1887.] its application to the solution of some equations. 105 
which, the terms 2n7, (2n+2)7, &c., give the same values for 
sine and cosine which are given by 0, 2a, 47, &c. Thus we 
obtain, as available for p, the values 0, z ae a , &e., a8 far as 
jf Hp) 
Die 
(i 2) . And we have, for values of «, 
cos (0), 
2 ee 
eae me Bee cel 
n n 
Airy — . 4a 
cos ae nik, an —, 
n n 
&c., as far as 
cos (Bae ay (ey gin CUT) 
nN 1 
But the condition sin (np) =0 is satisfied equally well by negative 
; : 2 
values of 27, 40, &c.; which will not alter the terms cos a 
4, : : Me é 
cos ae &c., but will change the signs of sin =, sin a. &e., and 
thus give for values of a, 
Qa — . I 
eds = = 21 sim = F 
n n 
Agr —. . 4 
cos — Seine) 
n n 
&e., as far as 
cos ens Be zis Jol. sin Pea) ; 
n n 
In order to form the series of terms whose continued product 
will be equal to the given quantity «” — 1, we must connect, 
negatively, each of these values of # with the symbol #, and multiply 
all together. The multiplicands, thus formed, are’ 
a—1, 
7 ere Ne 
1 — (cos T+ /=1.sin—), 
n n 
e225 Hh 
L— (cos +4/=T. sin ="), 
n n 
