STUDIES IN STATISTICAL REPRESENTATION. 479 
6. The logarithmic homologue.—The analytic value of 
taking the logarithm of a quantity depends upon the fact 
that the operation converts the products of quantities into 
sums, and the powers of quantities into products. In 
general before the logarithm is taken the quantity to be 
operated on must be in the form of a product or a power. 
Thus if Dake byt ON Opa See mann (7) 
B must if possible first be eliminated by some method other 
than mere subtraction of two values of y so that a new 
equation is obtained in the form, Ay’ denoting log y’ 
CRASS ES ls) BMS a (8) 
Pee —NG 1 G2, OF Say = Yar aaa. ht 022. (9) 
since, using Napierian logarithms, 4e = 1. Thus the equa- 
tion becomes linear: and this last expression may be said 
to be the logarithmic homologue of equation (8). 
In some cases the expression of the form y = k + f(x) 
is manageable by approximation. Thus the above equation 
(7) may be written 
ax B 
o = UE2S (has onee Pan hre eN (10) 
hence Aa NC er) a NL bo) Sree (11) 
which may be quite satisfactory, if B be so small, that 
roughly approximate values of the denominator are suffici- 
ent, forasmuch as the expression is small. When the term 
in B is very small, it is sometimes convenient to calculate 
it by the formula 
Mate BF 8 
6 denoting 4/Ce*, 
Or yet again, when B has any value whatsoever, we may 
proceed in the following way :— 
Col 
CO 
Go 
| 
oO 
et 
© 
“~ 
— 
bo 
— 
Take the values of y1, ¥2; Ys, corresponding to three points 
2,2 +kand« + 2k: then we have identically, from (7) 
Ys — Yo _ CO fer(x+2k) — eax+k)] 
Yo yi.  C[#at ex | 
