472 
THE REV. B. BRONWIN ON THE SOLUTION OF 
From the last we find 
I *+5’'® 
We easily find c= therefore p= 
The other condition alluded to is 
From the values of tt^, we find 
VnU — + (m — w) ^(D)X' (^) . 
Therefore we must have 
(/.) 
py.(D) 
— flr^'-*=(p(D)X'(D), or — as"*' ?'(D)“'^=^(D)? v.'(D) ; 
and as in the former case, 
-MD)<;MD)=rf{f(D)V(D)}. 
Therefore, also, 
h — 2 ^(D)^=<p(D)x'(D), and ^ 
dT) 
b-~KW 
which is the required relation between ?i(D) and X(D). 
/6_1x(D)2^ 
where we easily perceive that c= — a. Whence f^=j 
Application of the four last Theorems and Formulae. 
Let T^7r„M+/?g'M=X (18.) 
A single case of this equation was solved by Mr. Boole in No. 7? New Series of 
the Cambridge Mathematical Journal, the quantities <p and A being given functions 
of the independent variable, and having a given relation, T being nothing. In the 
Philosophical Magazine, vol. xxxii. p. 257, 1 gave two similar solutions, but contrived 
to introduce an arbitrary function of by which the solution was very greatly ex- 
tended. Here that method is superseded and the solution rendered much more 
general. To solve the above, make ; then 
by substitution, 
by (c.), 
+1 w, = ^;;; 'X, 
= by (e.), 1). 
