602 
Proceedings of the Royal Society 
ently of all previous computations, and tlius we come to revise 
critically the deduction of the formulse themselves, and of the series 
resulting from them. 
It is pointed out that if, in the series for the cosine of an arc, 
viz. : — 
cos a = 1 
+ 
1.2 1.2.3.4 1 
+ &c. 
we substitute for «, the length g-, of a quadrant, the cosine becomes 
zero; and that, according to the law for equations, the series is 
divisible by 1 - — . But the cosine is also zero for a = -q, hence 
the series being divisible also by 1 + - , is divisible by the product 
cd 
1 — 2 • same way, since the cosine is zero at each odd 
(P cd o? 
quadrant, the terms 1 - , 1 - j 1 ~ so on, are all 
divisors of the series ; whence it is concluded that the cosine is 
the continued product of all these divisors to infinity, or that 
cos a 
cd \ 
TVV’ 
By similar arguments it is shown that 
sin ct = a\ 1 
_^Yi __^Yi _ 
2VA 
a 
Q2g2 
Taking now the logarithm of each side of the equation, and 
using the formula 
/y» /v»0 
vU frO tA/ tA/ 
log (1 -x)= - J 
collecting also the terms containing the like powers of a, we get 
, ^2 f 1 1 1 1 , ) 
logseca= I p + -+^+^ + &c. I . 
-;N 
/ 1 1 1 1 , I 
O-l ■ 2 i + 34 + 5I + 74 + f 
