by a Method of Differences. 281 
b sine of qy &c, ; we should have had the series reduced to 
the form a' cos. of py -f b’ cos. of qy &c. which is of the 2d 
form of the theorem, and consequently from it is deduced the 
sum of each of the series, 1st. a' cos. of pw . cos. of pv -J- b' cos. 
of qzv . cos. of qv &c. that is, of the series a cos. of pw . cos. 
of pv . sine of px -|- b cos. of qw . cos. of qv . sine of qx & c. in 
terms of v, w, and x, which is indeed similar in form to the 
series found by the other substitution ; and 2d. the sum of the 
series a! sine of pw , sine of pv -f- b’ sine of qw . sine of qv &c. 
or its equal the sum of the series a sine of pw . sine of px . sine 
of pv -|— b sine of qzv . sine of qx . sine of qv -f- &c. in terms of 
v, w, and x. And in a similar manner, from the sum of the 
series a cos. of pz -\-b cos. of qz &c. having found the sum of 
the series a cos. of px . cos. of py + b cos. of qx . cos. of qy &c. 
we may find the sum of the series a cos. of pw . cos. of pv . cos . 
of px + b cos. of qw . cos. of qv . cos. of qx -f- &c. in terms of w, 
v, and x, and likewise the sum of the series a cos. of pzv . sine 
of pv . sine of px b cos. of qzv . sine of qv . sine of qx &c. 
And in a similar manner also may w r e proceed by degrees to 
more complicated cases. 
Example 1. Because (from Example 1. Scholium 2. after 
Example 3. Theorem IV. ) we have the sum of the r first 
terms of the series, cos. of nz -j- cos. of n-\-q.z - f- cos. of 
n -j- 2 q . z &c. = [sine of n -J- r — ~.q . z — sine of n — -~q . sfj 
; 2 sine of ^qz : if x — y and x y be written for z, then 
the half sum and half difference of the resulting expressions, 
by case 2 of this theorem, will give the r first terms of the 
series cos. of nx . cos. of ny -j- cos. of n + q . x . cos. of 
