464 
THE REV. B. BRONWIN ON THE SOLUTION OF 
Let y’(zE7)'nT (ra- -f (■sfT + +72/](^) (ra" + = X, 
and 
f{'uj){‘UT-\-nk) {■uT-{-nk-\-k) {7!r-\-nk-\-2k)u-\-pf^{zs)7sr'^’‘u=.yi. 
The first of these, by making 
M— (c7+3A’) {rs-\-Ak) .. . .{;uj-\-{n-\-2)k)v^ 
will reduce to 
f{7;s){7s-\-nk-\-k){7AS-\-nk-\-2k)v -\-'pT^’‘v = P| q^| X, 
and the second, by putting 
(cj + 3/c) . (ot + (w + 2) 
reduces to 
/W(=r+A:)(=T+2A:)«+;)r**t, = p|[”“*)'^jx. 
It will be observed that the equations in the two last examples are of the third 
order at the lowest, and those to which they are reduced of the second ; and if we 
were to continue the series, they would rise an order at every step. But we will here 
leave them and proceed to give a few particular examples. 
In (1.) and (2.) make 
/(^)=^+«', /W = i ; 
and they become 
7!s{7s-\-a)u-{-p{'is-\-nli)7'^ii=\. (9.) 
(z;r+a)?;+pr'^z;=p|^^j X. 
The last gives 
X, 
and therefore 
«=p|f }(®+«+pr‘)-p{f }■' X. 
It is not necessary to reduce the value of u any further, as the mode of doing it has 
been made sufficiently plain, and moreover it is quite as convenient as it stands. We 
shall only observe that if, to abridge, we make 
we have 
( 73" + a -j- r*) “ ^ = (D + ~ ^ = 2 ~ . 
If we wish to see (9.) under the ordinary form, it is easily reduced, first to 
+ l)A’j9r*'M=X ; 
and then by substituting for to 
(fjyu-\-(p{(p' -\-2'K-\-a-\-pr'‘)Du-\- {(p}^ {a-\-pT’‘)'K-\' {n-\-\)kpr'‘)u='K. 
If we would deduce particular integrable equations from this, we may assume 
1 
r and A any functions of x at pleasure, and the relation will give It will be 
