PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 263 
where C' is a constant, a function of C and the as, and <£ 2 consists of the following- 
parts :— 
(i.) A quadratic function of the 17 ’s from rj l to 
(ii.) A quadratic function of the e’s from e K+1 to e m , 
(iii.) A series of functions of the type : 
0( + I ( bl , n + lV \ 4“ &2,n+l Vi + • • • + b re, re+ U?«)> 
e , t +2 ( bl , 're+ 2^1 + b 2 , re+ 2^2 + • ■ • + ^re, re + 2 ^ 7 «)’ 
{b], m r }\ + b 2 m 7]^ + . • • -f b^nT],,), 
where some of the V s may be zero. 
Now if P' be integrated for the values from — co to + co of all the contributory 
causes e„ +1 , e „ +2 . . . e m we shall have the whole chance of a complex with organs falling 
between 7) l and rj l -j- Srj^ y) 2 and rj 2 + Srj 2 . . . r\ a and y] n + But every time we 
integrate with regard to an e, e„ +u say, we alter the constants of eacli contributory part 
of (f) 2 , but do not alter the triple constitution of <f> 2 , except to cause one e to disappear 
from its (ii.) and (iii.) constituents. At the same time we alter C' without intro¬ 
ducing into it any terms in 77 . Thus, finally, after m — n integrations, <^> 3 is reduced 
to its first constituent, or we conclude that the chance of a complex of organs between 
r) 1 and rj 1 -j- Brj l , r) 2 and y]. 2 + Sr/. z . . . rj ri and 7], t -f- Sr] n occurring is given by 
P = 0e~^ x ~ S , > 7 1 Srjn . . . S rj n .(iii-) 
where y 3 is a quadratic function of the 77 ’s. This is the law of frequency for the 
complex. 
Now our deduction of (iii.) seems to have considerable justification in case of 
heredity. We allow for an indefinite number of quite inappreciable and unascer- 
tainable independent contributory causes. We suppose that some of these causes 
are common to parent and offspring ; how many and in what degree we make no 
pretence at saying. We assume, however, that the action of these causes does not 
differ very widely in intensity throughout the special range of organisms from which 
our complexes are drawn, and further, that the variation in intensity of any con¬ 
tributory cause follows that law of frequency, which we know- to be at any rate 
approximately true, for distributions of physical and organic variation similar in 
character to those in which we may with a high degree of probability suppose the 
phenomena of heredity to ultimately have their origin. 
Having thus deduced, with special reference to our particular topic, Bravais’ law 
of frequency, I propose to consider its characteristics in two special cases, as it is 
needful to deduce for our present purposes one or two, I believe, novel results. 
