Motion of the Dynamics of Continuous Media. 799 



In the same way, 



Qzu = Quz = U • . U. 



If be expressed in terms or! the pressure components, q 1 , 

 of the second system, it will be of the form described 

 by (5) and (6), where all the quantities except the i's are 

 distinguished by a dash. 



Thus 



q'* = ii . 9 . ii = ii . <£> . . <!>' . i x 



= i x . (kixii. + MviiU-) • (Jcii + kivi^ii,i\ 



= k 2 (i 1 .6.i 1 + 2ivU.0.i 1 -v 2 U.6 .i 4 ) 



= ^(q zx +2kivq vx —v 2 q U u) (11) 



As a final example, we will find how the energy, w 9 

 transforms. 



q f uu = k.0.U= U.&.0 .<£>'. U 



= k 2 ( — v 2 q xx + 2ivq ux + q uu ), 



or 



w = k 2 (v 2 q xx — 2ivq ux + iv). . . . (12) 



If we begin with a medium at rest, 



q xx = pxx and q ux = 0, 



so that 



w' = k 2 (v 2 p xx + ic ) ; 



and when the pressure system is hydrostatic, p xx =p, 



io = k 2 (v 2 p -f w ) , 



a formula which shows how the pressures contribute to the 

 energy of the strained medium. 



It may be added that the function is self-conjugate, 

 provided that is self-conjugate. This is to be anticipated 

 on account of the connexion between the self-conjugate 

 property and the principle of energy-mass, but it may be 

 proved formally from the fact that 



A.0. B = B.0.A 



provided that = <&. , <§' with self-conjugate. 



