24 
Indiana University Studies 
and an initial velocity Lo 0 - Then from (11) we shall have 
0(t) = 6 0 + « 0 t - X f (t -s) sin 0(s) ds, X = g/L. 
J o 
This integral equation, is non-linear, but could be solved by 
methods already existing in the literature. Assuming, however, 
that for small values of 0, sin 6 may be replaced by 6, we shall have 
a Volterra equation of second kind, 
0(t) = 0 Q + Wo t - X f (t - s) 0(s) ds. 
J o 
The resolvent kernel, for this equation is easily calculated to be 
k(t,s) = sin Vx (t -s), 
Vx 
so that the solution appears in the form, 
0(t) = + w o t - Vx f sinVx (t -s) (0 + w o s) ds, 
J o 
Vxel + <ol _ _ Vx«„ 
sin (V X t + 7), where 7 = arc sin— — . 
Vx V x»j + 
We have already indicated how Fredholm was led to his dis- 
coveries from a study of the Dirichlet problem in the theory of 
potential functions. This problem may be briefly stated as follows: 
Along a curve C, enclosing a region S, a continuous function f(s) 
is given. A function u(x,y) is sought which, together with its 
partial derivatives of first and second orders, is continuous within the 
region S and satisfies Laplace’s equation 
d 2 U d 2 ll 
— + — = 0 , 
dx 2 dy 2 
and upon the boundary C, assumes the given value f(s). The 
existence and uniqueness of a solution for this problem are the 
essential points. 
The associated Neumann problem is that of finding a function 
v(x,y), satisfying the same conditions as u(x,y) with the exception 
of the last one, which is replaced by the condition that the normal 
derivative along C shall assume the given value f(s). 
Complexities arise when the character of the curve C is modified 
and we have already mentioned the investigations of Kellogg and 
Hilbert in connection with curves admitting corner points. G. Bert- 
rand, 26 G. Lauricella, 50 P. Levy, 51 E. Picard, 57 J. Horn, 10 H. B. 
Heywood, 8 M. Frechet, 8 and J. Plemelj 64 are among others who have 
made important contributions to this subject. 
