AT THE SURFACE OF AN UNCRYSTALLIZED BODY. 



(A). 



q da 



V dt ' 



da 



dz 



s da 

 V dt 



Then we have the following relations, viz. 



'' da p da da 



da; V dt dy 



and similar equations connecting the partial differential coefficients of /3, 7, and V, 

 _ or any function of these quantities. 

 It is easy to prove these equations, without assuming the integrals of the equations of 

 motion, in the following manner : 



Let P be the point (xyz), AP the wave-surface which contains P at any time t. A' P' the 

 position of this wave-surface at the time t + dt, PP' the normal 

 to the wave at P, PQ' a line parallel to the axis of x 

 meeting the wave A'P' in Q'. Then assuming dx to represent 

 PQ', we have 



PP' = PQ' cos P'PQ' = pdx. 

 Also, since the space PP" is described in the time dt 

 with the velocity v, PP" = vdt; hence we have 



vdf =pdx (1). 



Now at the time(< + dt), any point of the wave A'P is in 

 the same phase of vibration as any point of the wave AP at 

 the time (t) ; therefore a, /3, 7, V, or any function of these 

 quantities will not be altered by putting x + dx, and t + dt, 

 for X and t. We have therefore 



-— dx A dt = 0, which by (l) becomes 



dx dt 



da 



dx 



p da 



V dt 



-r- = 7T , and 



s da 

 vli 



In the same way we may shew that 

 da g dc 



dy V dt' dz 



and thus the truth of the equations (A) is proved. 



(2.) Suppose the wave-surface to be a cylindrical surface perpendicular to the plane of xz, 

 the vibrations to take place parallel to that plane, and therefore /3 = 0, 9 = 0, and a and 7 

 independent of y : then we have the following relations between V, v, and the partial differential 

 coefficients of a and 7, viz. 



da „ dy 



~ = Vp, -^ = Vs 



dt ^ dt 



(B). 



(O. 



.for normal vibrations. 



.for transversal vibrations. 



B 



