4] IN SERIES OF MULTIPLE SINES AND COSINES. 
21 
where T (X x , X 2 ...) being expansible in the form 
I (X x , X 2 ...) = X_ 00 ... Ag u S2 V'V* ••• \A, lt s2 ... = A_ Su _g t ] (15), 
F (Xj, X 2 ...) =2 0 £ 0 ••• 2^-4«,, Xi^X/a..., (16), 
N being the number of exponents which vanish. 
The equations (13) and (14) may also be written in the forms 
y{ e wV(-D} =£_” cos rm 2 cos ?'%X ———^ ^ 2 e (^ + ^ (17^ 
Jl — e 2 
^ X 
yjfWiVl-iq gi/aVl-D . #< J 
= n (cos rm) n |2 cos r x \ J ' 5 '' X + X ~‘ )i }/(V, (18). 
^ 
As examples of these formulae, we may assume 
X {e w ^ (_1) } = m = u — esin u (19). 
Hence, putting 
+ a. (20), 
and observing the equation 
J — 1 [e w * ,(-1) } = 1 — ecos u (21), 
the equation (17) becomes 
/(e „vi-.,) = 24 cosrmA.lLzii^+pE A (22). 
V ✓ 
Thus, if e-v = cos- 1 - C0S ^ ~- (23), 
1 — e cos u 
assuming / {e V( *} = ^ _ cQg — (24), 
cos (0 — ot) = 2_ * y--r=^ cos m [1 — (X + X -1 )} {£ (X + X -1 ) — A r .. .(25), 
the term corresponding to r = 0 being 
— > - 1 = {2X — 2e — e (X 2 +1) + 2e 2 X], = -e (26). 
2 Jl - e 21 
Again, assuming 
f ( 6 uV(-D| = ¿0. = ~ e * (27), 
^ ^ 1 dm (1 — ecosw) 2