m 
by a Method of Differences. 
help of Cor. ». find P in terms of p, q, and *, and particular 
values, namely, the sum of the series a cos. of pz-\-b cos. of 
p-\-q.z &c. and in a similar manner the contrary is proved. 
Q. E. D. 
Theorem VII. 
If we have the sum of the series, a sine of pz-\-b sine of 
qz-\-c sine of rz-\- Sec. expressed generally by z; we have 
likewise the sum of the series, a cos. of px . sine of py-\-b cos. 
of qx . sine of qy-\-c cos. of rx sine of ry &c. expressed gene- 
rally by x and y. And if we have the sum of the series, a 
cos. of pz-\-b cos. of qz-\-c cos. of rz See. expressed generally 
by z; we have likewise the sum of the series, a cos. ofpx . 
cos. of py+b cos. of qx . cos. of qy-\-c cos. of rx . cos. of ry 
Sec. ; and also the sum of the series, a sine of px . sine of py 
-J-6 sine of qx . sine of qy-\-c sine of rx sine of ry Sec. ex- 
pressed generally in terms of x andy. 
First ; if we have the sum of the series, a sine of pz-\-b 
sine of qz Sec. expressed in terms of z, by writing x-\-y in the 
room of z throughout, we shall have the sum of the series, 
a sine of p.x-\-y-\*b sine of q.x-\-y-\-c sine of r.x-\-y Sec. ex- 
pressed in terms of x and y and in like manner by writing 
x — y for % we shall have the sum of the series, a sine of 
p.x — y-\-b sine of q.x — -y-\-c sine of r.x — y Sec . expressed in 
terms of x and y, therefore the half difference of these two, 
that is ci s ' ne s " me P’ x — y }) sme Q' x ~t~y — sine — v 
2 2 
, sine of r.x-\-y— sine of r.x — y 0 ., . . , , T ir 
. — — — — - &c. or its equal by lemma No. II, 
a cos. of px. sine of py-\-b cos . of qx. sine of qy-j-c cos. of rx. 
A a 2 
