MR. R. C. ROWE ON ABEL’S THEOREM. 
731 
Section II. 
12. The expression (in a form algebraic or logarithmic) of the sum Xf ~Kdx having 
been shown to exist, and having in fact been found, Abel proceeds, in his art. 5, to 
investigate the condition that this expression should become a constant. Of the pos¬ 
sibility of this we have been assured by the result of the first example and of the first 
case of the second example of art. 11. This investigation, as subordinate to the main 
purpose, may be conveniently postponed to the second principal inquiry with which 
the memoir is concerned. 
This inquiry presents itself in two forms. 
I. Mention was made at tlie outset of the “requisite algebraical laws” which 
connect the variables when the summation desired can be effected. And in the case 
of the elliptic functions we have found that in order to express the sum of three func¬ 
tions it is requisite that the variables should be connected by a single relation. We 
are naturally led to investigate the number of relations necessary for the same effect 
in the case of more complicated forms. This number, it must be said, depends not at 
all on the number of the functions we consider but only on their form. 
II. We ma} 7 also consider the matter thus 
Representing by \jj(x) the integral JXc/x, we have shown how to express, by the use 
of an operative symbol ®, the sum 
H x i) + '/'(%) + • • • +#*>) 
where aq, aq, . . . ay are the roots of an equation 
F(x)=0. 
Now this equation involves a number, a, of arbitrary quantities cq, ci 2 , a s . . . 
Its [x roots are functions of these a quantities. We can then find expressions for 
cq, cq, . . . , in terms of a of these roots, say aq, aq, . . . x a ; and substituting these 
expressions in those which give x a+l . . . ay shall have these g — a. roots determined 
as functions of the other a* 
The result then is an expression for the sum of a series of functions 
#*h)+ • • • +#&«), 
* This is most conveniently effected by 
(1) solving for <q, a 2 , . . . the « equations—linear in a’s— 
%i)=°> % 3 ) = ° • • • » 0 (Va) = O, 
where the equation &(y 1 ) is the factor of E which supplies the factor x—x 1 to F(;r), and 
(2) substituting* the values so obtained in F(®), which then becomes divisible by 
0 — * 2 ) • • • (pS — Xa), 
and gives as quotient an equation of the degree /<,—« whose coefficients are rational integral functions 
of Oq, l/i), &c., and whose roots are the quantities x a+l , x a+z , . . . x jL which it is required to determine, 
5 b 2 
