1 d 2 a . ld 2 a d 2 a d 2 a \ m, „ > 



0r ' m ~dt 2 - A [d? + W + ~d*) ~ m 



■ (F), 



IN THE INTERIOR OF TRANSPARENT BODIES. 417 



Hence the equations (2>) evidently become 



1 d*a . d°a _ (d 2 a d 2 a\ ' . _. ld*a d 2 a\ m, -, 



1 d 2 a M fd 2 a d*a . dV 



W 



and similarly, 



m dt 2 \dx 2 dy 2 dz 2 1 m 



m df \dx l r ety* + «?*V «i V 



which are the equations (D) adapted to the case of normal vibrations. 

 It is remarkable, that these equations should be of exactly the same 

 form as the equations (E), differing only in having A in the place B. 



It is evident that the equations of vibratory motion cannot assume 

 the form 



1 d 2 a (d 2 a d 2 (S d*y\ ' d'(3 . d°y 



m-3f = A [d* + d? + d*)' and similar ex P ressions for -j? and w>> 



unless the vibrations be either altogether normal, or altogether transversal. 



§ 22. To adapt the equations (E) to the case of plane waves. 

 The equation to the wave-surface in this case will be 



pn + qy + sz = u. 



When u is the perpendicular from the origin on the surface, and 

 p, q, s the cosines of the angles it makes with the co-ordinate planes, 

 and therefore p 2 + q 2 + s* = 1. 



Now in this case, since a is a function of u and /, we have 



da dadu da , ,, . d 2 a .d i a 



S"S3S^3S' andtherefore a?-*^ 



A • ,l d*a 4 C? 2 a d 2 a d*a 



and in the same manner, — — . = o* _— , -y~ = s" — ; 



dy 1 * du 2 dz" du* 



