Mr Weatherburn, On the Hydro(hjnamics of Relativity 75 



On scalar multiplication of (11) by w, it follows in virtue 

 of (10) that 



w.V// = 0, 



showing- that H is constant along a vortex line. It can also be 

 shown that H is independent of x, y, z and is therefore a function 

 of i only. For taking the curl of (11) we deduce 



On scalar multiplication by Vx it follows, by (10), that 

 and similarly that 



w.Vl'^fl^O, 



From these we deduce as in the old theory* that 



^ = ^ = ...(13). 



dt dt ^ 



Thus the first two terms disappeai- from (11), which becomes 

 simply VH = 0, showing that H is constant in space and is 

 therefore a function of t only ; or 



(j) + \/l + V + c^fc = H (t) (14). 



From (13) it is clear that the surfaces \ = const, and /x = const., 

 and therefore also the vortex lines which are their lines of inter- 

 section, are always composed of the same particles of fluid. 



§ 4. Steady motion. When the motion is steady partial 

 derivatives with respect to t are zero. If then the impressed 

 force is derivable from a potential V, (8') becomes 



2v X w = V ( K + c-k) (15), 



and the equation of continuity 



div(/.v) = (16). 



If we multiply (15) scalarly by v the first member vanishes, 

 showing that 



vV( l^ + c-/^) = 0. 



Thus the function V + c-k is constant along a line of flow. 

 Similarly scalar multiplication of (15) by w gives 



w.V(F+c-^/c) = 0, 

 * Cf. Biis-et, lor. cit. p. 29. 



