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Theodore R. (Theodore Rudolph) Running.

Empirical formulas online

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-0.0977


— . 0644


3


1910


3.1304




8


3


0594


8


-0.1356


-0.1695


-0.1344


3


1938


3.0604




9


2.9759


9


— 0. 2191


-0.2434


— 0. 2196


3.1955


2.9752












I


3)31.9476


















a


= 3


1948





The numbers in column 6 were found after b 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



>' = 3.i948 + .44X-76x2



(4)



22



EMPIRICAL FORMULAS



II.






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


s


6


7


8


<J


*


y


I _

X


X


y


Ay


aV


2


Com-
puted y


I.O
1.2

1-4
1.6
1.8

2.0
2.2
2.4


4.000
2.889
2.163
1.656
1.284
1. 000
0.777
O.S97


1.0

0.9
0.8
0.7
0.6

0.5
0.4

03


1. 000
i.m
1.250
1.429
1.667
2.000
2.500
3-333


4.00
3-32
2.68
2.07

I-5I
1. 00
0.52
0.08


-0.68
— 0.64
-0.61
-0.56
-0.51
-0.48
-0.44


0.04
0.03
0.05
0.05
0.03
0.04


2.00
1.50
1. 14
0.87
0.67
0.50
0.36
0.25


4.000
2.889
2.163
1.656
1.284
1. 000
0.777
O.S97



In column 3 are given values of - in arithmetical series

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

^-+*©+<^^)- • • • • (5)

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

X

may then be stated :

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

X

are taken in arithmetical series the nih. diferences of the corre-



DETERMINATION OF CONSTANTS



23



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



II



= +-+-+— +

X X^ 0(^






Values of -^
.3 .4 .5 .6 .7



1 1.2 1.4 L6



1.8 2.0 2.

Values of x



Fig. 4.



If in equation (5) - be replaced by X, then



and



.9 1



K
















































































\


































>


y




N


\






























/


/








\


s


























/














\


\,






















/


















^


N
















^


^
























^


V,










y
































^


"^


• >


r




































/


^


^


^-^
































/


/


































/




















_j



lA 2.6 2.8 3.0



/y = a+6Z+cX2,

y^Ly = a^b{X^LX) +c(X+AX)2.
By subtracting (6) from this equation

A3; = tAX+2c(AX)Z+c(AX)2; . . .
and from (7)

A3;+A2>; = MX + 2c(AX)(X+AZ)+c(AX)2. .

Subtracting (7) from (8)

A23; = 2c(AX)2;

A2y
"" 2(AZ)2-



(6)



. (7)
. (8)



24



EMPIRICAL FORMULAS



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



_ 0.04
Writing the equation in the form



y-h 7


/


^


_/


t


Jl


1


7


i^


\ 17


7


j2


1


7


t


1


t


J2


17


I



A .5 .6 .7 .S .9 1.0

Fig. 5.



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



x"



are the coordinates. From



Fig. 5 & is found to be 3
and a to be — i. The for-
mula is



x2/'



>'=-i+3(-)+2(-



The last column 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, F in volts.



A

Observed V . .
z

I

Computed V .



1.96
50.25



.5102
50.52



2.46
48.70

.4065
48.79



2.97
47-90

• 3367
47.62



3-45
47-50

.2899

46.84



3-96
46.80

•2525
46.22



4-97
45-70



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 on 6. 7. The intercept on the V — axis
is 42. This gives for the relation between V and A



F = 42-+



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.



— I 1 — I — I — I — I — 1 — I 1 1 — >



.3
Fig. 6.



III. - = a-^bx-\-cx^+do(^+ . . . +gx".



y



nl



Values of x form an arithmetical series and A - constant



// two variables, x and y, are so related that when values of
X are taken in an arithmetical series the n\h differences of the cor-
responding values of - are constant, the law connecting the variables

y

is expressed by the equation



III



= a+bx-\-cx'^ -\-dx^ -{-



+gx".



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



IV.



i2_



a-\-bx-\-cx^+dx^-{- . . . -\-qx^.



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

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



26 EMPIRICAL FORMULAS

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

IV 'f = a-^bx-{-co(^^-dx^-\- . . . -^qo^.

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



senes.



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

If the equation be written in the form
\ogy = \oga^-([ogb)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-
pound 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, /x,
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 /z, when a is the angle of contact
between the belt and pulley measured in radians.



a


2


27r
3


6


TT


77r
6


47r
3


3![

2


5^
3


IITT

~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 graphically by first writing
the equation in the form

log P = \oga -\-a log h

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



2J0

an












^■^^














log a =0.4750,


















^


■^


'^




log ^>= 0.1733;


i








-—




















l^
























'


« = 3,








\


^TT


F


[G.


TT

alue
7.


sof


a


r^ir


raT


^Ke^


& = i.49-



or



The formula expressing the relation between the variables is
i^ = 3(i49r,

VI. y = a-Vhe.

Values of x form an arithmetical series and the values of Ay 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



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

VI y = a^-bc^.

