of certain differential Expressions , &c. 227 
in the last case may be found, with nearly equal convenience, by 
either of the methods in the two preceding cases. 
For the sake of conciseness, I employ the symbol d to denote 
the numeral coefficients of the terms arising from the expansion 
of (1 — x) m ; thus, di* signifies m ; d 3 i m , m . m — 1 ; D 3 i m , 
n . m— I . m.(m — i) (m — z). „ . -r- m.(m — i) (m—z) ..m — n+i) 
; D i m , — — - » JT i m signifies,— — — — — ? 
i . 2 ’ c 1.2. 3 C ° ’1.2. 3 ... n 
and consequently, in particular values of m and n, d 3 iI signifies 
■ ?* 1 i signifies - 5 ?* f ~ l signifies - : 
2.4.6 
signifies See. 
Employing, therefore, this notation in the expansion of 
v/(i-e'i s ),weha \edx (—73^) — 2 | if — diI e* x' 
-f-D* lie 4 x* — D s lie 6 x 6 + &c. j, and the (w + i)th term is 
, „ x‘ ln . dx 
Y^ e ' ■ VIT^r 
No W> rf(*-V(i-^))-(?«’0 hence, 
f?ZJ. l - = — ‘ v/(i-r) 4 
J 2» V ' ' ' 
‘v/(i— j;-) — 
271 
zn— 
2 71 . 2)1 — 2 
1 rx zn — z dx 
n J 
x' n ~W(i—x') 
+ 
(2«— i) . f2W— 3) 
?— 3) fx Zn —* dx ' 
27i . {zn— 2 ) ’ J x a ) ’ 
consequently, continuing the reduction, 
(2K-l) 
/"x 2 " dx , , s r x 7 ’ 1 — 1 . 
J 7TT37T)=-* / (i-.r){— + 
„2«-3 
(zw— 1) . (2M — 3) &C 
&c - 5-3-i r 
6.4.2 J s 
zn . (2 w — 2) 
dx / \ 
(*)• 
+ &C.} 
4 . 2 ,y 3/(i-x 2 ) 
Hence, putting for rc the several values o, 1, 2, 3, &c. we have 
— Dll £*{ 
-X3/(I— x 1 ) 
1 } 
+ D- liA ■ LjLa j 
1 c L 4 4.2 ' 2 . 4 r J 
4 
4 . 2 
