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N)\ < T V ^ \LAM^M;N)\ δ*α where Lx(M', M; N) is a continuous function of M', M and N in B, vanishing when M' coincides with M. Hence it follows that a positive number η exists, which does not depend on the position of the point M and which can be assumed not greater than δ, such that J \K(M'i N) - K(M; N)\ dcoN < ·£- , (108) if the distance | MM' | is not greater than η. On taking (107) into account, we get \v{M') - v(M)\ < e, if \ΜΜ'\<η, which proves the continuity of v(M) in B. e. with fixed Gx we get a family of equicontinuous functions v(M) for all the functions u(N) satisfying condition (104).

We observe t h a t the 32 [8 INTEGEAL EQUATIONS absolute term in the expansion of D0(s, t; X) in powers of (λ — λ0) is a function of (s, t). I t may vanish for certain particular values of s and t, but is not identically equal to zero, since if this were so λ = λ0 would be a zero of D(s, t; λ) of multiplicity greater t h a n I. We can formulate the theorem just proved more strictly as: there exist values of s and t for which λ = λ0 is a pole of the resolvent. We have shown t h a t every zero A0 of the function Ό(λ) is a pole of the resolvent.

Throughout the and write it as fP {x) - fq (*) = Up (a) - fP (*')] + [fp (*') - U (*')] + + Uq(x')-fq(x)l (a) where x' is one of the points of the above-mentioned set, everywhere dense in [a, &]. Let ε be any given positive number and η the number corresponding to it in the definition of equicontinuity. We take a finite set r'consisting of points xk and such t h a t the points of the finite set divide the interval [a, b] into subintervals, the lengths of which are < //. This is obviously possible, since the set of all the points xk is everywhere dense in [a, 6].