ME. W. H. L. EUSSELL ON THE CALCULUS OE SYMBOLS. 
255 
For this purpose, let us assume 
W+f-’C.W+...+«.(x))». 
From this we find, equating the coefiicients of f”-, 
+ W) = <p2„(7r). 
from whence 
or 
whence 
, — (P2„(7r — 3w)<p2„(7r — 5«)... ^ 
~ '^■InkF — ‘in) <p2n (tT — 4ra) <p2n (’T — 6«) . . .’ 
Again, equating the coefiicients of ^ we shall have 
+ l)^„_i(7r) = (p2„_i(7r)...; ..... (A.) 
^„(:r), ^ 2 n-i('^) known rational functions of (tt), wherefore assume 
<^„(7r) = a + J tt + CTT^ + . . . + l'7r% 
where a, b, c, ... a, (^, y, . .. are known, and 
^„_i(7r)= A+B tt fi-C7r^+ . . . +K'r’“'', 
where A, B, C are known. They may be easily found by equating the coefiicients of 
(t) in equation (A.) and thus determined. 
If we equate the coefiicients of we shall have 
from which in ay be determined in like manner. By this method we may in all 
possible cases reduce a proposed differential equation. 
to the form 
i?''u 
X.^.+X,,_/;b?^:+...+X 
dx^’’ 
.,g+x.«=x. 
£_ 
dx^ 
“1“ ^r-l 
£^+...+E.}«=X, 
when H^, E^_„ &c. are rational functions of {x). 
The methods for finding the highest common divisor in ordinary algebra apply equally 
to the present Calculus, as will be seen by the following examples : — 
To find the highest common internal divisor of the symbolical functions 
and 
2 M 2 
