82 ME. W. H. L. EUSSELL OX THE CALCULUS OF SYMBOLS, 
is an internal factor of the symbolical function 
— 1)-|- 2^(77 — l)(7r — 3)-j-7r(7r — 3)(7r — 4) ; 
■wherefore, effecting the internal division, the equation becomes 
{^(tt — l)®+7r(7r — 4} {g'(7r — l) + (7r — 3 )}m=X. 
This equation may be written 
in which the inverse calculations are all practicable. 
As a final example we take the differential equation 
(.r ’ + 4:W* (2^* + 3^^ + — Qx) ^-\-{x-\-iyu=X.. 
The symbolical form of this equation is 
g>V(7r+l)M+^^(47r^ — 7r+l)2J+g>(57r^ — 57r-f-2)^^-^-(7r — — 1 )m=X. 
If we put u—T7V^ the equation becomes 
{g-\-l)(7r — l){§{7r — l)-\-7r){§Tr-\-(27r — l))v=X, 
in which the inverse calculations necessary for the solution of the equation are all 
practicable. 
In cases where the assumption M=(7r+|)y does not lead to the solution of the equa- 
tion, we may assume u—{Tr-\-^y){'7r^^y)v, and proceed as before. 
We may also treat linear differential equations by ascertaining the condition that 
•>4,(c)7r4--v|/o(f) inay be an internal factor of this symbolical expression, 
-f ^,(^)7r4-<po(f). 
I have shown how this is to be effected when n=-2 or 3. 
For higher degrees the investigation would be very laborious. In all cases in which 
the second member of the differential equation is zero, this internal factor, supposing 
it to exist, would conduct us to a particular integral. 
