( 369 ) 



2. hi the first place the frequency-formula known as Type I of 

 Prof. Pearson's formulae finds an application : 



which, for the assumed conditions and choice of origin, takes the 

 simple form : 



u = 21 (1 -f- »)« (1 — xf (1) 



and can be regarded as a generalization of the condition that the 

 function vanishes for x = ± 1. 



Its constants may be calculated from the following relations: 



1 r(a + b -f 2) 

 51 = LJ1L_T (2) 



2a+6+i r(a + l) r(b 4- i) 



(f + iy«) = 2 (a + w) 

 (ft + l)(«-0 o-f 6+w+l * 



where 



(fi 4- l) r ") ft„ + n fi„— l + — ^7-. Mm— 2 4- etc. 



and f*„ represents the mean of the n th order. 



As, besides 21, by which the area of the curve is defined as equal 

 to unity, only two constants appear in formula (J) as characteristics 

 of the curve, it is sufficient to calculate the means of' the first and 

 second order fc x and f« 2 . 



Putting 



1 — fi, 



p = 1 — fi 1? q 



1 + Mi 



we find 



(2-p), yg .... (4) 



2(p- 9 )' x 2(p-j) 



This formula offers the advantage that, the constants a and b 

 being known, a simple expression can be given for the situation of' 

 the maximum-value : 



a — b 



In the second place we have to consider the expression in series- 

 form proposed by the author in an earlier publication *), which may 

 be regarded as a generalized zonal function, modified according to 

 the condition : 



i) These Proceedings Vol. X. (799-817). 



