MR. R. C. ROWE ON ABEL’S THEOREM. 
719 
Now (using as before y or y x indifferently for the root with which we are concerned), 
we have 6(y 1 ) = 0: whence if \(x, y) be any rational function 
\(x, y)h E=\(x, y)tj^y§0(y r ) 
E 
= H X > 2/i)^y S %i) 
all the other terms in % vanishing, 
=3 x - y ' ] m S0(yr) 
if we introduce a set of vanishing terms. 
We have then obtained an expression for dx and a convenient modification of the 
result when the differential is multiplied by any function A. of x and y. 
So, finally, 
f(x, y)dx 
A(x,y) . 
x'(y)M x ) F o(®) F '( a 0 
i_Nd /iOo y,) _ E 
/a(®) r o(®) F '(®)r=i x(y r ) 6{y r ) 
«%r). 
8. From this point a symbol and theorem due to Boole"' furnish a short path to 
the result. The symbol is thus defined :— 
“ If <f)(x)f(x) be any function of x composed of two factors <p(x), f (x), whereof <f>(x) 
is rational, let ®[<£(x)]/(cc) denote the result obtained by successively developing the 
function in ascending powers of each simple factor x — a in the denominator of (f>(x), 
taking in each development the coefficient of adding together the coefficients 
thus obtained and subtracting from the result the coefficient of ^ in the development 
of the same function $(x)f(x) in descending powers of as.”t 
Boole’s theorem is the following :— 
“If <f>(x) be any rational function of x and if E = 0 be any equation, rational and 
integral with respect to x, by which x is connected with a new set of variables oq, a. 2 , . . . 
then, provided that cfj(x) does not become infinite when E = 0, we have 
. ,, . cl lop; E ” 
%<l>(x)=-®[<f)(x)] ~ x - 
the % indicating summation for the various roots of the equation E = 0. 
* Phil. Trans, for 1857, pp. 751, 757. 
f Cauchy employs in his ‘ Calcnl des Residus ’ a symbol £ only differing from Boole’s 9 by not 
including the subtractive term last mentioned. Any theorem can be instantly transferred from the one 
-notation to the other. 
