ME. W. H. L. EUSSELL ON THE CALCHLUS OF SYMBOLS. 
79 
which is independent of (tt), is 
+ (25,(aw-2}25.(aw— 1)29, (a?^)^-23l(a^^— 2)25, (an) 
+ 29,(an - 1)29, (an) + 293(an) )§“"-'+ . . . 
SECTioJf III. On the Solution of Linear Differential Equations with Variable Coefficients. 
The general linear differential equation 
dru 
dXr-i 
“ fX,_,£^+&c.=X, 
where X,, X,.., are rational and entire functions of (.r), may, as Professor Boole has 
shown, be always expressed in the symbolical form 
?”<3„(7r)zi + ‘ <p„_ ,(7r)2t 4- . . . f (p , (7r)ti + <po(7r)lf = X, 
where 
§ — ^’5 
and 7r=^ 
A 
dx' 
and <p„(’*')? are rational and entire functions of (tt). 
Suppose that by using the methods explained in this paper, we are able to reduce tins 
equation to the form 
Assume 
(7r)n„_,+4W (7r}u„_,=X 
^4;"- *^(7r)zf„_2 + 4^”- ^^( 7 r ) u „_ 2 = ««- 1 
§4/',”-2^(7r)M„_3 + 4/'”-">(7r )n„_3 = n„_2 
&c. =&c. 
f4i(7r) U +4o (7r)n =tCy 
We thus reduce the proposed differential equation to forms already treated of by Pro- 
fessor Boole. 
We may much simplify the process already explained for treating the symbolical 
quantity f<p„(7r) -f &c. -f (tt), by remarking that 4,(7r) must be sought among tlie divisors 
of <p„(7r), 4o(’*') among the divisors of <Po(’'') J and we shall make use of this principle in 
the following application of the preceding theory to the solution of differential equa- 
tions. 
We shall denominate the equation deduced in the former part of this memoir. 
<Po(7r) 
<p,(7r — 1) 
\f/,(7r — l)4/,('!r— 2) 
<p2(7r — 2) — &C. = 0, 
the criterion of the factor c\}/,(7r)-|-4/o(’*')- 
* It may be proper to remind tbe reader that r{/'^®’(7r), ao reference to the functions , 
derived from by differentiation. 
