
10.10 



the abscissa of the maximum point is foimd from (6). 

^^ (io.o5)(~6.75) + (io.i4)(i3) + (io.io)(~6.25) 
2[(io.o5)(-.s)+(io.i4)(i) + (io.io)(-.s) 



= 6.596 



y = 10.1424, 



CHAPTER VIII 
NUMERICAL INTEGRATION 

Areas 

An area bounded by the curve, y=f{x), the axis of x, and 
two given ordinates is represented by the definite integral 



=r 



ydXy 



where the ordinates are taken at x = a and x=n. It may be 
said that the definite integral represents the area imder the 
curve, or that the area imder the curve represents the value of 
the definite integral. 

If a function is given by its graph, it is possible, by means 
of the planimeter, to find roughly the area boimded by the curve, 
two given ordinates and the jc— axis, or, what amoimts to the 
same thing, the area enclosed by a curve. This method is used 
in finding the are^ of the indicator diagrams of steam, gas or 
oil engines, and various other diagrams. The approximations 
in these cases are close enough to satisfy the requirements. 

If, however, considerable accuracy is sought, or whenever 
the function is defined by a table of numerical values another 
method must be employed. 

Mechanical Quadrature or Numerical Integration is the method 
of evaluating the definite integral of a fimction when the fimc- 
tion is given by a series of numerical values. Even when the 
function is defined by an analytical expression but which can- 
not be integrated in terms of known functions by the method 
of the integral calculus, numerical integration must be resorted 
to for its evaluation. 

The formulas employed in numerical integration are derived 
from those established for interpolation. 

114 



NUMERICAL INTEGRATION 115 

In interpolation it was found that the order of differences 
which must be taken into accoxint depends upon the rapidity 
with which the differences decrease as the order increases. 
This is also true of numerical integration. It is the same as 
saying that if the series employed does not converge the process 
will give unsatisfactory results. An illustration will be given 
later. 

Formulas for numerical integration will be derived from (i) 
of Chapter VII. 

In this formula it was assumed that the ordinates are given 
at equal intervals. 

yz=yo+xAyQ+^ — ^A^yoH- — | -A^yo 



^ x(x-i)(x-2){x'-s) ^. ^ x(x-i){x-2)(x-s)(x-4) ^.^ 

k ° Is 



^ x(x-i){x-2)ix-s)(x-4)(x-5) ^f, ^ ^^>^ 

Integrating the right-hand member, 

J\ yxdx=yQ I dx+Ayo I ocdx-\ — p^ I x{x — i)dx 
Jo Jo \2 J 

- I x{x — i){x-'2)dx 
Jo 






+ 



4 
Asyo 



■ I x(x-'i)(x-'2)(x — ^)dx 
Jo 

J I x{x—i){x'-2)(x—:y)(x—4)dx 




\-^jJx(x^i)ix--2)(x-3)ix-4){x^5)dx+ . . . 



=wyo 



2 \3 2/ |2 \4 / I3 



\S 2 3 / I4 



116 EMPIRICAL FORMULAS 

\6 4 3 /Is 



+ 



\7 2 4 3 / P 



The data given in any particular problem will enable us 
to compute the successive differences of yo up to A"yo. On the 
assumption that all succeeding differences are so small as to 
be negligible the above formula gives an approximate value 
of the integral. It is only necessary to assign particular values 

Let w=2, then 

I yzd:i: = 2yo+2Ayo+iA2yo, 

A23;o=Ayi-Ayo=y2-yi-yi+yo, 

=3'2-2yi+yo. 
Substituting these values in the above integral it becomes 

I yxdx = 2yo+2yi-2yo+iy2'-iyi+iyoy 

_yo+4yi+y2 

3 

This is equivalent to assuming that the curve coincides with 
a parabola of the second degree. 

If the common distance between the ordinates is h, the 
value becomes 

ydx=ih(yQ+4yi+y2) (7) 



X 



If ;^=3 

'3 

ydx=syo+^^yQ+l^^yQ+i^^yo, 


Ayo=yi— yo, 
A^yo=Ayi-Ayo=y2-'2yi+yo, 



i 



NUMERICAL INTEGRATION 117 

A^yo = A^yi — A^yo = Ay 2 — Ay 1 — Ay 1 +Ayo 
=y3-3>'2+3)'i-yo. 

Substituting these values in the equations, 

1 ydx=syo+h^i-h^o+iy2-hi+ho+hz-h^2+h^i-ho, 

=lyo+lyi+ly2+ly3, 
=1(^0+3^1+3^2+^3). 

If the common distance between the ordinates is h the 
formula becomes 



XZh 
=U(yo+3yi+3y2+ys) 



(8) 



This is equivalent to asstmiing that the curve coincides with 
a parabola of the third degree. 

If there are five equidistant ordinates, h representing the 
distance between successive ordinates 



r 



y^^=^4(y»+>^4)+64(yi+y3) + 24y2^^ ... (9) 
b 45 



If the area is divided into six parts boxmded by seven equi- 
distant ordinates the integral becomes 

I y(/:i: = 6yo+i8Ayo+27A2yo+24A3yo+ WA^yo 
+HA5yo+ittrA«yo. 

Sine 3 the last coefficient, t^, differs but slightly from -nr 
and by the assumption that A^yo is small the error will be slight 
if the last coefficient is replaced by tu* 

Doing this and replacing 

Ayobyyi-yo, 
A^yo by y2 - 2yi +yo, 
A3yo by y^-^yz+Syi-yoy 



118 



EMPIRICAL FORMULAS 



A^jb by y4-4)'3+6y2 -4)'i+yo, 
^^yo by ys - 5^4 + loya - ioy2 + syi - yo, 
A^yo by /-6y5+iSy4-2oy3+iS)'2-6yi+yo, 
gives the formula 



Jrth 
I- 



y(/:i: = ^%o+y2+y4+y6+s(yi+)'5)+6y3]. . (lo) 




The application of these 
formulas is illustrated by 
finding the area in Fig. 27. 



