470 
THE REV. B. BRONWIN ON THE SOLUTION OF 
where Dj operates upon u, Dg upon x. Thus we find 
(p(D)xu=-x(p{T))u-\-<p' (D)u 
(p(D)x<p(D)xu=x^(p(Dyu-\-3x(p(D)(p' (D)u-\-<p{D)<p" (D)u-{-<p' (Dyu. 
But this reduction would lead to resulting equations of considerable length, unless 
we give to 0(D) a particular form. This form should be such that negative powers 
of D may disappear. Or we may make m=0(D)z, and so take away such powers, 
and at the same time may introduce more arbitrary constants into the equation. 
Thus we should find integrable equations of the second order, having coefficients of 
the form a-\-hx-\-cx^, of considerable generality, owing to the large number of con- 
stants which they would contain. The method will be found on trial well adapted 
to the integration of such equations. 
By changing h into we shall have 
r’-»=ryD+r, ?.(D) = -2 «^(d+Q, MD) = -2i(D+^)(m+|®), 
® }) • 
With these values we derive from (14.), 
{Tss'-{-nk){^'-\-n]i-\-h)u-\-pT!s'{qD-yr)u=.^ ( 17 .) 
Other examples might be given under this head, but I shall now proceed to the 
solution of two equations somewhat similar to some of those which have been given, 
but which cannot be solved in the same manner. 
III. THIRD AND FOURTH GENERAL THEOREMS. 
Make 
and consequently 
5r,„=(pD-fT-l-mX, 
‘T„=(pD-l-T-|-wX. 
We easily verify the equation 
by substituting for ^ and their values, and performing the operations indi- 
cated in the result. Therefore by the process followed in the investigation of (a.), 
we find 
ff{'^m + n)u=f{'?r^)fU (C.) 
To establish the other general theorem, we make 
^L=9(D)x+T(D)-l-mX(D), 
and therefore also 
■3-^ = ^ (D)x -b T(D) -f wA(D) . 
