PuifSER — On Ether Stress, Gravitational and ElectrostaticaL 199 



This represents a normal stress («) for plane containing radius for which 



X cos A + y cos B + z cos C = 0, 

 (b) for plane perpendicular to radius, with which we are at present concerned, 



and for which cos A, cos B, cos C are -, -, - respectively. Putting m 



r r r 



these values, we find for unital pressure 



At the surface this has the value 



4:fjigpcc 



(X + 2/x)15* 



It is to be noted (1) that this expression for the stress does not admit of 

 being evaluated until the constants X, fj. of the ether, or at least their ratio, 

 are determined. The Maxwellian stress, on the other hand, is independent 

 of these constants. Hence, (2) if we suppose the ether q, y, incompressible, 

 i.e., fx very small compared with X, the stress required may be very small, in 

 place of the Maxwellian 4000 tons on the square inch. 



Consider now the case in which the sphere is very small — i.e. where the 

 ethereal stress is due to the presence of a small particle of matter, or, in the 

 electric problem, of the presence of a small electron. 



The displacement is then, as we saw, radial. Its amount is finite for the 



go a^ 

 interior, and for the exterior, with which we are now concerned, = - tv- — . 



' 15 r^ 



Consider now the general case of gravitating matter distributed homo- 

 geneously in space. Then, denoting the constant density by p, the Kelvin 

 expressions for the displacements are the following : — 



u = 



tTTfX 



dq — + idq — A X' -- -^ Y -— + Z — ] 

 r ax \ ax ay clzj 



with analogues for v, iv, where 



£ = - |(X + fx)l\ + 2fX. 



These will be found to yield the following displacement expressions for any 

 point P in the dielectric, x, y, z, r now denoting the coordinates and distance 

 from F of any point in the matter region : — 



47r// 



u = (1 + e) 



— -V = (! + £) 

 P 



4:^^^ 



i^ = (1 + e) 



dq dV 

 r dx 



dq dV 

 r dy 



dq dV 

 r dz 



X dV , 

 ?-d?^^^ 



■ydV , 

 ?'- dr 



'z dV , 



