FINITE DIFFERENCES AND LINEAR DIFFERENTIAL EQUATIONS. 
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obtain another linear diflferential expression (p(D, — the solution of which will 
be m=- 4/(D,— j;)X. 
In the application of this theorem, care must be taken that the first-mentioned 
solution is so written that the operations included under the function 4^ are not sup- 
pressed ; and it must also be borne in mind that the expressions obtained by this in- 
terchange of symbols will not in all cases be obviously interpretable. 
For applications of this singular analytical process I beg leave to refer to the me- 
moir above cited, where it is employed for the solution, in finite terms, of extensive 
classes of linear differential equations, and equations in finite differences. So far as 
the process is legitimate, it is to be observed that it is founded on reasoning of a 
purely analytical character. It does not in any manner whatever flow from the 
calculus of operations, or depend for its validity upon the soundness of the logical 
basis on which this calculus rests. 
Now it is a remarkable property of this mechanical interchange of symbols, that it 
instantaneously converts a linear equation in finite differences into a linear differen- 
tial equation ; so that wherever the former is soluble, the latter is soluble also, pro- 
vided the result be intelligible, a condition always satisfied when the functions em- 
ployed are rational algebraical functions. 
As an instance worthy of notice, let us take the example last above given (£x. 6.). 
Bearing in mind that the proposed interchange gives the equation (writing 
<px for PJ, 
u+a,<p(D) {fu) -f a^(p(D)(p(D — 1 ) -f . . -f a„<p(-D) . . <p(D — wfl- 1 ) = G ; 
whose solution, therefore, depends upon that of 
v—a<p(D) = a““*AG, 
a proposition established by Mr. Boole by the methods of the Calculus of Opera- 
tions. 
I propose, therefore, now to employ this theorem of the interchange of symbols for 
the purpose of converting the forms of solution, above given, of equations in finite 
differences into the particular solutions of some general forms of differential equa- 
tions ; viz. those equations whose factors do not contain any irrational or transcen- 
dental functions of x, or contain them only in the form of series of ascending powers 
of X. 
11. Mr. Boole, in his General Method in Analysis, has shown that expressions of 
this character may be placed in the form 
/o(D) w +/ (D) { b ^ u ) -f/,(D) -1- . . = U, 
by changing the independent variable from x to its logarithm 0, and making use of 
the relation, being 
B(B — l)..{D-n-^l)u=x^(J^'^ u. . .. ' 
