by a Method of Differences, 163 
sequently will the sum of the series 
DDL .p+^\ m+l &c. be — the expression — x.pA-\-pBr-\-pCr % 
&c. — qA'r — qB'r 2 See. of finite terms. This being proved, it 
follows, because the sum of the series p m — r .p-{-q\ K -\-r . 
pj^veff See. when m is equal to o, is equal to i-r-j- r . -y- 
Scc.=, by the binomial theorem, A } whatever r may be, that 
the sum of the series p — r ■p+q+ r - L r L I ) -D < 2 q &c. namely, the 
said series when m is equal to 1 , may be expressed by A x a 
series of pure powers of r of a finite number of terms what- 
ever r may be, and comes out by the bye A x/> — \qr, the 
same as above, and consequently by writing 1, 2, 3, 4, 5, 6, 
See. one after the other for, m, we shall find that the sum of 
the series p m — r.p~~Dq\ m -\~r. ’-t~ p-\-2q{ n Si c. may always be ex- 
pressed by A multiplied by a series of finite terms in the form 
A-fBr-fO* &c. A, B, C, See. p, q, m, See. being independent 
of r; and m a whole positive number. And these will, we shall 
prove without running through all the infinite cases, be the 
very same expressions as those given above, by comparing 
the coefficients of the homologous powers of %. In order to 
this we observe, since we have just proved that the sum of 
the said series, whatever r may be, may be expressed by A x 
series A-f-Br-J-O* & c. of a finite number of terms, and from 
the comparison of the homologous powers, that when r is a 
whole number it may be expressed by A x value a-]-br-\- cr* 
&c. of a finite number of terms, it follows that when r is any 
