ME. R. C. ROWE ON ABEL’S THEOREM. 
723 
(ii.) The assumption (in Boole’s theorem) that <j> is not rendered infinite by the 
values which satisfy the equation E=0; and the assumption (in art. 8) that F(x) and 
F'(a;) have no common factor. 
These assumptions are identical: for </> is rendered infinite by the vanishing of 
f. 2 (x) F 0 (x)F , (x), and, since the roots of F are all functions of the a’s, they cannot satisfy 
the equations f 2 (x) = 0, F 0 (t) = 0, into which no a enters. 
If then F and F' have no common factor, the first assumption is justified. 
We assert in this that F = 0 is not an equation possessing equal roots— i.e., that 
x Y , x 2 ,. . . x /x are all unequal. Suppose, on the contrary, that we have equal roots—say 
x ] =x 2 =x 3 / / ' If then y 1} y 2 , y s are the corresponding roots of y we shall have 
%i) = °> % 3 ) = o 5 % 3 ) = ° 
for the same value x } of x ; and therefore in the expression of 
__1_ jX'SJr) ' E 
fi(x)'F 0 {x)F , {x) x\Vr) 6 Vr Jr 
we have a term of the form viz. : that due to the root x=x x , and it will be three 
times repeated. 
We see then the character of the difficulty introduced by the equality of roots. It 
does not altogether vitiate the solution ; it only requires that we should modify it by 
using, instead of the equations 6(^y 1 ) = 0, 0(y 2 ) = 0, 6(y s ) = 0, the equations 
%i) = 0, 
T6y x 
dx 2 
The manner in which all the steps of the analysis and the final result are aftected 
by this change is obvious. 
11. It will now be natural to give examples of the application of the general 
theorem, and those are chosen the results of which are well-known, as furnishing 
comparison between this and other methods of research among transcendants. The 
second and third are treated by Boole, in the paper frequently referred to, as examples 
of his less general theorem. 
I. The function sin 1 x. 
Let 
and take 
X 
v/(l-* 2 ) 
x(y)=y 2 + x2 ~ 1 > 
* The reasoning will be applicable to any other number of equalities among the roots. 
5 A 2 