Fig. 27. 



By (7) 



4=1^(^0+4^1 + 2^2+4^3 + 2^4+4^6 + 2^6+4^7+2^8+4^9 
+ 2yi0+4)'ll+yi2). 

By (8) 

4=fA(yo+3)'i+3y2+2y3+3y4+3)'6+2y6+3>^+3y8+2y9 
+3yio+3>'ii+yi2). 

By (9) 

-4=AA[i4(yo+2y4+2y8+yi2)+64(yi+y3+y6+y7+y9+yii) 

+ 24(3/2 +y6+y 10)]. 
By (10) 

-4=iir%o+y2+y4+2y6+y8+yio+yi2+s(yi+)'6+y7+yii) 
+6(3/3+^9)]. 

I. A rough comparison of the approximations by the use 
of these formulas will be obtained by finding the value of 






— . The value of this definite integral is log 13 = 2.565. It 
1 X 

is also equal to the area imder the curve 



y= 



X 



NUMERICAL INTEGRATION 119 

from x = i to a;«=i3. Dividing the area up into 12 strips of 
unit width by 13 ordinates the corresponding values of x and 
y are 



X 


I 


2 


3 


4 


s 


6 


7 


8 


9 


10 


II 


12 


13 


y 


I 


i 


i 


i 


i 


i 


i 


i 


1 


■h 


1 
11 


1^ 


1^ 



By (7) 

i4=Mi+2+f+i+f+f+f+Ht+f+A+i+A] 
= 2.578, error .5%; 

By (8), ^ = 2.585, error .8%; 

By (9), -4 = 2.573, error .3%; 

By (10), A = 2.572, error .3%. 

2. The accuracy of the approximation is much increased 

by taking the ordinates nearer together, as is shown by the 

following evaluation of 

^^ dx 



r 



i+x 



The value of this integral is equal to the area under the 

curve 

_ I 

from a;=o ta x = i. Dividing the area into twelve parts by 

thirteen equidistant ordinates the value of I — -— is foimd to be 

Jo i+x 

By (7), 0.69314866, error 0.00000148; 

By (8), 0.69315046, error 0.00000328; 

By (9), 0.69314725, error 0.00000007; 

By (10), 0.69314722, error 0.00000004. 

The correct value is, of course, log^ 2, which is 0.693 14718. 
Formulas (7) and (8) are Simpson's Rules, (10) is Weddle's 
Rule. 



120 



EMPIRICAL FORMULAS 



3. Apply the above formulas to the area of that part of the 
semi-ellipse included between the two perpendiculars erected at 
the middle points of the semi-major axes. Let this area be 
divided into twelve parts by equidistant ordinates. 

Since the equation of the ellipse is 

^ y'f — 

these ordinates are 

By (7), -4=0.9566099^6; 

By (8), i4 =0.9566080^6; 

By (9), -4=0.956611406; 

By (10), A =0.956611406. 

The correct value to seven 
places is 0.95661 1506. 

In the application of these 
formulas it is highly desirable 
to avoid large differences among 
the ordinates. For that reason 
the formulas do not give so 
good results when applied to 
the quadrant of the ellipse. 

4. The area under the curve, 
Fig. 28, determined by the 
following sets of values: 

.6 .8 



3.0 

2.5 
2.0 

1.6 

1.0 

0.5 





. 


















/ 




V 








J 


Y 




N 


^ 


1 




/ 












/ 










































i 


1 i 


1 


S A 


3 1. 


1. 


2 3) 



X 



Fig. 28. 
.2 



.4 



i.o 



1.2 



1.0 1.5 2.2 2.7 2.6 2.3 2.1 

is by (7) 

A =i-Ki-o+6.o+4.4+io.8+s.2+9.2+2.i) = 2.58, 
and by (8), 

4 =1-1(1.0+4.5+6.6+5.4+7.8+6.9+2.1) = 2.5725. 



NUMERICAL INTEGRATION 



121 



This area is represented by the definite integral | ydx. 



: 



The area found is therefore the approximate value of this 
integral 

5. Find the area xinder the curve determined by the points 

x\i 1.5 1.9 2.3 2.8 3.2 3.6 4.0 4.6 4.8 5.0 

— I' _^^_— — — ^-— ^-^— ^— ^^— ^— ^^-^— ^— — ^-^— ^— ^-^— ^^— ^— ^— — .^— — — ^^^— __ 

y\o .40 1.08 1.82 2.06 2.20 2.30 2.25 2.00 1.80 1.5 

The points located by the above sets of values are plotted 
in Fig. 29 and a smooth curve drawn through them. The area 




Fig. 29. 

is divided into strips each having a width of .4. Rectangles 
are formed with the same area as the corresponding strips. 
The eye is a very good judge of the position of the upper boimd- 
ary of each rectangle. Adding the lengths of these rectangles 
and multiplying the sum by .4 the area is foimd to be 6.644. 
By Simpson's Rule, formula (7), are found 



for h = ,2, 
A = .4, 



i4 =6.639, 
A =6.645. 



The graphical determination of areas can be made to yield 
a close approximation by taking narrow strips, and where the 
points are given at irregular intervals the area can be found 
more rapidly than by the application of Simpson's Rules. 



122 EMPIRICAL FORMULAS 

6. A gas expands from volume 2 to volume 10, so that its 
pressure p and volume v satisfy the equation pv = ioo. Find 
the average pressure between v = 2 and z; = 10. 

