1876.] Evaluation of a class of Definite Integrals. 11 



§ 7. In this example the integral contains an infinite element 

 corresponding to a; = /^a, and the value obtained is the principal 

 value of the integral. This is generally true of all results derived 

 from (2), as can be easily proved. It is interesting to notice that 

 in transforming 



into 





we suppose that the latter has its principal value for a;=l, if 

 ^ {x — x"^) should become infinite when x — 1: for writing the 

 former integral 



i/rv:j(--^)<^-^)t ■ 



{\, \' infinitesimal), it becomes by transformation 



that is, the principal value of the integral is to be taken. 



§ 8. As an example, in which the integral contains no infinite 

 element, let 



Q,n-x =j ^'"~' [^ - ^) cosech (x - ^ dx, 



/•OO 



then P^r-i—] ^^'"*'^cosechi»i?i», 



so that the value of ^^n-i is 



I (2'- 1) 5.'^'«- + f^^] {V-V^Bya-^... 



2n ^ ' "• 2/1+2 



Addition — added Novemher 7, 1876. 

 The following proof of the fundamental theorem that, / being 



even, 



i ■^r~^)^^"'j ■'f'^^^^^' 