By the conditions stated the wth value of x will be

Xn = Xi+{n-l) ^x,

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

Lyij Lyir, Ayir^, Ayir^ ^^- "^ a.. .,n-2

The values of y will form the series
yu >'i+A>'i, yi+Ayi+rAyi,

yi-{-Ayi-YrAyi-\-r'^Ayi-\-r^Ayi-\- . . . +r""^A>'i.
The wth value of y will be represented by



Ayir^ . . . Ayir"
yi+rAyi-\-r^Ayi . . .



yn=yi-{-Ayi



i—r



From the nth. value of x



n — i=-



Xn X\

Ax



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

Xn —XI



yn=yi-\-Ayi-
= a+bc\



i—r



where a stands for 71 H — —, b for ~— r ^ , and c for r^x.

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





. I


. 2


■3


-4


•5


.6


• 7


.8


•9


I.O


y

Ay

y


1.300
0. 140
1.300


1.440

O.IS7
1-439


1-597
0.177

1-597


1-774
0. 200
1-774


1-974
0. 224

1-973


2.198

0.254
2.198


2.452
0.285
2.452


2.737
0.323
2.738


3.060
0.363
3 . 059


3-423
0.407
3-421


3-830
3 830



30 EMPIRICAL FORMULAS

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

y — a'\-hc'.

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 = hc'\

which may be written in the logarithmic form

log (;y-a) =log h^x log c,

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

y = a^h{c'),

whose slope is h 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.

R={xQ^-2^x, a+h(f'c^^).
Select also two more points S and T such that

S^{xQ-\-^x,a-\-h(f')\

T={xQ-\-2^x, a-\-h(f'c^).
The equation of the line passing through Q and R is

y= ^ Lx ^ -{xo+Axj+a-^bcfc^. (i)

Ax Ax



DETERMINATION OF CONSTANTS



31



The equation of the Hne through the points S and T is



y==



Ax



A:*:



{x,^+^x)+a+hc'\ , (2)



These Hnes 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)=\ogb-\-x\ogc.

In Fig. 9 log {y — .2) is plotted to x as abscissa. The slcpe of
the line is 0.5185 which is the value of logc, hence c is equal to
S-T,. The intercept is the ordinate of the first point or 0.0414,
which is the logarithm of b, hence b is equal to i.i.
The formula is

v = o.2 + i.i(3.3)^



32



EMPIRICAL FORMULAS



or



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

Second Method. For any point {x^y) the relation between
X and y is expressed by

^ y^a-^-hc',

and for any other point (jc+Aic, y+^y) by

y-^^y = a-\-h(fc^.

From these two equations is obtained

^y^h(f{c^-'l)

log A3; = log h{c^* — i)-\-x log c,

6

If now log i^y be
plotted to X as
abscissa a straight
line is obtained
whose slope is log c.
The value of c hav-
ing been determined,
the relation

y=^a+h{(f)

will represent a
straight line pro-
vided y is plotted
to c* as abscissa.




Fig. 9.



The slope of this line is h and its intercept a,

VII. \ogy = a^hc\

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 equation

VII log y = a-^bc\

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 i^'^y form a
geometrical series.

// two 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 nih. value of x is represented by

Xn=xi + {n — i)d^x.
The values of y and the first and second differences may be



m colu


imns

yi


^yi






y2


Ay2


A2yi




^3


Ays


A2y2




y^


Ay4


A2y3




y^


Ays


A2y4




y^


etc.


etc.




etc.







34 EMPIRICAL FORMULAS

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

A^yi, r^^yu r^^^yl, r^^^yl . . . r^-^A^yi.

The series of first differences will then be

A>'i , Aji +d?yi , A^'i +^^yl ■^r^^yl , ^yl -\-L^yi ■\-rd?yi -\-r^^^yi

^yl+^^yl-{-r^^yl-\-r^^^yl-\- . . . -{■r^'-^L^yi.

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

yn=yi

-\•^yl

+^yl-\-^^yl

+A>'i -{-d^yi -{-rA^yi

+A3'i + A2;yi 4-rA2;yi +r2A2;yi

+ A^/i + A^yi + r^^yi + r^^^yi + r^^^yi



■\-^yi-\-^^yi-\-rL^yi-\-r'^^^y\+r^i^^yi-\- . . . -{-r'^'^^^yi.
Adding gives

I / \a I a9 fi— ^ I I— ^^ , I— ^ , I— r^

yn=yl + {n-l)^yl-{-^^yl\ \ 1 \-

Li— r i—r i—r i—r

+lz^+ . . . +l^I^\
I—r i—r J

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



DETERMINATION OF CONSTANTS



35



yn^yi-{-(n-i)Ayi+^n-2)-'^^(r+r^+r^+ . . . +r - ')



i—r



i—r



=yi-\-(n-i)Ayi



A2y,



-{n-i) ^

i—r i—r I—r



=.4+5(w-i)+Cr»-';

where

A^y



A=y, - ^^^,B=^Ay,+^,^ndC =



A^y,



A^yi



(i-r)



I—r



(i-r)'-



From the value of Xn is obtained

Xn—Xl



n—i =-



Ax



Substituting this in the value of yn it is found

xn-xi

y„=A+E^^^^^^^+Cr ^'
Ax

= a-\-hXn-\-cd'''.

