152 On a Dynamical Theory of 



formulse for the transformation of the quantities X^ T^ Z^ will 

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

 themselves. The like will be true of the quantities X^, F //} Z tl , 

 if we put 



T ='-' r ='-' z -'- 



" dz dy* " das 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 #', y', the 

 quantities ', *j', ' will be independent of the co-ordinates #', y', 

 and will be functions of z only. This consideration reduces 

 formulae (D) to the following : 



v </n' d& 



JL = -r-. cos o , cos a , 

 dz dz 



,_ drf 



Z = dz 7 



>'~ 



in which it is remarkable that the normal displacement % does 

 not appear. If ' = 0, these formulae become 



tr d-v TT- di\ n dr{ . . 



X = -r 1 , cos a, Y =, cos p, Z = -77 cos 7 ; (r) 



dz dz dz 



or if TJ' = 0, then we have 



X = -f oosa', r=-gcos(3', Z~*S. (O 



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

 to 0, b, c, there be two rectangular diameters, one making with 



