184 Mr. Gompertz on Series which may be summed 
sider the subject more minutely, to which purpose Scholium ir. 
at the end of Theorem IV. will afford great assistance : from 
that it appears, that the sum of the r first terms of the series 
cos. of nz -f- cos. of n -f- q . z -f~ cos. of n zq . z See. = 
sine oC n-\-r—\q.z— sine of n—\qz j u • -i .1 . ^ 
22 — — — • and by similar means that the 
2 sine or ’ J 
sum of the r first terms of the series sine of nz- j- sine of 
n-\-q.z-\- sine of n-\-2,qz-{- &c 
— cos. of n + r — \q z-f cos. of n—\q 
2 sine or \qz 
now it is plain that if qz were either equal to o or a multiple 
of 360°, sine of \qz would be equal to o, and because r is a 
whole number, rqz would either be equal to o or a multiple of 
360°, and consequently the sine of n-\-r — \.q.z— sine of 
n — -q.z and the cosine of n-\-r — ±.q.z= cos. of n — q.z , and 
therefore the sum of the series, cos. of nz-\- cos. of n + q.z 
&c. = ( when qz=o or some multiple of 360°) - - - - 
sine of «— \q z~ sine of n— o j r • r i c 
— 21 — CL- — — and of sine of nz 4 - sine of 
o o’ 1 
n-\-q.z &c.=§ whatever r may, whether finite or infinite. 
Indeed the determining the value of §, depends on the value 
of r; but if qz be any thing but o or a multiple of 360°, the 
value of the sine or cosine of n-\-r — \ q.z will depend on the 
value of r, and may then be varied from positive to negative 
and from negative to positive, by merely increasing r, and 
consequently when r is infinite, there being no reason for its 
being positive rather than negative, or negative rather than 
positive it should be considered o ; and therefore the sum of 
the infinite series, cos. of nz- f- cos. of n-\-q.z &c. = — 
■ -7 - 1 — • anc! of sine of nz 4 - sine of n 4 - q .z etc. =ss 
sine or 1 1 * 
