AbstractsMathematics

I. The principal object of the following paper is the discussion of a Neumann problem, with reference to a potential of a single layer which is based on a general distribution of matter on a simple closed plane boundary. Such potentials were introduced by Plemelj. The result obtained here is of the same order of generality for these boundaries as that obtained by G. C. Evans with the aid of conformal transformations, but the present method is entirely different, and simpler. The problem is equivalent to a Stieltjes integral equation, which is solved by reduction to the classical Fredholm type. II. The potential due to the most general distribution of finite positive and negative mass deposited in a single layer on a closed surface S may be written in the form vM=S 1MPdm&parl0;e p&parr0;, 1 where the mass function me is a completely additive function of point sets e on S. The most general distribution of mass in a double layer on S yields similarly the potential uM=S cosM P,np MP2dn&parl0;ep&parr0;, 2 where ne is likewise a completely additive function; here np denotes the direction of the interior normal to S at P.