422 Mr. J. W. L. Glaisher on a Class of Definite Integrals. 

 series from the double of the series so formed, we obtain* 



= < e 4«2 C0S ^ e 4a2 C os |-...r- • (29) 



a L a a J x ' 



Integrating these series between the limits x and 0, we find 



Erfc a? + Erfc (x— a) -f Erfc (x + a) + ... 



X\ZtT , 1 f _!L 2 . 27T.2? 1 _lii . 4?™? , ~\ /OAN 



= V -r-< e « 2 sm +„e « 2 sm — + . .-. k (30) 



Erfca?— Erfc (x— «) - Erfc (^ + «) + ... 



= -r-Se 4a2 Sm — _j_ e a 2 Sin h...r« (31) 



V 7r L a 3 a J - 



These formula?, however, involve an ambiguity ; for on the 

 left-hand side the terms at a great distance from the com- 

 mencement of the series take the form -=— (1 — 1 + 1 — . . . ) ; 



and if we substitute —= Erf a? for Erfca? &c, we still intro- 



duce the same indeterminate series. The difficulty would not 

 be avoided by integrating between the limits oo and x instead 

 of a? and 0; for then, on the right-hand side, we should intro- 

 duce sin go into both formulae, and in addition an infinite term 

 in (30). 



We should, however, from the results of similar inquiries, be 

 inclined to suspect that in point of fact we must, with the ex- 

 ception of the first term, take the terms in pairs, so as not to 

 end with a term Erfc (x+na) without including also Erfc(# + 7M) 

 (n infinite); and the following independent investigation will 

 show that this is the case. 



From the integral (21) of the previous paper we find that 



I 



<*> ■ , , , m 2nirx a _«2jt2 



Erf a? sin dx = - — -, — (1 — e <# ), 



whence 



I- {Erf* + "Erf {x-a) + Erf \x+ a) + . . . } sin ^^-dx 

 Jo a 



(1— e » 2 ); 



2nv7r 



* The series (29) is given by Sir"W. Thomson (Quarterly Journal of 

 Mathematics, vol. i. p. 316) ; and (31) is deduced by integration ; the am- 

 biguity, however, is not noticed. 



