Mr. Gompertz on Series which may he summed 
*x-\-c cos. ofp-\-Qq.x-}- &c.=2B cos. of p — ^ x ; half the dif- 
ference of these two series gives, 
a cos. of^\r-f& cos. of p-\-q.x-\-c cos. of p-\-zq. x -f- &c.=B 
cos. of p—^.x — A sine of^> — expressed generally and 
distinctly in terms of p,q, and x ; and consequently by writing 
z for x throughout, we have the sum of the series, a cos. of 
pz-\-b cos. of p-f-q.z-{-c cos. of p-f-zq.z^ &c. expressed ge- 
nerally and distinctly in terms z,p, and q. Q. E. D. 
Cor. i. It is evident that p and q may be taken any numbers 
either positive or negative, but v. ought not to be equal to o, 
for we could not then effect the substitution %= — . 
Cor. ii. Putting, a cos. of pz-j-b cos. of p^q.z &c.=P, and 
a sine of pz-\-b sine of p-^-q.z & c.— Q, and also B' and A' for 
the values that B and A become, by writing z for x in those 
values, that is, z for-^ in the given expressions B and A we 
shall have P=B' cos. of p — ^ z—A' sine of p — ^ . z } and Q 
=B' sine of p — ~ . z-f-A 1 cos. of p — 
Cor. hi. Hence we may again prove, that if we have the 
sum of the series, a sine of pz^-b sine of p-\-q.z-\- &c. ex- 
pressed generally in terms of p, q, and z , we may find the 
series, a cos. of pz-\-b. cos. of p-\-q.z-\~ &c. expressed gene- 
rally in terms of p, q, and z, and the contrary. For having 
the sum of the first by writing tt for p, y. for q , we shall have 
the sum of the series, a sine of nz-f-b sine of n-\-K.z-\- &c. 
==A, expressed by z, and particular values ; in which writing 
~ for z, we get A', therefore having A' and Q, we may, by 
