160 Mr. Gompertz on Series which may be summed 
and consequently we shall get new series, the coefficients of 
whose terms are formed from the coefficients in the preceding 
series, by addition instead of subtraction ; and may be of good 
purpose on some occasions. And if we alternately use these 
theorems the operation will be performed by alternately 
taking sums and differences ; and this will amount to the 
same as taking the differences of the alternate terms, begin- 
ning always with two noughts : but, for the more readily 
comprehending this, we shall offer a theorem which more- 
over is the first of these theorems I discovered, but previously 
thereto shall propose 
Example 2. Let the series be either of these, sine of pz-\-r 
sine ot p-\-q.z-\-r.l~ sine of p-\-%q . z-\-r . r -~ . r -~- sine of 
p-\-$q-%-\- &c., cos. of pz-{-r cos. of p-\-q . z-\-r. . cos. of 
p-\-2q.z-\- &c., sine of pz—r sine of p-\-q . z-\-r . sine of 
p-j-2q.z— &c. or, cos. of pz — r cos. of p-\-q.z-\-r. ~~ cos. 
of p+v-q . z-~ See. r being a whole positive number, the terms 
in the two first series all positive, and in the two last alter- 
nately positive and negative. 
The coefficients being, 
the first differences 
r— 1 r—i r r— i r r-{- 1 r— i r r-f i r+z 
1 * 1 * ~T~ • 1 e 2 * 3 5 1*2*3" + 
2d differences 
r . — . — 
2 3 
1 
r — 2 r — 2 r— 1 r—z r- 
z 
2 
t r—z r— 1 
1 
1 
3 
1 
2 
