i$o Mr. Gompertz on Series which may he summed 
p" and s" for p and s,) it follows that if we put b" — a"— a'", 
c"—b''=b'", d" — c"=:c"',&.c. p n — %q=p J ",s". 2 sine of \ qz~s 
that we shall have, a cos. of p"'z-\-a"' cos. of p ,,, -\-q-Z’\- b"', 
cos. oi p'" ~\- c iq. z Scc.zz=s'", which is exactly similar to the se- 
cond equation ; (if a'", b"', c"\ &c. p'" and s'" be written for 
a' ,b', c' , &c.^> 7 and s' in that,) and therefore putting b"' — a"'=a lv , 
c ,n — b'"=b iv ,d'" — c'"=c iy Sec.p"'—±q=p iy , — s'". 2 sine of \qz 
=-s xy , we get a sine of p iy z-\-a iy sine of p lY -{-q ■ z-{-b iy . sine of 
p lv -\-zq . z, Sec.=s iy , again, similar to the first, by putting a iy , 
b iy , c iv , &c. p iv , s lv in that equation for 6, c , d, &c. p and s, and 
thus do we continually get equations in form similar to the 
first and second equations QED. 
Cor. 1. s'*— — s'. 2 sine of ^qz= — s . 2 sine of \qz^, s"'=s''. 
2 sine of \qz — — s' . 2 sine of ±qz? = — s . 2 sine of ±qz\\ 
s iy = —s'"-. 2 sine of ±qz=s' . 2 sine of i qz\ 3 —s . 2 sine of ^qz^, 
and in general put $(*■) to represent the 7rth successive value 
of s, and we shall have $(«■)= + s 7 . 2" sine of \ qz ^ ±s . 
2 sine of ±qz\ v , the upper sign to be taken when % being di- 
vided by 4 leaves o or 1, the under when it leaves 2 or 3. 
Tz-th successive value of p—p—v-iq, note the values s', s", s'", 
§ 
&c. I call successive sums of .9, and 5 = + — — — = 4- 
2 sine of 
(■«r— l) 
s 
2 sine of \qz ^ 1 
Corollory n. If A, B, C, &c. A 7 , B 7 , C 7 , &c. A 7 ', B 77 , C 7/ , See. 
Sec. be put for the series of the ist, 2d, 3d differences See. of 
the series a, b, c, Sec. taken according to the common method 
of differences, we shall have the series 
