484 EEPORT— 1880. 



cot X = + + + jr- + — + &C. 



X X —TT X + TT X — ZTT X + 2'rr 



•which are the fractional series referred to in the title. 



The formulae (1) and (2) may he estahlished hy the ordinary process for re- 

 solving an expression into partial fractions ; or hy means of the theorem : — If 



X X + \ X + 2 X + n 



where Aq, A,, . . . A„, are any coefficients independent of x, he denoted by <^ (.r), 

 then 



- .r<^(.r) <^(- ,r) = ^ + A,<^(1) , ^ _ ^ , ^ ^ ^ 





+ rA, (r) |-l_+_i-[ 



.r — r X + r 



X — n X + n 



+ nK^in) \ -i--+--i-. [. 



This thelorem applied to the expansions 



1 = ± Jl ^l_ i__ 1 1 



x{x + \) . . . {x + n) n\ X (w-l)' x + \ 2!(w-2)! x + 2 



n\ X + n 



( 2.r + 1) (2r + 3) . . . (2.r + 2w - 1 )_ 1 (2w) ! 1 1 2 ! (2w - 2) ! 1 

 .r(.r + 1) . . . {x + n) 2" (w !)'^ x "^ 2" (1 ! (w - 1) !)■' x + 1 



+ i. (2 r) ! (2w - 2r) ! 1 ^ }_ (2w) ! 1 



2" (r ! (n - r) !)'^ x + r' " 2" (« IV^ .r + n 

 gives at once the formulae (1) and (2).* 



15. Note on a Trigonometrical Identity involvinq products of Four Sines. 

 By J. W. L. Glaisher, M.A.', F.B.8. 



In a paper in the * Messenger of Mathematics ' (vol. x. p. 26), the author had 

 drawn attention to the following identity :— 



sin a sin b sin c sin d 

 = sin «' sin b' sin c' sin d' 

 + sin a" sin 6" sin c" sin d" 



where a, b, c, d are any four quantities, and a', b', c' . d', a" , b" , c", d" eight quan- 

 tities derived fi'om them by the equations 



a' = ^ (— o + 6 + c + <f), a" = ^{a + b + c + d), 



6' = i ( a-b + c + d), b" = i.(a + b - c - d), 



c' = ^( a + b — c + d), c" =• ^{a - b + c — d), 



d' = i I 'a + b + c — d), d" = ^{a — b - c + ct), 



and in this note he pointed out that this was a particular case of a more general 

 formula involving products of foui- sines. 

 The foregomg identity may he written 



sin rt sin 6 sin c sin d 

 = sin (o- — a) sin (o- — 6) sin (a — e) sin {a —d) 

 + sin a- sin (o- — 6 — c) sin (o- — 6 — (?) sin (o- — c — c?) (1) 



* The paper is printed in extenso in the Quarterly Journal of Mathematics, voL 

 xvii, pp. 211-226. 



