Colloquium publications.

FUNCTIONALS AND THEIR APPLICATIONS.

117

If now, as in Lecture III, equation (23), we define
flU2 = 1, U1 < U C U2,
= 0, otherwise, etc.,
and also define the new symbol
22 b U 2 b
(25) JtuK(St) = T[fu2], JtuK(st) = T[fau],
U1 a a a
then we have
U2 q42
(26) a(su2) - a(sul) - JtuK(St), a(S2) = JtuK(St),
u1 a
which gives us a in terms of K.
Let us now divide the interval ab into sub-intervals by points
a = u0, ui,., un = b, and let si be a point in the region uji< si < ui. On account of the continuity of the linear functional, the following equation may be deduced:
n
(27) JtuK(ut) = lim Zi(a(SiUi)- a(sii-1)).
n=oo 1
The left-hand member of (27) is one of the constituents of the
b
Fredholm determinant, corresponding to f K(tt)dt in the theory
of integral equations.*
~ 3. THE THEORY OF PERMUTABLE FUNCTIONS, AND
COMBINATIONS OF INTEGRALS
71. The Associative Combinations of the First and Second
Kinds. In this second form of generalization of the theory of
integral equations, what is of primary importance is an associative combination of integrals. In fact, Volterra's theory rests
upon the identity
b rb
) dsiKl(rsi) I K2(sls2)K3(s2s)ds2
(28) rb rb
= i j[b Ki(rsl)K2(sls2)dsl K3(s2s)ds2,
* See footnote at end of Art. 62.