448 TRANSACTIONS OF SECTION A. 
is the unknown function, An important question is whether there is more than 
one continuous function #(¢) for which the equation is satisfied, supposing it to be 
soluble ; and this leads us to the consideration of the homogeneous equation 
o-[ GG, op(oat™ Le 
If no continuous solution (other than @(¢)=0) of this equation exists, the charac- 
teristic function g(s, t) is said to be perfect tor the ranges (c, d) and (a, 6). 
§ 2. The function ¢(s, ¢) in equation (1) is in general not asymmetrical function 
of its arguments, but if we multiply both sides of the equation by g(s, 2’) and 
pais fe with regard to s between c and d we obtain an integral equation of the 
first kin 
Aw, NES 5c See ead se 
a 
with 
d ad 
fAa)= | Heyl, «ds, (2, o-| ie; oats, tia 
in which the characteristic function «(2, ¢) is a symmetrical function, so that the 
problem is reduced to one of a simpler character. 
§ 3. Starting from equation (3), Professor Hilbert has indicated a very general 
class of symmetrical function «(s, ¢) for which the solution of (8) is certainly 
unique. ‘he characteristic property is that if o() is any function which is con- 
tinuous in the interval (a, b), the double integral 
1 K(3, v)w(so(a)dsde =... ) 
ava 
is essentially positive. A function x(s, 2’) which satisfies this condition is said to 
be definite. ‘The function x(x, ¢) given by equation (4) is easily seen to be definite 
if g(s, t) is perfect for the ranges (c, d) and (a, 6). 
When one definite function is known it is easy to construct any number of 
others; for instance, if x(a, ¢) is definite, and 
A(x, t)=e(a, Daft).  . . . « © 
U(x, o-|| K(8, Ys, ay, thdsdy « « (@) 
ava 
the function A(2, ¢) is definite, and (x, ¢) is definite if f(s, 2) is perfect. 
A criterion for determining whether a function x(2,¢) is definite or not is 
furnished by the following theorem due to Hilbert :-—— 
‘If x(a, t) is perfect, the necessary and sufficient condition that it should be 
definite is that the singwar values of d for which the homogeneous integral equation 
of the second kind 
ve) -a| x(s, tp(t)dt=0 ey 
can be satisfied should be all real and posztive.’ 
An elementary proof of this theorem may be based upon the fact that if K(s, ¢) 
is the solving function of the integral equation of the second kind 
b 
fe)=x)- nls, *)y@)dt teu, Oe 
a 
