452 Correlation Function of any Number of Variates. 



But according to well-known theorems on determinants, 

 we hare 





and hence 

 For the sum 



a s — Zr s tm. 



)^± 2* Z, Opq OCp a.q 

 -« p q 



we may now write 



^ 2 2 S^ %r pt ic t %r sq w s . 



Changing the order of summation, we have this equal to 



2g 2,%w s wt2,r pt %8 pq r sr 

 But 



2S^ = /° when ^*> 



« I b „ p = s. 



Thus the quadruple sum is equal to 



— 22,ic s w t r sf $, 



or 



~~ 2, Z, fopqUp &q — 9 ^ ZiTpqWpWq. 



As Up — ia^,, we have finally 



<)0 2,2* ^HpqUpStq — ZZrpqWpWq. 



-& 2> q "pi 



Q. E. D. 



