796 PEOFBSSOE BOOLE ON THE COMPAEISON OE TEANSCEXDENTS, WITH 
late. We can do little more than address ourselves patiently to follow the tracks which 
open before us, without attempting to prescribe their direction or conjecture their end. 
The interpretation of the formulae which have been arrived at in this section is so far 
distinct from the general course and design of the paper, that I have thrown into a note 
such observations as I have to offer upon the subject. And I have the rather taken this 
course, because those observations have been to some extent anticipated in a former pub- 
lication. See Note B. 
I have not attempted the further extension of the theory of multiple integrals, which 
would seem to be involved in the general theorem of Art. 32, when other values than 
unity are assigned to the function Neither have I attempted to extend that 
theorem to the case in which is rational — -a case evidently of some importance 
from its formal connexion with the last member of (5.). 
Note A. 
On the Connexion between the Symbol 6 and Cauchy’s Symbol t , employed in 
the Calculus of Mesidues. 
It has been explained that the operation denoted by the symbol B is equivalent to 
that portion of the operation represented by 0 which depends upon the ascending 
developments of the subject function. There is, however, a difference in the mode of 
statement. 
Thus B f 7 ^ 7 — 
L(a? — aY{w~ 
in ascending powers of x. This would be the same as the sum of the coefficients of 
and in the respective ascending developments of the primitive function 
coefficients of - in the developments of the respective functions f and , — A 
X X 1 ^ "P Cl D ) ( ^ *y“ 0 d ) 
would denote, according to Cauchy’s definition, the sum of the 
5 ) ? ™ ascending powers of x — a and x—b respectively. The operation 0 would 
add to the above the coefficient with changed sign of ~ in the development of the same 
function in descending powers of x. 
We shall perhaps best exemplify the connexion thus established between the two 
symbols, by applying the new symbol 0 to the solution of some of the problems which 
Cauchy has treated by the Calculus of Residues. I select for the purpose, — 1st, the 
problem of the integration of linear differential equations with constant coefficients; 
2ndly, the problem of the integration of rational fractions. 
Both these applications depend on a transformation of the rational fimction f(v). 
1st. It follows from the subsidiary theorem (6.), Art. 13, that if we have 
^=0 ( 1 .) 
as an equation connecting x and v, then 
2/W=0[/W]A^- 
