380 MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
(iv.) It remains to say a few words about the integral 
G(r, ») = | sin’ e* a0. 
0 
Provided r > 1, we find a formula of reduction 
G(r, vy) = od G(r — 2, r). 
Thus the value of the integral from 7 = 0 to * = 2 only will be required for diverse 
values of v. The integral does not yet appear to have been studied at length or 
tabulated. Dr. A. R. Forsyra* has kindly answered my inquiry for a fairly easy 
method of reducing G (7, v) for purposes of calculation, by sending me the formula 

2-7 ei II (7) 
Cy) a Garda ea): 
where II is Gauss’s function such that 
Il (xn) =T (nm + 1). 
Taking as definition of II that 
Uo a)a nt 
(2+1)(@+2)...@+2) 

II (z) = limit of 
when 1 is infinite, we can reduce the above expression to the form 
2-v gel T(r +1) 
n= yp ria 
Product ed (1 + ie) 
Here, since 7 can always be supposed to lie between 0 and 2, when »v is small a few 
terms of the product will generally suffice for the calculation of G(r, v) to the degree 
of accuracy required in statistical practice. 
On the other hand when 7 is large, 7.e., generally in cases of slight skewness, I find 
if tan ¢ = v/r 
(Br 

CG 2) 
ue © 20 = gr tan gd 
ee ON 
cos 2 cos : 

vol 
| 
bole 
ys 
a 
q 
— 
wie 
> 
a 
bole 
S 
>. 
— 
I 
very nearly. 
Hence 
cos? & 1 
as — — —¢r tan¢d- 
127 or $ 
1 = aa t per eR ETT EET 
Yo a Qa (cos @)’*1 

very nearly. 
* “Ryaluation of two Definite Integrals,” ‘Quarterly Journal of Mathematics,’ January, 1895. 
