Davis: Present Status of Integral Equations 25 
G. Bertrand* and H. Poincare have applied integral equations to 
the theory of the tides and J. Hadamard to water waves. Poincar6 
studied) also, the diffraction of Herzian waves by this method, and 
T. Hayashi applied it to the Cauchy problem of the telegraphy 
equation. G. Herglotz and P. Hertz used integral equations in the 
theory of electrons. Thermoelasticity has been formulated in terms 
of integral equations by L. Koschmieder and H. Laudien, and as 
early as 1837 J. Liouville used them in the phenomena of thermo- 
mechanics. 
J. A. Carson in a series of important papers 493 , 494 has renewed 
interest in the operational calculus of 0. Heavyside which has 
been used effectively but with admitted lack of rigor in circuit 
problems in electricity. Carson’s contribution has been in con- 
necting the theory of this calculus with an integral equation of 
Laplace type. The Heavyside theory depends essentially upon an 
operational equation of the form h=l/H(p), where p is symbolically 
d 
equivalent to the differential operator — . Carson has shown that 
dx 
this equation is equivalent to an integral equation of the form 
f 00 
l/pH(p) = I h(t) e pt dt and has found its solution for a variety of 
o 
functions. One interesting feature of the theory is the occurrence of 
asymptotic series representing the solution of the original circuit 
problem. It seems that this application of integral equations is 
likely to renew interest in the formal operational calculus. Indi- 
cations of this trend are to be seen in the use of fractional derivatives, 
and integrals in the solution of integral equations 79 and in recent 
important contributions to the subject of general operators and the 
Heavyside Calculus by N. Wiener f and E. G. Berg.J 
E. Picard and R. Marcolongo have written general articles on 
the applications, and A. Kneser, in his treatise, has studied exten- 
sively problems in the conduction of heat and the theory of 
vibrations. To the first of these problems an important addition 
was made by H. S. Carslaw in 1912, and to the latter H. Weyl, in 
1912, and E. Trefftz, in 1922, contributed the results of new re- 
searches. In 1906 Fredholm founded a theory of spectrum lines, 
reducing the problem to one in integral equations. Three years 
later C. Schaefer modified the theory in some essential details. 
The study of H. Bateman in a mathematical theory of retail 
*For references see section I of the Bibliography. 
tThe Operational Calculus. Math. Annalen, vol. 95 (1926), pp. 557-584. 
JHeavyside’s Operators in Engineering and Physics. Journal Franklin Institute, vol. 198 (1924) 
pp. 647-702. 
