84 Mr Weatherbarn, On the Hydrodyncmdcs of Relativity 



ACP are two different paths, the flux across the complete boundary 

 ACPBA is 



h-v . ndfi — I div (7bV) dr = 0, 



as is also obvious because the motion is steady. The lines 

 ■y^r = const, are the actual stream lines : for if P moves subject 

 to this condition there is no flux across the path traced out by 

 that point. 



The above is true whether the motion is irrotational or vortical. 

 The vorticity w is equal to 





,(34). 



and therefore for irrotational motion -^ must satisfy Laplace's 

 equation 



V^f = (33). 



If this relation is satisfied there is a velocity potential (f), and (32) 

 may then be expressed in the form 



• dcf) _ d-\fr I 



dx dy I 



dy dx ) 



These are identical with the relations subsisting between the 

 stream function and the velocity potential in the classical theory 

 of the two-dimensional irrotational motion of a liquid. They are 

 the conditions that (jy + iyfr should be a function of the complex 

 variable x + iy. The theory of such functions may then be used 

 as in the theory referred to*, to give various possible forms of 

 stream lines and lines of equal velocity potential. 



§ 13. Source, sink and doublet. Similarly the irrotational 

 motion of a fluid of minimum compressibility defined by the 

 velocity potential 



/-^■l («^>' 



where r is the distance from a fixed point 0, corresponds to the 

 assumption of a continual creation of matter at the point 0, 

 of amount 4<7rm per unit time. For 



so that ^•v = — . 



r^ 



* Cf. Lamb, loc. cit. chap. iv. 



i 



