by a Method of Differences. 151 
a , a\ b c', &c. the same as the series a, A, B, C, D, &c. 
a, a", b", c", dec. a, a". A', B', C', &c. 
a, a"', b"', c"', &c. a, a'", b"', A", B", &c. 
a, <z iv , # v , c iv , &c. a, a iv , 6 iv , c iv , A!", &c. 
&c. &c. &c. &c. &c. &c. &c. &c. &c. &c. &c. 
This is evident by taking the differences by both methods,, 
and comparing them. 
Cor. hi. Likewise if A, B, C, &c. A', B', C, &c. A", B", 
C", &c. &c. be put for the series of the 1st, 2d, 3d, &c. dif- 
ferences of the series a, a 1 , b', c\ &c. found by the common 
method of differences, then shall the series 
a, 
a", b". 
c". 
d"y 
&c. 
= a, 
A, 
B, 
c. 
& c. 
a , 
d\ 
c", 
d'"y 
&c. 
= a, 
in 
a , 
A', 
B', 
&c. 
a, 
bf 
c iy > 
d iy , 
&c. 
— 
a iv , 
b". 
A", 
&c. 
dec. 
dec. dec. 
de c. 
& c. 
&c. 
&c. 
&c. 
dec. 
dec. 
&c. 
These things being known, we shall now propose some 
examples of their use. 
Example 1. Required the sum of the infinite series sine of 
pz-{- sine of p-\-q.z-{- sine of/>-J-2^-f sine of f-fgq.z Sec. 
Here a, b, c, &c. = 1, 1, 1, 1, 1, &c. ".therefore f or ^2 sine of 
a,a',b',& c.=i, o, o, o, o, &c. Jj-qz==cos. ofp'z=cos. of 
p — ^qz the sum s— cos ; of ^ \f z 
1 2 1 2 sine of \qz 
Cor. 1. If p and q were each =1, we should have, sine of 
£-j- sine of 2%-f sine of 3% &c.== cotangent of \z. 
Cor. 11. If^> were — \q, we should have sine of pz-\- sine of 
SP Z + sine of &c. 
cos. of p — p.Z 
2 sine of pz 
7 = 7 cosecant 
2 sine of pz 2 
