30 MR. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND 



Introduce a new function W, defined by 



w = v,.-v 



at all points of space, then we must have 



V 2 W = - 



at all points of space, and, at the boundary, W = and : = 

 These last two conditions are equivalent to 



dW dW 



dx dy dz 



= (3) 



at the boundary, together with one further condition. Equations (3) require that W 

 shall have a constant value all over the boundary ; the further condition is that this 

 constant value shall be zero. 



4. Let F (x, y, z, X) = be the equation of a family of surfaces obtained by varying 

 the parameter X, and such that the boundary of the solid is the surface X = 0. The 

 surfaces of this family will divide up the solid into a series of thin shells. There will 

 be a contribution from each shell to V, and also to V,,. Thus W may be regarded as 

 the sum of a number of contributions, one from each shell. 



Let the thicknesses of the separate thin shells be determined by small increments 

 in X, say d\ lt d\.,, ... , starting from the boundary X = 0. Then we may write 



:<&,)+ - (4) 



where Y,-(f7Xj) represents the contribution to V; from the shell d\ t , and so on. 

 Similarly 



V. = V,(dX 1 )+V (dX i )+ : (5) 



Suppose that V,-, V , and W are being evaluated at a point x', y', z' on the shell d!X, 

 at which the value of X is X'. Then if d\ t is any shell inside the shell d\ s , the contribu- 

 tions to V< and V from the shell d\ t will be the same ; we have V,- (d\ t ) =- V (d\ t ). 

 Hence from equations (l), (4), and (5), 



. W- V,- V = (V.^xO-V^x,)} + {V,.(dx 3 )- V (d\ a )} + ... + (V^x.)- V (dx.)}, (6) 

 or expressed as an integral, 



W = ?'<S>(x',y',z',\)d\ .......... (7) 



Jo 



5. This form for W satisfies automatically the last of the conditions of 3, namely, 

 that W shall vanish at the boundary. We proceed to determine 4> so as to satisfy the 

 remaining conditions which are expressed by equations (2) and (3). 



