246 Prof. Dixon, Expressions for the remainders when 0, 6%, 
sin'@ | 2sin'@ | 2.4sin°d 
2S A BO aos 
4, ie sin2” @ 
..(2n — 1) * an +2n0| p0000c (4). 
3. In (1) (2) (3) Be we oe the expansions of sin 6, cos k0, 
@ and 6? in ascending powers of sin 6, with expressions for the 
remainders. ; 
Now ior 2.4< 32 
3.5 < 4? 4.6 < 5%, 
(2n —1)(2n+1)< (2n)?2, 2n(2n—2) < (2n—1)?. 
Hence by multiplication the coefficient of %,4, in (3) is 
< /(2n +1) and that of v, m (4) is<2,/n. 
Thus the remainders in (3) (4) tend to zero if Un, n does so, 
when no. 
Now if @ is between 0 and = inclusive 6 —£ does not exceed 
2, 
a=! which does not exceed cot 7. 
8 
Hence Din, K | sin” ¢cos¢dt or sin” @/n, 
0 
Urn < sin” O/r/n 
which tends to zero when » increases even when @= 47. 
Thus the infinite series given by (3) (4) converge to 6 and 
30° respectively when @ is any real angle between + $7 inclusive. 
4, In the series (1) (2) when & is real, if the thot: 
0 (i I) anne el) 
is put in the place of k? — m? in the coefficients of tn41, Kuen they 
reduce to those of Us;41:, Ven in (8) (4). Hence these coefficients 
in (1) (2) increase in a less ratio than those in (3) (4), at any rate 
when n> 3h. 
Aiso |sin k(@—t)+k|< (0 —2), 
so that Oh, S80 
Hence the validity of the infinite expansions (1) (2) follows 
when & is real and @ a real angle between + 47 inclusive. 
5. The above is the elementary case. When & is not real, 
but @ is restricted as before, we can say that 
| sin k (@ —t)|< «(0 —2), 
where « is a finite quantity, but generally is >|%|; then 
| 5 |S one 
