98 The Asymmetrical Probability- Curve. 



Thus the required expression for y is 



which, when c 2 is substituted for 2k, coincides with the 

 expression given above. 



This solution may be completed by observing that it satisfies 

 the fundamental equations (1) and (2) unconditionally, as well 

 as (3) when account is taken of condition (a). 



A further verification of the theory is afforded by showing 

 that if the sum of m independent elements obeys the law of 

 frequency y = F(x, k u j x ), F having the form which has been 

 found, and k x and j\ being the sum of the respective mean 

 squares of error and mean cubes of error for the m elements ; 

 and likewise the sum of another set of m independent elements, 

 n in number, obeys the similarly defined law of frequency, 

 y = F(x, k 2 ,j 2 ); then the sum of {in + n) elements of which m 

 are of the first class and n of the second obeys, as it should, 

 the law 



y = F(x,k l + k 2) j l +j 2 )*. 



A particularly interesting case of the asymmetrical proba- 

 bility-curve is that in which an element has only two possible 

 values, say zero and unity, occurring with the respective 

 probabilities p and q — the case considered in a former number 

 of the Philosophical Magazine (vol. xxi. p. 318, 1886). 

 Observing that the mean square of error for this elementary 

 locus is pq 2 + qp 2 =pq{p + q)=pq, and the mean cube of 

 error = pq (p — q) , we have by the general formula for the 

 curve representing tire law of frequency for the sum of n such 

 elements, an expression in terms of those constants which, 

 mutatis mutandis, proves to be identical with the expression 

 which Todhunter, after Laplace, has obtained by a method 

 peculiar to the Binomial f. 



The general or multinomial probability-curve, involving 

 (in addition to the centre of gravity) only two constants k 

 and j, may always be replaced by a binomial ; through the 

 equations 



npqi 2 = k, npq (p — q) i 3 ==/, 



where i is the length of each element J. There are thus only 



two equations for three quantities, n, i, andjo-i-^ (p + q—^)- 



When it is proposed to construct a binomial from a given 



* The work is given in the original paper, 

 t History of Probabilities, Art. 993. 

 X Cf. above, note to p. 92. 



