22 
ON CERTAIN EXPANSIONS, 
[4 
and integrating the resulting equation with respect to m, 
a v „sin rm^ ^sin rm^ . 
6 — -33-= A r = ra + 22 A r (28), 
X v. ./ 1 X 
a formula given in the fifth No. of the Mathematical Journal, and which suggested the 
present paper. 
As another example, let 
uji-d) ta s/l - e 2 (cosw - e) . 
/{«^ «},«■(»-w)^- (1 _, c03M y (29). 
Then integrating with respect to m, there is a term 
< 3 °)’ 
which it is evident, à priori, must vanish. Equating it to zero, and reducing, we obtain 
e — A + \ 1 
1 - e 2 l-^X + X" 1 ) K h 
that is j= X + g (X“ + 1) + — (X 3 + 3X) + — (X 4 + 4X- + 3) + (32), 
a singular formula, which may be verified by substituting for X its value : we then obtain 
sin <*-- )-— -VÎ (33). 
V y 
The expansions of sin k{6 — ct), cos k (0 — -&), are in like manner given by the 
formulae 
cos k(6 — tn-) = A r L' cos kL cos rm (34), 
V y 
y N 
• 7//) \ K? 7 J- Sill f Tli)X /or\ 
sin k(6 — *&) = 2 -00 A r y— cos kL (35), 
rCV V 
N S 
where, to abbreviate, we have written 
U0S Ai J 4«<>+*-‘) :=/ ' (36) - 
{1 - (X + X—))=» _ r , (vrt 
J 1-e 2 ’ 
Forming the analogous expressions for 
cos k (6' — or'), sin k [6' — •ex'),