i8s Mr. Gompertz on Series which may he summed 
n q . y See 
+ 
sine of «-|-r — \ . q . x — y — sine of n — \q . x — y 
4 sine of \q . x — y 
sine of w 4. r — . x -f v — sine of«-k j + r 
— ^ ~ J — - — — ; and the sum of the 
4 sine of \q . x -J- y 
r first terms of the series sine of nx . sine of ny + sine of n + q 
,x. sine of n + q-y -f- sine of n -f- 2q. x . sine of n 2q.y Sec. 
sine of n-\~r —\q .x—y — sine of w — \q . x—y . r ' T 11 
= — — — 77-= Lsme of n + r— - \. q 
4 sine of \q x — y * i. 
. x + y — sine of n — \q .x + y] 4 sine of ±q . x -f y. 
Cor. If rx and ry be both multiples of the whole circumfe- 
rence the said two values will be equal to o. 
Example 2. Because (from Cor. 1. Example 2. Theorem II.) 
we have sine of pz — r sine of p-\- q . z -f- r . sine of p -f- 2q 
. z See. = ~ -- f p ~n- qr ' * . we have by this theorem case 1. cos#' 
2 cos. of \qz [>• 
of px . sine of py — r cos. of p -{• q . x . sine of p -f- q ■ y + r. 
cos. of p -f 2 q.x . sine of p 2q . y cec. = v_l . _ 
2 . 2 cos. of £q-x+yl r 
! I 2 1 0 JL P =J± r And because by the same cos. of pz — r 
2 . 2 cos. of . x— 
. of p 4- q . z 4- r . cos . of p-\-2q . % &c. = .fczi SL^L 
* ' f 1 2 2 cos. of \qz\ 
we have by case 2 of this theorem cos. of px . cos. of py — r 
f±l 
2 
cos 
cos. of p -J- q.x. cos. of p q.y -j- r cos. of ^ -j- sq . x . 
t:os. of p — \qr . x—y _j_ cos, of p — \ qr . x -f y 
2 . 2 cos. of \q.lc~y\ 2 . 2 cos . of \q . x+y\ 
cos. of p-\-2q.y Sec. 
and sine of px . sine of py —r sine of p + q.x. sine of p + q 
1 o„ cos. of J~— \qr . x — y cos, of p — lq r ■ x + y 
O "* 1™ ' V^wCo _ ' ~ | y L m ' '' ■ ' ” If* 
2 . 2 cos. of \q . x—y\ 2 . 2 cos. of jq . .s-Fj’l 
