156 Mr. Gompertz on Series which may be summed 
We have therefore 
a,b, c, d, e, &c.=i, g, g\ g\ g\ 
&c. 
a, a', b ', c d',&c—i,g—-i,gxg—i,g*xg—i,g 3 xg—i,8ic- 
a, a", b ", c", d",&c.=i,g— 2 , g— 1 *, g .g— ij% g\g— 7|b& c -- 
Consequently, s" or — s . 2 sine of ±qz\*— sine of p"z-{-g — 2 . 
sine of p" - j- q. z-\ - g — 1)*. sine of p"- f- 2 q.z -j- g — 0*.g sine of 
p"-\~3q z &c.=sine of p — q-z-\-g — 2. sine of pz-\-g— 1| 2 . sine of 
p-\-q-Z-\-g — l] 1 .^ sine of p-\-zq .z-^-’g—i} 1 . g* sine of -f-Qq.z 
&c. but, s=sine of pz-\-g sine of p -j- q . z-\-g 2 . sine of p J - 2 q . z 
&c. Consequently, by multiplication, division, and transposi- 
tion, -Tl 2 sine of p-f-q-Z+g — i| 2 -g sine of p-f-zq.z-f-g — i) a g a 
sine of p-\-3q.z &c.==s.€ll H — ~~ • sine of pz , consequently 
the above equation becomes by substitution s" or — ■ 
s. 2. sine of ^%] 2 :==sine of p — q.z-\-g — 2 sine of pz-\- g ~ l 
g 
. s- 
cr j y 
-■ ■ . sine of pz, therefore, 5 the sum required = 
sine of p-q,z-{-g — z—g — fl 1 . sine of pz 
g 
— g sine of p — q-z-\- sine of pz 
— zg cos. of qz 3 
and 
—g—ig —2 sine of {qz[ 
g 
by similar means, we have the sum of the series, cos. of pz 
-f g cos. of p -j- q . z -j- g* cos. of p -}- 2q.z &c. = 
— g-cos. of p — q z+ cos. of pz 
g z + 1 — 2g. cos. of qz 
Scholium in. Hitherto we have been considering, a series 
of sines and cosines, whose terms have all the same signs ; 
but if the terms of a series proposed were alternately positive 
and negative, it would be necessary to divide them into two 
series, the one of the positive term and the other of the nega- 
