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PUBLISHED BY
JOHN WILEY & SONS, Inc., NEW YORK. 
CHAPMAN & HALL, Limited, LONDON. 1
MATHEMATICAL MONOGRAPHS
EDITED BY
MANSFIELD MERRIMAN and ROBERT S. WOODWARD
No. 19
EMPIRICAL FORMULAS
BY
THEODORE R. RUNNING
w
Associate Professor of Mathematics, University of Michigan
FIRST EDITION
NEW YORK
JOHN WILEY & SONS, Inc.
London: CHAPMAN & HALL, Limited
1917
^^,
Copyright, 19 17,
BY
THEODORE R. RUNNING
PRESS OF
BRAUNWORTH & CO.
BOOK MANUrACTURERS
OBOOKLVN. N. y.
or
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
groi^rid 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
ing approximate values.
For valuable suggestions and criticisms the author here
expresses his thanks to Professors Alexander Ziwet and Horace
W. King.
T. R. R.
University of Michigan, 191 7.
366666
CONTENTS
PAGE
Introduction 9
CHAPTER I
I. y = a\bx\cx^{dx^{ . . . ^qx" 13
Values of x form an arithmetical series and A"y constant.
II. y = a+:^+iL+4+ . . . +1 â€ž
X x^ x^ x^
Values of  form an arithmetical series and A^y constant.
X
III.  = a{bx+cx^+dx^+ . . . +gx'Â» 25
y
Values of x form an arithmetical series and A" constant.
y
IV. y'^ = a\bx{cx^\dx^{ . . . ^qx" 25
Values of x form an arithmetical series and A"y^ constant.
CHAPTER II
V. y=ab^ 27
Values of x form an arithmetical series and the values of y
form a geometrical series.
VI. y^a+bc"". 28
Values of x form an arithmetical series and the values of Ay
form a geometrical series.
VII. log y = a\b(f 32
Values of x form an arithmetical series and the values of
A log y form a geometrical series.
VLU. y = a\bx\cd=' ss
Values of x form an arithmetical series and the values of
A^y form a geometrical series.
IX. y = ioa + f>x + cxi 37
Values of x form an arithmetical series and the values of
A2 log y constant.
5
6 CONTENTS
PAGB
X.yks'^' 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 2 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\b log x\c log^^c 44
Values of x form a geometrical series and A^y constant.
XIV. y=a+bx^ 45
Values of x form a geometrical series and the values of Ay
form a geometrical series.
XV. y=aio*^ .. 49
Values of x form a geometrical series and the values of
A log y form a geometrical series.
CHAPTER IV
XVI. {x+a)iy+b) =c 53
Pointl^epresented by (xâ€”xk,  â€” â€”\ lie on a straight line.
ft
XVIo. y=aio*+'' 56
Points represented by ( log â€” , log â€” ) lie on a
\xâ€”xic y ykj
straight line.
XVII. y=aâ‚¬^^+6^^ 58
Values of x form an arithmetical series and the points repre
sented by ( ^^\ ^^^^ ) lie on a straight line whose slope,
\yic yn I <
M, is positive and whose intercept, B, is negative, and also
M'^\^B is positive.
XVIII. y =eÂ°^(c cos bx\d sin bx) * 61
Values of x form an arithmetical series and the points repre
sented by ( ^^\ ^^^\ lie on a straight line whose slope,
\ yk yk I
M, and intercept, B, have such values that M'^\4B
is negative.
CONTENTS 7
PAGE
XIX. y = ax'+bx^ 65
Values of x form a geometrical series and the points repre
sented by /â€” \ ^^^ lie on a straight line whose slope,
\ yk yk /
M, is positive and intercept, B, negative, and also
M^+4B positive.
XlXa. y = ax^c'' 72
Values of x form a geometrical series and the points repre
sented by Ixn, log ^^^^ ) lie on a straight line.
\ 3'Â» /
CHAPTER V
XX. y=aQ}aiCosx\a2COS2x\a3 COSSX+ . . . +ar cos rx 74
\b1smx\b2sin 2x\b3sin sx\ . . . \br sin rx
Values of y periodic.
