Motion oj the Dynamics of Continuous Media. 797 



so we may write 



6. = A 1 i 1 . + A 2 i 2 . + A 3 i3. + A 4 i4., -. . ' . (5) 



where 



Ai = qxxk + qxyh + gxzU + qxuU, 

 A 2 = qyzii+ywU + qyzU + qyuU, 

 A 3 = qzxii + 'qzyU + qzzU + q'zuia 



a 4 = Quxii + q uy U + qmk + quuU- 



(6) 



The precise physical significance of the q's is given in 

 above places. Two particular cases will be required for 

 the purpose of illustration, viz. q xx and q uu ; these are 

 equivalent to (%>xx + Uxffx) an( l (~ w ) respectively. p xx is 

 the pressure component, u x is the component velocity at 

 a point of the medium, and g x and w are respectively the 

 component of momentum and the energy per unit volume. 



In this case 6 is assumed to be self-conjugate, and the 

 assumption is consistent with the principle of energy-mass. 



For a medium in equilibrium under a force, f, per unit 

 volume we have 



4> ■ V = f, (7) 



where V is the Hamiltonian vector operator : 



ox oy o? 



It occupies its position on account of its vector character, 

 but it operates on the p's : i. e., 



$ .V = (pxxi+pxyj +p xz 'k)i .V + etc. 



A similar equation represents the dynamics of the medium 

 if the four-dimensional notation is employed. Thus the 

 equations of ch. xiv. § 1 of Cunningham's Treatise are 

 equivalent to the very compact single equation 



6 . D = P. ■ (8) 



D is the operator 



ox dy os du 



and F is the force four-vector. 



The problem is to determine how 6 changes in passing 

 from one system of co-ordinates to another : i, <\, how it 



