CEETAIN APPLICATIONS TO THE THEORY OF DEFINITE INTEORALS. 747 
summary and criticism of the chief contributions which have been made, both by Englisli 
and by foreign mathematicians, to the theory of transcendental integrals in all its 
branches. I had completed my investigations on that particular branch of the theoiy^ 
which relates to the comparison of transcendents before I had the opportunity of studying 
Mr, Ellis’s report ; but I know of but a single point, and that of no real importance, in 
which I should be disposed to dissent from the general views which he has expressed on 
this subject. 
The fundamental idea upon which the doctrine of the comparison of transcendents 
rests may be thus stated. We know that where we cannot express in finite terms the 
values of the roots of an algebraic equation, we can nevertheless finitely determine the 
values of various symmetrical functions of the roots, e. g. their sum, the sum of their 
squares, the sum of then* binary products, &c. Those values will be expressed in terms 
of the coefficients of the equation, or, speaking more generally, of the independent con- 
stants involved in any way in the equation. If we represent the equation in the form 
E(.r, ( 2 i, « 2 , ..a,.)=0, (1.) 
E being a functional symbol, and the independent constants, and if we repre- 
sent the roots of the equation by ..x^, then whatever the interpretation of the 
functional symbol ip may be, the expression 
p{x,)-\-p{x,)..-\-p{x„) (2.) 
will denote a symmetrical function of the roots, whose value, in terms of «i, a^, . . a,, can 
always be determined when p{Xi) is a rational function of Xi, and can very often be 
determined when p(Xi) is an u’rational function of Xi. Thus the value of the sum 2p{x), 
where the different values of x are the roots of an algebraic equation, can often be found 
when the values of the separate terms of which that sum is composed cannot be found. 
Now this suggests to us the question whether it is not possible in certain cases to find 
the value of the integral-sum 'L^'K.dx, where we cannot find the value of the separate 
integrals involved in that sum. What we usually mean by finding the value of the 
single integral is the expressing of the value of that integral in terms of its superior 
hmit, an arbitrary constant being annexed, and consists, in fact, in determining the 
function p{x^ in the equation 
r‘x&=?(®,)+c (3.) 
The suggested problem is the finding of the value of the integral-sum 
J '*! 
X(Z^+i X(f^..-fl ^dx, (4.) 
where x^^ x^-, ..x„ are the roots of an algebraic equation, which we will suppose tobe(l). 
Representing (4.) by 2^^ ILdx, the solution of the problem will, according to the nature 
of the analysis employed, assume one of the two following equivalent forms, viz. either 
..«,)+C, ....... (5.) 
6 E 2 
