8 
Indiana University Studies 
series, and his summation formula is essentially a solution of the 
Laplace equation when S is the real axis from 0 to °° . P. Humbert 216 
published results for a more general type of equation in which e xt 
is replaced by f(x,t), and J. R. Carson 78 recently found an applica- 
tion for the Laplace equation in the theory of electric circuits, in- 
dicating its intimate connection with the operational calculus of 
Heavyside. M. Lerch* in 1892 discussed the homogeneous case 
where f(x) is identically zero and the integral is taken along the real 
axis from zero to infinity. 
We are also indebted to Laplace for the integral equation 
f(x| = f t x_1 ll(t) dt, 
J s 
which Riemannf used in his classical researches on the distribution 
of prime numbers and which is important in investigations on the law 
of error. J. Liouville 482 as early as 1837 considered the existence 
theorem for the homogeneous problem 
f x" u (x) dx = 0, n = 1, 2, 3, . . . . 
J a 
which is of importance in connection with the uniqueness of solutions 
of the non-homogeneous equation. This equation was further studied 
by C. Severini 238 who made use of it in connection with the closure 
property of orthogonal functions. 488 
After the work of Laplace one of the most noteworthy discoveries 
in the history of integral equations in spite of its appearance so 
early in the development of the theory was made by J. Fourier in his 
memoir on the “Theorie du mouvement de la chaleur dans les corps 
solides” which, altho crowned by the French Academy in 1812, was 
not published until ten years later. Here for the first time appears 
the integral equation 
r 
f(x) = I cos xt u(t) dt, 
and, with proper restrictions on f(x)t, the remarkable inversion 
formula 
2 /•<*> 
u(t) = — I cos xt f(x) dx. 
X J o 
To enumerate the various researches to which these equations 
have led would be to review a large chapter in modern mathematics, 
♦Acta Math., vol. 27 (1903), pp. 339-351. 
vol. 1, no. 33. 
fWerke (Weber), Leipzig (1892), p. 149. 
tFor example, 
/: 
f f (x) I dx c 
First published, Rozpravy ceske Akadamie, 2d class, 
