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Indiana University Studies 
du probleme de Dirichlet” 34 and was followed three years later by an 
elaboration of these results in a second memoir: “Sur une classe 
d’equations fonctionnelles” which is now classic in the theory of the 
Fredholm equation. 137 The equation solved by Fredholm presented a 
difficulty which did not appear in Volterra’s problem since, in the 
former case, the determinant of the algebraic system was not always 
different from zero and required, necessarily, that the limiting form 
of the solution should be a meromorphic function of the parameter. 
Thus the solution of the Fredholm equation (4) appeared in the form 
f(t) dt 
where D(x,t;x), called Fredholm’s first minor, and D(x), called 
Fredholm’s determinant, are entire functions of X. 
It was Fredholm’s notable discovery that corresponding to those 
values Xi, the so called characteristic values of the equation, for 
which D(x) -= 0, there existed solutions of the homogeneous equation 
and these solutions were actually D(x,t;Xi). In a paper of consider- 
able elegance L. Tocchi 169 has shown that D(x,t;Xi) can always be 
factored into two factors v(x) w(t) so that the extra parameter is a 
parameter only in appearance. The results of both Fredholm and 
Tocchi are readily extended to the case where D(x) = 0 has multiple 
roots.* 
The proof of the convergence of the X — series representation for 
D(x) which Fredholm obtained depended essentially upon a theorem 
due to J. Hadamard on the maximum value of a determinant. 280 
This theorem has played an important part in much of the work 
involving infinite determinants, and a number of independent prccfs 
have been given for it. 
Shortly after the appearance of the Fredholm papers^ D. Hilbert 
commenced the publication of a series of memoirs in the Gottinger 
Nachrichten on the “Foundations of a General Theory of Integral 
Equations. ’ ’ The first of these appeared in 1 904 and the last in 1 9 1 0. 9 
One of the most noteworthy achievements of these papers was the 
formulation of the Sturm-Liouville boundary value problem of 
differential equations in terms of integral equations and another the 
remarkable connection established between integral equations and 
bilinear and quadratic forms in an infinite number of variables. This 
subject was extensively developed by M. Mason 273 in his thesis pub- 
lished in 1903. 
One of the important extensions of the work of Hilbert in con- 
nection with the Green’s functions of differential equation systems 
*See Heywood-Frecbet: L’equation de Fredholm, ref. 8, footnote p. 73. 
