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THE FUTURE OF MATHEMATICS—POINCARE. 135 
determination of the number of limited cycles. It is very sensitive 
and what will help us is the analogy with the method of the deter- 
mination of the number of real roots of an algebraic equation; when- 
ever any step whatever shows the real status of this analogy we may 
be sure of a very great advance. 
IV. EQUATIONS WITH PARTIAL DERIVATIVES. 
Our knowledge of equations containing partial derivatives has 
taken recently a very considerable step in advance by means of the 
discoveries of M. Fredholm. If we examine closely the basis of these 
discoveries we will find- that this difficult theory is modeled upon an- 
other more simple, that of determinants and of systems of the first 
degree. In the greater part of the problems of mathematical physics 
the equations to be integrated are linear; they serve to determine un- 
known functions of several variables, functions which are continuous. 
Why? Because we have made the equations in conformity with the 
supposition that matter is continuous. But matter is not continuous; 
it is formed of atoms; had we wished to write equations as they 
should be for an observer whose sight is sufficiently keen to see these 
atoms, we would not have had a small number of differential equa- 
tions serving to determine certain unknown functions; we would 
have had a very great number of algebraic equations for determining 
a great number of unknown constants. And these algebraic equa- 
tions would have been linear and of such a nature that with infinite 
patience we could have applied directly to them the methods of 
determinants. 
But, since the brevity of our lives will not allow us this luxury 
of infinite patience, we must proceed otherwise; we must pass to the 
limit and suppose matter continuous. There are two ways of gen- 
eralizing the theory of equations of the first degree in passing to 
the limit. We can consider an infinity of separate equations with 
an infinity, equally independent of unknowns. This has been done, 
for example, by Hill in his theory of the moon. We will then have 
infinite determinants which are to ordinary determinants as series 
are to finite sums. 
We can take an equation of partial derivatives representing, we 
may say, a continuous infinity of equations, and use them to de- 
termine an unknown function representing a continuous infinity of 
unknowns. We then have other infinite determinants which are to 
ordinary determinants as integrals are to finite sums. Fredholm 
used this method; his success moreover came from his utilization of 
the following fact: If, in a determinant, the elements of the prin- 
cipal diagonal are equal to unity and the other elements are 
homogeneous and of the first order, we can arrange the development 
45745°—sm 1909——10 
