by a Method of Differences . 
cos- of n—\q.x 
2 sine of qz 
in every case, ( the same as in Example to Theorem 
I. ) except when qz~ o or some multiple of 360° ; on account 
of there being something else to be taken into consideration, 
in that case. Again, it appears by Example 2 of the said 
scholium, that the sum of the r first terms of the series, 
cos. of nz — -cos. of n-\-q.z-{- cos. of n-\-<zq ,% — - &c. = 
taken according as r is odd or even. And by similar means 
the sum of the r first terms of the series, sine of nz— sine of 
n-\-q • sine of n-\-zq . z— sine of n-\-^q.z &c. is found = 
odd but the under if even. Here if qz— 180°, or any odd 
multiple thereof, the cosine of \qz will be = o ; and if r be 
even at the same time the cos. of n-\-r—~.q.z will be equal to 
the cosine of n — %-q.z and the sine of nfr—fq.z~ sine of 
n ~ §• qz ; hut if r be odd we shall have the cos. of n -r — q.z 
— — COS. of n—^q.z and sine of n+r—±q.z= — the sine 
of r—~q.z; consequently by substituting these values in the 
above expressions for the sum, due regard being had to 
the signs, we shall find that, whatever r be, the sum of either 
series will be expressed by § : but if % be any other value it 
appears that ± the sine or cosine of n +r—\.q.% depends on 
the value of r, and may be either positive or negative, by 
varying r; and consequently should as above when r is infi- 
nite be considered ~ o. And the sum of the series, sine of 
B b 
cos. of n — \ q.zdb cos. of « + r 
2 cos. of iqz 
■, the upper or under sine to be 
•, the upper sign to be taken if r is 
2 cos. of 
MDCCCVI, 
