134 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1909. 
is limited, the demonstration of which Hilbert has so happily sim- 
plified, it seems to me that it leads to a problem much more general: 
If we have an infinity of whole polynomials, depending algebraically 
from a finite number among them, can we always deduce them from a 
finite number among them by addition and multiplication ? 
We must not believe that the task of algebra is finished because we 
have found rules for all the possible combinations. We have still to 
search out the interesting combinations, those which satisfy such and 
such conditions. Thus there will be established a sort of indeter- 
minate analysis in which the unknowns will not be whole numbers 
but polynomials. Then in this case algebra will model itself upon 
arithmetic and take as a guide the analogy of the whole number, 
either as a whole polynomial of any coefficients whatever or as a 
whole polynomial of whole coefficients. 
IiI. DIFFERENTIAL EQUATIONS. 
Much has already been done for linear differential equations and 
it remains to perfect what has been commenced. But with nonlinear 
differential equations there has been much less advance. The hope of 
an integration by the aid of known functions has been given up long 
since; therefore we must study for themselves the functions defined 
by these differential equations and then attempt a systematic classi- 
fication; the study of the mode of change in the neighborhood of 
singular points doubtless will furnish the first elements of such a 
classification, but we will be satisfied only when we shall have found 
a group of transformations (for instance, the transformations of 
Cremona) which will play with respect to the differential equations 
the same role as the group of birational transformations does for 
the algebraic equation. We can then group in the same class all the 
transformations of the same equation. We shall have for our guide 
the analogy with a theory already made—that of birational trans- 
formations and the genus of an algebraic curve. 
We may propose to lead back the study of these functions to 
that of uniform functions, and this in two ways: We know that if 
y=f(«), we can, whatever may be the f/(#), express y and # by 
uniform functions of an auxiliary variable ¢; but, if f(a) is the solu- 
tion of a differential equation, in what case will the uniform auxil- 
iary functions themselves satisfy the differential equation? We do 
not know; neither do we know in what cases the general integral can 
be put in the form F (a, y)=arbitrary constant, where F (a, y) 
is a uniform function. 
I will urge the qualitative discussion of the curves defined by dif- 
ferential equations. In the simplest case, that in which the equation 
is of the first order and the first degree, this discussion leads to the 
