32 
MR. HARGREAVE ON THE SOLUTION OF 
therefore true ; and the correctness of the result thus derived from this interchange of 
symbols leads to the inference, that if in any linear diflFerential equation capable of 
being reduced to the form (1.) or (2.), and its symbolical solution, x be changed into D 
and D into —x, we shall obtain another form accompanied by its symbolical solution. 
Possibly the form so obtained and its solution may not be interpretable ; but in every 
case in which they are interpretable, they will be found to be true ; and if, by any 
transformation, meanings can be attached to those forms which appear to be unin- 
telligible, they also will be found to be true. 
It is an essential condition to be observed in all operations in which this process 
is used, that the solutions are to be preserved in a symbolical form ; or, in other 
words, that the operations are not to be performed or suppressed. It would mani- 
festly be a source of error to write zero for (J7 ').0, if in a subsequent stage x is to be 
converted into D. 
The process may be conveniently exemplified by applying it to the general equa- 
tion of the first order, 
(pxJ)u-]-’4^x.u=X. ; 
of which the solution, (the processes being preserved,) is 
If we make the proposed conversions, we have for the solution of 
— (p(D){a^M}+'4/(D)M=X (3.) 
w= — £~^^{x~h^ ((pD)“*X}. 
But equation (3.), by (1.), is equivalent to 
j:’^(D)m-1-(?5'(D)— 'v^(D))m= — X or Xo- 
Let <?'(!>)— 4^(0) =A(D), or 4/(D)=(p'(D)-xD. 
Then x{0)=/^dJ)=iog<p{D)-/^dO; 
and the solution assumes the form 
M=(^(D)) 'g'' ip(D; Xo} (4.) 
the equation to be solved being 
M -j- X(D ) M = Xo . 
This solution was first given by Mr. Boole in the Philosophical Magazine for 
February 1847- It indicates in a striking manner the interchange of symbols which 
is here proposed as a general theory; and leads naturally to the inquiry, whether 
such a conversion may not be extended to other forms. 
I am not prepared to assert that the considerations stated above actually establish 
its validity as a theoretical process ; but it possesses considerable practical utility, 
when applied to a subject in which the value of the result, if true, is in a great 
measure independent of the validity of the process. 