CHAPTER VI
Method of Least Squares 90
AppHcation to Linear Observation Equations.
Application to Nonlinear Observation Equations.
CHAPTER VII
Interpolation m0. 100
Dififerentiation of Tabulated Functions.
CHAPTER VIII
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 problem 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 found 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
^^
U
y.
/
X
^
y
^
V
r
^
^
/
/
a
/
C^
A
\
#
Fig. I.
or 6? 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 determining 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, 75 â€¢â€¢ â€¢
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 â€” 2.2.
Geometrical Series. A series of numbers 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
*
y
Ay
AÂ«y
AÂ»y
A^
I
I0.2
2
II. I
0.9
I.I
0.2
0.0
3
12.2
0.2
1.2
4
5
6
I3S
l6.2
i8.o
1.3
2.7
1,8
I.O
14
0.9
0.8
1.2
2.3
0.1
2.8
35
2.4
2.7
7
19. o
2.0
8
22.0
30
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 t^y. 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+hx+cx^Vdx'\ . . . +qx''
Values of x form an arithmetical series and A"y constant.
In a tensile test of a mild steel bar, the folbwing observa
tions were made (Low's Applied Mechanics, p. i88): Diameter
of bar, unloaded, 0.748 inch, pr = 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
0.0027
0.0013
0.0040
0.0015
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 hne.* 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 ^^^ 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
0.0014
0.0027
0.00137
0.00274
3
4
5
6
0.0040
0.0055
0.0068
0.0082
0.0041 I
. 00548
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
y
,/
/
.0072
0068
/
/
/
MMUl
/
/
.0056
.0062
/
/
A
/
.0018
(WUi
/
/
/
0036
/
y
.0032
.0028
.0021
.0020
^16
nnio
J
/
/
/
/
/
/
Y
0008
/
/
.0001
/
/
6 10
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
y = mx]rk
Starting from any value of x, give to x an increment, ^x, and
y will have a corresponding increment, A3'.
y\^y = m{x\^x)\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 = mW+k,
where m 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 under
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 = 2â€”;^x\x^.
If an increment be assigned to x, 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 ^y from its succeeding
value. The values of x have the common difference 0.5.
X â–
o.S
I.O
iS
2.0
2.5
30
35
4.0
y
0.7S
0.00
0.25
0.00
0.7s
2.00
375
6.00
Ay
0.7S
0.2s
0.25
0.7s
I 25
I 75
2.25
AÂ«y
0.50
0.50
0.50
0.50
0.50
0.50
The values A^^;, which we call the second differences, are
constant.
These differences could equally . well have been computed
as follows:
yjA); = 2â€”2,{x\^x) + {x\^x)^,
Ay= â€” T,{^x) \ 2x{/:^x) { {^xY y
^y+^^y= 3(A:c) + 2(ic+Ax)(Ax) + (Aa;)2,
A2y= 2(Ax)2=o.5 since A:*; = 0.5.
From this it is seen that whatever the value of Aic (in
y = 2 â€” T,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.
y^
Ayi
y2
A2yi
Ay2
A3yi
etc.,
3^3
A2y2
A^yi
A^/s
A3y2
etc..
3'4
A2y3
A4y2
Ay4
A3y3
etc.,
yo
Ay5
A2y4
...
y&
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.
y^ = y3+Ay3,
= );i + 2 A3/1 + A^^'i + A(3'i + 2A3;i + A^^/i),
=yi+^Ayi+s^^yi+^^yi
y5=y4.+AyA,
= yi \s^yi +3^^yi +^^yi +A{yi +sAyi +$A^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 yjc+i.
By hypothesis
1.^ ^^ 11 ^A3>;i+etc (i)
If this equation is true, then
yi.+i=yi + {ki)Ayi+^tll^^t:^A^yi
2
f :^ il L^ ^A^^;! + etc.
