330 Mr. K. Hargreaves on Radiation 



This completes the more important algebraical connexions 

 between the two types of variables for a plane wave which are 

 useful for the interpretation of formulae, and in reflexion and 

 refraction. We may note also that by (II.), 



kxx(x / -x)=y- i 2X( V7 -^) 



= Y- t %*(wY-vZ)=M.%(cJ-*)a, 

 so that 



K2XX' = MW whenever K2X 2 = M£« 2 , 



as for plane waves. 



§ 11. We proceed to the equations for the movement of 

 energy, first within the medium, and then across a surface of 

 separation. We have 



K 2 x!( X '-X) = V- 12X (^- W f) 



-V-^(.T-^)-lC6j(^> 



and again 



M2*i («'-*) =Ksg(x'-X> 



Ilence, by addition, 



KSX^-' + WZ« d 4 = K2rf+ MS*''*" 

 at at at at 



and therefore 



%■ f^SXX' + ^tJ\ = 2K & + SM p a ' 

 dt\2 2 / at at 



= W rf /-f)x' + V2(^-fV 



\dy dz ) \dz ay ) 



S^^(X^-YV)=0; . . . (33) 



i. e., change in S is due to the flux of which V(X'|3 / -Y / « / ) is 

 the ^-component. For a plane wave this is ?S by (31), 

 £ being ^-component of ray-velocity, and then (33) is 

 expressed by 



The flux across the ^r-face contains precisely those quantities 



