12 Mr. S. H. Burbury on the Law of Probability 



_2 



which differs from (15) only in s p 2 being replaced by s for 

 every p, and s^T q by ^ ^ for evei T P and 2- "Whence 



and therefore finally 



2(s-r)0V-l 



*('V.. rj^jj...^...^,/ 



_2 __ 



i2(s 2 -s )02+22(ss'-ss')00' 



X0 ' v ' _ , . . . . (17) 



because in restoring the exponential we may again neglect 

 powers and products of 1 ... 6 n above the second degree. 



21. I now simplify this expression by assuming every s to. 

 be zero, and will point out in art. 37 how this becomes 

 important. Our equation then becomes 



- (£2s 2 2 + 22ss'00') . - 2r0 V _ 1 



0( n . . ■'•»)=^||. ■ .«Wi.. .*?. «" (i&ai 



us now write 5j 2 = a l5 





5 2 = <^2j 





&C, 





SpSg — b pq = 6^, for every p and g ; 



and let the index so 



Let 



and 

 obtained, or 



ia 1 6\ 2 + MA + i« 2 2 2 + &c. +KV=Qfc 

 so that 



+ 00 



#(r l ...r.)=2:JJ..^...«W,»-^- :wVZI • < 18 ) 



22. Before integrating for 1 ...6 n , it is necessary to 

 introduce the conjugate functions ^^ L ...•^/ n . 



The coefficients ~a, b are a property of the given system, 

 being determinate if *!...*„ are given in form. Let us then 



