606 Dr. T. H. Havelock on Surfaces of 



The wave-front S is defined as the position at a given time 

 of a surface given in three-dimensional space by 



f(.i;y, Z ,t) = Q (4) 



Now consider x, y, z, t as space coordinates in a four- 

 dimensional space : and let 2 be the multiplicity given by 

 equation (4). Then the various positions of 8 are the 

 sections of 2 by planes t = constant. 



Now the relations of identity were obtained by considering 

 variations on the surface S, and in an exactly similar manner 

 it is clear that the conditions of compatibility are to be 

 obtained by the same process applied to 2. And in general, 

 for a discontinuity of order n in a function <3>, we obtain 

 instead of (1) the relation 



(£]- '£]*♦ B>+ &]*)'• 



Thus for a discontinuity of order n in the displacement 

 (? V> ?) a single vector (\, a, v) arbitrarily given over 2 is 

 sufficient to define the variations in all the derivatives of 

 order n, provided the wave-front is persistent. The relations 

 can be written in the form 



([»> + S]^[s> + El>)«*o 



= (X, ft v)(g 8 ., + | 8 , + |- &+ | & )I . (6) 

 which is to hold for all values of &c, By, Bz, Bt. 



§ 6. The^Ether as a Rotationally Elastic Medium. 



Having obtained the kinematical relations, we must con- 

 sider now the dynamical equations of the particular medium. 

 Let (X, Y, Z) be the electric force measured electrostatically, 

 and (a, /3, y) the magnetic force in electromagnetic units; 

 then we have the usual circuital relations in the form 



^(X,Y,Z)=Curl(«,/3,7) j 



} (7) 



- ;|(«,A7) =Curl(X,Y,Z) 



