by a Method of Differences , 
173 
. . . • n i , sine of « 4 -r— i.q.z 
writing n-\-rq m the room of n, is equal to 2 - s v ne ZfnT~ > 
consequently the sum of the r first terms = 
sine of M + r — \q z — sine of n—\q z 
2 sine of \qz 
Cor . 1. If n=q= 1, and rz the whole circumference of the 
circle, we shall have cos. of z-{- cos. of 2 z-\- cos. of 3% 
+ cos. of rz= 
sine of 360 °-\-n—\q.z — sine of n — \q.z 
2 sine of \qz 
o, a theorem 
said to be used by Le Gendre in his inscription of a polygon 
of 17 sides ; and if we have rqz— to the whole circumference, 
we likewise have in general cos. of nz-\- cos. of n-\-q.z 
+ cos. of n-\-r— 1 .q.z— o, and if n=±q, we have in general 
cos. of nz- 1- cos. of $nz-\- cos. °f 5 nZmi r &c. -• cos. °f 
2r— 1 . nz— 
sine of 2 rnz 
2 sine of nz 
Example 2, Required the sum of the series, cos. of nz- 
cos. of n-\-q.z-\- cos. of n-\-<2.q.z + cos of n-\-r— 1 .q.z 
the upper sign to be taken if r be odd, and the under sign if 
even. 
The series is evidently the difference between the series 
cos. of nz — cos. of n-\-q.z-\- &c. ad infinitum and + cos. of 
n-\-rq.z + cos. of n-\-r-\-i .q.z &c. ad infinitum, by proper 
substitution in Example 1, Theorem II. we have their respective 
sums and - + 
2 cos. of ±qz 1 2 cos. of \qz 
, and the difference = 
cos. of n — \q.z± cos. of n-{-r — \q-z , ■, , i . c 
cos ' " of ' \qz — ” — “ — » the u PP er S1 g n to fi e taken if r 
be odd and the under if even. 
Example^ > Required the sum of the r first terms of the series, 
