60-4 Dr. T. H. Haveiock on Surfaces of 



These latter quantities are not absolutely independent. 

 For if we vary <I> along a path on the surface S, first in region 

 1 and then in region 2, we have 



where the variations &p, hy, Sz are connected by the relation 



d.y By J B^ 

 Hence we have the relations 



rB^I kf = rB$"i l~bf_ = rB^-i \bf 

 LB^ J/b« L^y J/ By La* J IBs' 



Again, if we suppose <3> and its first derivatives continuous, 

 but the second derivatives discontinuous on S, we find in a 

 similar manner the following relations among the second- 

 order discontinuities — A being an arbitrary parameter : 



m=<n [Pmp m~m> 



LByB^J "dy'dz* !_B~B^J 'dz'dx' LB^ByJ "d^c'df 



And, in general, if <1> and its derivatives are continuous up 

 to order n — 1, the relations between the discontinuities of 

 order n can be written in the form 



(Cs>- [|>+ [&M* 



§ 3. The Order of a Discontinuity. 



We are to consider #, ?/, # as the actual coordinates of a 

 particle of an elastic medium ; thus if a, b, c be the initial 

 or natural coordinates of any particle, we have 



(a?, y, s) = (a + f, 5 + 77, c + f), 



the displacement (f , 77, f) being considered small. 



We shall consider discontinuities in (f, 7;, f) and their 

 various derivatives with respect to the time t and the co- 

 ordinates (a 9 y,z). An absolute discontinuity is one which 

 occurs in (f , 77, £) itself ; and if n be the order of a derivative 



