74 PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
Multiply equation (17) above by .r, and x^ in succession, and integrate the results 
respectively between tbe limits a and — We find 
(^1 — {^^ 4 : ^2) ^2.(/^)j 
{ 3 <(^ - h.^ + 6,3 - L;} c, = {S<T.^{h - h) + W - ¥1 ^ 2 , 
reducing by aid of (oc) and (yS) to 
2 >cri — 6,63 = ScTo® — h.^h^ .(y), 
and 
[ 150 - 1 ^( 6 , - 63) + 10o-,3(6,3 - 638) _p 6,5 _ 635] c, 
= {150-3'^ (6^ — 6^) + lOcr/ ( 6 q 3 — 6 /) + 6 / — 6 y] Co, 
reducing by aid of (a), (yS), and (y) to the two forms, 
2 ( 7 / + 8 c 7,2 + 36,2 37,^2 _p ^0.(8), 
2 ( 7,2 + 8 ( 7 / + 36 / + 36/ + 46 ^ 6 , = 0 .(e). 
Equations (a), (yS), (8), and (e) are four independent equations, which sufiBce to 
determine 6„ 6^, 63, 6,^, as definite functions of (7,, (73, c,, and C3. But 6„ 6, are in 
general independent of (r„ (To, c,, and (?3 ; hence it follows that (a) cannot in general 
be true, or we must have 6, = 63 and 60 = 6,^. That is, a curve which breaks up into 
two normal components can break up in one way, and one way only. 
Now it is clear that in actual statisticaJ practice our abnormal frequency-curve will 
never be tbe absolutely true sum of two normal-curves ; indeed, if it be not a mixture, 
but an asymmetrical frequency-curve, it is not necessarily a very close approach to 
the sum of tivo frequency-curves of normal type,—it may be the limit to an 
asymmetrical binomial.''' We must not, therefore, be surprised if more than one 
solution be given by any method of dissection. A mathematical criterion for dis¬ 
criminating the “ true” solution might easily be given. For example, in the method 
of the present paj^er, we might define that as the “ true,” or at any rate the “ best,” 
solution which gave for the compound-curve a sixth moment, nearest in value to that 
of the observation-curve. Such a theoretical criterion, however, may not have much 
* The values of the successive monieuts of the uormal-curve are given in § 5 of this paper, and 
permit of these integrations being performed at once. 
f The general form of the limit to asymmetrical binomials is 
where C, c, and (3 are constants, and x is to have positive values only. /3 is always positive. [A 
slightly fuller form is given in the abstract of this paper, ‘ Roy. Soc. Proc.,’ vol. 54, p. 331.] 
