248 Prof. Dixon, Expressions for the remainders, etc. 
This step is further examined below (§ 8). Thus 
sin $k or (1 —cos krr)/2 sin thr 
is the limit when n—20 of 
he (2? — he) (4° — he) ... (An? — We) 1 2/ “sin” tdt 
0 
2n! 
: k(2?— ke)... (4? -—k*?)a7 1.3.5...(Qn—1) 
ee On! a a 4.6 
ke ke 
or of 57 (1 5) = aC - 75): 
Also cos $km or sin as +2sin thr 
is the limit when n—20 of 
(1? — k*) (3?— Fe)... {(Qn + 1)? — #*} a 4.6... 2n 
(2n + 1)! “3.5 Tea 
or of (1-5) (1-3) . : 
8. To justify the statement made as to wu, when n>, take 
the difference 
Tr 
Un — Sin} ler [ sin” tdt 
0 
and write 7 —tfortin u,. The difference is then 
if sin kt — sing kar 
0 Gos ¢ 
= kt —sind kr sin?! ay 
sin” ¢ cost dt 
cos t fics Tl 
d sinkt—sint kr 
n+l 
-—/ se dt cos t dt. 
The terms at the limits vanish, and thus the difference bears 
Beg h , 
to ' sin” tdt a ratio less than : where h is the greatest abso- 
cae 
+1 
lute value of fal a ish Sb 9) SD between the limits. Since A is 
dt cost 
finite, this proves the statement, unless & is an even integer, in 
which case the product expressions for sin $ka and cos $ka are 
evidently true. 
