CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 
183 
numerical factors are functions only of the standard deviations and the correlation 
coefficients, and will accordingly be unchanged if these be unchanged. 
Let Oi and O-, be any organs and Mj and M, their means, and iio their numbers, 
and their coefficieut of correlation. Suppose that any hygrometric changes, 
different method of measurement, amount of animal matter in the organs at time of 
measurement, etc., cause us to measure Oi and yi 02 -fi 72 = 0 ^ instead 
of Oi and Oo, and let o-'j, cr' 2 , M'l, Mb, and r\o be the resulting characters, then 
clearly, S standing for summation :— 
M'i= / 3 iMi + ;8.2, M'2 = 71M2 4 - 70, 
a-'? = S (O'j - M\y = 131^ (Oi - = /3lcrl, or 
cr'l = S (Oh — Mh)^ = 7^8 (O 2 — Mj)" = yWli or o-h = 7 icr 2 , 
, S (0\-M\) (Oh-Mh) S (Oi - Md (0, - M,) 
12— _/ _/ — P271 , , — 7 
/ / 
cr icr 2 
(T tCr 
12 * 
2 
Thus a correlation coefficient will be quite unchanged. A regression coefficient will 
be changed or not according as the ratio of two standard deviations is changed or 
not, or according as to whether j8,/yi sensibly differs from unity. Now in stature 
or any of the long bones with which we have to deal quantities corresponding to 
A) 72 may amount to 1 per cent, of the value of Oj or Oo, but the multipliers like 
A and 7 i are not only quantities differing in the second order from unity, but 
probably very nearly equal to each other. Hence it is reasonable to suppose that 
changes in the condition of the bones, and stature measured on the living or on the 
corpse, while sensibly affecting Mg, Mf, Mh, M^, and Mr will produce little or no 
effect on the numerical constants of the regression formulm (a) to (k). We shall find 
that this d priori conclusion is borne out by actual measurements. Hence we 
conclude that Tables Y. and VI. may be applied to stature measured on the living 
or the corpse, to bones measured humid or dry, with or without the cartilage, 
provided proper modifications are made in the values of the five means. We might 
even go so far as to predict that provided Mj be properly altered, the stature from 
tibia reconstruction formulae will not be much modified, even if the tibia be measured 
with instead of without the spine. The change, however, in the regression formulae 
when the femur is measured in the oblique position is more likely to be of import¬ 
ance, and the correlation between stature and oblique femur has accordingly been 
worked out. If F' denote oblique femur we have :— 
Male Msf' = 44-938, o-p = 2-331, rgF-= '8025, 
Female Msf'= 41-240, crp,= 2'205, 7yp, = '8007, 
whence for [a) we find : 
Male S - Ms = 1-894 (F' - Mf-), 
Female S — Mg = 1-979 (F' — Mf-). 
