20 
Indiana University Studies 
Volterra used permutable functions in a problem in integro- 
differential equations in 1910, 394 and in a set of lectures delivered 
at Rice Institute in 1919 490 he formulated a calculus of such functions 
whose application extended far beyond the linear equation which 
suggested it. 
For an elegant exposition and correlation of these various gen- 
eralizations the reader is referred to the Cambridge Colloquium 
lectures of G. C. Evans delivered to the American Mathematical 
Society in 1916.* 
Another natural development in the theory arose from the 
problem presented by the singular integral equation. The ordinary 
methods of Volterra and Fredholm were found inadequate to solve 
various problems which naturally arose and to explain curious 
phenomena which appeared in them. As early as 1896, W. Wirt- 
ingerf had pointed out the existence of band spectra, or lines of values 
of the characteristic parameter for which there existed solutions of a 
differential equation with linear boundary conditions. This theory 
was investigated from the standpoint of integral equations by 
Hilbert in 1906 9 and its application extended to differential equations 
by E. Hilb, 479 M. Plancherel, 424 and H. Weyl.t E. Hellinger in 
1909§ extended Hilbert’s work by considering the point and band 
spectra of a general quadratic form. 
Important contributions to the subject of the singular equation 
of Volterra type were made by G. C. Evans 436 in a series of papers 
which appeared from 1909 to 1911 in which existence theorems were 
stated with considerable generality for various kinds of kernels. 
E. Picard 448 performed a similar service for the Fredholm equation, 
and the case of the Volterra equation of first kind was treated in 
an elegant manner by the combined work of V. Volterra, E. 
Holmgren, 213 T. Lalesco, 287 and J. Horn. 215 The work of the latter 
was devoted to a problem in the inversion of integrals which is 
analogous to the problem presented by differential equations with 
irregular singular points. 
In spite of the researches in this direction, however, the pos- 
sibilities of the singular equation are still far from exhausted. In 
most of this work special kinds of solutions are specified in advance, 
and as these restrictions are removed, new and novel diffculties 
*Functionals and their Applications. Selected Topics, including Integral Equations. Cambridge 
Colloquium Lectures. Amer. Math. Soc. (1918). 
fMath. Annalen, vol. 48 (1896), p. 387. 
JUber gewohnliche Differentialgleichungen mit Singularitaten und die zugehorigen Entwicklungen 
willkurlicher Funktionen. Math. Annalen, vol. 68 (1910), pp. 220-269. 
§Neue Begriindung der Theorie quadratischer Formen von unendlichvielen Veranderlichen. 
Journal fur Math., vol. 136 (1909), pp. 210-271. 
