EQUATION AND ITS TRANSFORMATIONS. 
803 
and 
so that 
t£)^ 2i ^(p-^ip- 3 ) • • • (p-2i+l)xP~ 2i , 
x dx) 
(l d\ i+ 1 . 
- — xP—r>\ r> — 
l~£) x2i+ %ic) xP =p{p- 1 )(p - 2 ) • • • {p-2i)x r ~ oj 
d \*+1 
and therefore, 
dx 
^ v+1 
a; 21 ; 
a; dx 
x dx 
d \2i+l 
^ 6 ’ 
= a 2i+ V' r . 
The preceding investigation shows also that, if <f)(x) denotes any function of x, then 
■ ■ (26); 
for this theorem has been proved to be true when <f>(x) is of the form Aaf B.PCx c 
+&c.; and as it merely asserts an identical relation between the derived functions of 
<f)(x), it must hold good universally, since the truth of such a relation could not be 
dependent on the fact of whether (f>(x) was or was not expressible in any particular form. 
33. The general property upon which the theorem (26) depends is that the symbols 
of operation 
d d 
ry I”«- ryO- ryl~& - ryP p 
tAJ -j • t AJ tAj a Uv V • 
dx dx 
are convertible as regards order*; that is to say, operating with such symbols upon 
4>(x), the result is the same in whatever order the operations are performed. This is 
evident, for 
X l ~ a ^X a .X p = ( p-\-a)x p , 
dx 
so that the result of the operations upon x l \ and therefore upon (f>(x), is independent of 
the order in which they are performed. 
Now the left-hand side of (25) multiplied by x~ ,+0 ' is 
d d 
/y>2i+l_/y»—2 i rJh—\ _ 2t+2 
t As iAj • 1 tv 
dx 
dx 
rySHLryS \l r Y SHA%^_ry-%i-\ ryZi^Lry-^i+1 
• iAy tAy j | t Aj • iAx T t Ay 
d 
'__ 
dx 
dx 
dx 
. X 2 yX l ^jX(f)(x), 
# See Cayley, ‘ Proceedings of the London Mathematical Society,’ vol. viii. (1876), p. 51, and also 
‘ Solutions of the Cambridge Senate-House Problems and Riders’ for 1878, pp. 99, 100, 
5 L 2 
