276 Prof. W. McF. Orr on 



in any homogeneous region, we accordingly have 



DW =#( D f- D z) + K D I- D S)+H4l- D S)}^ 



Replacing dm by pdv, where dv denotes an element of 

 volume, and integrating by parts, this is equivalent to 



f{ [a*p(ng-mk) - ZA 2 A] D£ + US 



where {I, m, n) represents the direction of the normal to the 

 element of surface dS. Replacing A 2 A by p the vanishing 

 of the volume-integral in this expression for all possible types 

 of variation of £, 77, £ leads to the equation 



V dy dz J dx 7 



and two analogues. And the consideration of the surface- 

 integral leads to the conditions that 



cfip (ng — mh) — Ip 



and its two analogues should be coutinuous across any sur- 

 face of discontinuity. These surface conditions express that 

 p should be continuous, and that 



(Pi«i 2 /i —PstoVaffl = (pi«iVi —P^g^/m — {p&\K — p2a 2 %)/n, 



where the suffixes refer to the two sides of the interface. 



The equations obtained from the vanishing of the volume- 

 integral lead by differentiation to 



vv=° 



throughout any homogeneous region ; and by integration give 



^$ = tfp(fdx+9dy + hdz) : 



wherein the left-hand member is taken over any open surface 

 which lies altogether in a homogeneous region, and the right- 

 hand member is taken along the bounding edge. Hence the 

 integral on the left vanishes for any closed surface which does 

 not intersect any surface of discontinuity but may completely 

 enclose any number of such. 



