754 PEOFESSOE BOOLE ON THE COMPAEISON OF TEANSCENDENTS, WITK 
will be determined as above by Tatloe’s theorem. The 72 + 1th term will involve the 
operation denoted by the symbol Cj^, and it is by this term only that the interpretation 
X 
of 0 differs from that of Cauchy’s symbol, T . We have in fact 
( 10 .) 
X 
the complete interpretation of 0 invohing two distinct elements. 
It happens in certain problems that one of those elements alone presents itself, the 
symbol 0 becoming equivalent either to T or to — In other problems they both 
X 
appear, but I am not aware of any problems in which the vanishing of one of the 
elements is not due to some special circumstance of restriction or hniitation affecting the 
interpretation of the symbol 0 in the case supposed. In a Xote to this paper I have 
endeavoured to illustrate the above remark by employing the symbol 0 in certain pro- 
blems, in which Cauchy has made use of the symbol of residues, and to that Xote the 
reader who is interested in the comparison is referred. 
11. The properties of the symbol 0 are now to be considered. The two following are 
the most important of them : — 
1st. The operation 0 is distributive as respects both the function without and the function 
within the brackets, provided that those functions do not become together infinite for any 
finite value of x. 
Peoof. — First, as respects the function without the brackets, we have the theorem 
Q[<P(-^)](/iW+/2(^)- • +fn{x)) = e[(p{x)'\f{x) + Q\(p{x)'\fl^^ ; . (1.) 
for the coefficient of a particular term, as — ? -, &c. in the development of a function 
X — a X 
is equal to the sum of the coefficients of all the correspondhig terms in the development 
of the several component functions from which the proposed function is formed by 
addition. 
Secondly, as respects the function within the brackets, we have the theorem 
Here, it is to be observed that <p. 2 {x), . . <P„(a) represent any rational fractions into 
which the rational fraction (p(x) within the brackets [ ] is resolved. Thus the theorem 
might be written in the form 
^l‘Pix)]f{x}=eMx)]f{x)+el(p,{uff(x)..+el^^^^^ . . . (3.) 
wherein 
=<p,{a:)+ 
To prove this theorem I shall show that it is necessarily true for each of the opera- 
tions into which 0 is resolvable. Let one of the distinct factors of the denominator of 
(p{x) be X — a, then one of the component operations in 0 consists in developing in 
ascending powers of x — a, and taking in the development the coefficient of — . Now 
CC 
