404 Prof. A. Gray on the Attraction of 



density (amount of attracting or repelling matter per unit 

 area) in the neighbourhood. In the language of the potential 

 this is expressed by the equation 



where a is the surface-density, and dY/dn denotes the rate 

 of variation of potential per unit of distance outwards along 

 the normal. This equation is at once- transferable to gravi- 

 tational attraction, and a becomes the density of a thin 

 stratum of ordinary matter. Here cr of course denotes the 

 surface-density of ordinary matter ; e. g. the mass ^ppdk/k of 

 a homoeoid taken per unit area at an element just outside 

 which the normal force —dV/dn is taken. 



25. I shall not discuss the properties of level surfaces 

 here; but merely apply some of the properties I have 

 mentioned to the problem of the ellipsoid. But a theorem 

 of Bertrand may be referred to of which our process will 

 afford an illustration. Let there be a family of surfaces 

 represented by the equation 



f(x,y, :) = u (33) 



where a is a variable parameter ; and let them be such that 

 if a distribution of matter be placed on the surface, S say, 

 characterized by any chosen value of a, and be made of 

 surface-density inversely proportional to the distance from 

 that surface to an adjacent one of the family, the whole space 

 within the surface is at uniform potential. Then the surfaces 

 external to S are level surfaces for the distribution specified. 

 The truth of this theorem may be seen as follows. Let 

 the distribution specified be made on an inner surface, 8! 

 say, of the family : the space within is at uniform potential, 

 and therefore so also are all points of the surface. But if cr 

 be the density at any point of the surface, then just outside 



~^ Y -47ra- (34) 



dn v 7 



Now the step from the surface S x to an adjacent one S 2 

 may be taken as dn, and being inversely proportional to cr 

 gives a constant difference of potential between Sx and S 2 . 

 Hence S 2 is a level surface for the distribution on S x . Let 

 now the distribution be transferred to S 2 , and be made 

 according to the law set forth in (32) for the values of dYjdn 

 which exist at 8 2 with the distribution on S r Since S 2 is a 

 level surface for the distribution on S 1? the transference thus 

 effected will bring the whole space within S 2 to uniform 