The average pressure is equal to the work done divided by 

8. The work is equal to the area under the curve />=— from 

V 

z; = 2 to z; = 10, which is 



£ 



That this area represents the work done in expanding the 
volume from 2 to 10 becomes evident in the following way. 
Let s represent the surface inclosing the gas, ps will then be 
the total pressure on that surface. The element of work will 
then be 

dW=psdnj 

when dn lepresents the element along the normal. 

W=fpsdn. 

But 

sdn=dVj 
and 

W=j^pdv. 

This is the equation above. The average pressure over the 
change of volume from 2 to 10 is 

160.944-^8 = 20.118. 

7. Find the mean value of sin^ x from x=o to x = 2w. Plot 
the curve y = sm? x by the following values of x and y: 

X 



X 






12 


TT 

6 


4 


■K 

3 


Sir 
12 


TT 
2 





.0670 


.2500 


.5000 


.7500 


•9330 


I. 0000 


^l^ 


2t 


J^ 


S^ 


inr 






12 


3 


4 


6 


12 







•9330 -7500 .5000 .2500 .0670 



NUMERICAL INTEGRATION 



123 



X 



T 


I37r 

12 


77r 
6 


4 


47r 
3 


I77r 
12 


3^ 
2 


O 


.0670 


.2500 


.5000 


.7500 


•9330 


I. 0000 



a; 



19F 
12 



3 



Ttt 
4 



IITT 



135 
12 



27r 



•9330 -7500 .5000 .2500 .0670 O 

Applying Simpson's Rule, formula (7), the area is found to 
be TT. The mean value is the area divided by 27r or .5. 

8. A body weighing 100 lb. moves along a straight line 
without rotating, so that its velocity v at time / is given by the 
following table: 

/sec 



V ft./sec 



1.47 1.58 ^.67 1.76 1.86 

Find the. mean value of its kinetic energy from / = i to / = 9. 



/ 


I 


3 


5 


7 


9 


v^ .' 


2.1609 


2 . 4964 


2 . 7889 


3.0976 


3 4596 




Kinetic energy . 


3. 355 


3.876 


4-331 


4.810 


5-372 



Plotting kinetic energy to /, the area under the curve is 
34.755. This divided by 8 gives the mean kinetic energy as 

4.357- 

Volumes 

Fig. 30 explains the ap- 
plication of the formulas 
, to the problem of finding 
the approximate volume of 
an irregular figure. The area 
of the sections at right angles 
to the axis of x are: 

Ai = lk(yi+4y5+y4^y 

A2=lk(y6+4yg+y8)y 

^3 = ^0^2+4^7+^3). Fig. 30. 




124 EMPIRICAL FORMULAS 

If the areas of these sections be looked upon as ordinates, 
h being the distance between two adjacent ones, it is evident 
that the volume may be represented by the area under the 
curve drawn through the extremities of these ordmates. 

V^\h{Ai+^2+Az) 

Substituting the'values of A\, A2, and Az in this equation, 
the volume becomes 

F = JA[P(yi+43'5+)'4) +^^Cv6+4y9+y8) +P(y2+4y7+y3)] 
=iA%i+y2+y3+y4+40'5+y6+y7+y8)+i6>'9] 

In order to apply formulas (8), (9) and (10), the solid would 
have to be divided differently, but the method of application 
is at once evident from the above and needs no further discussion. 

1. The following arp values of the area in square feet of the 
cross-section of a railway cutting taken at intervals of 6 ft. 
How many cubic feet of earth must be removed in making the 
cutting between the two end sections given? 

91, 95, 100, 102, 98, 90, 79. 

These cross-section areas were obtained by the apphcation 
of Simpson^s Rules. 

By (7), 

7=^-6(91+380+200+408+196+360+79) =3428; 
By (8), 

7=1-6(91+285+300+204+294+270+79) =3426.8. 

2. -4 is the area of the surface of the water in a reservoir 
when full to a depth A. 



hit, . . 



A sq.ft. 



30 25 20 15 10 



26,700 22,400 19,000 16,500 14,000 10,000 5,000 



NUMERICAL INTEGRATION 125 

Find (a) the volume of water in the reservoir, (b) the work 
done in pumping water out of the reservoir to a height of loo ft. 
above the bottom until the remaining water has a depth of 
lO ft. 

F =1(26,700+89,600+38,000+66,000+28,000+40,000+5,000) 
=488,833 cu. ft. 

rzo 
Work=w I A{ioo—h)dhy where w= weight of i cu.ft. of water 

= 62.3 lb. The value of this integral will be approximately the 
area under the curve determined by the points 

h 



30 25 20 15 10 

— — • — t 

i4(ioo— A), j 1,869,000 1,680,000 1,520,000 1,402,500 1,260,000 

multiplied by 62.3. 
This area is equal to 

1^(1,869,000+6,920,000+3,040,000+5,610,000+1,260,000) 
= 31,165,000. 

Multiplying this by 62.3 gives the work equal to 1,941,579,500 
ft.-lb. 

3. When the curve in Fig. 29 revolves about the ic-axis, 
find the volume generated. 

The areas of the cross-sections corresponding to the given 
values of x are given in the following table: 



X 





.2 


.4 


.6 


.8 


I.O 


1.2 


^ .... 


IT 


2.257r 


4.847r 


7.297r 


6.767r 


5.297r 


4.4i7r 



By (7) F = 5.8627r = 18.416. 

By (8) F = 5.8o37r= 18.231. 

4. When the curve in Fig. 30 revolves about the nc-axis, 
find the volume generated from x = i toa;=4.2. From the curve 
the following sets of values are obtained: 



126 EMPIRICAL FORMULAS 

:t;|i.oi.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 



3^0 .11 .29 .53 .87 1.37 1. 71 1.90 2.01 



'fX O .012 .084 .281 .757 1.877 2.924 3.610 4.040 



X 

y_ 
f 



2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 



2.06 2.12 2.2 2.27 2.30 2.28 2.25 2.20 



4.2444.4944.84 5.153 5.290 5.198 5.062 4.84 

The volume is by (7) 

^•ii(i49-oo4)=3i.2 cu. units. 