+ A [3/1 + (y^  1) A:yi + ^^ij^^^A2>;i
(.x)a.)(.3)^3,,^etc.]
y,^kAy,+^=AA2y^ ^(k^^(k^) ^Sy
2 \3
14
18 EMPIRICAL FORMULAS
This is the same law as expressed in the former equation, and
therefore, if the law holds for yu, 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 y5 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)
yt=yiiiki)Ayi.
If the second differences are constant the third and higher
differences will be zero, and it follows from (i) that
2
In general, then, if the wth differences are constant
yt^yi^{ki)Ayi\  f^ ^A^^;!^^ ^^ â€” i^ â€” ^a^a/i
11 13
+ . . . \ (ki)(k2){ks)(k4) â– â™¦ . (kn) ^n ..
The law requires that the values of x form an arithmetical series,
and hence
Xt=xi\{ki)Ax;
from which follows
j^^^^^Xi . (3)
Ax ^'^^
Substituting this value of k in equation (2) it is found that
the righthand member becomes a rational integral function of
Xt of the nth degree. Equation (2) takes the form
yt = a\bXk]cx.j^hdx^^\ . . . +qXk*.
Since Xt and yt are any two corresponding values of x and y
the subscripts may be dropped and there results the following
law:
DETERMINATION OF CONSTANTS
19
// two variables, x and y, are so related that when values of x
are taken in an arithmetical series the nXh differences of the cor
responding values of y are constant, the law connecting the variables
is expressed by the equation
y = a\bx{cx'^\do(^{
^qo<r.
The wth differences of the values of y obtained from observa
tions are seldom if ever constant. If, however, the wth 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 unity.
Ah
At surface.
0.1 depth,
0.2
0.3
0.4
o.S
0.6
0.7
0.8
0.9
1950
2299
2532
2611
2516
2282
1807
1266
0594
9759
+349
+ 233
+ 79
 95
234
475
541
â€” 672
835
116
154
174
139
241
 66
131
163
 38
 20
+ 35
 102
+ 175
 65
 32
+ 18
+ 55
137
+ 277
â€” 240
+ Z3
+ 37
â€” 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^,
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+bxjcx^ (i)
from the data given proceed in the following way:
Let it:=Xfa;o,
y=Y\yo,
where ocq 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\bxo\ cx{? + (6 + 2cx^X\ cX^.
Y=^{h+2cxo)X+cX^] (2)
since yo = a\'hxo\coco^. Dividing (2) by X it becomes
â€” = h^2C3CQ\cX (3)
Y
This represents a straight line when X and â€” are taken as
X
coordinates. The slope of the line is the value of c and the
intercept the value of h\2C0CQ. The numerical work is shown
in the table and the points represented by \X, â€”\ are seen
DETERMINATION OF CONSTANTS
21
in Fig. 3. The value of c is found to be â€”0.76. When xq = o^
the intercept, 0.44 is the value of h. For x=X, the value of
.5
X
A"
.
.
.
1
.4
.3
^^
'^
L
=^
^
â€”
â€”
â€” â–
â€”
â€”
â€”
â€”
â€”
â€”
.2
"^
â€”
K
â€”
^
K.
â– ^
>
'^
^>
â– *>.
H
"^
f^
"
.1 .2 .3 ,4 .5 .6 .7 .8 .9
]
Fig
3.
X
3/0 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.
X
y
X
F
Y
X
.44x.i6x^
a=y.44x

Computed
y
.0
3 1950
0.0000
. 0000
3.1950
3.1948
I
3
2299
I
0.0349
0.3490
0.0364
3
1935
3.2312
2
3
2532
2
0.0582
0.2910
0.0576
3
1956
3.2524
3
3
2611
3
0.0661
0.2203
0.0636
3
1975
32584
4
3
2516
4
0.0566
0.1415
0.0544
3
1972
3.2492
5
3
2282
5
0.0332
0.0664
. 0300
3
1982
3.2248
6
3
1807
6
0.0143
â€” 0.0238
â€” 0.0096
3
1903
3.1852
7
3
1266
7
â€” 0.0684