406 Prof. A. Gray on the Attraction of 



From (37) it follows that the work done against attraction 

 in carrying a unit particle outwards through a small distance 

 dn along the normal is 



- dX =lA^ dn < M > 



27. If then dn be taken for different points of the surface 

 so that pdn is constant, there will be the same step of 

 potential at every point, and the surface on which lie the 

 extremities of these elements, dn, of the normals will be, 

 like the surface of the homceoid, an equipotential or level 

 surface. The distance between this surface and that of the 

 homceoid is inversely proportional to p. It can in fact be 

 shown very easily that the shell of space between the surfaces 

 is nfocaloid, to use the name given by Thomson and Tait to 

 a space bounded by confocal ellipsoidal surfaces. For the 

 equation of an ellipsoidal surface external to and near to the 

 external surface of the homceoid %(x 2 /a 2> ) = k, is 



a* + </«~ ' 



where du is small, and ,i\ y, z are the coordinates of a point 

 on the new surface. If for oc, y, z in the last equation we 

 write %+dx, y + dy, z + dz, and subtract from the result 

 1(x 2 /a 2 ) = k, we obtain for the thickness dn at a, y, z of the 

 shell of space 



P z t(^)=dn^du (39) 



k \ a 2 J p 



Thus the equipotential surface given by dn thus chosen is 

 confocal with the surface of the homceoid. 



28. Let us now imagine the mass of the homceoidal shell 

 carried out along the normal at each point, and distributed 

 on the near confocal surface, so that the mass on any element 

 ds of the shell is placed on the element ds ! which is marked 

 off by normals drawn from the periphery of ds. The shell 

 thus formed will, by Green's principle of equivalent distri- 

 butions, be a new homceoid which will give the same field 

 external to itself as was produced by the original shell. 

 For, take the tubular space marked out by normals drawn 

 from the periphery of c/s, and terminated by two caps, one 

 iust inside <r/s_, but otherwise coinciding with it, the other 

 outside rfs', and fitting closely to that element. The surface 

 of this portion of space is the lateral surface, the inner cap, 

 and the outer. Now take the integral of normal force /"N^s 

 over the whole surface of this space. The inner cap contri- 

 butes nothing to it since there is no force within the homceoid, 



