478 
THE REV. B. BRONWIN ON THE SOLUTION OF 
If cI)=o, the equation is integrable once, and we shall have 
D~^X=d>iM-{-Dd>2M+D^d>3Md- — 
If both d>=0, and d>i = 0, it is integrable twice, and we have 
D-"X=<b2M+D03M+.... 
Let us take as an example the following equation of the second order, 
-b + C^i - =X. 
The transformed gives 
Dd>22<+OiZi=D"'X, 
or 
and 
u 
= (D4 
^ " V3-'D-^X= 
This example was given by Mr, Hargreave in the Philosophical Transactions, 1848. 
But the proposed equation may at once be put under the form 
D { + ('ki — d' 2) } = X, 
which is immediately integrable, giving 
('kj — 4"2 )m=D'^X, 
as before found, but this is far from being a solitary example. 
V. FIFTH AND SIXTH GENERAL THEOREMS. 
The theorems now to be given bear no resemblance to those which have been here- 
tofore investigated, nor is their mode of application at all similar. Making 
9'x!=Xi> ^Xi=X2, <PX2—Xz^ &c., 
7l\7l 1 1 
= — &c (g.) 
will be verified by putting for zs its value ^D-f-X, and actually performing the opera- 
tions denoted by D, which may be readily done by the method which has now been 
repeatedly explained. 
Again, making 
f(D)x'(D)=x,(D), f(D)x,(D)=x,(D), «)(D)x;(D)=x,(D), &c., 
we shall have 
This also may be verified by putting for m' its value ip(D)ji'-{-X(D), and performing 
the operations D, where required, by the formula 
/(D)=/(D,) +D,/(D.) +iD;/"(D,) + &c., 
and afterwards dropping the marks of distinction. 