Centroids 

Let the coordinates of the centroid of an area be represented 
by X and y. Then from the calculus 

rxydx 
^ 



1 ydx 
- I y^dx 

y=P — " 

I ydx 



The integral in the numerator of the value of x may be 
represented by the area bounded by the curve Y=xyy the x-auds 
and the two ordinates a; = a and x = b. The original area is 
bounded by the curve whose ordinates are represented by y, 
the ic-axis and the two ordinates x = a and x = b. The integral 
in the numerator of the value of y may be represented by the 
area bounded by the curve Y=y^, the ic-axis and the two 
ordinates x = a and x = b. 

For a volume generated by revolving a given area about the 
x-2ods • 



TT I y^xdx 
TT I y^dx 



NUMERICAL INTEGRATION 



127 



/ If 

10 . 



1 ir 



idu 



repKSSi- 



tie 
le 



When the volume is irregular 






Axdx 



f. 



Adx 



The process of finding the coordinates of the centroid of the 
area in Fig. 28 is shown in the table: 



X 





.2 


.4 


.6 


.8 


1.0 


1.2 


y 


I.O 


IS 


2.2 


2.7 


2.6 


2.3 


2.1 


xy 


0.00 


0.30 


0.88 


1.62 


2.08 


2.30 


2.52 


y' 


1. 00 


2.25 


4.84 


7.29 


6.76 


S.29 


4.41 


yH 


0.000 


0.450 


1.936 


4.374 


S.408 


5.290 


5.292 



The area under the curve F=a;y is 

i^[o.oo+i.2o+i.76+6.48+4.i6+9.2o+2.52] = 1.688; 

- 1.688 . 

^ = —-^ = •654. 
2.55 

The area under the curve F=J3^ is 

^[1.00+9.00+9.68+29.16+13.52+21.16+4.41] = 2.931 

- 2.931 , 

As was pointed out before, large changes in the ordinates 
must be avoided. 

For the volume generated by revolving the area about 
the ic-axis 

-_7rA^[o.ooo+ 1.800+3.872 + 17.496+ 10.816+21. 160+5. 292] 



X — 



7rr5[i.oo+9.oo+9.68+29.i6+ 13.52 +21. 16+4.41] 



=^=.687. 
8793 



128 



EMPIRICAL FORMULAS 



Moments of Inerila 

The expression for the moment of inertia of an area about 
the y-axis is 

Iy= I x^ydx. 

About the x-axis 

Ix= I xy^dy. 

When the equation of the curve is known these integrals 
can be calculated at once, but when this is not the case approxi- 
mate methods must be resorted to. 

I. The process of finding the approximate values of these 
integrals is shown in the table below. The values of x and y 
are taken from Fig. 28. 



X 





.2 


.4 


.6 


.8 


1.0 


1.2 


y 


I.O 


15 


2.2 


2.7 


2.6 


2.3 


2.1 


x^ 


O.CXX) 


0.060 


0.335 


0.972 


1.664 


2.300 


3.024 


Jy' 


0-333 


1. 125 


3. 549 


6.561 


5.859 


4.056 


3.087 



If the values of x^y be plotted to x we will have a curve 
imder which the area represents the moment of inertia of the 
area in Fig. 28 about the y-axis. 

Dividing this by the area foimd before, there results for the 
radius of gyration about the y-axis 



if/ = .526. 

Plotting ^ to X and finding the area under the curve so 
determmed 

/x= 4.6136, 
and 

22x2 = 1.788. 



NUMERICAL INTEGRATION 



129 



2. The form of a quarter section of a hollow pillar, Fig. 31, 
is given by the following table. Find the moment of inertia 
of the section about the axes of x and y. 



y 

.6 


I 












•i: 


^ 


r-^ 


^. 








A 


'^^ 


^ 




^v. 


[N 




N 


w. 




\ 








Ji 


N 


'v 




\ 






\ 






\ 






.2 


\ 




\ 


k, 








\ 






\ 




a 




\ 




\ 










\ 




\ 














\ 







.2 



.3 

Fig. 31. 



.4 



X 



X 


F 


x^Y 


y 


X 


y^X 


.00 


.050 


.00000 


.00 


.100 


.00000 


•05 


.055 


.00014 


•05 


.108 


.00027 


.10 


.068 


.00068 


.10 


.116 


.00116 


■ 15 


.078 


.00175 


IS 


.120 


.00270 


.20 


.096 


.00384 


.20 


.125 


.00500 


• 25 


.116 


.00725 


.25 


.130 


.00812 


.30 


.148 


.01332 


.30 


• 133 


.01197 


.35 


.2CX5 


.02450 


•35 


.140 


.01715 


.40 


.300 


.04800 


.40 


.150 


.02400 


•45 


•215 


•043S4 


•45 


.215 


.04354 


•50 


.000 


.00000 


.50 


.000 


.00000 



In the above table X stands for the width of the area parallel 
to the x-aixis and Y for the width parallel to the y-axis. The 
area is 0.066. 

The moment of inertia about the y-axis is 



r- 



.066 



130 EMPIRICAL FORMULAS 

The moment of inertia about the ^-axis is 



where if stands for the radius of gyration. 

The values of the above integrals were computed by for- 
mula (7). 



APPENDIX 



If a chart could be constructed with all the dlfiferent forms 
of curves together with their equations which may arise in 
representing different sets of data it would be a comparatively 
simple matter to select from the curves so constructed the one 
best suited for any particular set. Useful as such a chart would 
be its construction is clearly out of the question. The most 
that can be done of such a nature is to draw a number of curves 
represented by each one of the simpler equations. 

