ME. W. H. L. EUSSELL ON THE CALCULUS OE SYMBOLS. 
75 
Then 
■ 2 (§). 
+ 
Put, for the sake of simplicity, 
— ;i;^ { ‘ ^ ~ ^ ), 
4'i(?V+*'o(?)=;j;||(4'i?-^+4'of)+‘i'o(f)— '4'o(i)- 
Hence the condition that 4'i(g)T+4'o? may be an internal factor of 
^p3(§V+<Pl(fK+^5l(?K4-<Po(?) 
is equivalent to the equation 
"i(§)4'o(§)— "o(?Hi(f)=0- 
Hence, substituting for ci^i(§) and their values, we have 
or, again substituting for 5o(§), 5i(§), we have 
Had we wished to ascertain the condition that 4/i(g)7r-|-\}/o(§) may be an internal factor of 
<p4(§ V + <p3(?)’r" + cp,(f )7r" + (? i(^ )7r + <po(§ ), 
we must have calculated the value of 7r^<?3(§). It is evident that for every increase in the 
degree of the highest power of (tt), the labour of the investigation becomes immensely 
greater, and the result far more complicated. It is, however, of considerable utility in 
the integration of differential equations, and we shall refer to it again at the close of 
this paper. 
