Google
This is a digital copy of a book that was preserved for generations on library shelves before it was carefully scanned by Google as part of a project
to make the world's books discoverable online.
It has survived long enough for the copyright to expire and the book to enter the public domain. A public domain book is one that was never subject
to copyright or whose legal copyright term has expired. Whether a book is in the public domain may vary country to country. Public domain books
are our gateways to the past, representing a wealth of history, culture and knowledge that's often difficult to discover.
Marks, notations and other maiginalia present in the original volume will appear in this file  a reminder of this book's long journey from the
publisher to a library and finally to you.
Usage guidelines
Google is proud to partner with libraries to digitize public domain materials and make them widely accessible. Public domain books belong to the
public and we are merely their custodians. Nevertheless, this work is expensive, so in order to keep providing tliis resource, we liave taken steps to
prevent abuse by commercial parties, including placing technical restrictions on automated querying.
We also ask that you:
+ Make noncommercial use of the files We designed Google Book Search for use by individuals, and we request that you use these files for
personal, noncommercial purposes.
+ Refrain fivm automated querying Do not send automated queries of any sort to Google's system: If you are conducting research on machine
translation, optical character recognition or other areas where access to a large amount of text is helpful, please contact us. We encourage the
use of public domain materials for these purposes and may be able to help.
+ Maintain attributionTht GoogXt "watermark" you see on each file is essential for in forming people about this project and helping them find
additional materials through Google Book Search. Please do not remove it.
+ Keep it legal Whatever your use, remember that you are responsible for ensuring that what you are doing is legal. Do not assume that just
because we believe a book is in the public domain for users in the United States, that the work is also in the public domain for users in other
countries. Whether a book is still in copyright varies from country to country, and we can't offer guidance on whether any specific use of
any specific book is allowed. Please do not assume that a book's appearance in Google Book Search means it can be used in any manner
anywhere in the world. Copyright infringement liabili^ can be quite severe.
About Google Book Search
Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers
discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web
at http: //books .google .com/I
I
. ,,, j^. ^Tf^
\
t
f
I
MATHEMATICAL MONOGRAPHS
EDITED BY
Mansfield Merriman and Robert S. Woodward.
Octavo, Cloth.
No. 1.
No. 2.
No. 3.
No. 4.
No. S.
No. 6.
No. 7.
No. 8.
No. 9.
No. 10.
No. 11.
No. 12.
No. 13.
No. 14.
No. 15.
No. 16.
No. 17.
No. 18.
No. 19.
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.
1 1. 00 net,
Hyper'jolic Functions. By James Mc
Mabon. %i.oo net.
Harmonic Functions.
Byerly. Ii.oo net.
By WnxiAM B.
Qrassmann's Space Analysis. By Edward
W. Hyde. Ii.oo net.
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
Merriman. i.oo net.
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.
Carmichael. i.oo net.
Algebraic Invariants. By Lbonasd B.
Dickson. I1.25 net.
Mortality Laws and Statistics. By Robert
Henderson. I1.25 net.
Diophantine Analysis. By Robert D.
Carmichael. I1.25 net.
Ten British Mathematicians. By Alex
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. yalf • 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 Nonlinear 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 132143
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
iS
2.0
25
30
35
4.0
y
0.7s
0.00
0.25
O.CX)
0.7S
2.00
375
6.00
Ay
0.7s
—0.25
0.25
0.7S
I2S
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
selfexplanatory.
Vi
Ay\
y2
A2yi
Ay2
A3yi
etc..
yz
A2y2
A^yi
Ayz
A3y2
etc..
y^
A^ya
A^y2
Ay4
A^ys
etc.,
yo
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 + ()fei)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+iki)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+{ki)Ayi+ p A^yi+ ^^ —  — —A^yi
\l 13
. (^x)(^~2)(^3)(^~4) . . . (kn) .. , .
+ . . . + 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 righthand 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
32532
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.i948f.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
iSi
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
3333
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=42f
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 giaphically by first writing
the equation in the form
log P = log a+a log b
and plotting the values of a and P on semilogarithmic paper;
or, using ordinary crosssection 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+{ni) 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
ir
,nl
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
1439
1.597
0.177
1597
1.774
0.200
1.774
1974
0.224
1973
2.198
0.254
2.198
2.452
0.285
2.452
2.737
0.323
2.738
3.060
0.363
3059
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 {ya)=^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
>'0.2f 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 fr^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 + (wi)Ayi+A2yi ? — ^+ —  + ^ } — ?
