by a Method of Differences. 1 63 
"by pure powers, and by similar means we may from the 
equation, cos. of pz—r. cos. of p-\-q . z-\-r. r -~ cos. of p-j-eq • » 
&c. = obtain the series i-r+r. * HT 
2 cos. of iq«\ r 223 
&c.— />•— &c. ~ -H> 4 — r^+gf4-r. 
1 if o 1 2 4 o. 1 P— iqr\ z — 1.2. A s ; 1 , 
-/’ + 2?|‘-&C.|. t — -&C.= p— fj . — + 
r_j_ 
2 
* J± A -P ^ yr l 1+ l t --,'. 34 ' B . — ; and consequently by again 
7* j I 
comparing the homologous terms, we find i—r-j-r.— -r. 
. fii &c.= ~ } as it is well known to be, p'—r . p-\-q\ L -\-r * 
l±ip+ 2q \*— &c.= ■ • A ,/>w./»+gl 4 +y • + 2 rf 
8tc. = - — — 2 ' /r ' a nd so for the other even 
2 r ? 
powers, r being only concerned in these expressions by pure 
powers. 
Hence r being a whole positive number, the sum of the 
series, p”-r.p-\-q\ m + r. ./» + agf— r . . r -~p + 3 q\ m &c , 
m likewise being a whole positive number, may be always 
expressed by ~ x by a series of finite terms of pure powers 
of r whose coefficients are given, of the form a-^br-^cr 1 See. 
p, q, and m being given values, and a , h, c. Sec. determinate 
values independent of r ; merely by comparing the coefficients 
of the homologous powers, of z, in the two equations of the 
series above. Now if we can prove that the same expressions, 
derived from the comparison of the coefficients of the homo- 
logous powers of z, give the sum of the series p”’-—r.p-\-q\ m 
Hh r . — — ,p-\-2q m Sec. whether r be a whole positive number 
Y 9 
