MR. MURPHY ON THE THEORY OF ANALYTICAL OPERATIONS. 
209 
jn —2 
d u 
d u 
+ n(n- \){n- 2 ) ...1 
k n 
in the product of both which series the coefficient of being sought will give the re- 
<Tu 
quired value of and this is manifestly of the form 
. dJ 1 u . d n ~ l u . d n ~ 2 u du 
Ai • ~~r n "T* Ao ~ ~ 7 Ao • ” 9 • • • + A . j 
1 dx 1 2 dx n ~ l 3 dx n ~ 2 1 n dx 
k n k n 
A 2 is the coefficient of in n h n ~ l S, or of ^ in n S 
k n lc 11 
A 2 of in n {n — 1) h n ~ 2 S, or of^^y in n (n — 1) . S, &c. 
Now if we observe that Y contains h as a factor it follows that the coefficient of k n 
in the series S is of the form 
*" + h n ~ l + h n ~ 2 + 
and consequently 
A l — ncc l A 2 = n(n— \)cc 2 A 3 = w(w— 1) (rc—2)a 3 ....A re = n(w-— 1) (n~ 2 ) 1 .a n . 
Therefore 
in— 1 
d u 
d n ~ 2 u 
cT u d n u . . w 
df = n a i dM + n “ 2 - + n(n- 1) (n - 2) a 3 +, &c. 
By taking in fact the coefficient of k n in S and multiplying it by h n we find that the 
product, viz. + a 2 h -f a 3 h 2 + . . . . a n h n ~ 1 +, &c. is equivalent to 
_L (- H\~ n y | n + 1 ( d y\~ n y* (n+l)(n + 2 ) (dy\~ n V3 
n ‘ \d x) \d x) ' 2 ’ \d x) ' 2.3 ° \d x) 
= i . (&)- _ h (iiy 1 - 1 . ; pi a * + & . * +, fa.. i 
n \d x/ \d xj d x 4 2 1 d X s 2 . 3 1 5 j 
Hence 
n \d x/ 
+ r ~ 1 h 2 
(n + 
-f- &C. 
dy\ ~ n ~ 2 C dAy 
{^1 _L . £y A &c 1 2 
[ d x s ’ 2 ' d x 3 ' C Z. 3 ’ J 
OO + 2 ) 73 /^y \" w ~ 3 f d?y 1 d?y h •) Z 
2.3 " \dx) 1 2 ' e^ 3 ‘ 2.3’ f 
«0 = 
n + 
— n — 1 
d*y l 
dx 2 ' 2 
-n — 2 
2 /^a-”- 2 f <?y V 
‘ \dxj \2 d x V \cf x/ 
— n— 1 
d 3 y 
* 2 . 3 d x 3 