Li— r I— r i—r i —
+— ^+
I— r
, if"' 1
I—r J
The first two terms on the righthand 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
wi =
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
03
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.5o.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
5997
.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,y7Sa+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 + {ni) log r,
loga;nloga;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 castiron tube subject to a tensile stress; x represents
the stress in kilogrammes per square centimeter of crosssection
and y the elongation in terms of ^ cm. as unit.
X
979
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+{ni) 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\
iR
log xi / 1 \\ogxn
. . log XI / 1 \
^ iR iR \ 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(y2.72)
.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 (yb^)
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
\ yyk/
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
yy^
=p{x—X))\q.
xx:,=p{xx^{y'yi)\'q{yy^.
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
149
8.50
3.96
56.00
2.00
—11.00
4.97
53.50
3.0I
1350
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 = 767.00.
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,
&= 4478,
{;= 46.89.
56 EMPIRICAL FORMULAS
These values give
(.4+o.i5i)(F44.78)=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.
xxu 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
\xxic 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 xxt 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 ylog yi){x''Xi) = w(log ylog y*)+&(apa;*),
or
log y{xxtm) = (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
BAC
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+^^,
•
(logyloga)(a;+c)=6, .
Let
logy=logF+logy*,
and
Then follows
(log F+log y,log a){X+Xi,+c) =6,
X log F flog Y{xk \c) fZOog yu log a) + (log y* log a) (a;* Hc) = 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 alog y*.
Replacing log F and X by their values
log ^ =  {xi,+c)—^ log — +log alog y*.
From this it is seen that if log — be plotted to log ~
y* ococfc ^'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^ie'^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
gA12x
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 = 2e2*^*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
toe^*^. 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 30J: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+if'y* =bx^(r^r'), (4)
yt+2r'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_
=386.
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
gVM*
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
14333
.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
{yn2—iyni+ 8y«+i3/»+2),
y"n = p iyn2  i(>yni +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. ydaf(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
16
?8
o
o
•
.9
1.0
o
— n
.1 .2 .3
Values of y
Fig. 26.
.4
J5
From the first equation
log ynXnlogc=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+ . . . fOrCosfX
46isinx+ftisin 2x46jsin3x+ . . . 4ftr_iSin (r— i)x.
Values of y periodic.
The righthand 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
17
2.0
1.8
iS
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''la2as{ — ^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 12ordinate 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+1^11^2 — ^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
15
0.9
Vo
2.8
32
2.9
Wo
.8
.2
I.I
2.8
32
t
.8
.2
Po
2.8
29
6.1
ro
I.I
.8
13
»
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.
8ORDiNATE 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
37
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 \ y3y4+: — 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
— —ysy^ — — 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
3 312
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^.
ioOrdinate 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,
qoqi+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 loordinate one, only
the results will be given.
Sum
Difference
12Ordinate 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
16ORDINATE 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+ClflC2r3,
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^°.
20Ordinate 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°.
24ORDiNATE 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,
^qoq^+W^iqiqzqb),
= ho+^hiy
 qo+C2qi  hVJq2  W^qz +^^4 +Ci js,
=/o,
=?oC2giV3g2+^V2?3+^g4Cig6,
■■qoq^+W^iqi+qz+qb),
■nio+^fn2^Vsmi,
=goCigi+iV3g2^Vjg3+ig4C2g6,
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^V2r3V3r4+C2r5+r6,
I2J6 = ^l~"^3,
i2&7 = Ciri^r2V2r3+^V3r4+C2r5r6,
I2&8 = iV3(wiW2),
i2ft9=r6r2+jV2(ri+r3r5),
I2ftl0=53+i(^l+^5)V3(52+54),
i2Jii = C2riir2+§V2r3^V3r4+Cir5r6,
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 X3.410 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 12ordinate 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
II5
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
345
215
10
.1
5
(.8
85
0.
2
I.S 
17.2 —
17
•5
— I]
t5 ■
7.1
^0
27.2
36.0
4.3 ■
74
— ]
[7
14
0.
