10 
Indiana University Studies 
year, while discussing Laplace’s differential equation, P.v du Bois- 
Reymond* reduced his problem to one in integral equations and then 
made the following remark: 
I write down these equations not as if they solve the problem or even 
carry it nearer to a solution; they serve only as examples of the fact that in 
the boundary value problem of linear partial differential equations one is 
continually faced by this type of problem which still, for the analysis of today, 
presents in general insurmountable difficulties. I propose to give to these 
very useful problems the name of integral equations. 
5. The Modern Development. With V. Volterra, T. Levi- 
Civita, and J. Le Roux begins the modern development of integral 
equations which has made such rapid strides under the impetus of 
their work. Le Roux 221 in 1895 gave a solution of the equation 
c 
with certain restrictions on the kernel, obtaining his results by 
means of successive approximations. 
The same year Levi-Civita 222 considered the case of the solution 
of both the Fredholm and Volterra equations of first kind with a 
kernel of the form K(x — y). In the latter case he obtained the 
elegant result that, with proper restrictions on the kernel and the 
known function, the solution is of the form, 
where we have 
o 
Altho Volterraf eleven years before had commenced his study 
of integral equations, his work did not mature before 1896 when, in 
a series of papers celebrated both for the completeness with which 
they discharged the problem and the novelty of the methods em- 
ployed, he gave the first general treatment of the problem of inversion 
of integrals. 251 ’ 252 It appears to have been Volterra who, in these 
memoirs, first pointed out the remarkable connection which exists 
between the theories of linear algebraic and linear functional equa- 
tions. Altho Sturm in his great memoir of 1836 in the first volume 
of Liouville’s Journal was led to some of the results on the properties 
of solutions of a linear differential equation by considering the 
limiting form taken by the solution of a difference equation 
L u “ +M u + N it = 0, (i -= 0, 1, 2,. . . .) 
i i-f-1 i i i i — 1. 
*Journal fur Mathematik, vol. 103 (1888), p. 228. 
fSopra un problema di elettrostatica, Atti dei Lineei, vol. 8 (3), (1884), pp. 315-318. 
