370 Prof. Karl Pearson on Testing the 



Further, A stands for the determinant 



l n, 2 n, 3 ... 

 r 2 ,i 1 ^2, 3 ■ • • 



^1, ?-l 



^2, '2-1 



»"«-], l' r Q-l,2 ^-1,3- • • 



1 



and Aw is the minor of the constituent Tw- 

 in the paper already referred to, I found ithe value of 



the exponential power by actually reducing the above 



determinant and its minors. I have since deduced other 



proofs, of. which the present seems for lecture purposes the 



briefest. 



In the expression (i.) above we must include only q — 1 



variate frequencies, as the qth. frequency follows at once 



from these by the relation 



N = n ± + n 2 + . . . + n Q , 



the size of the sample N being fixed. Accordingly, Si is a 

 sum for s from 1 to q — 1, and S2 a sum of products for all 

 unlike pairs s and s' out of 1, 2, 3 ... 5 ... s' . . . q — 1. 



Now consider what happens when we fix the variates 

 # l9 %2 . . , Xq-i, or all but x s ; then the equation to the 

 frequency of x s will be given by 



A.« (xs — hsf 



z = z a e 



all the other terms being constant and falling into z '. 



Xs = cr s 2 A/A gs will give the standard deviation of this 



distribution. We can to determine S s give the other a?'s 



their mean values, in which case we shall have samples 



of constant size n s + n q to be taken out of the population. 



The standard deviation of such samples will then be 



given by 



<2 ,- , ~n m s (1 m s \ 



Z s = (n 8 + n q ) (1 ■ I 



m s -tm q \ m s + m q J 



1- . ~ ^ H * ft ^ \ 

 = (n s + n ) - „ 1 1— - - I 



n s + n. 



Accordingly 



Act, 



_1 



1 1 



n s n q 



(ii.) 



Thus the coefficient of x* in our exponential is determined. 