2
W\
po
33.0
27.2
0.2
38.7
36.0
14
27
.6
21
43
17
[3
15.6
74
27.4
37.4
2.6
74
33
15.6
27.0
38.7
21.3
34.6
27.6
6.0
27.4
2.6
374
74
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+9994 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
31950
■5
3.2282
.1
3.2299
.6
3.1807
.2
32532
7
3.1266
•3
3.261 1
.8
30594
•4
32516
9
29759
* Three wellknown 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 righthand 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 = 244
= 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 = 24C = 2.85. Also the sum of the numbers in the column
headed 4iV = 24iV=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
^=+319513;
. 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
32532
3 2530
+ .0002
.00000004
.3
3.261I
3 2590
+ .0031
.00000441
■4
3 2516
3 2497
+ .0019
.00000361
.5
3.2282
32251
+ .0031
.00000961
6
3.1807
3.1851
.0044
.00001936
7
3.1266
3.1299
.0033
.00001089
8
3 0594
3.0594
.OCXX5
.00000000
9
29759
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 righthand 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^ +4A5+^Am+^A/^.
9^0 9^0
9^0
9wo
The observation equations will be of the form
a/
AA
dAo dBo
^'^ ■AB+^Afn+^An=yyo.
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.32iAjB4.637Aw+ .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
\ lLting 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 Textbook of the Institute of Actuaries, part II (ist ed. 1887, 2nd ed.
1902), p. 434; Encyklopadie der Mathematischen Wissenchaften, Vol. I,
pp. 799820; 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 righthand 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{xl)(x2) . . . ^^''^'^^\ 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)(.i592) ^ 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 righthand 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(xa){xb)(xd) . .
. {xk)
+D{x'a){x—b){x—c) . ,
. {xk)
+ etc. to n terms.
105
Each one of the n terms on the righthand 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 righthand 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
{ba){bc){bd) . . . (bk)'
Proceeding in the same way we obtain values for all of the
constants and, finally,
{xb)(x—c)(x—d) . . . (x — k)
yx=ya
+yi
+y*
(ab){a—c){a—d) .
{x—a){x—c){xd) .
{ba){bc){bd) .
{x—a)(x—b)(xd) .
+yd
{c—a){c—b){c—d) .
{x—a){x—b){x—c) .
{d^a){db){dc) :
+y\
{x'a){x—b){x'c)
(ka){kb){kc)
. (a — k)
. (x — k)
. (bk)
. (x—k)
. (x — k)
. (dk)
(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. 642693.
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+{xi)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^ (abXacXadXaey^' {ba){bc){bd){be)
, {x—a){x'b)(x—d){x—e). (x'a)(x'b){x'c)(x—e)
'^^' {ca){cb){cd){ce) ^^"^ {daXdbXdcXde)
{xa)(xb)(xc){xd) , v
■^^^ (ea){eb){ec){ed) ^ ^
Selecting the points at equal intervals and letting
e—d=d'C=c—b = b—a=hf
and differentiation
ya=^[25ya+48y636yc+i6yd3yJ,
/&=— t[ syaioyt+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'« = ^[3Syaio43;6+ii4yc S6yd+iiyj,
y"6=m5[iiya 203^6+ 6yc+ 4yd>'J,
12^2
y'c=^^[ya+ i6y6 3oyc+ i6ydyJ,
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)
^ (xb)(xc) (xaXxc) (xa)(xb)
^' ^"^ {abXacy^'ib^aXbcy^' (ca)(c6)'
Equating to zero the first derivative with respect to x
y.
=ya
2X'b'C
4^6
2X—a
x =
{abXac) ' '^ (baXbc)
ya(b^  c") +y,{c^  a^) +yc{a'  ^ )
2\ya{b  c) +yj,{c  a) +yc{a  b)]
fyc
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(xi)(x2){x's) ^. ^ x(xi){x2)(xs)(x4) ^.^
k ° Is
^ x(xi){x2)ixs)(x4)(x5) ^f, ^ ^^>^
Integrating the righthand 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)ix2)(x3)ix4){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=AyiAyo=y2yiyi+yo,
=3'22yi+yo.
Substituting these values in the above integral it becomes
I yxdx = 2yo+2yi2yo+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=AyiAyo=y2'2yi+yo,
i
NUMERICAL INTEGRATION 117
A^yo = A^yi — A^yo = Ay 2 — Ay 1 — Ay 1 +Ayo
=y33>'2+3)'iyo.
Substituting these values in the equations,
1 ydx=syo+h^ih^o+iy2hi+ho+hzh^2+h^iho,
=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
Ayobyyiyo,
A^yo by y2  2yi +yo,
A3yo by y^^yz+Syiyoy
118
EMPIRICAL FORMULAS
A^jb by y44)'3+6y2 4)'i+yo,
^^yo by ys  5^4 + loya  ioy2 + syi  yo,
A^yo by /6y5+iSy42oy3+iS)'26yi+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
semiellipse included between the two perpendiculars erected at
the middle points of the semimajor 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 =iKio+6.o+4.4+io.8+s.2+9.2+2.i) = 2.58,
and by (8),
4 =11(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
4331
4.810
5372
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
crosssection 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 crosssection 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=16(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 icaxis,
find the volume generated.