Since Xn and yn stand for any set of corresponding values
of X and y the resulting formula is

YIII y = a-\-hx+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.



*


y


Ay


A^y


log A^y


(2.00)* y — I.


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


1.8


3.082


418








3.482

4.000 —


435
540


3.085


2.0


3.500










3 506







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-\-cd'.
Taking the second difference

log A^y = log c(d^ - 1)2+ (log d)x.



or



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



-1.0
-1.1

o
§4.4



































^


^


























^


C4






^


^


















^


^














X


^










y


[^








00














^


X


J


H


x










n —•
















^^


X.


X












V












^


[y










X








J.










^
















hN












^


>^'






















\^




y


^































^ ^



1.0 1.2 L4 1.6 1-8 2jO
Yaluea of <»



Fig.



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

— 1.6500, is equal to log c(d^' — iy.

Since

d = 2,
.02239 = ^(2-^-1)2,
C = I.OII.



DETERMINATION OF CONSTANTS 37

Plotting 3;— (1.01)2* to X, Fig. lo, the values of a and b

are found to be

a= 0.5,

J= -0.515.

The formula derived from the data is

3; = o.5-o.5i5:j;+(i.oi)2^

In the last column 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 ^^ to a fair approximation.

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

// 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

IX y = id'+^^+'='\

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

\o%y = a-\-bx-\-cx^,

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

X. y = ks'g^.
Values of -x form an arithmetical series and values of A^ log y form a
geometrical series.

// 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



38 EMPIRICAL FORMULAS

This becomes evident by taking the logarithms of both sides
and comparing the equations thus obtained with VIII. X
becomes

log >; = log ^ + (log s)x + (log g)d''.

This is the same as VIII when y is replaced by log y, a by
log k, b by log 5, and c by log g*

XL y =



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



// 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 oj - are constant, the relation between the

y

variables is expressed by the equation



XI



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

y

the law stated above becomes evident.
If a is zero XI becomes

_ I

^~b-\-cx

which, by clearing of fractions and dividing by y, reduces to

- = b-{-cx,

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

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.







X


X


X ,,






Y X




Com-


X


y




A—


A2- X




Y




-2.5^2








y


y


y






X y




puted y


o


o.ooo




...












0.000


.1


1-333


0.075


100


.050 —


9


- 2 . 703


3


003


050


1.329




2


I -143


0^175


150


.050 —


8


— 2 . 603


3


254


075


1. 140




3


0.923


0.325


200


• 050 —


7


-2.453


3


504


100


0.929




4


0.762


0.52s


250


.050 —


6


-2.253


3


755


125


0.760




5


0.645


0.775


300


-051 -


5


— 2.003


4


006


150


0.644




6


0.558


I -075


351


-049 -


4


- 1 . 703


4


257


175


0.558




7


0.491


1.426


400


.047 -


3


-1352


4


507


201


0.491




8


0.438


1.826


447


.058 —


2


-0.952


4


760


226


0.438




9


0.396


2.273


505


.040 —


I


-0.503


5


030


248


0.395




°


0.360


2.778


545


•054





0.000








0.360




I


0.331


3 323


599


.056


I


0.545


5


450


298


0.331




2


0.306


3.922


655


-051


2


1. 144


5


720


332


0.305




3


0.284


4.577


706


-035


3


1.799


5


997


352


0.284




4


0.265


5-283


741




4


2.505


6


262


383


0.265




5


0.249


6.024







5


3.246


6


492


399


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=



a-\-bx-^cx^^



or



= a-\-bx-\-cx^.



This represents a parabola when - is plotted to x.

y

Let X = x—i,



Y = -

y



.778.



40 EMPIRICAL FORMULAS

From these equations are obtained

- = F+2.778.

y

The formula becomes

Since the new origin lies on the curve

a-\-h-\-c = 2.^^2>,




the equation reduces to

Y={h+2c)X+cX^,
Y



or



X



= b+2c-{-cX.



Y .



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

value obtained for c from P'ig. ii 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 — 2.5ii;2 to x. In this way is obtained
the line

— 2.<x^ = a+bx.

y

From the lower part of Fig. ii the values of a and b are
found to be

a = .o2S,

^ = •2525.

Substituting the values of the constants in XI the formula
becomes

X



y=



.025 + .25250:+ 2. 5x2"



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.



yc



c \



-M



^\c^



CHAPTER ra
XII. y^a^.



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

// two variables^ x and y, are so related that 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 {b) are obtained.

Xn=xir''-'^, {a)

■ yn=yiR!'-\ {b)

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

log Xn = log xi + in-i) log r,

logr

Also by substituting this value of n — i in the value of yn in
equation (b),

log XT —log XI

yn=yiR '°^^

_ log XI / 1 \ log Xn



DETERMINATION OF CONSTANTS



43



where
and



log xn



a = yiR '°er



Io^=i^log^



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



X


9-79


20.02


40.47


60.92


81.37


101.82


204 . 00


y


0.33


0.695


I 530


2.410


3.29s


4.185


2 4 5 6 7 8

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