154 Mr. Gompertz on Series which may he summed 
what has been just shown, T = r —~ . g . - — , T= 
t 
*+i 
r—t 
~t~ 
r — t—b 
t-\-b 
g* 
t .... t 
2 >+ 2 
•, T= 
r—t r—t—b r—t— ib 
t 
t+b ' t-\-zb 
g x 
t 
t 
3 "+3 
and the eth increment or difference == r ~~ t r ~ l ~ b r ~ t — zl > 
t 
, &c, 
& c. 
x r — £ ~ 1 • b g x - 
t-\-c — i . b 
t 
»+« 
t + b ‘ t+2b 
which, it is evident, will be 
equal to o. If r—t=e — l .h whatever v may be, that is, what- 
ever term of the eth order of difference be sought it will be 
found equal to o ; the truth of this will be likewise evinced 
in particular cases by the following examples. 
Example 4, Required the sum of the infinite series, -I sine 
of pz-\- ~ sine of p-\-q . z-\- sine of p-\-q . z &c. 
Here a, 6, c, d, &c.=-f,A±, &c.'j 
«. d , V , C , & c.= 3> &c. [ 
a,a",b",c",%LC.=Q > , o, 1, 1, &c. ( 
a,a“', b'\ c% &c.=3,— 3, 1, o, &c.J 
therefore 5"' or — 5. 2 sine of ±qz\=, 3 cos. of p^z — 3 cos. of 
p'“-\-q . »+ cos. of p m -\-<zqz s the sum = £3 cos. of p w z — • 
3 cos. of . 2 + cos. p‘" -\-zq . sQ-f- — 2 sine of ^ . z| 3 == 
3 cos. of />— 1«7 z— 3 cos of /> — I9.Z + cos. of p-\- ±q.z 
• — 2 sine • 
jtyote. The series might have been written thus, 3 sine of 
pz-\-6 sine of p-\-q . 2+10 sine of p-\- qz . &c. 
Example 5, Required the sum of the infinite series , j 1 - cos. of 
cos. of n COS. of 
3 4-5 
