482 THE REV. B. BRONWIN ON LINEAR DIFFERENTIAL EQUATIONS, 
which is of a remarkable form. Also 
The equation (27.) might very well be reduced to the ordinary form without parti- 
cularizing the arbitrary functions. 
If we make 
Xil))=^m 0(D=(p(D), 
and change into 'i', we shall have 
With these values (26.) will become, operating with 4"(D)“^ on both members, 
»»«+(pP(D)+?>'(D))»'«-|®(;4{^’«>(D)'>'(D)+®(®(D)'I'(D))}«=^'(D)-'X, (28.) 
which is of a form not a little singular, and which might be put under the common 
form, retaining the arbitrary funetions. 
The value of u in this example is 
w=(p(D)"'T(D)2'’"+^^’'a?-' {<?(D)'T(D) } 
In the two last examples we see from the expression of the value of u that they are 
practicable, or that their solutions can be interpreted. 
It may be well to observe, that if we make 
<p(D)=a{B)x{'D), T^(D)=/3(D)a>(D), 
which give 
MD)=a(D)(m:<;(D)+%'(D)), ■y‘(D)=“(D)/3(D)yiD)(p<I>(D)+4>'(D)) ; 
and if we eliminate both 9(D) and X(D) from the value of ¥, and T'(D), ‘'T^(D) from 
(26.), our equation by the substitution of these values will then contain the four 
arbitrary functions a(D), (3(D), %(D) and 0(D). These should then be made integer 
functions of D ; but after actual substitution the resulting equation would be very 
complicated, unless we first give particular forms to the arbitrary functions. If this 
be done, and as many constants as possible be introduced, partieular integrable 
equations of interest and value, and of sufficient simplicity, may be obtained. 
I shall terminate this paper by observing, that in our attempts at integration, we 
are apt to seek for equations which immediately present themselves under simple 
forms, and by this means fall chiefly, or altogether, upon such as could previously be 
integrated. 
Gunthwaite Hall, near Barnsley, Yorkshire, 
February Ath, 1850. 
