152 On a Dynamical Theory of 



formulae for the transformation of the quantities X# F y , Z^ will 

 be similar to those for the transformation of the co-ordinates 

 themselves. The like will be true of the quantities X //5 F^, ^ /y , 

 if we put 



_ V = - -' 7 = .< - 



dz dy' " dx dz' " dy dx 



and so on successively. 



It is to be observed that, in this Lemma, the displacement 

 is not limited by any restriction whatever. Each of its com- 

 ponents may be any function of the co-ordinates. But the 

 displacements produced by a system of plane waves are re- 

 stricted by our definition of such waves ; they must be the 

 same for all particles situated in the same wave plane. If 

 the waves be parallel, for instance, to the plane of x ', ^', the 

 quantities ', i/, % will be independent of the co-ordinates x ', y ', 

 and will be functions of z' only. This consideration reduces 

 formulae (D) to the following : 



dri d% 



X = -7-7 COS a - , COS a , 

 dz dz 



F-J'eosp-gooBp', :() 



dr,' 



in which it is remarkable that the normal displacement ?' does 

 not appear. If ' = 0, these formulae become 



-r- dr{ ~ dr\ , . 



or if r\ = 0, then we have 



(F') 



Lemma III. If, in an ellipsoid whose semiaxes are equal 

 to a, by c, there be two rectangular diameters, one making with 



