^isconsir^^maemy of /Sciences, Arts^ma^Smers. 



)■ 



(5) 



The part of the kinetic energy in unit of volume, expressed by 

 this triple integral, may be written 



\ dt dt dt 



where m, v, w are the components of the electric current. 



It appears from this that our hypothesis is equivalent to the as- 

 sumption that the velocity of the particle of the [medium whose 



components are — — ^, — , is a quantity which may enter into 

 dt dt dt 



combination with the electric current whose components are m, 

 v, tu. 



Returning to the expression under the sign of triple integration 

 in (4), substituting for the value of a, /9, y, those of a', /?', /, as 

 given by equation (1), and writing 



d { d , Q d , d 



— for a — + p — + T — 

 dh dx dy dz 



(6) 



the expression under the sign of integration becomes 



\dt dhSdy dz) dt dh\dz dx' dt dh\dx duj) 



In the case of waves in planes normal to the axis of z the dis- 



d d 

 placements are functions of z and t only so that — = y , and 



dh dz 



this expression is reduced to 



' Idz' dt dz" dt f ^^ 



The kinetic energy per unit of volume, so far as it depends on 

 the velocities of displacement, may now be written 



T = i, 4+,^ + 



, ri ) d^i d-f] d^-/] d$ 



^ ^d? df d^di 



(9) 



where p is the density of the medium. 



The components, X and Y, of the impressed force, referred to 

 unit of volume, may be deduced from this by Lagrange's equations 



x= p'^-Gr-^'-^- 



dt^ 



Y=p^ 



di^ 



Cr 



dz^-di 



(10) 



(11) 



