158 Mr. Gompertz on Series which may he summed 
And if the series be, 
a cos. of pz—b cos. of p-\-q . z-f " c cos - °f p-\~ 2C t • z &c ,=s } 
then shall, 
a cos. of p'z—al cos. of p'-\-q .z-\~b r cos. o f p'-\-2q.z &c.—s' 
a cos. of p n z~— a” cos. of p"-{-q.z-{-b" cos. ofp"-\-2q.z &c .==s" 
&c. & c. &c. &c. 
a\ a"> a'", &c. b', b'\ b"\ &c. c\ c ", c"', &c. &c. being formed 
from a , 6 , c, d, e, &c. as in Theorem I. p\ p" ', p'" , &c. likewise 
as in Theorem I. s'=2.? . cos. of . \qz, s"=2s'. cos. of .\qz, s'" 
=2 s". cos. of ±qz, &c. 
First, if a sine of pz — b sine of p-\-q .z-\-c sine ofp-\-q.z &c. 
= by multiplying by 2 cos. of \qz, by lemma No. II. we 
shall have a. sine of p — ±q.z-{-a . sine of p-\-^q . z — b . sine of 
p-\-\ q-z-~b. sine of p-\-j-q .z-\-c .sine of p-\-±q .z &c.=s. 2 cos. 
of \qz; consequently, putting b — a—a\ c — b—b', &c. p — \q 
=/>', s'— 2 s cos. of \qz, we have, a sine of p'z — a! sine of 
p'-f-q . z-\-b' sine of p ,J f-2q . z &c.=/, which being exactly si- 
milar in form to the original series, the other series will be de- 
duced from this by continually proceeding in the same method. 
Again, if a . cos. of pz — b. cos. of p-\-q .z-\-c. cos. of p-\-2q.z 
&cc.=s, we have by multiplying by 2 cos. of \qz by the help 
of lemma No. III., a cos. of p'z — a' cos. of p'-\- q. z -f b' cos. of 
p'-\~2q.z 8 cc.=s, which being exactly similar in form, to the 
original, we may obtain the other series, which are likewise 
similar in form by the same mode of proceeding. 
Cor. the 7 rth successive value of s=s.2 cos. of \qz}\, the 7 rth 
. , n . . 1 1 wfh successive value of s 
successive value of p=p — 7 r .j and — • 
2 cos.oi 
Example 1, Required the sum of the series, sine of pz — • 