A word of caution is, however, necessary here. A particular 
curve may seem to the eye to be the one best suited for a given 
set of data, and yet, when the test is applied, it may be found 
to be a very poor fit. It is of some aid, nevertheless, to have 
before the eye a few of the curves represented by a given formula. 

The purpose of the following figures is to illustrate the 
changes in the form of curves produced by slight changes in 
the constants. Figs. I, II, III, and IV show changes produced 
by the addition of terms, Figs. V to XIX changes in form 
produced by changes in the values of the constants, and Fig. 
XX the changes in form brought about by varying both the 
values of the constants and the nimaber of terms. 

A discussion of all the figures is imnecessary. A few words 
in regard to one will suffice. Formula XIV, for example, 
y = a+bx'', an equation which can be made to express fairly 
well the quantity of water flowing in many streams if x 
stands for mean depth and y for the discharge per second, 
represents a family of triply infinite number of curves. Fixing 
the values of b and c and varying the value of a does not 
change the form of the curve, but only moves it up or down 

131 



132 



EMPIRICAL FORMULAS 



along the y-axis. Keeping the values of a and h constant and 
varying the value of c, the formula will represent an infinite 
number of curves all cutting the y-axis in the same point. In 
the same way, keeping the value of a and c constant and vary- 
ing the value of J, an infinite number of curves is obtained, 
all of which cut the y-axis in a fixed point. In Fig. XIV the 
quantity a is constant and equal to unity, while h and c vary. 

To one trained in the theory of curves the illustrations are, 
of course, of no essential value, but to one not so trained they 
may be of considerable help. 

The text should be consulted in connection with the curves 
in any figure. The figures are designated to correspond to the 
formulas discussed in the first five chapters. 



y 

1 ft 


















































/ 




1.0 

lA 
L4 
L2 

1 
0.8 
























/ 


























// 
























h 


/ 


























X 






«^ 


?^^ 










(22^ 




f 


r 










^ 








— ^ 


(6) 














^ 




0.6 

0.4 

0L2 



-OJ! 

•0.4 


?^ 












"^ 




W^ 


^ 






















tD 


^ 


V, 


^ 


\ 
























^ 


\ 




























^ 






• 




















\ 


\ 




-08 
























V 


























\ 





(1) i/=l-.lx 

(2) i/=l-.la;+.01a;2 

(3) i/=l-.la;+.01x*-.001x5 

(4) y=l-.lx4-.01a;2-.001a;'-L 

.OOOlx* 

(5) y=l-.lx+.01x2-.001x'+ 

.0001x*-.00001z* 

(6) y=-l-.lx4-.01x2_.ooix«-f- 

.OOOlx*- .OOOOlx'+.OOOOOlx*' 

See formula I, page 13 



8 9 10- u u m 



Fig. I. 



APPENDIX 



133 



y 

1.4 

1 


\\ 




















































\ 
























i 












=«« 


— 










0.8 

0.6 

0.4 

OJt 



-0.2 

-0.4 

-ae 

-03 




w^ 










fftH? 


^ 


^ 




















15W 


/ 


























































































































* 




' 




















































1 


























» 


I 


i' I 


} i 


\ ( 


I ( 


i 3 


r J 


i f 


) 1 


1 


1 L 


e » 



(1) i/=l-l/a; 

(2) y=l-l/x+l/x^ 

(3) i/=l-l/a;+l/x*-l/x» 

(4) y=l-l/x+l/x^-l/x^'\-l/x* 

(5) i/=l-l/x+l/x2-l/x»+ 

l/x*-l/x* 

(6) i/=l-l/x+l/x*-l/x«4- 

l/x*-l/x*+l/x« 

See formula II, page 22 



Fig. n. 



8, 

2.8 


V 












/ 


\ 
























/ 


\ 












&6 












1 


/ 


/ 


1 










2.4 












J 




J 












2.2 












1 


(8)/ 














2 










/ 




// 


f 












1.8 








,/ 


/ 


/; 


/ 














1.0 


- 




J 


/ 


4 


/^ 






N^ 










1.4 




/ 


^ 


^ 


^0^ 




-^ 


^N 


^ 


\ 








1.2 




^ 
















^ 








4 






















^ 


V^ 


• 


M 






















\ 


s> 




0.6 
























\ 




0.4 

0.2 


























































I 


I \ 


i ' 




^ ( 


J 


K 


1 


I 


I 


i I 


i h 



(1) -«l_.la; 

y 

(2) -=l-.lx+.01x* 

{S) -=l-.lx+.01x*-.001x' 

y 

(4) -=l-.lx+.01x2-.001a8i- 

y 

.OOOlx* 
(6) -=l-.lx+.01x«-.001x3+ 

' y 

.0001x*-.00001x* 



— 1 



(6) -=l-.lx+.01x2-.001x5+ 

y 

.OOOlx*- .OOOOlx^+.OOOOOlx* 
See formula III, page 25 



Fig. m. 



134 



EMPIRICAL FORMULAS 



y, 




























t 

L8 






















































•u 
















































y. 


^ 




1 


-= 


^ 




^ 






(2L 







^ 


^ 






0i8 
08 








*^ 


■^ 






■^ 


— i«v. 


















