362 Mr. W. H. L. Russell on [Nov. 20, 



Manifestly therefore when (n) is infinite S * n (2^ + 3)0 _ ^ ano ther 



sin (2% + 1)0 



form of the same theorem. 



Moreover a. sin + « 3 sin 30 + . . . = - — (^"^"^' ) sm ^ when a is less 



1 — 2« 3 cos 26> + a 4 



than unity. 



As ex. approaches nnity in the limit, 



sin0 + sin 30 + sin 50 . . . =—- — , 



2sin<9 



a well-known theorem. 

 Bnt also 



• a , • Q n ,. • /o t i\n 2 sin 0— sin (2;z + 3)0 + sin (2^ + 1)0 



sm0 + sm 30 + . . . sm (2/^ + 1)0 = ^— — — y ' — i — i- . 



2(1 — cos 20) 



Making (n) infinite and subtracting the last eqnation we have again 

 sin (2^ + 3)0— sin (2»+l)fr=0. 



T have considered this subject at length on account of a remark of 

 Professor Stokes, that sin (2t2, + 3)0— sin (2^ + 1)0 cannot converge to 

 zero in the ordinary manner, on account of the. perpetual fluctuation of 

 the signs of sin (2u + 3)0 and sin (2?i + 1)9. There can only I think 

 be one explanation of this, namely, that the sine of an infinite angle 

 is zero. Following up then the theorem we find 



(86.) f*-^-«"°« , «sin (*sin0)=-(e«- e -*). 

 J o sin 9 2 



(87.) \ —— cos * sin 6>( e aSill(? -e-* sin 0) = 2tt sin a . 

 J o sin 9 



d9 sin (2n + l)9__ 

 o sin 9 



We shall also obtain the following: — 



(89.) [" d0e°™° cob (a; sin + 0) log 6 (1 + 2a cos + <* 3 ) = ^ f^ - 1 ) ; 



J «6 a;C 



sr\r\ \ [ n ln e xcosd cos (x sin 0— 0) + cos , /i o /i i ^ 



(90.) d0.— — r — - loge(l — 2«cos + « 3 ) 



v ; J o e>* c * sd + 2e* cos e cos (« sin 0) + 1 ° v } 



=-lor ^ + 1 



(91.) \\l9. g±*>™^ 



J0 (l + 2«COS + a")" 1 — a" 



^2 ^3 ^4 



Since — — + __ + ___ + . . =x+ (1 — x) log e (1 — x). 



1.2 2.3 3.4 ' & ; 



when 03 is any quantity from to 1, inclusive, we shall have 



