154 Mr. S. H. Burbury on the 



when the associations are not all independent of each other, 

 provided we can choose in many ways a group of them con- 

 taining a great number which are independent of each other. 

 Let there be ISlq systems, where N" is very large and q a 

 positive integer to which we may assign any value. Let 

 them be called A l5 A 2 . . . A^ g . If it be not true that every 

 one of the N^ systems is independent of every other, never- 

 theless it may be true, and we will assume that it is true, that 

 the N systems A ly A i+g , Ai +2g , &c. are independent of each 

 other. And, similarly, the IS systems A 2 , A 2+2 , &c. may be 

 independent of each other, and so on for other groups. On 

 this assumption that A,, Ai+ g , &c. are mutually independent, 

 we may apply our method to them. And let Fj (c x . . . c n ) 

 dci . . . dc n be the chance that the sum of the ^s in these N 

 systems, each divided by VN ? shall be q . . . c x + dc if &c. 



nS 



Then F(c x . . . c n ) = Ce~ 2T . Also let F 2 ( Cl . . c n ) be the corre- 

 sponding chance for the group A 2 , A 2+q , &c. All the systems 

 being supposed similar in character, evidently 



F^Ci . . . C n ) = ¥ 2 (c 1 .... G n ) = &G. 



But the result for any one of the groups of N systems cannot 

 differ from that for the whole Nq systems. Whence it follows 

 that F(ci . . . Cn) for the Ng systems 



= Fi(Ci . . . C n ) = F 2 (C! . . . Cn) = &C. 

 _wS 



= Ce 2T . 



We are therefore at liberty to apply our method to the l$q 

 systems, provided that, although they are not all independent 

 of each other, yet we can divide them into groups, each of N 

 systems, the members of which are independent of one another. 



(14) Again, in order that the N systems A ly Ai +2 , &c. 

 may be independent of each other, it will be sufficient if we 

 make every variable, as x, in A 1 independent of every 



variable x' in A 1+ff , &c, that is 7 / must be zero or 



" ax 



negligible. 



Let us therefore consider a series of single magnitudes 



# 1? x 2 , &c which may be the variables in many systems of n, 



and let the chance of their simultaneously having assigned 



values be 



C 



-|p(ai2) 2 +taZ3+&c.), 



dffx-,) 

 and let it be required to find the condition that 'J J shall 