' 




> 


^ 


3 


\ 








02 

• 




















\ 








«-04 
•06 




















-^ 


























' 








1 


1 
i 


1 1 


I i 


d 


> C 


1 1 


f 


1 i 


1 II 


I I 


L L 


r'Sj 



(1) i/«=l-.lx 

(2) i/*-l-.lx+.01x* 

(3) |/»-l-.la:+.01x«-.001x» 

(4) y«=l-.lx+.01x'-.001x»+ 

.OOOlx* 

(5) y«=l-.lx+.01x2_.ooix»+ 

.0001x*-.00001x* 

(6) y«=l-.lx+.01x2-.001x'+ 

.0001x*-.00001x^+.000001x« 

See formula IV, page 25 



Fig. IV. 



s 

M 

tA 

U 

ti 

S 














/ 










./ 


1 




r 












/ 










/ 


/ 


















/ 










/ 


















/ 


4' 








/ 




















/ 








/ 


r 


















J 


/ 






f\ 


/ 












^ 








/ 






/ 


r 










^ 


-^ 








/ 


r 




/ 












^ 








lA 




/ 


/ 


y 


/ 






^ 


^ 


"^ 












1.4 




6 


L/ 




^ 


-" 






















ti^ 




r 










(»). 







— 








— 


1 


^ 








— ■ 







-(7)- 













-__ 


08 
06 










^ 


s^ 


^ 


^~ 




-^ 


^ 
















$^ 








^~ 




^ 


- 






■ — 








— 


M 




\ 


:v 


^ 


v-._ 


.1% 


■^ 




" " 




■— . 
















^ 


^ 


^ 


2*2.^ 







■*" 


r: 




— 


__ 


___ 




1 


I 1 


i 


1 i 


< 


\ 4 


\ ] 


( 


r 1 


> 11 


> 1 


1 L 


B 11 


S V 


\ u 



(1) i/-(.5)^ 

(2) l/=(.6)* 

(3) l/=(.7)^ 

(4) i/=(.8)^ 

(5) y=(.9)^ 

(6) y=(.95)^ 

(7) y=.99)* 

(8) y=(1.01)* 

(9) y=(1.05)* 

(10) y=(l.l)* 

(11) y=(1.2)« 

See formula, V, page 27 



Fig. V. 



APPENDIX 



135 



V 




























16 






3> 


?y< 


^ 


:::::: 




== 


, . 







_^ 




1.6 




y^ 


$; 


"" (a 


(4)^ 


^ 


-.^-' 


^-' 


-^ 










1.4 


^ 


^ 






^ 


m^ 




-^ 








^_ 




12 


>^ 


^ 


:^ 




■^ 


(7) 


0) 


— ' 


' — 


_ ^ 









1 
08 


1 


,^^ 


=== 


I^ 


— :;; 







— 


— 









^ 


^ 


== 


^ 





mr 
^1) 





— 


— 









06 




■ 


^ 


^ 


vS 


--^ 


(12) 


" " 


^^ 


-^ 








0.4 










v 


^ 


i^, 


"^ 


^-^ 


^ 




"^ 




0..2 














N 


N. 


N^ 






v^ 




•Oi2 
















s 




[\ 


V 
























\ 




\ 




-04 






















\ 


s 




-(L8 
























\ 




-U.B^ 


> 


I J 


1 I 


( 4 


I 


( ( 


\ 


r 


\ 


\ 11 


9 1] 


L L 


I OB 



(1) y=:2-(.5)* 

(2) y=2-(.6)* 

(3) i/=2-(.7)* 

(4) i/=2-(.8)* 

(5) i/=2-(.85)* 

(6) i/=2-(.9)*' 

(7) i/=2-(.95)* 

(8) y=2-(.97)* 

(9) y=2-(.99)* 

10) j/=2-(1.01)* 

11) i/=2-(1.03)* 

12) i/=2-(1.05)* 
13)j/=2-(1.07)* 
14) j/=2-(1.08)* 

See formula VI, page 28 



Fig. VI. 



y 

10 





























9 






J 


y^ 




"-^ 


_^ 












— 


— 


8 








V 


^ 


x^ 








"^ 














// 


'% 


y 




^^ 




















/ 


/ 


/ 




,^ 


















7 




h 


V 


/ 


^ 


r^ 








-^ 








« 




TT 


/— 


-p^ 


















"^ 


■ 


6 


1 


7/ 


y 


/ 




(5J^ 


''^ 
















Fa 


/ . 


r 














«i^ — 












// 


y 


^^ 


--^ 




(6) 


^^ 
















4 

8 


i 




— 


^ 




J7), 

=i2i 


' 














2 
1 




N 


% 


^ 


^ 


^ — 


.42> 


















\^ 


^S^2 


f->a 


)^ 


■^ 


, 






— 








_ 






~^ 







■ 


















• 






























D ] 


L 


1 { 


1 ' < 




5 ( 


1 ' 


r 1 


) 


» 1 


1 


I 1 


2 


m 



(1 

(2 
(3 
(4 
(5 

(6: 

(7 
(8 
(9 

(lo: 

(11 
(12 
(13 



logi/= 
log|/= 
log J/= 
log|/= 
log J/= 
logy= 
logi/= 
log J/= 
logy= 
logy= 
log J/= 
logy= 
logy= 
base= 



-.6(.6)^ 

-.5(.6)* 

-.5(.7)* 

-.5(.8)* 

-.5(.9)* 

-.5(.95)* 

-.5(.98)* 

-.5(1.02)* 

-.5(1.1)* 

-.5(1.2)* 

-.5(1.3)* 

-.5(1.5)* 

-.5(2)* 



See formula VII, page 32 



Fig. Vn. 



136 



EMPIRICAL FORMULAS 



y 

t 


































.^ 






(1) 
















1.A 




(2) 




\4i 




-^ 


^ 




■ 


J3)^ 


^ 














1.4 




^ 






■^^ 




^, 












U 






N^ 




S.I71 




^ 




^ 








1 


f 






N 


\ 


^ 




\ 


V 










M^ 












S(8> 


\ 


\ 


\ 


N 


•^ 






0.6 














\ 




\ 


s. 


