136 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1909. 
of the determinant by combining in a single group all the homoge- 
neous terms of the same degree. The infinite determinant of Fred- 
holm may be so arranged and it happens that we thus obtain a con- 
verging series. 
Has this analogy which certainly guided Fredholm given us all it 
ought to?, Certainly not. If his success came from the linear form 
of the equations we should be able to apply ideas of the same nature 
to all problems having equations of linear form, and, indeed, to 
ordinary differential equations, since their integration may be al- 
ways reduced to that of linear equations of partial derivatives of the 
first order. 
Recently the problem of Dirichlet and those connected with it have 
been approached by another method, returning to the original one of 
Dirichlet and searching for the minimum of a definite integral except 
that this is now done by rigorous processes. I do not doubt that 
_ these two methods without much difficulty will be made comparable 
and advantage taken of their mutual relationships. Nor do I doubt 
that both will have much to gain by such a comparison. Thanks to 
M. Hilbert, who has been doubly an initiator, we are already on that 
path. 
V.—THE ABELIAN FUNCTIONS. 
The principal question remaining to us for solution concerning 
Abelian functions we know. The Abelian functions begot by the 
integrals relative to an algebraic curve are not the most general ones; 
they belong only to a particular case, so we may call them special 
Abelian functions. What is their relationship to the general func- 
tions and how shall we classify these latter? But a short time ago 
the solution of these problems seemed far distant. I believe that it 
is virtually solved to-day, now that MM. Castelnuovo and Enriques 
have published their recent memoir upon the integrals of total differ- 
entials of the varieties of more than two dimensions. We know now 
that there are Abelian functions belonging to a curve and others to 
a surface, and that it will never be necessary to extend them to more 
than two dimensions. Combining this result with what we may ob- 
tain from the works of M. Wirtinger we will doubtless reach the 
end of all our difficulties. 
VI. THE THEORY OF FUNCTIONS. 
It is especially with regard to functions of two and of several 
variables that I wish to speak. The analogy with the functions of 
a single variable gives a valuable but insufficient guide; there is an 
essential difference between the two classes of functions, and every 
time a generalization is attempted by passing from one to the other 
