320 
SIE J. r. W. HEESCHEL OX A THEOEEM OE DE. BEIXEXET. 
stitute and we have 
a4 1” 0^- ^ A' V” 0^ • 
denoting by [^] the continued product 1.2 — w, by [x-\-n], 1 .2.0. ..{x-\-n), and by v 
the combination of operations expressed by that is to say, making 
V0'= &C.) 
o 10 “ 0!/1 5 
2 12 ' 24 
SO that if we put ^=0, we shall also have 
and^’=^ (A.) 
[pc} [ar + ra] [ar] 
t 
This premised, putting for the coefficient of If in j , Dr. Beixklet’s general 
formula for is as follows : — 
. ra + 2.n + 3....ra + a? 1 AO'*'+' , — n + 3....w + a7 1 A^CK"^* „ 
1.2 (a;-l) T’l.2...(a? + 1) +^^-^ + 1 • i ,2 . .. .(a?-2) '172 ‘ 1 .2 (x+2)~'^^^'^ 
or in our abbreviated notation, 
. 1 A0'* + ’ 1 \xA-n\ A^CH"^'^ „ 
■* l.(ra+l) [n— l].[a;— 1] [a?+ 1] ' 1.2.(^ + 2) [ra— l][a? — 2] [a7 + 2] 
which, in consequence of the equation (A.), will be transformed to 
A.= - 
[a; + n] 
[n — 1] . [a:’]'[[a:’ — 1] ’ [?i + 1] 
Now we have 
(l-vr-=l-^V+ <^+’‘»; - t'‘~^V . &c. ; 
and therefore, putting S„ for the sum of the n first terms of this series, 
(-\ S ( 1 ^ ^ T'ji + i ['*^ + ”1 1 v^H+ai C-p 1 
and consequently 
whence it appears that 
A.-( l)”-[„-iV[.i.]-0v) { 
0^ 
Dr. Beinkley’s expression is therefore noAV divested of its form in which successive 
different powers of 0"^ occur, and reduced to one in which, according to the s})irit of the 
algorithm adopted in my system, successive poAvers of A, or of» some functions of A 
* Pliilosophioal Traiisactious, 1807, p. 125. 
(B. 
