142 THE LAWS OF 



will be expressed by 



'--- , 1V 



2 . 4 ...... 



5-l ...... *+ 2' 25- 



Hence the portion of this coefficient containing the function (/), 

 when the various powers of //. have been replaced by their values 

 in functions of this kind agreeably to the preceding observation 

 will be found, by means of the equation (9), to be 



/\ 2^-1 .Ti + 1 7& + 2/+4J' 25-3 



^ ~2 '. 4 2/+4'-2s 



i + 2t' 25 . / + 2' 2.9 1 /+ 1 



x 



2. 4. ..2'- 25X2/4- 2*' -25 + 1 . 2/ + 2*'- 25- 1...2^-f 3 



oi^-2./ iv + 2*'-s.* + 2*'-5-l 

 1.2.3 



2.4.6 ...... 



+ 1 . i + 2 . "+ 3 . ^+4 ...... t + 2^'- 



. ... ^ (- 1) 8 . Ti-1 .n + l.n+3 

 "' 



3.5.7 ...... 



1.2.3 ...... i 



^ ' 

 :2 "" 



2.4.6 



2.4.6 ...... 25 



where all the finite integrals may evidently be extended from 

 .-s = to 5 = co , and it is clear that the last of these integrals is 

 'equal to the coefficient of x v in the product 





 x ] 



