AN INTERRUPTED MEDIUM. 527 



Denote fy + ct by 6, and when x=Q 



l=i cos 6, K=r cos (6 + a), ~R X = p cos (0 + /3) 

 Ta, = * cos (0 + /3'), R y = a cos (6 + 7), T,, = r cos 



The equations (1), (2), (4) and (5) become 



(i cos 6 r cos + a) sin = * cos (0 + /3') > cos (0 + /3) ....... (1) 



(i cos 6 + r cos + a) cos (j)r cos 8 + 7' <r cos + 7 ........ (2) 



ap cos 0+/3 + b t cos 6 + (3' (i sin + r sin + a) sin <p + g sin 6 + 7 tan 



<rr sin + 7' tan = . (3) 



ff cos 6 + y + br cos + 7' ( sin r sin + a) cos + p sin + /3 tan <f> 



stsin (0 + /3') tan = . (4) 



Now is indeterminate ; hence, equating to zero the co-efficients respectively of 

 cos and of sin we obtain : 



(i r cos a) sin (p = t cos /3' p cos /3 ...... , ..... . . (1) 



r sin a sin = / sin /3' + > sin /3 .............. (2) 



( + r cos a) cos (f) = v cos 7' <r cos 7 .............. (3) 



r sin a cos 0= r sin y + ff sin 7 .............. (4) 



a> cos /3+6 t cos /3' r sin a sin 0+trsin 7 tan * r sin y tan (p = . . (5) 



a p sin /3 6 ^ sin /3' (z' + r cos a) sin + <r cos 7tan * r cos y tan (p = Q (6) 

 a<r cos 7 + 5 r cos Y + r sin a cos + p sin /3 tan ^sin /3' tan = . . (7) 



a e sin 76 r sin 7' (' r cos a) cos + p cos /3 tan (p st cos /3' tan 0=0 (8) 



Also of f T* represent the co-efficients of vibration of the incident and reflected 

 wave parallel to the axis of z ; and if the actual vibrations make respectively the 

 angles w, 8 with the plane of xy, i'=i tan w, /=r tan <5. If, however, the reflected 

 vibration parallel to z suffer a different retardation from that which is in the place 

 of x y, we must simply denote it by ^ cos (0 + a'). Let also R Z =T cos (0 + t), 

 T z =4 cos (0 + e'). Then we "have the following equations : 



f r' cos a'=-4/ cos e' v cos f ............. , . . (9) 



/ sin a'= -^ sin e' + TT sin e ................. (10) 



a-TT cos + 6-^ cos e' r 7 sin a' = ............... (11) 



air sin e 6 -^ sin e' (i' + r' cos a')=0 ............ (12) 



These are the twelve equations of motion : and the first eight of them con- 

 tain ten unknown quantities ; the last four, six ; which have no apparent de- 

 pendence on the others. We will first solve them separately by means of parti- 

 cular hypotheses. 



We commence with waves in the plane of incidence. 1. Let us suppose 

 that the lost vibrations in the same direction have the same phase, i. e., that 

 )S=/3',7=Y- 



VOL. XV. PART IV. 7 C 



