262 PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
of any kind, but the tie is to remain the same for every complex, whether it be the 
result of mating or parentage, or flow from any physiological or social relation, &c. 
• We shall now assume that the sizes of this complex of organs are determined by a 
great variety of independent contributory causes, for example, magnitudes of other 
organs not in the complex, variations in environment, climate, nourishment, physical 
training, various ancestral influences, and innumerable other causes, which cannot be 
individually observed or their effects measured. Let these causes be m in number, 
m being generally much greater than n, and let their deviations from their mean 
intensities be e ls e 3 , e 3 , . . . e m , then 77^ 7? 2 , yj 3 , . . . 77,, will be functions of e 1} e 3 , e 3 ,. . . e m . 
Further, certain of the e’s will appear only in certain of the 77’s, and the e’s will not 
be fully determined for a given 77 complex. 
We shall in the next place assume that the variations in intensity of the contri¬ 
butory causes are small as compared with their absolute intensity, and that these 
variations follow the normal law of distribution. # The mean complex being reached 
with the mean intensities of contributory causes, we have by the principle of the 
superposition of small quantities : 
% 
dn 
Here any of the system of as may be zero. 
Further, the chance that we have a conjunction of contributory causes lying 
between e x and <q + Se l5 e 3 and e 3 -p Se 3 . 
. . e m and e ni -p §e m will be given by 
_ _ + zt +-S1+.. 
P = Ce W 2k s 2 
, Cm 2 \ 
2k '" 2 ' X Se^Scg, . . . S,„, .... 
. (ii) 
where the standard deviations of the variation distributions for e l5 e 3 , e 3 , . . . e„ are 
respectively /q, k 2 , k 3 , . . . k h , and C is a constant. 
Now by aid of the equations (i.) let n of the variables e, say, the first n, be replaced 
by the variables 77 , then the probability that we have a complex with organs lying 
between and 77 x -j- 877 ^ rj. 2 and -f- 877 ^ ... 77 ,, and rj n -p 877 ,,, together with a series 
of contributory causes lying between e ;i+] and e„ +1 + 8 e„ +1 , e , i+2 and e„ +2 + Se, /+2 . . . e* 
and e m + Se m will be 
P' = S'>7 1 877.3 . . . 877,, Se„ + i Se )1+2 . . . Se, ;i 
* This may be taken at any rate as a first approximation. It is at this point that the theory of skew- 
correlation diverges from our present treatment. 
a ll e l + a 12 e -2 + «l3 e 3 + • 
• • \~ "''j 
1 
+ «o 3 e 2 + a 23 e 3 + . 
• • “1“ ^2 m^m .9 ^ 
a «i e i -p a„ 2 e 3 "P a « 2 e s “P • • 
• ”1“ 
