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LXVII. On Surfaces of Discontinuity in a Rotationally Elastic 

 Medium. By T. H. Havelock, M.A., D.Sc, Fellow of 

 St. John s College, Cambridge* \ 



§ 1. Introduction. 



THE kinematics o£ wave propagation have been studied in 

 great detail recently by Duhem f and Hadamard %. Their 

 method consists in following the motion of surfaces of dis- 

 continuity in the fluid and in developing in connexion there- 

 with the idea of compatibility or persistence, due originally 

 to Hugoniot. 



In §§ 2-5 of the following paper a short account of this 

 method is given, and the rest of the paper consists of an 

 application to the electric sether when this is considered as a 

 rotationally elastic medium ; the production and mode of 

 propagation of discontinuities of various orders are discussed, 

 together with the relations of various vectors at the surface of 

 discontinuity. 



§ 2. Surfaces of Discontinuity in general. 



The discontinuities which we shall consider are not perfectly 

 general, but are limited by the condition that they are situated 

 on isolated surfaces. Let <I> be any function of #, y, z, t and 

 the derivatives of the coordinates with respect to the time, 

 and let S be a surface on which <J> is discontinuous at any 

 time ; let S be given by 



The surface S divides the space into two regions 1 and 2. 

 At each point of the region 1, <l> is continuous and takes a 

 definite value <!>! ; further, at each point (# , y , z ) on the 

 surface S, <£> takes the definite value <£>J and is said to be con- 

 tinuous provided the path by which (x, y, z) approaches the 

 point (x , y , z ) lies entirely within the region 1. 



Similar lv for the region 2 there is a limiting value <3>° 

 which the function <I> takes at the point (x , y , z ) when 

 approached by a path lying entirely in the region 2. 



The difference &° 2 — <I>f is the measure of the discontinuity 

 of <3> on the surface S and is written [<!>]. 



Next suppose that <E> itself is continuous on S, but that its 

 first derivatives are discontinuous ; that is, 



m- [Hi' m- 8?]*° 



* Communicated by the Author. 



t Duhem, Recherches sur V Hydrodynamique, Paris, 1004. 



X Hadamard, Lecon* sur la propagation des Ondes, Paris. 1003. 



