480 
THE REV. B. BRONWIN ON THE SOLUTION OF 
If (24.) be reduced to the ordinary form, it becomes 
<p' + 2A 4^1 
<P 
+ 
^2 + ^ 2^2 + ^^2 JU — ^2\Jrr 
This would give a very large number of integrable equations by assigning parti- 
cular forms to the arbitrary functions (p, X, 4^’, and And indeed every one of the 
equations treated of in this paper, by giving particular forms to the arbitrary functions, 
would furnish a large number of particular ones. This circumstance makes the 
chance of our being able to put a given equation under some one of these forms the 
greater, and consequently in this respect enhances the value of the methods employed. 
Application of the Theorem (h.). 
The equations to be reduced by {h.) are of the form 
X="k(D)t<-l-4''(D)OT'M-l-4'^(D)c7'^M-|- &c (25.) 
This is deduced from (23.) by the interchange of the symbols D and x, but the 
theorems (g.) and (A.) by which the transformations are effected cannot be thus 
deduced the one from the other. 
The above will be transformed by (A.) into 
X=cI>(D)w-f n7'0.(D)M-ft^'^0,(D)w-f , 
where 
O (D)='k(D)+'k;(D)-f 4^:(D)-f4^^(D) + ... 
<I>,(D) =THD) 4- 2>k?(D) + 34^:(D) -f . . . . 
a),(D)='k^(D)-f3T?(D)-f 
0,(D)=4'4D)+...&c. 
If 
O (D)=T(D)-l-4';(D)-h4':(D)-l- ... --0, 
or 
4'(D)=-4';(D)-4^:(D)-...., 
we shall have 
z;7'->X= tI>,(D)M-h t;r'02(D)M-f . . . . 
We might take as examples equations of a higher order than the second, but as 
these last are of the greater importance I have hitherto selected them, and shall take 
here the equation 
•T4D)^'^M4-T4D)tJ7'M-(T;(D)+T^(D))M=X, 
which by putting for T|(D) and 'T 2 (D) their values, the figures here at the top and 
bottom of T signifying the same as in the former case, becomes 
4^(D)z:7'^a-hT^(D)t«T'M-(p(D){T‘'(D)-f?)'(D)T^'(D)-|-^(D)T^"(D)}w=X. . (26.) 
The transformed equation gives 
t3-'02(D)M-l-^i(D)w = ar'~'X ; 
