MR. MURPHY ON THE THEORY OF ANALYTICAL OPERATIONS. 
207 
and similarly if we put in the general formula v 1 = 1, and write v 1 for v 2 , v 2 for v 3 , &c. 
and then multiply by v n , finally making all equal to v, we obtain 
dx 
(d x vy=d;v n + 
+ d x 
n - 2 | (” — 2) (w — 1) w (3 n - 5) J)W _ 2 drf j_ (” - 2) • (» - 1)» a ,n - 1 
2.3.4 
d. 
+ 
2.3 
d 
2 _v 1 
x 1 ] 
+,&c. 
Put now — for v in this formula, whence 
(d x v-r=d;v 
(n — 1) . n 
2 
. d n ~ l v~ n - 1 
d v 
dx 
, /ra -2 f(«-2)(«- l)n{n+ 1) _ 2 dv 3 (»-2)(»-])n ,_ A d 2 v} 0 
+ d x | 274 dF ~ 2.3 • d7 5 /+’ &C - 
30. Change of the independent variable. 
When u is a function of y, and y of x, then it is easily shown that 
du ~~ ~Toc • ("dl) > ov dy = d x . omitting the subjects; hence by substitu- 
dy 
tion in the preceding general formula we have 
7 n (dy\~ n (i n-l).n x ( dy\ 
W = d * -Kdi) 2 — • d ■ ■ Xii) 
f (« — g)(» — i). »(»+!) (dy\- n -' i (£y\ 
* x 2.4 \dx) \dx V 
+ 5 & C - 
Thus, for example, if w = 3, 
dtf 
and 
— ft— 1 
2 (w— 2).(«— 1) . w (dy\~ n 1 d 3 j 
' 2 . 3 
! / dy\ n 1 d?y 
\d x) ' d x 3 J 
_ d 3 ^ /djA -3 o ( dy\ ~ 4 d 3 ?/ d?< f / dpv ~ 5 / d 2 jA 2 /djA~ 4 Fj/ 1 
dx a '\dx/ 'dx 2 \dx/ ' dx 2 ' d x\_° \dxj \dx V Vdx/ •dx 3 j ’ 
d 6 x_ fdy\~ 5 ( d 2 y \ 2 / dj/\ ~ 4 d 3 ?/ 
dp \d<r/ '\dx 2 / \dx) ’ dx?' 
As I am not aware that any formula has heretofore been given for the general 
change of the independent variable, I shall here add a reinvestigation of the same 
subject on simple principles. 
When x is changed into x + h, suppose y to become y + k, and u to be changed 
into u + /. 
Now u + l may be expressed by Taylor’s theorem in two ways, 
, j . du j d 2 u Ji 3 d 3 u h 3 
+ — + ^—2 . F 72 + Tx 3 * TTTTs &c - 
P 
Hence 
d7 
dy n ’ \ .0, .. .n 
du 1 cPu F d? u .. _ 
W + dy ‘ * + dp • 171 + dp - 1.2.3 &C ‘ 
is the coeflicient of k n in l, that is, in the expression 
du 1 , cPu ft 3 , d?u 
F 
d n u 
h n 
