MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
245 
Equations of the second order comprised in the above general class are of the 
form 
ffifi fin 
{A x )} 2 ^+f( x )(fX x )+v)fa+q u=v > ( 22 -) 
as is found by writing in the equation 
whence t=J fay 
cfi'U dii « o 
The equation (1 +tf£ 2 )^+aj^+w 2 M= 0 is a particular case of (22.), and its solu- 
tion is determined by the system 
'=/ 
dx dru . 9 
-■ --- a -^7o-\-n'u = 0. 
v'l +ax 2 dt- — 
The symbolical form of the equation just considered is 
u 
a(D — 2 ) 2 + n 2 
D(D — 1) 
£ 26 U = 0 ; 
( 23 .) 
to which we shall have occasion to refer. 
In the employment of the general symbolical form of the linear differential equa- 
tion, two principal cases will be considered ; the first comprising such equations as 
are reducible to a system of an inferior order, by a method of resolution similar to 
that which is employed in the solution of linear differential equations with constant 
coefficients ; the second including those whose solution depends on a transformation 
affecting the dependent variable u. A more general method of resolution will be 
explained in the sequel. 
Proposition 1. — The equation 
M-j-a i ©(D)^w+(/ 2 < ? ) (D)<p(D— l)£ 2 *w...4-a ri <p(D)<p(D— l)..<p(D— n-\- l)g«*w=U 
may be resolved into a system of equations of the form 
u— qcp(D)&*u=U, 
the values of q being determined by the equation 
/+0i9 ra-1 +tf 2 9" -2 ...+a B =O, 
For 
(XII.) 
<p(D)p(D — l)s 2 ^u—(p(D)^<p(D)z*u= {<p(D)^} 2 w, 
and in general 
<p(D)<p(D— l)..<p(D— n-\- \)z n 6 u= {<p(D)s^} n u ; 
so that if we represent the symbol ^(D)^ by §, the equation in question becomes 
(l+«ff + a 2 g ‘ 2 • • • + a /) u = U ; 
M= (l 4-a 1 g > +« 2 £ 2 — + 
= {Ni(l-^)- 1 +N 2 ( 1 -^ 2 f)“ 1 ”+N w (l-^)* , }U, 
