32 Proceedings of the Royal Irish Academy. 



moment in an element of area is normal to the surface. In the former case 

 we have what is generally called a double-sheet, whose strength is the normal 

 moment per area. The possibility of the latter case leads to the conception 

 of a tangential doublet distribution in a surface, in which the moment in an 

 element of area is tangential to the surface, and the strength is a tangential 

 vector. It will be worth while to inquire later into the properties of a 

 tangential distribution, but the double-sheet and its bearing on applications 

 of potential theory will be considered first. 



3. Uniqueness Theorems for a Double-Sheet. — Let 8 be a closed surface, 

 and let r 1; t 2 be the strengths of two double-sheets in the surface S which 

 separately produce the same field of potential in all space outside S ; then r t 

 and r 2 can differ only by a constant. For if we put t x - r 2 = r, we see by 

 superposition that a double-sheet of strength r produces zero potential in the 

 region outside S ; thus r produces no normal force outside S, and as normal 

 force is continuous in crossing a double-sheet, there is no normal force just 

 inside S, and therefore no force at all. So r produces constant potential 

 inside S and, as the discontinuity of potential in crossing *S is 4ttt, t must be 

 a constant. That this constant is not necessarily zero is corroborated by the 

 well-known fact that a double-sheet of uniform strength in a closed surface 

 produces zero potential at outside points. 



If S be a closed surface, and r x , r 2 the strengths of two double-sheets in 

 the surface S which separately produce the same field of potential in the 

 space inside S, then must t 1 = r 2 . For if we put r, - r 2 = r a double-sheet of 

 strength r produces zero potential in the inside region, so there is no normal 

 force just inside and therefore no normal force just outside. Thus there is 

 no force and therefore constant potential outside, so the discontinuity of 

 potential 4ttt must be constant. But it is known that a double-sheet of 

 uniform strength r produces potential + 4ttt at points inside, and as the 

 potential inside is known to be zero we must have t = 0, i.e. r, - r 2 . 



4. Notation. — It will make for brevity to introduce a special notation. 

 Let V(p, q, r, s) stand for the potential at a variable point P due to a 

 combination of gravitational distributions of different kinds represented 

 symbolically by the letters p, q, r, s. Thus if we have a surface density <?, 

 a volume density p, a double-sheet of strength r, and particles typified by m, 

 the potential due to all these simultaneously at a point P is denoted by 

 V (a, p, t, m). 



In general we shall have to do with a surface S which divides space into 

 two regions ; one of these we shall call the " relevant " region, and we shall 

 distinguish the potential and other functions associated with this region by 

 the suffix (,) ; the other we shall call the " irrelevant " region, and distinguish 



