1879.] Mr. W. H. L. Russell on certain Definite Integrals. 361 



cos (arc tan q— arc tan q 5 + arc tan q 5 — &c.) 

 _ l — q 6 —q l0 +q^ + q^- &c> 



(l+ 2 8)*(l + 2 «)*...l-2*.l-. 2 8... 



sin (arc tan q— arc tan g 3 + arc tan 2 5 — &c.) 

 _ q-qS-q^ + q^ + q^- &c. 



(10), 



(11), 



(l + 2 3 )*(l + 2 6 )*. • • 1~2 4 ■ 1-2 8 • • • 



the exponents being the triangular numbers. These formulae can be 

 readily identified with (8) and (9) ; for 



^T^q-h 



(l+g.l+s»...) l! 



(l_ ?2 .l_ 2 4... ) = 2iK_Wj 2 -A, 



so that, replacing q by q- and putting for 



arc tan q k — arc tan q§ + arc tan qi— &c, 

 its value \9, (10) and (11) become 



which are in fact (8) and (9). 



X. " On certain Definite Integrals." No. 5. By W. H. L. 

 Russell, F.R.S. Received October 13, 1879. 



The following is a continuation of four papers inserted in the 

 " Proceedings of the Royal Society." In my last paper I gave a proof 

 that 



sin (2^ + 3)0— sin (2rc + l)0=O 



when (n) is infinite. We may consider the subject also thus. Taking* 

 the expression 



(2,+3)0-^+^-V+ . . . +J>±*)^e^ 



v } 1.2.3 -1.2... 2w + l 



(2n+i)g-(g!Lhl)V ...+ (w^L^i 



} 1.2.3 -1.2.3.2m+l 



we see manifestly that as (n) increases to infinity the fraction becomes 

 unity. And this is true however large (in) is considered to be. 

 vol. xsix. 2 c 



