Davis: Present Status of Integral Equations 17 
The first of these equations will be seen to be an example of the 
type which Volterra classified under the name of int egro-diff&r ent i al 
equations. Important extensions, suggested by problems in elas- 
ticity, hereditary mechanics, electrodynamics, etc., have been made 
by Volterra 325 in the form of equations in several variables, as for 
example, 
d 2 u(t) d 2 u(t) d 2 u(t) /»t d 2 u(s) 
1 j 1_ I s q (t >s ) ds = f(x,y,x,t)' 
. dx 2 dy 2 dz 2 • o ax 2 
a 2 u(s) a 2 u(s) 
where u(t) EE u(x,y,z,t) and S f x (t,s) = fi (t,s) 
dx 2 dx 2 
5 2 u(s) < 3 2 u(s) 
d f 2 (t,s) H — f 3 (t.s). 
dy 2 dz 2 
In the exploration of this field the Italians, under the stimulus 
of Volterra, have taken a leading part. 
Another development closely connected with the subject of 
functional equations in general, and integral equations in particular, 
is that which has arisen from the idea of “functions of lines.” This 
line of thought was first set forth by Volterra in 1887 in a paper 
under the title “Sulle funzioni che dipendono da altre funzioni”* 
and has reached maturity in a volume published in 1913. 191 In 
Volterra’s words the idea may be explained as follows :f 
An ancient problem, that of Zenodorus, has been to find among the plane 
curves of given length that which encloses the largest area. Now, if we study 
this isoperimetric problem and regard a plane area as depending upon the 
curve which encloses it, we have a quantity which depends upon the form of a 
curve, or, as we say today, a function of a line. Since a line is able to be repre- 
sented by an ordinary function, the area can be regarded as a quantity which 
depends upon all the values of a function. It is evidently a function of an 
infinite number of variables. In fact, we are able to look upon it as the limiting 
case of a function of several variables by supposing that their number increases 
without limit, in the same way as a curve can be regarded as the limiting case 
of a polygon whose number of sides increases to infinity. 
In the analytical treatment of problems involving functions of 
lines, as for example elastic and magnetic hysteresis, Volterra was 
led in general to integral and integr o-diff er ential equations. 
T. Lalesco in 1908 made an original contribution to integral 
equations which deserves special mention. In the second part of his 
thesis, “Sur Y equation de Volterra” 287 which appeared in the Journal 
de Mathematique, he has pointed out the connection existing between 
integral equations and differential equations of infinite order. 
*Atti dei Lincei (18871, (2), pp. 97-105, 141-146, 153-158. 
tRef. 191, p. 14. 
