1881.] certain Definite Integrals. 331 



And again expanding in terms of b, and equating coefficients of b m , we 

 shall have 



j"^2aj/ 2 (w, m, c . . x) = (p 2 (n, m, c . . ). 



And thus we may proceed in general until we arrive at a simple de- 

 finite integral containing only one arbitrary constant and the indices 

 n, m, . . . 



Conversely we may obtain a complicated definite integral in many 

 cases from a simple one, by multiplying it by constants raised to the 

 powers of certain quantities contained as indices in the integral, as- 

 signing successive values to those indices, and then summing the re- 

 sulting series. Thus the integral (123), 



rd0 s0 cos 0(j3 ~\~ cos tan $) — A. sin sin tan 

 J C ° S ' (1 + 2/3 cos tan + /3 2 ) (,\ 2 sin 2 d + ff cos 2 0)' 



was obtained from the definite integral 



2 cos'^OdO cos tan + (n — 1) 0) 



by a process of double summation. These considerations show us why 

 the method of summation is of such great importance in the evalua- 

 tion of definite integrals. 



I now hope to prove, as I stated in the last paper, that every func- 

 tion of an algebraical magnitude may be regarded as a centre from 

 which systems of definite integrals emanate in all directions like rays 

 from a star, in such a manner that the value of each integral is 

 equivalent to the original function transformed by a known symbol. 



Let /(a?) = A + A x aj + A 2 a 3 + A 3 a 3 + . . . 



Then 



4 . 3A 4 a3 2 + 5 . 4 . A 6 a> 3 + 6 . 5 . Aje*+ . . 2 A 2 - 2 . 3 . A s x ; 



or since n(n — 1) = 1 dO cos" cos (n— 4)0, 



^ Jo 



we shall have 



-^-\ d0{A 4 cos 4 0+A 5 cos 5 0. cos6> . 2a;+AgCos 5 0cos2<9(2£) 2 -|- ... .} 

 =f'( x )-2A 2 -6A,x; 



■or putting aj==i — d#{A 4 cos 4 + A s cos 5 6e i0 + A 6 cos 6 0e 2id + . . .} 

 ""Jo 



, cos 4 0+A 5 cos 5 0e~ ie + A 6 cos 6 06-™ + .}=/" (k) - 2 Ag - 3 A 3 . 



~\ 2 dO{A^ 



