74 ifr Weatherburn, On the Hydrodynamics of Relativity 

 and the equation of motion takes the ver}^ convenient form 



^(a:v) + c-V/c + 2w X v= F (8), 



where we have written 



2w = curl (kv). 



In cases where the impressed force F admits a potential, so 

 that F = — V F, our equation reduces to 



^(«v) + V(t;-/c+F) + 2wxv = (8'). 



§ 3. Glehsch's transformation^. The equation of motion may 

 be expressed in terms of functions analogous to those of Clebsch 

 if we write 



KV = V(f) + A.V/X (9), 



(ji, \, fi being three independent functions of x, ?/, z and t. Taking 

 the curl of both members we find immediately that 



2w = VXx V;^ ^0). 



The function w = i curl (kv) plays the same part in the present 

 analysis as |- curl v in classical hydrodynamics. It will therefore, by 

 analogy, be called the vorticity ; and a line whose direction at any 

 point IS the direction of w at that point, a vertex line. Since 

 by (10) w is perpendicular to both VX and V/x it is clear that the 

 vortex lines are the intersections of the surfaces 



A, = const., /J, = const. 



Using then dots to denote partial differentiation with respect 

 to t, and assuming the existence of a force potential, we may write 

 (8') as -^ 



- V ( F+ c-a:) = V(j) + X/i) + XV/i - ^Vx 



+ (v . VX) VyLi - (V . V/x) VX 



which may be neatly expressed in the form 



§v,-|vx + Vi.= o (u), 



where the function H is given by the equation 



If= (f) + \jiL + V+C"K (12). 



_ * ^f; Basset, Treatise on Hydrodytiamics, Vol. 1, p. 28 ; also Silberstcin, Vectorial 

 Meciianics, p. 146. 



