196 Proceedings of the Royal Irish Academy. - 



proportional to the square R- of resultant electric force along the lines of 

 force, with an equal tension in all directions perpendicular to them. In 

 the gravitation problem the stress will consist of a pressure along the line of 

 resultant force, with an equal tension in all directions perpendicular thereto. 

 This stress then will account, if a possible one, for the gravitating force on 

 a particle of matter, or, in the electrical, for the electrical force on a particle 

 of free electricity in the dielectric, and hence for the normal stress at the 

 surface of conductors. It is, however, impossible to find a system of strains 

 corresponding to this stress. 



In the case of a homogeneous isotropic medium the attempt, in fact, to 

 connect this system of stresses with strains has been shown by Dr. "Williamson 

 to- lead to the absurdity of making R - constant where there is no gravitating 

 matter in the one case, and, therefore, where there is no free electricity in the 

 other.* More generally, however, let the ether be supposed of the general eolo- 

 tropic form with Green's twenty-one constants. Xow the strains «, &, c,/, ^, h 

 are expressed as Linear functions of A,B,C,F, G, H, containing the twenty-one 

 constants. IMoreover, it is known that a, h, c, f, g, h satisfy six linear 

 equations in their second differential coefficients. A corresponding set of six 

 equations obtains then in A, B, C, F, G, H, these having the values written 

 above. This would then lead to the absurdity of conditioning the distribution 

 of matter by the elastic constants of the ether. 



AVe must then, in attempting to solve our problem, whether in the 

 electrical or gravitational form, commence with a system of strains or, which 

 is the same, of displacements. 



It is now at once seen that we have only to avail ourselves of the solution 

 given in Thomson and Tait's "Xatural Philosophy," Part II., Art. 731,t of 

 the problem of determining the displacements in an infinite solid, to a finite 

 part of which given bodily forces are applied — a problem fundamentally 

 identical with our present one. 



Before, however, applying these formula; in their general form, it is con- 

 venient to discuss directly the gravitational case in which all the gravitating 

 matter is confined to a sphere, in the interior of which it is uniformly 

 distributed. This is, in fact, the case of the ether strains produced by the 

 gravitation of the Earth. It is now obvious that we may assume for the 

 displacements in the ether the form u = Rx, v = By, v: = Bz, where i? is a 

 function of the distance from the centre. 



* See Williamson, "Elasticity," p. 86. 



t Compare also Love, "Elasticity," vol. i., p. 258, 1st edition, where the equations are given in 

 a slightly different form. 



