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IV. 



ON DOUBLET DISTRIBUTIONS IN POTENTIAL THEOEY. 

 By J. CI. LEATHEM. 



Read May 11. Published September 3, 1914. 



1. In the study of Potential Theory there is a tendency to regard doublet- 

 distributions as of little practical importance except in so far as they are 

 relevant to the theory of magnetism. When the student has gone through 

 the investigation of the force in cavities of various shapes cut in magnetized 

 matter, and the discontinuity of the potential due to a double-sheet, he hears 

 little more of doublet-distributions. As a matter of fact such distributions 

 present themselves, if only as interpretations of purely mathematical ex- 

 pressions, in the formulation of other physical problems which are susceptible 

 of treatment by the analysis of potential theory ; but it is perhaps the view 

 of many that it is unprofitable to reformulate problems on the subject of 

 whose solution the mathematician feels that he is not likely to be able, with 

 the weapons at present available to him, to add to the existing body of exact 

 knowledge. 



The present writer, confessing at the outset that he has little to offer in 

 the way of new results, nevertheless thinks it worth while to fill a gap in the 

 current presentations of potential theory by examining some of the properties 

 of doublet-distributions corresponding to familiar properties of ordinary 

 surface and volume densities. In doing this it has seemed useful, for 

 suggestion and illustration, to indicate how, in the case of the application of 

 the theory to the motion of a liquid, the fundamental problem presents itself 

 simply as a double-sheet problem, and to study the matter in this aspect. 

 Though in hydrodynamics the ground has been too thoroughly explored to 

 leave any hope of obtaining fresh exact results by old methods, there is 

 always a chance that a new and concise formulation of a standard problem 

 may put some student on the track of an approximate solution of practical 

 importance. Considering how simple is the fundamental property of liquid 

 flow, the failure of the mathematician to obtain (except in the very simplest 

 cases) even roughly approximate specifications of the flow corresponding to 

 boundaries of given form and motion, is a striking exemplification of the 

 limitations of modern analysis. 



A surface concentration of doublets may or may not be such that the 



K.I. A. PROC, VOL. XXXII, SECT. A. [5] 



