by a Method of Differences. 
*53 
Scholium 1. It is evident from Cor. ir. and in. Theorem I. 
that if the coefficients of the sines (a, b, c, &c.) or of the co- 
sines ( a , a', b c', See . ) be such that any order of differences 
taken according to the common method becomes = o, we 
shall then have the corresponding value, of the successive 
values of 5, s', s", Sec. expressed in finite terms, and we shall 
consequently get the value of the series sought expressed in 
finite terms, and likewise all the intermediate values of s', s", 
s'", See. contained between s and the said corresponding suc- 
cessive value of s, expressed in finite terms ; hence if the 
values of a, b, c, &c. or of a, a', b', c', See. be respectively equal 
to ~r S’ — g, r ' &c. r, h, t, being all affirmative 
* 6 t . t + h 3 t.t + b. t+2b ’ ’ ’ 3 
values, and r — t a multiple of h, we may obtain the sum of 
the series. 
In order to prove this, I shall put r, r, r, &c. to represent 
« a * 
r-\-h, r-\-vh, r-j-g/i, &c. t, t, t , Sec. for t-\-h , t-{-2h, t-\-fh, &c. 
« a 3 
then will the increment of 
r r r 
1 2 
r r 
1 
r 
,-f 1 
t t t t 
t E-f It-J-2 £-f ‘ 
£ + "+ I 
r r . 
1 
r 
t t 
e e v 
r 1 
__ v -{* * t -f- * 4- I 
(1/ being supposed the only variable quantity) 
— ; itis like- 
r . r . r .... r 
12 v-J- 1 
t .... 
£+1 
E + V-J- I 
r — t—ih 
t -j- ib 
rr 
1 
t t 
£-f- I E-J-I/-J- I 
wise evident that the v-f-ith term of the series proposed may 
r r r r 
be expressed by — ■ * * ^ * . g, (v being a whole positive 
123 V 
number,) this term we will call T, therefore we have, from 
mdcccvi. X 
