for a System of Correlated Variables. 9 



16. It comes next in order to prove that (n being very 

 great) we may in evaluating the above integral neglect 

 powers and products of 1 ... n above the second degree. 



Let X= \\ ...ds l ...ds n (j>(s l ... sje"* 

 so that 



Then 



X =jj . . . ds 1 . '. . ds n cj> («!... s n ) (cos Zs0 + s/ ~1 sin tsff) = * + fty/^T, 



if a denote the real, and ft V~~ 1 the imaginary part of the 

 last expression for X. Since |T... cp(s 1 ... s n )ds 1 ... ds n =l, 

 evidently if every = 0, ot=l, and ft = 0, and a 2 + /3 2 =l. 

 And we can now prove 



Proposition II. 



That if any differs from zero, a 2 + /3 2 contains the product 

 of ft factors, each of which, unless # = 0, is numerically, and 

 generally in a finite ratio, less than unity. 



For 



- 2 + /3 2 =\^...ds 1 ...ds n <l>(s 1 ...sJcos(sA+...+^A l )\ 2 

 + {§...d Sl ... ds^s, . . . O sin A + . . . + s n ej } 2 , . (9) 



that is, the sum of the squares of the integrals. Or replacing 

 the square of each integral by the product of two similar 

 integrals between the same limits, a 2 + ft 2 



JJ . . . A, . . . dsj, { Sl . . . sj COS (2*0) X jj . . . dsj . . . chjfisy' . . . O COS (tJff) 



jj .. . dsj . . . dsj>(sj . . . sj sin (ts6) X jj . . . <**i' . . . <fc„"K s i' • • • O sin ~ s ' 6 - ( 10 ) 



jj . . . d Sl . . . ds n dsx' . . . dsj<j>(s t . . . »J4>W- ■ •*»') cos (*i — V#i 



+ S2 - i vU + ...+^7",X) . . (11) 

 17. Again, 



cos (s l -s l '0 l -\-s 2 —s 2 / 2 -h ... + $ n —s n '0 n ) 



consists, when expanded, of the term 



COS S 1 — s/^1 . COS S 2 — S 2 '0 2 ••• coss n ~' S n^ni 



and other terms each of which contains one or more of the 

 factors sin s L — s±0, sin s 2 — s 2 '0. : , &c, each in the first degree. 



