3h^ Weatherhurn, On the Hydrodynamics of Relativity 81 



It has been proved already that a mrtex filament consists 

 alvmys of the same particles of fluid, though this can also be now 

 deduced from (24) and (25), using the invariability of yu.. 



IV. Fluid of Minimum Compressibility*. 



§ 10. According to the theory of Relativity no velocity can 

 exceed that of light. Hence there is no such thing as an incom- 

 pressible fluid ; for such a fluid would admit a wave propagation 

 with infinite velocity. A fluid of minimum compressibility is one 

 in which a wave can attain a velocity equal to that of light ; and 

 for such a fluid the quantity k is directly proportional to the 

 densityf 



K = k/kJ, K^k'jk; (27), 



where A:,,' is a constant representing the normal rest-density, i.e. the 

 rest-density corresponding to the normal pressure p^. 



For a fluid of minimum compressibility the equations of motion, 

 energy and continuity may by (27) be expressed in terms of the 

 velocity v and the rest-density k'. The equation of motion, viz. 



becomes on substitution 



, dv dk c'-„,, J ,„ 

 ^' -ZtT + V ,^ + - V^' = k,'F. 

 dt at y 



Dividing by 7 and using the equation of continuity to transform 

 the second term, we have at once 



k' (^ - V div v) + (c^ - v-^) Vk' = A-o'F/7 (28), 



which is the equation of motion in the required form. 



Multiplying this equation scalarly by v, and transfortning 

 V • V/t', we obtain 



,,/lrfv^ , ,. \ ,, ,, fdk' dk'\ /co'F.v 



.11.' .1 //. a/^J _ ^,2\ 



Now 



* Latnla, loc. cit. p. 788 ; Laue, loc. cit. § 37. f Laue, loc. cit. p. 241. 



