Discontinuity in a Rotationally Elastic Medium. 605 

 such as 



at 



UJ~ x d. c ; 



L37J - x w 



rBfi x b/. 



fto-l-Jf. 





r^i_ b/. 



rBri b/ 





LbJ Be' 



v ^-^T775 where p + q-t-r + s — n, 



then the order of a discontinuity is the lowest of the orders of 

 the derivatives in which discontinuity occurs. 



§ 4. Relations of Identity. 



Consider a discontinuity of the first order in (f, rj, J). 

 Then from the general results in § 2 we have 



(2) 



where \, fi, and v are arbitrary parameters. Thus a single 

 vector (X, p, v), given over the surface S, is sufficient to 

 define these nine discontinuities of the first order. 



Similarly for a discontinuity of the second order ; and in 

 general for one of the nth order the relations between the 

 derivatives not involving the time can be expressed by means 

 of a vector (X, fi, v) in the form 



([s> + [|V ft] «-)■«*» 



= ( w)(|js,«+gs J+ g&):' . . (3) 



§ 5. Conditions of Compatibility. 



The notion of compatibility was first introduced by 

 Hugoniot into the study of fluid motions in two regions 

 separated by a surface of discontinuity, and the same idea 

 is used by Duhem under the name of " persistence." 



The relations of identity found in the previous section are 

 those which hold at a given surface of discontinuity S at a 

 given time t Q . The motions in the two regions separated by 

 S are said to be compatible if the surface S is unique not only 

 at the instant t but also at times immediately before and after 

 that instant ; the moving surface S is then a wave-front and 

 is said to constitute a persistent wave. There are certain 

 kinematical relations which are satisfied whatever be the 

 dynamical equations of the particular medium. 



