70 
ME. W. H. L. EUSSELL ON THE CALCIJLIJS OE STIVIBOLS. 
Section I. On the Principles of Symbolical Algebra. 
Let (§) and (tt) be two functional symbols combining according to the law ^f{y)u 
=/’(7r— where (w) is the subject. We shall suppose throughout this paper that 
d 
Let P, Q, and R be three functions of (tt) and (§), such that PQ acting on any sub- 
ject is equivalent to R acting on the same subject, or PQ=:R. We shall say that P 
externally multiplies Q, and is an external factor of R. In like manner we shall say 
that Q internally multiplies P, and is an internal factor of R. We shall also say that R 
is externally divisible by P, internally by Q. 
We easily see the truth of the following symbolical equations depending on the laws 
of combination assumed above ; — 
We shall commence with instances of symbolical multiplication and division, when 
the multipliers and divisors are monomials. 
The following is an instance of external multiplication : — 
-f §7r + TT^) = 2 ) + §*( 7 ^^ + tt) + f ’T® ; 
the following is an instance of internal multiplication — 
(2§^ — 3§7r^-|-(7r^-}-7r))§7r^ = 2gV^ — 3g^(7r^ + :r)® + t’*'X^4-l}(’»‘ + 2) ; 
the following of external dhision : — 
— 2§V(7r-l-l)-l-3§V®) = ^^ — 2§7r-l-37r^ ; 
the following of internal division : — 
(§V-f- 3^^(7r^-l-7r) 4" f 7r(7r-J- lX)(^7r)“' = ^^-1- TT*. 
We shall now consider cases where the multipliers and divisors are polynomials. 
The follo-\\ing are instances of external multiplication : — 
^ — X 
— g(x4-l)— x^ 
§X^ (x-1-1) 
§x — x^ 
^^(7r-^l)7r'^—^7r(7r-^Tj 
— g(7r4-l )V^4 -xX7r4-l ) 
g^(x -b 1 )7r^ — gx(x -f- 1 Xtt^ “j- X -p 1 ) xX tt -p 1 ). 
