254 
ME. W. H. L. EUSSELL ON THE CALCULHS OE SYMBOLS. 
the given symbolical function, 
Again, the conditions that g>®4'2('^)+'4'o(‘^) bs an external factor of the same 
symbolical function will be given by the equations 
/ \ 'Po^ / \ I 'Po'^'Pn(‘^ + / \ n 
r„.v I ^o(^ + i)^o(^ + 3) . , 
We may also find in like manner the conditions that g'^'vf/3(^)+4^o(’*')3 f‘‘4^4(^)+'l^o(’r') 
may be internal or external factors of the given symbolical function : in every case the 
number of equations of condition will be equal to the degree of (§) in the given factor. 
By applying the method of divisors, as explained in the former paper, we may ascer- 
tain the forms of ■4^o(’^) in order that ^^\(/2(7r)-b4/o(7r) may be an internal factor of 
the given symbolical function. In the present case, however, ^^/2(7^) must be a divisor 
both of <p„(7r) and <p„_i(7r), 4 ^ 0 ( 7 ^) n divisor both of <Pi(7r) and — n consideration which 
will greatly simplify the process. We proceed, in like manner, should there be no internal 
factors of the form §^4/2(7r)-j-4'o(7r), to ascertain if there are any of the fonu 
+ ; and continuing the investigation as before, we are able, in all 
possible cases, to resolve the given symbolical function 
f • • •+^oW 
into binomial factors. 
Hence, in any linear difierential equation. 
if we put 
d'^u 
dx^ 
d^-h 
dx' 
^ d^~'^u 
,.=x, 
5=ir, 
we shall be able in all possible cases to reduce it to a series of equations : — 
&c. —See., 
(7r)u + 4/0 (7r)u =W„ 
a series of binomial equations, each of which may be treated by the methods due to 
Professor Boole. I shall now explain a process analogous to that denominated ‘ evolu- 
tion ’ in ordinary algebra. To resolve the symbolical function 
• • +<PoM 
into two equal factors. 
