i68 Mr. Gompertz on Series which may he summed 
for t and s, we shall have the results determined above. 
IVTany more corollaries may be derived from these. 
Theorem III. 
If there be formed a series of terms 
a, 
b, 
c , d, 
e > fy 
&c. 
a, 
b, 
a', b'. 
d, d\ 
&c. 
a, 
b, 
a", b". 
c'\ d", 
& c. 
a. 
b, 
a'", b"', 
c'", d"'. 
&c. 
&c, &c, &c, &c, &c, &c, &c. 
The terms of each series being formed from those immedi- 
ately above, by taking the alternate differences of the terms, 
always beginning with o, o; that is, taking o from the ist 
term, o from the 2d term, 1st term from 3d term, 2d term 
from 4th term, &c. in any of the series, for 1st, 2d, 3d, &c. 
terms of the next series. And^' be put — p — q , p"—p' — q> 
p'"— p' ! — q, &c. s’— s . 2 sine of qz, s"= — s'. 2 sine of qz, 
s'" — s" . 2 sine of qz , s iv = — s'" . 2 sine of qz, &c. I say if 
there be a series a sine of pz 4 - h sine of p -f- q . z -f- c sine of 
p -j- zq . z -f d sine of p + $q . z, & c. = s, we shall have, 
a cos. of p' z-j-b cos. of p' -\-q.z a' cos. of p' + 2 q.z -f- b ' cos. 
of/» + 3q.z, &c. = s' 
it sine of p"z -j- b sine of p" q.z-\- a " sine of p"-f- 2 q.z b " sine 
ofp-\- <$q.z, &c. = s", &c. 
For multiplying the first of these by 2 sine of qz, by help 
of lemma No. I. we shall have, a . cos. of p — q . z — a. cos. of 
P + q.z -f b . cos. of pz — b . cos. of p -j- zq . % -f c cos. of p + q 
,z— -ccos. of p -j- $q . z -f- d cos. of p + 2q.z~-dcos. of p-\-^q 
