76 Mr Weatherhurn, On the Hijdrodynamics of Relativity 



and therefore V + C'k is constant also along a vortex line. This 

 is a particular case of the more general theorem, proved in the 

 preceding section, that H is constant along a vortex line. Thus 

 the surface 



V + c'-V = const, 

 is composed of a double system of vortex lines and lines of flow. 



II. Irrotational Motion. 



§ 5. When the vorticity i curl («v) is zero the motion will be 

 termed irrotational or non-vortical, being analogous to the motion 

 of that name in the older theory. In this case «v can be expressed 

 as the gradient of a scalar function 0,, which may be called the 

 velocity potential : i.e. 



'cv = V(f} : (17). 



The lines of flow are orthogonal to the surfaces of equal velocity 

 potential. 



The equation of motion can always be integrated when a force 

 and a velocity potential exist. For (8') then becomes 

 V (<^ + c'k +V) = 0. 



The function in brackets is therefore constant throughout the 

 liquid, and will be a function of t only; i.e. 



4> + c'K+V=f(t) (18). 



This is the required integral of the equation of motion. An 

 arbitrary function of t may, however, be incorporated in the 

 velocity potential cf), and this equation then written without loss 

 of generality 



(f) + c-K+ F=0 (18'y 



When the irrotational motion is steady (c'^k + V) is constant 

 throughout the liquid, and is also invariable in time. In the 

 preceding section, where w was not assumed to be zero this 

 function was only proved constant along vortex lines and lines of 

 flow. 



The equation of continuity (6), or as it may be written 



dk , -. 



^ + ^-divv = 0, 



may be expressed in terms of 0, if we write kvJk for v, and expand 

 the divergence of the quotient. The equation then becomes 



|logA- + v(l).Vc/, + lv^0=.O (19). 



