42 
MR. HARGREAVE ON THE SOLUTION OF 
And if <px be made equal to or we have for the solution of 
D% + QDm + (Q' + QT -T2 - T')m = R, 
a form easily arrived at by ordinary processes. 
IV. Application to Partial Differejitial Equations. 
Most of the soluble forms above deduced are readily convertible into analogous 
forms of partial differential equations, by substituting for the constants any function 
of D', where D' denotes differentiation with regard to a new independent variable. 
Thus if in (9.) for Q we write f{x, D') and for c we write \/ — lA:D', we have the 
symbolical solution of 
dx^ ' a \ ’ dyJ dx diA ' 
where P may be a function of x and y. 
u = P, 
For example, if be of the form “^5 the equation becomes 
dy 
d~u ^ 2n (t^u 
• \x^ j diA 
dy 
\d% n du m[m — \) 
x^ dy 
u=-4^{x,y ) ; 
dx‘^' X dxdy 
of which the solution is 
Now ^) = -v^(j;, y+wlog J?), and y)='>\^{x, y—n\og x). 
The question, therefore, is reduced to the solution of 
dH d‘^u 
D2m — a2M=T(a’, y), writing a for A-D' ; 
«/)— y). 
or 
whence 
2u 
1 
~ 2/cD''' 
kxwj^ (x, y — hx) dx — [x, y + kx) dx, 
which is 
3 /) +"^2(^5 y))-\-HyA^^^)+iJ^{y—'kx), 
where X and ytj are arbitrary, and and derived from T as follows; iov y in T 
wi'ite y — kx, integrate to x % change ?/ intoj/+A:x, and integrate toy, this gives Tj, 
from which 'T '2 formed by changing the sign of k. 
If f (^x, be of the form the equation becomes 
d^u d^u 
dxdy 
y) j 
( (?vj :)2 _ + ^ixx .^ + (X'-r 4- 2Xx.(/.x)^+ (^.^)2 -f 
u 