X 


s 




a4 
















\ 




X 


s 


^^ 




a2 


















\ 




^-^ 


\ 























\ 








02 






















\ 






a4 






















\ 









i 


a 


4 


5 


fl 


> 


fl 


fl 


IC 


> u 


L U 


1 « 



(1) y=2-.01a>-(.5)* 

(2) i/=2-.03a!-(.5)* 

(3) y=2-.05x-(.5)* 

(4) y»2-.08a;-(.5)* 

(5) y*2-.lx-(.5)* 

(6) y=2-.l2z-(.5)* 

(7) i/=.2-.15i-(.5)* 

(8) i/=2-.2x -(.5)* 

See formula VIII, page 33 



Fig. VIII 



V 
























^■■■■B 




e 






















' 








i 


























6 
























































4 






1 


















/ 






\ 








- 














/ 




8 


\\ 






















// 






V) 


\ 


















\ 


/ 




2 


\ 


\ 


















/ 


/ 






\" 


X! 


















1/ 


/ 




1 




::^ 




^ 














r—- 


^ 









I6i^ 












1 




:^ 


^ 







1 


% 


a 


4 


6 


« 


1 


8 


8 


U 


1 1 


L U 


1 _• 



(1) j/s=lO-81--3to + .03a;* 

(2) y =10.54-. 24a; +.02z* 

(3) j/=10-27-.12a;+.01a;« 

(4) i/= 10-135 -.Oftr+. 005a;* 

(5) i/=10--135 + .06x- OOSx* 

(6) y=10--54+.24a;-.02z« 

See formula IX, page 37 



Fig. IX. 



APPENDIX 



137 



V 


' 






















































i 


\ 


























L6 


\ 








' 


















L6 


\ 


















3^ 


(3) 






1.4 
















^ 


=^ 






1.8 


\ 


\r 


— 





■** 


















1 




\ 




















/ / 




a8 




> 


V 
















J 


7 




0.0 






\ 


^ 














/ 


/ 




0.4 
02 








^ 


^ 








y^ 


(5/ 




















— " 




J2- 

















I i 


1 


i ^ 


1 1 


5 


6 


7 


i 


9 1 


1 


1 i 


2 w 



(1) i/=(1.01)* (1.05)(l-2)^ 

(2) y=(1.01)* (1.05)<1-1®^^ 

(3) y=(1.01)* (1.05)<1-1S^* 

(4) y=(.6)* (2) (1-24)^ 

(5) i/=(.5)* (2)a-23)^ 

(6) y=(.5)^ (2)Cl-2)^ 

See formula X, page 37 



Fig. X. 



1 


i 


























10 
9 




f 


\ 














" 








8 

7 
6 




f 


\ 


X 


V 






















\ 




\ 


















5 




f^ 




\ 


\ 


^ 




















\ 


\ 


^. 


\ 


s. 














4 






\ 




X 




v^ 














8 




-^^ 




s^ 


N 


N, 


^ 
















\ 






<v 




s,,^ 














2 
1 




\ 


Kt 




\ 


^ 


■^ 






- 


--^ 








N 






.^^ 




"^ 






;;^ 


^— 


-^ 








■■^••. 


^ 


- 


— 






^^"^"" 


1 — 


— 


' 



















c 

i 


! I 


4 


6 


fl 


1 


8 


1 S 


) 1 


} I 


L U 


8 '« 



^^^ ^ .2-.lx+.05x* 

^^^ ^=.2-.lx+.07? 

<'> «'=.2-.J+.lx« 

<*> ''-.2-.1X+.2X' 

<^> ''=.2-.J+.4x' 

See formula XI, page 38 



Fig. XI. 



138 



EMPIRICAL FORMULAS 



y 




























10 












y 
























y 


y 
















• 






r\ 


/ 












— 












/ 


r 






. 



















/ 


^ 


-I8V- 




— 








— 


— 










^ 


-^ 


"^«lLl 























J£L 


— __ 























\s 


"^ 


/iti 


— 




























*2L. 


-- 


■ 


_ 






— 












































» 




















1 
m 




t i 


J 


i (1 


i 1 


« 


1 S 


u 


J 1 


I a 


I » 



(i) y=5x-* 

(2) i/=5z-2 

(3) y«5x-l 

(4) y=5x""*^ 

(5) i/=5x--2 

(6) y-Sx--* 

See formula XII, page 42 



Fig. Xn. 




(1) y=l+log x+.l log* x; 

1/=— 1.5 (min.) when log x=— 5 

(2) i/=H-log x+.011og« x; 

l/=— 24 (min.) when log x=— 50 

(3) l/=H-.21ogx+.31og*x 

(4) j/=l— logx+log*x 

(5) y=l— logx+.51og*x 

See formula XIII, page 44 



10 11 U 13 u u 



Fig. Xm. 



APPENDIX 



139 



»i 




























i 














































































5^ 


^ 






















^ 


;;^ 




"^ 












^ 








■ — 










1 




'^ 










. 




ft) 
























"^ 


^ 


























X 


N, 


^^ 


^^ 
























> 


S(to 




























\ 









1 i 


i i 


> < 


i i 


i ( 


) 


r 


9 


9 1 


1 


1 1 


i « 



(1) i/«H-.008xl-'' 

(2) y^l+.007x^-^ 

(3) i/«l+.006a;l-'» 

(4) i/«l-.002a^ 

(5) y=l-.003a:2.1 

(6) j/=l-.004a:2.2 

See formula XIV, page 45 



Fig. XIV, 



y, 


1 


























s 






























^ 


s 




























\ 


























^: 




\ 


























\ 




\ 






















s^ 




\ 


\ 


\ 
















I 


^ 


^ 


\ 


V 


\ 


\ 


\, 


















^ 


\ 




\ 


\ 


\ 


V 
















^ 


'^ 


^s, 




::^: 




s^ 


s. 


















