acted upon by the Undulations of an Elastic Fluid. 347 



w=mf sin q£— Q ^ g ^ ^ cos 2q£, 



'JZm-fcg 

 Sa(fc 2 -1) 



m df „ m 2 /e da . n ^ 



= -j- cos g £ — - — r -g — — -f sin 2^f, 



<? dr * 3gtf(«; 2 — 1) dr * 



2s 



>2„2, 



cr=-/ S m g r~ 3 , v ^ 1) co S 2^ 



+ 2a 2 J Bin <^ 2?2fl9 ^ cos «. 



As f=^ — «7+ c, the velocity of propagation is a r , for determin- 

 ing which we have the equation a' — fc'a. The complete value of 

 h! involves, as we have seen, the constant m ; and it might hence 

 be supposed that the rate of propagation is different for different 

 sets of vibrations. But as there are no a priori conditions which 

 determine the value of m to be different for different simple vibra- 

 tions, it may be assumed to have the same value for all ; and 

 consequently different composite vibrations, the components of 

 which have all the same value of q, although they may differ in 

 magnitude according to the number and phases of the compo- 

 nents, will be propagated with the same constant velocity /c'a. 

 Also composite vibrations differing as to the value of q will be 

 propagated with that velocity. The last two inferences are readily 

 seen to be true, if only the first power of m be taken into account 

 and the equations are consequently linear with constant coeffi- 

 cients. We have now to inquire how far they are true when 

 terms of the second order are included. 

 For this purpose let us assume that 



f = -m . 2 . p cos q{\ - m?A . % . \j- sin 2g?l + ro*Q, 



the first symbol 2 embracing an unlimited number of terms for 

 which/, q, and c may be different, and the second an unlimited 

 number of terms for which g, q, and c may be different. On 

 substituting this value of f in the equation 



\dx* + dy* + dz>) dt* 

 _ gjr d*f df d*f df d*f 



dx dxdt dy dydt dz dzdt 



(C) 



it will be found that that equation is satisfied if the values of/) 

 g, and Q be determined by the following conditions : — 



