172 On a Dynamical Theory of 



Hence, 



n i / / / 



P = rs cos a 2 + r s cos a 2 , 

 az 2 dz z 



(24) 



The quantities s, s' are, as appears by the last section, the 

 normal velocities with which the two sets of refracted waves are 

 propagated. The velocity with which the incident and reflected 

 waves are propagated is taken as unity. Therefore, if TI, T\ be 

 the transversals, or amplitudes of vibration in the incident and 

 reflected waves, and r 2 , r' 2 the transversals of the refracted waves, 

 the lengths of the latter waves being denoted by A 2 , A' 2) and the 

 length of an incident or reflected wave by AI, and if we put 



fr=??L (t-z l + Vl ) ^ l= ,^L( t - z \ + v \\ 



A 2 A 2 



where vi, 1/1, v 2 , v' 2 are constants, and IT is the ratio of the cir- 

 cumference to the diameter of a circle, we may write 



rii = TI COS 0i, t/i = T'I COS 0'i, IJ 2 = T 2 COS 2 , i/ 2 = / 2 COS 0' 2 . (26) 



By means of these values the formulse (23) and (24) become 

 X' = Y~ (TI cos ai sin 0i + T'I cos a\ sin 0'J, 



F 7 ^,:: (TI cos/3i sin 0i + /i cos|3'i sin^'i), 



P = 2ir (r- TZ co&az sin 2 + -p- r\ cos a\ sin r 2 ), 

 \A 2 A 2 J 



Q = 2ir ( ^ T 2 cos /3 3 sin 2 + TT- r' a cos^ r 2 sin 0' 2 ]. 

 \A 2 A 2 / 



The angles 0i, 0'i, 2 , 0' 2 are the phases of vibration in the 



(27) 