"^ 


^ 




.^ 


V 


sX 


^ 























^ 


^- 




1 


i 


1 


I i 


' 


i < 


i 1 


r c 


\ ( 


) I 


D L 


1 . I 


e » 



(1) i/=(2.0) 10- -01^ 

(2) |/=(1.6) 10--02ajl-^ 

(3) |/=(1.2) l0--03x^-* 

(4) y=(1.0) 10- O*^^-^* 

(5) i/=(0.8) lO--0635^'2* 

(6) y=(0.6) l0--0««^"^^ 

See formula XV, page 49 



Fig. XV, 



140 



EMPIRICAL FORMULAS 



«! 


1 
























"^■^ 


t 






























V 






i\\- ' 










-^ 


= 












^ 




\ 




^ 


"Vp: 


^ 


















\ 


x^ 


^ 


^ 
























y^ 
























1 


// 


K 


N 


s. 






















// 




\ 


<: 


\ 


//» 


















/ 




V 


\ 


V 


< 


^^ 
















f 








^ 


^ 






^ 


"^ 




...^ 




f 
















-- 


->^ 




— 




J' 


























1 


2 


a 


4 


6 


6 


7 


8 


1 


» U 


» u 


1 u 


i » 



(1) (1/-2) (x-.5) = -l 

(2) (i/-2) (x+.75) = -1.5 

(3) (y-2) (x+1) 2 

(4) (1/+.1) (a:+4)=8.2 

(5) (y+.l) (x+3)=6.3 

(6) (y+.l) (x+2)=4.2 

See formula XVI, page 53 



Fig. XVI. 




9 i 



6 6 7 8 9 

Fig. XVIfl. 



W U IS 



24 

(1) y=J 10a?+24 

-12 

(2) y=J 10a;-24 

2 

(3) v=i 10a:+2 

2^ 

(4) y=To 10^ 

.5 
(6) j/=-10af+l 

See formula XVIa, page 56 



APPENDIX 



141 




(1) j/=.5e.01^+e-05a; 

(2) j/=:2e.05«_.5e-l* 

(3) i/=2.25e-05a;_.75e.lit 

(4) y=1.8e-01a;_.3e-l« 

(5) v=1.92e--^^-A2e- 

(6) j/=2e--0^-e--01aJ 
^=4.2e--2a5-3.5e--25af 

j-.2z_4.ie-.25ar 
,-.01«_j3e-.15x 



.-.OU 



(7) l/= 

(8) y=4.5e- 

(9) i/=.25e- 



(10) i/=e- l«_l.le-.2x 

(11) i/=.27e--01a;-.77e--25a; 

(12) y=e-'^-2e--^^ 

See formula XVII, page 68 



10 u ja « 



Fig. XVII. 



V 


k 


























2 




























1^ 




























1.0 


P- 


























1.4 


\ 




'^ 




^ 


















1.2 


\ 








V 


^ 
















1 


") 


XiSi 










\ 


N. 












08 








J2L 








V 


\ 










Oifl 








(5) 












\ 








ai 


2 




\ 








J-"— 


•^_"^ 




N 


\— 




0.2 



f 




R' 


^'-^ 






(.«y 


/ 


N 


< 




:;^ 








\ 


\ 




\/ 


/ 


/ 


y 


S 




^ 




o.i 








v 




/^ 


> 


/ 












06 








\ 






/ 


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i 


i 


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6 


7 


9 t 


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(1) y=e«01a;(1.5 cos .Ix— .5 sin .Ix) 

(2) y=e"~'23J(1.5cos.5z— .5sin.5x) 

(3) j/=e~'^(.6 cos .lx+.8 sin .Iz) 

(4) y=e'^{.2 cos .3x— .1 sin .3x) 

(5) y=e-02x(.4cos.l6x+.17sin.l6x) 

(6) 2/=.5e~'^ sin X 

See formula XVIII, page 61 



Fig. XVni 



142 



EMPIRICAL FORMULAS 



t 
u 

14 
1^ 


k 












y 


r 





— — 


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(fl)/ 


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14 

1 




lU- 


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(1) y=-:2x-l-x-2 

(2) y«3x-5-2.2x-6 

(3) y=2.3a;-8-2x-85 

(4) y=.lx-l+.5x-2 

(5) 1/=. 33a;- .0012x3 

(6) y=.25x-5+.05x-8 

See formula XIX, page 65 



h 

K 

3 
9 

1 

-1 



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(1) y=15xl.5(.4)* 

(2) y=3x2(.5)* 

(3) j/=3x- 2(1.5)* 

(4) y=-.5xl-5(.75)* 

See formula XlXa, page 72 



Fig. XlXa. 



APPENDIX 



143 



y 

200 
190 
180 
170 
160 
150 
140 
130 
120 
110 
100 























































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»r 



X- 



FiG. XX. 

(1) i/= 166.25-14.5 cos a;-2.75 cos 2a;-10 sin « 

(2) y= 167.83-20 cos x-4.33 cos 2x+o.5 cos 3x- 13.28 sin «- 17.32 sin 2x 

(3) j/=167.62-17.5 cos x-2.75 cos 2x+3 cos 3a;-1.38 cos 4x-12.42 sin x-18 sin 2x- 

2.42 sin 3x 

(4) i/=167.08-17.22cosa;— 3.5 cos 2x+5.5 cos 3x— 0.83 cos 4x— 2.78 cos 5x+ 

0.76 cos 6a;— 12.14 sin a;— 19.05 sin 2x— sin 3x— 1.73 sin 4x+1.14 sin 5x 



See formula XX, page 74