for a System of Correlated Variables. 11 



expand the exponential, neglecting in accordance with the 

 last article powers and products of 1 ... 6 n above the second 



degree. Then e s ~ r is replaced by 



l + Ss-rV-l + i^^V- 1 ) 2 ' • • ( 13 ) 

 To this expression we may now apply art. 10, and so obtain 



jj...^...^^...*,,)^ 8 ^ 



= §...ds 1 ...ds n <f>(s i ...s n ) 



+ jy ... ds! . .ds^Qi! ... »j2*j^r0y/—i 

 - i jy .. . dn . . . dijKfx ■ ■ • s„)(2>~^) 2 



= l+2"(«-r)0/^l 



-i^...ds l ...ds^(s l ...sJ(t" i J^rOy-. . . (14) 

 20. Now, 



§...ds 1 ...ds n 4>(s 1 ...s n )(t n i i=te)* 



= jj... if,.. d, n <j>( H . ••0^(V-«V» + r F , )V 



for all positive integral values of p 



+2$... d Sl ...ds n <f>( Sl ... oCiOw+v-vr^)^ 



the last term including all pairs of unequal integral values 

 of p and q 



-K*W-*v,+W 



by art. 10. 



Also (2,(s—r)0)* 



+*ODvt+Vt-Vt-vi) tf A • • ( 16 > 



