270 



EEPORT — 1879. 



Now 



sin (a — x) sin (a + .r) 



f *)t x* 



1- 



) 2 j (■'•"(a + Tr) 2 ] T (a-27r) 2 j ( x ~ (a + 2tt) 2 j 



»i-, ** 



so that if 



be denoted by <j>(p), then 



sin (a — px) sin (a + px) 

 sin 2 « 



.#-' 



l-» 



(a-nf n \ 



cp{p) . <Kp 5 ) . . . <Mp 4 - s )= jl-»^ 



Also 



sin (a — p.r) sin (a + px) _ x cos 2p.r — cos 2a 



i ; — s — 2 : 5 



sura sura 



and therefore 



. / » -tin i cos 2 Ax cosh 2B.r — cos 2a — i sin 2 Aa* sinh 2B.r 

 ^ v ' J sin 2 a 



H 1 --^.}---'" 



Now if (a 1 + «)8 1 ')(a 2 + i|8 2 ) . . . = (x 1 + iy 1 )(x 2 + iy 2 ) . . . . . 



(the number of factors on either side being arbitrary) 

 then 



arc tan c? + arc tan *-2 + &c. = arc tan -- 1 + arc tan ^ 2 + &c. . . . 



a, 



x n 



(2) 

 (3) 



(4) 



for, changing the sign of i in (3) 

 and therefore 



! (*i-*yi)(*a-*Vs) 



„ , a + iS _ •■ x + iy 



2 log £=2 log » 



a — «/5 .r — M/ 



which leads at once to (4) in -virtue of the formula 



, B 1 , A + z'B 



arctan_ = -log x _ 7B . 



Applying this theorem to (1) and (2), we find that 



x 

 arc tan -— + arc tan 



(a-n-)' 



+ arc tan 



x* 



(a + jr) 2 



= 2 



+ &C. 



- =1 sin (2 A y r) sinh (2 B^) 



s=o cos (2 Ag.r) cosh (2 ~B s x) — cos 2a 



, A /4s + 1 \ -d . /4* + l \ 



where A, = cos I — M — tv I, B, = sin ( — 3 — 7r i 



8 V An P ' \ An ) 



and this, on replacing x and a by ^- and -^-, becomes 



arc tan — 



tt T„ + arctan ( . rt _ 6)2n 

 ,r 2 « 



a. - 2„ , x 



2. — + arc tan 



+ arc tan - 



s=n— 1 



+ arc tan '" OI ,., M + &c. = 2 



(a-26) 2n s=o 



(a + 6) 2 "' "(a-26) 2 " 



(» ** O) 



sin 



cos [ — AJcoshf — BJ -cos _. 



