244 Prof. Dixon, Expressions for the remainders when 0, 6, — 
Expressions for the remainders when 0, 02, sin kO, cos k@ are 
expanded in ascending powers of sin @. By Prof. A. C. Dixon, 
F.R.S. 
[ Received 9 April 1913—Read 28 April 1913. ] 
6 
1. TAKE i sin k (6 —t) sin” tdt, say up, 
0 
and integrate twice by parts. Thus 
6 6 
Poy = E cos k (@ — t) sin” | —kn | cos k (6 —t) sin” ¢ cos tdt 
0 0 
8 
=ksin”@ + E sin k (@—t) sin” ¢ cos | 
0 
= rere 
6 
- nf sin k (6 — t){(n — 1) sin” °t cost — sin” ¢} dé — 
0 
=ksin"6— n(n — 1) uyn_ + NuUty. 
This holds when n = 2, 3, 4... but when n =1, 0 we have 
k°u, =ksin 6 — sin ké +m, 
k?u, =k (1 —cos k@). 
The general formula may be written 
G ae n? — ke? 
Up = need) sin” 6 + ina Wine 
Hence 
sinké =k sin 6 —(k? — 1) uy, 
as 2 42\(J2_ 22 
hein 6 = = eps Oe 
3! 3! 
SE ach are ee see lg) Pe b(k? — 12)... { — Qn —1P} § 
=ksin 0 > gee St 6+...4+(-1) reine 
x sine 9 | ate (ib), 
cosk@=1—ku, 
[ps pe Ue OP) 
=1— 5; sin OO eee rag Us 
esceecee 
