VOL. XLVII.] PHILOSOPHICAL TRANSACTIONS. l^Q 



Put P = ^+"- + ^+£-&C. 

 a ' c ' a 



m , n p , g s 



a' ' 6" ' e" ' <? 

 &C. &C. 



Then it will be 



^ = ,r X (p — Qx 4- ^ — s^ + Tz* — vx* &c.) 



Assume Z\ =. a -\- bx -\- cx'^ + ^^ + Ear'* &c. let this value, with that of ^, 

 be substituted in the last equation : whence, by comparing the homologous 

 terms, there will come out 



B =: PA 



PB — QA 



2 



PC — QB — RA 



c = 



D = 

 E = 

 P = 



6 "^ ^^ a tH ,• ajiiic. 



&c. ..ti,S <:./-._ ...'J 



(Where the law of continuation is manifest, and where it is also evident, that, 

 the value of A, the first term of the required series, must be a unit, because 

 when X = 0, then the given expression becomes l"" X 1" X 1' = 1. q. e. i. 



Carol. 1. — If a be taken = 1, and n, p, q, &c. each = O; then will p = m 

 a = m, R= m, &c. And therefore 

 A = ] ; B = m; 2c = mB — mx; 

 3d = rac — OTB + mA := mc — 2c ; 



4e = TOD — TOC -j- TOB — TOA := TOD — 3d 



&C. 



, m.m — 1 c X m — 2 m.m — l.m— 2 Exm— T 



Consequently c = — - — , d = = — , e = -±iL^_i_. 



2.3 



l.m— 2.m — 3o 'hUh. itir 



— &C. 1 



2.3.4. 

 TT iU- 1 I 1 "t.m — 1 a , m.m — l.m— 2 ,„ 



Hence, in this case, 1 + mx ■] — x^ -\ — *'&c. (= a +bx -f 



car^'&c.) = (1 -\- x)": which series is the same with that given by Sir Isaac 

 Newton. 



Carol. 1. — If a be taken = -, p = -, y =i -, &c. and z = i ; then will the 

 proposed expression be transformed to ■ i 



(1 + ^rx (1 +^)" x(i + j)^x (i + ^)'&c. ^j 



VOL. X. S 



