i6s Mr. Gompertz on Series which may he summed 
Corollary. Because sine of pz—pz— ^ 4- &c. sine 
_____ 1 _ 2-3 1 2-34-5 
P + <h s 
of p -j- qz ~ p-\-q . Z 
P+*q\ 
2-3 
•s 3 + 
2-3-4-5 
z 5 , &c. sine of p-{- 2 q.z 
--P + 2CJ. 
,3 + ■ s 5 , &c. &c., and 
’ ^ “ r ’ COS. ot 
, tV<7 + * 4 
1.2 
4 
1.2. 3.4 
;i c.[ 
2.3. 4.5 
1 + As 1 + B s 4 + Cs 8 , &C. 
^ here A, B, C, Sic. stand for the coefficients of the multi- 
nomial, 1 — + ~~ &c. raised to the — r power, and 
consequently r only concerned in them by pure powers ; 
hence this being multiplied byp — -kqr.z — 
&c. the value of sine ofp — \qr. z, we obtain from the equation 
•sine of pz — r. sine of p-\-q . Z-\-r . -j-1 sine of p-j-sry . %, Sic. 
= — e ° fy '~ ' /? ‘ " , the series^ — r.p-\-q-\-r . — — .p-f eq — &c.\ z 
2 COS. of l^jr r r 1 i 1 2 2 * i 1 
— p 3 — r’p+^-K. h±±p + ^j\ 3 — &C.]. — - + f— r.p + qf+r 
■ -77+^ A- •/>—&'-]• ~- s See., the 
law of continuation being evident in both series, consequently 
by comparing the homologous terms we obtain the sum of 
the series, p—r. P+q+r/-^- .p+ 2 q-r. r -±t r ±i.p+ 2q &c. 
r-f 1 r-4 2 
2 3 
3 
P -~, of p 3 —r . p+q] 3 -f r . — . p-fzq} Sic. — 
— -—r . p — ±qr, of the series p 5 - — r. p-H/f+7*- — -p-j-2^ s — &c. 
="£31 — . p — ^qi) 3 - f- -~ 3 -; 4 r — - p — hqr, and so for the 
other odd powers, r being only concerned in these expressions 
