

Ellipsoidal Shells and of Solid Ellipsoids. 389 



Now, if r be the distance of Q from any element of the 

 shell A, it is also the distance of P from the corresponding 

 element of B. If /3, ft' be constant multipliers, as already 

 explained, the potential at Q due to the matter at rfs is 

 fipds/r, and the potential at P due to the matter at d*' is 

 fi'p'd^/r *. The former bears to the latter the ratio t3abc/{3'a'b'c' . 

 Since this is true of every element the total potentials have 

 the ratio just stated, that is the ratio of the masses of the 

 shells. 



It follows from this theorem that if the potential at an 

 internal point is known for a thin elliptic homoeoid, the 

 potential at an external point can be found, and vice versa, by 

 considering a confocul shell. 



7. These results are true whatever the law of attraction 

 may be, if only it is a function of the distance, as may be seen 

 by substituting f(r) for l/> in the expression for the potential 

 of an element. In the case of ordinary matter the law of 

 attraction is that of the inverse square of the distance, and 

 the potential determined is commonly called the Newtonian 

 potential. 



It is well known (and it will be referred to again presently) 

 that the Newtonian potential of a homoeoid is the same at 

 every internal point, and therefore also at every point of the 

 shell, since the potential is not discontinuous at points within 

 attracting matter of finite volume density, or even at passage 

 across a surface of finite density. Thus in order to find the 

 potential at an external point P, due to a given elliptic 

 homoeoid, it is only necessary to imagine a confocal homoeoid 

 of equal mass constructed so as to have P on its surface, and 

 find the uniform potential which it produces at every internal 

 point. This is the potential required. 



8. It follows from this result that the external confocal 

 ellipsoidal surfaces are the equipotential surfaces of a uniform 

 homoeoid, and that such a shell is itself at uniform potential. 



* We take here as tlie specification of the potential 



dm 

 V=2-r 



w 



here r is the distance of the point for which the potential is defined 

 from an element dm of the attracting matter, and % denotes summation 

 for all elements. Here the unit of mass is that which concentrated at 

 unit distance from an equal mass, also concentrated at a point, is attracted 

 with unit force. When the ordinary unit of mass, the gramme, say, is 

 used, the right-hand side of the equation for V must be multiplied by the 

 value of the force of attraction which exists between two such units 

 placed at unit distance, a centimetre, say, apart. This multiplier is 

 called the " gravitation constant." It is generally omitted (that is taken 

 as unit} T ) in what follows : where it is inserted it is denoted by k. 



