19 
4] 
4. 
ON CERTAIN EXPANSIONS, IN SERIES OF MULTIPLE SINES 
AND COSINES. 
[From the Cambridge Mathematical Journal, vol. hi. (1842), pp. 162—167.] 
In the following paper we shall suppose e the base of the hyperbolic system of 
logarithms ; e a constant, such that its modulus, and also the modulus of — {1 - Jl - e 2 }, 
are each of them less than unity; %{e wV(-1) } a function of u, which, as u increases 
from 0 to 7r, passes continuously from the former of these values to the latter, without 
becoming a maximum in the interval, any function of u which remains finite 
and continuous for values of u included between the above limits. Hence, writing 
and considering the quantity 
X {e M ^( 11 j = m 
N /nU’ 2 /{e uV( ~ 1, l 
•(1), 
(2), 
J — 1 {e w ^ (-1) } (1 — e cos u) 
as a function of m, for values of m or u included between the limits 0 and nr, we have 
J1 — e 2 /{e MV( ~ 1) } _ 2 ^ * [* Vl — e 2 /{e M ^ (-1) } cos rmdm 
cos rmf -¡=-- " U J —— J -~U" —•••(3), 
Jo J — 1 x (1 — e cos u) 
J — 1 1( x {e“^ ( U (1 — ecosu) 7r 
(Poisson, Mec. tom. I. p. 650) ; which may also be written 
Jl-e 2 f{e u ^-v} _ 2 v _ f" JfU e \f\e u ^- l) } cos r X } du 
J 0 
cos rm 
1 — ecos u 
...(4); 
J — 1 e 1 '*' 1 xW 1 ^ (1 —ecosti) 71 
and if the first side of the equation be generally expansible in a series of multiple 
cosines of m, instead of being so in particular cases only, its expanded value will 
always be the one given by the second side of the preceding equation. 
3—2