210 
MR. MURPHY ON THE THEORY OP ANALYTICAL OPERATIONS. 
+ 1) (n + 2 ) / dy \~ n ~ 3 / d 2 y \3 
a 4 2.3 \d x) ’ \2 dx 2 ) 
■ ” + 1 / dy\~ n ~ 2 ( cPy \ ( d 3 y \ ^ / dy\- n ~' L <Py 
' 2 \dx) ’ ‘ \2 dx 2 ) \2 ,3dx 3 ) \dx) 'dx' 1 ’^. 
In general let 
(dry 1 , d 3 y li , d x y h % , 0 \ m _ , 7 , 
XdF'F + d^-^TS + 3 ? *074 +’ &C 7 — 
Then 
__ Q + 1) (n + 2) , . . (n + ot — 2) /dy\~ n ~ m + l 
' ' a >» 2 . 3 . . . [m — 1) \dx) ' y 
m— 1,0 
(?? + 1) (n + 2) . . . (n + to — 3) / “ w “ w + 2 
2. 3... (to — 2) ' \dx) ' ^ m- 2,1 
, (« + 1) (» + 2) . . . ( n + to — 4) (dy\ - w -w+3 
+ S.3...( M -3) • \£) -y«_ M 
+ (g) 
— n— 1 
* *^1, m—2’ 
m being > 1. 
Corollary. Put u — x, then 
l^ = n{n- l)(»-2)...l# n ., 
where 
xre _l _ (ft + 1) (ft + 2) . . . ( 2rc — 2) /rfy\ ~ 2w + 1 
* Xtx) 
2.3... w — 1 
Vn- 1,0 
(» + 1) (ft + 2) ... (2 ft — 3) 
2.3... (»-2) 
-2 ti+2 
Thus 
■3>.-%v &c - 
_ („ _ 1) (n + 1) . . . (2« - 3) . (g) 2 
+ ( b - 2) (» - 1) . . . ( 2 » - 4) (off -y„-z ,>• &c - } • (jf) 2 ”- 
*£ = f*y 7l I (*s\~‘ — (iiY 
dy dx *^°»° j ’ \dx) \dx) 
(J Y- f 2 ^ v 1 (*yY*—( dy Y s d * y 
dy*~ Y dx'y^]-\dx) -\dx) 'dF 
$={3.4.g^o-2.3.(^y., 1>I }(g) 
= 3.(g)‘ 5 . 
1 
sTi- 
&c. 
