du du du 

 dx' dy' dz' 



IN THE INTERIOR OF TRANSPARENT BODIES. 415 



Now -j- , -r- , -j-, are proportional to the cosines of the angles which 



the normal to the wave-surface makes with the axes of x, y, z, 



respectively ; also -rt , -£ » ~jI > are the velocities of any particle parallel 



respectively to the axes of x, y, z ; if the vibrations be transversal, the 

 sum of these velocities resolved along the normal to the wave-surface 

 must be zero ; i. e. we must have 



da du d& du dy du 



_L. —LI J L == 



dt dx dt dy dt dz 



hence we evidently obtain (multiplying (1), (2), (3) by 



respectively and adding) the following equation, 



da dfS dy 



dx dy dz ' 



which is the condition of the vibrations being transversal. 



$ 19. To shew that the conditions of the vibrations being normal, 



are 



da _ dfi dfi _ dy dy da 

 dy dx' dz ~ dy' dx ~ dz' 



If the vibrations be normal, it is evident that we must have 



vel. resolved parallel to axis of x _ cos angle made by normal and axis of # 

 do do y ~ do do y' 



da du 

 dt dx 

 dfi ~ du' 

 dt dy 



Hence we evidently have from the equations (1) and (2), 



da_du _ dfi du da _ d(3 



du dy du dx' dy dx ' 



and similarly we may prove that 



— = — and dy = — 



dz dy ' dx ' dz ' 



3 A2 



