sink@, cosk@ are expanded in ascending powers of sin @. 247 
The coefficient of wu, or ku, in (1) or (2) is that of v, im (3) 
2 
or (4) multiplied by the product of factors of the form 1 a 
these are known to form a convergent product. Hence the results 
are still true when & is imaginary. 
6. When @ is imaginary, u, and v, may be treated by the 
following method, which would also apply if @ were real. 
(7) 
Un =| sink (@ —t)sect.sin” ¢ cos tdt 
0 
@ 
= sin k (6 —t)sect.sin®*? ¢/(n + | 
bs 0 
i eeu Re Ob ac 
— N+1 4 _— fa i se 
aay | se ta {sin k (6 — t) sec t} dt. 
The terms at the limits vanish, and if g is the greatest value 
BE = {sink (@—t)sect} on the path of integration, which is 
finite, we have 
[wal< 9 6 sin™* 6 | 
provided that | sin @| is the greatest value of |siné| on the path 
of integration, which may be the straight line from 0 to @. 
Now if 0=¢$ +.y, | sin? 0|=sin?¢+ sinh? which increases 
with the numerical value of @ or w,so long as ¢ is between + $77. 
Thus the remainders in the series (1) (2) (3) (4) tend to zero 
when n— provided that |sin@|<1 and the real part of @ is 
between + 47. 
7. It is curious that the coefficient of sin?”** @ in (1) tends to 
equality with an? mT *k cos 4k and that of sin” @ in (2) to 
equality with —3n ~? hk sin $k. 
The expressions (1) (2) may be used to give a proof of the 
factorial expressions for the sine and cosine. In (1) (2) put 7 
for 6. Thus 
k (2? — k?) (42 — k?) ... (4m? — ke?) 
1 —cosk7 = Fa Wars 
: _(?—F) (8 — &)...{Qn+17-—F} 
sin kr = Cn)! Os Nite 
Now when n tends to infinity the elements of uw, for which ¢ 
is not nearly $7 may be neglected and thus u, tends to equality 
with 
sin 4 kar | sin” td. 
0 
