416 Mr. O'BRIEN, ON THE PROPAGATION OF LUMINOUS WAVES 



§ 20. To adapt the equations (D) to the case of transversal 

 vibrations. 



If the vibrations be transversal, we have by Article (18), 



dji dy__da therefore A(M ±A d?a, 



ay d% ax ax \dy d%! ar 



hence the first of the equations (D) becomes 



1 d'a . d*a „ (d*a d*a\ , . „, d'a m i „ 



mdF = A d^ + B {dt + d¥) - {A - B) dtf-m Ca > 



(d 2 a iPa 



dtf 



*• 



1 d'a „ /d'a d'a. d'a\ ni t „ x 



' m dP ~ \dx % dy* dz'l m 



and similarly, 



1 *0 nl^Ll^tl,^ Hhra 



m df - \& dtf + dz*J m ^ p 



1 ^z 



GE). 



which are the equations (Z>) adapted to the case of transversal vibrations. 

 Since the first of these equations does not contain /3 or y, nor the 

 second a or 7, nor the third a or /3, it is evident that each may be 

 integrated separately, and so a, (&, 7 may be found; which is a very 

 important simplification. 



$ 21. To adapt the equations (D) to the case of normal vibrations. 



If the vibrations be normal, we have by Art. (19), 



rf 2 /3 d_ (dji\ _c? (da\ _dja , . .. . d 2 y d-a 

 dxdy ~ dy \dxl ~ dy \dy) dy*' ^' dxdy ~~ d«" * 



and therefore ^ (_ + -?) = _ + _ 



