Definite Integrals, with Physical Applications. 403 



and suppose F(k) when arranged according to the powers of h to be 

 = A a + Aih + A i h t +... + A.i.h- 1 + A .Ji~- + &c. 



it remains to determine A . Putting for h the above roots successively 

 we get 



F(1) = A + A l + A 2 + ... +A- i + A.» + &c. 



A_ A 



F{a l ) = A + A l ai + A 2 ai : + ... + + — ^ +&c. 



a, a* 



F(at) = A B + A 1 a i + A i a i *+... — — + — ~ + &c. 



02 a 2 



&c. = &e. 

 adding all these equations and observing that 



1+a, + a 2 + &C. =0 

 1 + a? + a? + &C. = 0, 



1 + - + - + &c = 0, 



a, an 



&C. &C. 



we get A = ^tt) + *'fa)+*'(«.)+...*V-') . 



Secondly, when the number of terms in F{h) is infinite, we must put 

 ii='*, , and observing that generally 



A = -4= \F(e").^^H + F( e ^). WEI 

 2ttV — 1 n n 



n 

 Vol. IV. Part III. SF 



