LINEAR DIFFERENTIAL EQUATIONS. 
477 
Mr. Boole’s general equation 
X=M+q/’(D)s®M + /^’(D)y’(D — 
may be generalized and extended in the same way. Thus if 
X= M + af{7s) r^u + ^/(ra-)/(w , 
or 
X = M 4" . . . -5 
by making in the first, and ^ =zf{zs')r'''’’ in the second, we shall have 
X = ^^ + a§'u + h^'^u 4- = (1— —p^^) u. 
And thus each equation, by the method explained further back, will be reduced to a 
number of others, each of which is much more simple than the proposed. 
It must be observed, that the method which has been applied to the solution of 
each particular class of equations will not apply to either of the other classes. We 
see th*e same thing when employing other methods, and we see no reason to suppose 
that our means of integrating equations will ever be greatly extended otherwise than 
by the multiplication or aggregation of particular methods. Such methods therefore 
ought not to be considered as possessing little interest. The same thing may be 
inferred from the various conditions of integrability at which we arrive in those cases 
where we can treat the same sort of equations by different methods. 
Some of the examples which have been given in this paper, when reduced to the 
ordinary form, are very complex ; but when particular forms are assigned to the 
arbitrary functions, results sufficiently simple may be obtained. If we will accept 
none but such as at first sight present themselves under a very simple form, we must 
not expect any great extension of our present scanty means of integration ; for these 
equations are usually very easily integrated. The following example may serve as an 
illustration. 
The theorem 
(pD”M = — &c. 
is easily verified by performing the operations D", &c. in the second member by 
means of the formula 
. D"=D”4-/^D2D^*4-'^^^^DlDr^4- &c. 
Now let there be given the equation 
X=T-w4-T-iDw4-T2D'm4-T-3D"w4- 
and let this, by the preceding theorem, be transformed into 
X=0M4-B0iM4-D^02M4-D^d>3M4“ 
We shall have 
0= T- t; 4- T"- Tg 4- . . . 
= — 2T;+3'T;— .... 
<D2=^2-3t;4- 
&C. 
3 Q 
MDCCCLI. 
