1880.] and Refraction of light. 883 



if 



And we get from (8) 



cos 0' sin 6' [p sin ^ (/c sin 6 -^ k^ sin 6^ — p sin (/>' {/c' sin ^')] 

 + cos 6' \p sin ^ cos ^ {«: cos 6 — k^ cos ^ J 



-p'sin^'cos</)Vcos^']=0 (11). 



If we put 6 = -^ , i.e., consider only vibrations in the plane of 

 incidence, then it is clear, from symmetry, we must have 



«' = ^.=|- 



Hence from (11) 



p{k + kJ sin (f) = p K sin <^ . 

 But from (5) 



{k + /cj sin cf) — K sin <^', 



therefore P = p' (12), 



or the density is the same throughout the two media and (11) 

 reduces to 



sin 2^ {k cos 9 — k^ cos 0^ = sin 2^V cos 6'. 



But /c cos ^ + /Cj cos 0^ = K cos 0\ 



Hence k, cos 0, = k cos ^ ^ — ^-7 -. — '^, , 



^ ' sm 2</) + sm 2<^ 



tan((^-^') , , 



= KCOS -^ 777 (IS), 



tan (</) + (f)) ^ ^ 



, ni /I 1 -. tan ((i — (^" 



/c' cos ^ = /C COS 6' U + 



tan (^ + ^\ 

 a sin2(^ ., ., 



= « COS ^ ^ / , , ,'\ 7T T?N • • •• (14). 



sm {(p + cfi) cos {(p — (p) 



The four equations (9), (10), (13), (14) give the amplitudes and 

 directions of vibration for the reflected and refracted waves, and 

 have been found by MacCullagh {Irish Transactions, 1848), 

 assuming a form for the potential energy of the strained medium 

 which, as Stokes has shewn, is inconsistent with the Conservation of 

 Energy. It is perhaps worth remarking that if we denote by V 



