l j 6 Mr. Gompertz on Series which may be summed 
steps backwards, and substituting pz — 90° for pz throughout 
we get the second part Q. E. D. 
This theorem evidently supposes that the functional values 
of pz are distinct in the general expression for the sum of the 
series, before the substitution takes place, which may not be 
the case if p has any particular value, or even if p y q, and z 
have any relation to each other. 
Theorem VI. 
Given the sum of the series, a sine of 7 r2-f .6 sine of 
+c sine of *--1*2 x.z-\-d sine of *-4-3*. z-|- &c. and likewise of 
a cos. of irz-^-b cos. of *--[ -x.z-\-c cos. of 7 r-^- 2 x.z-j- & c. ex- 
pressed generally and distinctly in terms of z for any parti- 
cular values of *- and x, except x—o, *• and x having the same 
value in both series ; there will likewise be given the sum of 
the series, a sine of pz-\-b sine of p-{-q.z-\-c sine of p-\-2q.z 
&c. and likewise of, a cos. of pz-\-b cos. of p-j-q.z-j-c cos. of 
p-j~2q.z See. generally and distinctly in terms ofp, q, and z. 
For, calling the first series A and the second B, and put- 
ting z— we have by substitution, 
a sine of^ jc-f 6.sine of ~ -\-q.x-\-c . sine of -^-f 2*7. x -\-d. sine 
of ~ -f 3*7 . JC-fi &c. = A, and 
a cos. of ~JC-f6.cos.of~ -f ^.jc-f c.cos. of ^ -j-2 q . jc-f d . cos. 
of — -f 3 q ■ *v+ &c. = B. 
A and B being now expressed in general terms of q and x , 
and particular values ; multiply the first by, 2 cos. of p — ■ . jc 
