20 
ON CERTAIN EXPANSIONS, 
[4 
= 2_"2_* ... П cos rm П (2 cosrxA)/(A 1( A 2 ...). 
(14), 
Now, between the limits 0 and nr, the function 
f f e « V (—i) | cos | 6 mV(-1)| 
will always be expansible in a series of multiple cosines of u; and if by any algebraical 
process the function fp cos rxp can be expanded in the form 
fp cos rxp = 2 - * a g p 8 , (or* = a_ g ) (5); 
we have, in a convergent series, 
y{ 6 «V(—Dj cosr% {e MV(-1) } = a 0 + 22* a* cos su (6). 
Again, putting ^ {1 — Jl - e 2 } = A (T), 
we have z—-—-— = 1 + 22* A? cospu (8). 
1 — e cos u 1 
Multiplying these two series, and effecting the integration, we obtain 
1 ("Jl- e’f {e“*g>j cos «xi«“' 1 -» 1 du _ 2 „ 2 . , v) ) 
Э 1/ ^ 1 ' ' ' 
£ 
1 — e cos и 
(9), 
and the second side of this equation being obviously derived from the expansion of 
f\ cos r^A by rejecting negative powers of A and dividing by 2, the term independent 
of A may conveniently be represented by the notation 
2/A cos r%A (10) ; 
where in general, if ГА can be expanded in the form 
ГА = 2_* (M S A S )> [A- S = A S ] 
we have 
Гл. = £ M 0 + 2* M g A s 
(И), 
.(12). 
(By what has preceded, the expansion of TA in the above form is always possible 
in a certain sense; however, in the remainder of the present paper, TA will always 
be of a form to satisfy the equation F J = TA, except in cases which will afterwards 
be considered, where the condition A_ s = A g is unnecessary.) 
Hence, observing the equations (4), (9), (10), 
Jl - e a /{e wV <- 1) } 
J — 1 e w ^ (-1) x {e MV(-1) } (1 — e cos u) 
= 2_* cos m 2 cos r^A/A (13) ; 
from which, assuming a system of equations analogous to (1), and representing by n (d>) 
the product d^dA ... , it is easy to deduce 
Jl — e 2 
П 
( 
\J — 1 € i,V(-1) x {e ,tV(-1) }(l — e cos u) 
l/| 6 MiV(-D } e « 2 v 
) 
(-1)