The areas of the crosssections 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 ncaxis,
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(i49oo4)=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 xauds
and the two ordinates a; = a and x = b. The original area is
bounded by the curve whose ordinates are represented by y,
the icaxis 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 icaxis and the two
ordinates x = a and x = b.
For a volume generated by revolving a given area about the
x2ods •
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 icaxis
_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 yaxis is
Iy= I x^ydx.
About the xaxis
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'
0333
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 yaxis.
Dividing this by the area foimd before, there results for the
radius of gyration about the yaxis
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 xaixis and Y for the width parallel to the yaxis. The
area is 0.066.
The moment of inertia about the yaxis 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 yaxis. 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 yaxis 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 yaxis 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/=ll/a;
(2) y=ll/x+l/x^
(3) i/=ll/a;+l/x*l/x»
(4) y=ll/x+l/x^l/x^'\l/x*
(5) i/=ll/x+l/x2l/x»+
l/x*l/x*
(6) i/=ll/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=lO813to + .03a;*
(2) y =10.54. 24a; +.02z*
(3) j/=1027.12a;+.01a;«
(4) i/= 10135 .Oftr+. 005a;*
(5) i/=10135 + .06x OOSx*
(6) y=1054+.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)(l2)^
(2) y=(1.01)* (1.05)<11®^^
(3) y=(1.01)* (1.05)<11S^*
(4) y=(.6)* (2) (124)^
(5) i/=(.5)* (2)a23)^
(6) y=(.5)^ (2)Cl2)^
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/=5z2
(3) y«5xl
(4) y=5x""*^
(5) i/=5x2
(6) ySx*
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/=Hlog 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) 1002ajl^
(3) /=(1.2) l003x^*
(4) y=(1.0) 10 O*^^^*
(5) i/=(0.8) lO0635^'2*
(6) y=(0.6) l00««^"^^
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) (y2) (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^+e05a;
(2) j/=:2e.05«_.5el*
(3) i/=2.25e05a;_.75e.lit
(4) y=1.8e01a;_.3el«
(5) v=1.92e^^A2e
(6) j/=2e0^e01aJ
^=4.2e2a53.5e25af
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/=.27e01a;.77e25a;
(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
\
/
*">>
^
. 
i
i
J
S
6
7
9 t
• 1
1
1 1
« ^
(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=e02x(.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
— —
^
^
(fl)/
7^
/
14
1
lU
^ ^
/
/
^
^
■^
as
/
?i
r^
^
^

^
^
^
^
.^'^
(S)
.
^
a4
02
^
r
<
1 <
I
1 ]
XT
X.
> {
1 1
9 L
1 1
T&
(1) y=:2xlx2
(2) y«3x52.2x6
(3) y=2.3a;82x85
(4) y=.lxl+.5x2
(5) 1/=. 33a; .0012x3
(6) y=.25x5+.05x8
See formula XIX, page 65
h
K
3
9
1
1
^
r
k
1
A
V
\,
^
A
A
\
s.
^
/.^
/^
sW
\
<:
>
^^
^
^
.
<:l
^
^
.
1
\
4^
.
'^^^"
1
u s
t 1
\ i
, 1
> «
1
8
fl
K
) U
I U
{"^
(1) y=15xl.5(.4)*
(2) y=3x2(.5)*
(3) j/=3x 2(1.5)*
(4) y=.5xl5(.75)*
See formula XlXa, page 72
Fig. XlXa.
APPENDIX
143
y
200
190
180
170
160
150
140
130
120
110
100
A
i2l
^
.
*
^
W
^
\
rf*
^^
>a^^
.^,0^
C3)
?^^
Iff
>
^
=
^
/
^
<
s
(2r
^■
N
vl*i
y
X
N
N
^,
>
^
\
^
Vl
/
\
■^
f
"^
3^
y
\
i
T_
1
r
¥
»r
X
FiG. XX.
(1) i/= 166.2514.5 cos a;2.75 cos 2a;10 sin «
(2) y= 167.8320 cos x4.33 cos 2x+o.5 cos 3x 13.28 sin « 17.32 sin 2x
(3) j/=167.6217.5 cos x2.75 cos 2x+3 cos 3a;1.38 cos 4x12.42 sin x18 sin 2x
2.42 sin 3x
(4) i/=167.0817.22cosa;— 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