EQUATION AND ITS TRANSFORMATIONS. 
777 
The coefficient of h l+l in {^(x 2 -\-xh) — x} n therefore 
i + 3).. .(2- 
(7 + 1 — n)\ 
_ 1_ / y+i— w n ('i ~t 2)(7 + 3)... (27 +1 — n ) _1 
2 “ ' ' 
^i+1—— ii 
( -ir- (i + 2) ;i .(2»+i -«) 
^ ^ (& + !—%)! 
and the coefficient of h l+l in {^/(x 3 -}- -xh)-\-x}“ 
_ 2v ft(w-7-2)(ft-7-3) ■ . . + -27-1) 1 _ (-If (i + 2-n) . . . (27 + 1-+) .. ^ 
(*+!)! 
/^i+l/^i+1 /p+L-^i+1 
(* + !)! 
12. The coefficient of h i+l in that is in 
l+<x{ \/ (x 3 +x/i)— x] +;7f{ v/(x 3 +a+) —xj 2 +:q{ ^/(pt?- 3 rxh)— x} 3 +&c., 
is, by the last article, equal to 
+U L ** (»+ 3 >, : --.(«-!) 2 . 2% * 
4 !+ V +1 |_ 4! 2! 
(*—1>* 
,ffi 3 (* + 2)...(2t-3) 
‘3! (7 — 2)! 
yi + 1 
+(-y- 
1 V l o 
(7+1)! 
1,^.7+1 
for, when n is greater than 7+1, there is no term involving h l+l . 
This expression 
, ... (7 + 2) ... 27 1 f i a? i(i— 1 ) „ „ , .. a) 
= (~) i a ~ -UU-H 1—a—2x+ - 0 . 0 . 2 2 x 2 . .. + ( —V -7 
' 2 4\t! x l 1 2! 2i(2i— 1) _ v ' %\ i 
= \-.{ 1 — \ ax+ 7 
a;' % 
4*.7! a; 1 
7(7 — 1) «. 2 + 
7! 27(27-1)... (7+1) 
2+ 1, 
j ! N; 7(7—1) . . . (7 — (7 — 1)} <++ 
7(7- + 2! — . . . {7—i(7 —1)} Tl J ’ 
for the constant multiplier 
. / _ \i\ a ^ + 2 )---^ = / _ y_(201_ a=={ _ y 1-3-5 • • • (27— 1) 
A 12 4\7! ' ' (2.4.6... 27) 2 (27+2) v ' 2/ R 1 ON 5 
2.4.6 . . . (27+2) 
which is the quantity denoted by \ in art. 10. 
The coefficient of h l+l in the expansion of e a - v .e"i u^+.rn-.q j s therefore =AP'. 
13. The coefficient of h 1+l in e c '{' ,{j: ‘ l+xJl)+x \ that is in 
1 + a{ v /(x 2 +xA)+x} + -{ y 7 (x 3 +x/?,) + x} 2 + —{ v /(x 2 +x/i) + x} 3 + &c., 
is, by art. 10, equal to 
0 H 
MDCCCLXXXI. 
