Light from Transparent Matter, 89 



for 



5. dP drj d*6 d Cz cj) . , 2 mxX 



It is therefore hardly correct to call the surface-waves expressed 

 by <j> longitudinal. They are more allied to those motions with 

 which we have so much to do in hydrodynamics, which involve 

 neither rotation nor yet change of volume. 



Since - =tan#; — = tan p 



,2 



a? + b 2 ~ sin 2 ~ r 2 " tf 



which expresses the ordinary law giving the direction of the re- 

 fracted wave. 



We have now to satisfy the boundary conditions. From the 

 continuity of displacement, 



a'<j> + b{f + y) = fl /^ + bf t , 1 

 b (ji-a^ -Y)=b <t>- aft^J 



or, on introducing the values of a!, a/, and putting yjr' + \jr" = X, 

 i^ + ^^^-X, \ ^ 



Were we to ignore the surface- waves altogether and put </> = (/>,=(), 

 equations (8) would give us 



X=f„ Y=f*,; 



whence 



1_ a j 

 tJt"_ X-Y __ g _ sin (6>,-6>) 



t / "'X+Y"" 1+ « L - S in(^+^ 



FresnePs first expression. This is exactly what has been done 

 by Zech*, and is in fact merely a translation into analysis of 

 MacCullagh's fourth principle. The worthlessness of the argu- 

 ment is sufficiently shown by the consideration that no assump- 

 tion has yet been made as to the relations between n } n 1 , D, lV, 

 other than that implied in taking the ratio of the wave-velocities 

 equal to fi. It is as necessary to satisfy the second pair of 

 boundary conditions, expressing the continuity of stress, as the 



* P°gg- Ann. vol. cix. p. 60. 



