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456 Mr. G. B. Jeffery on the Two-Dimensional 



motion in two dimensions are 



Ot Ox dy p 0>€ dff; 



ov^ ov , ov lop av 

 ot ox oy /o oy oy 



Eliminating the pressure from these equations we have 



where _ O"^ _oty 



U ~ Ty ' "'fa' 



-v|r being Earnshaw's current function. 



Take a system of orthogonal curvilinear coordinates de- 

 lined by conjugate functions u, /3 of .r, y. The equation for 

 iir may then be written 



or if the motion is steady 



§ 1. Solutions for which the lines of constant vorticity 

 are a possible set of equipotential lines. 



The coordinates can be so chosen that the curves a = const. 

 are identical with any given set of equipotential lines in free 

 space. Hence the characteristic property of this type of 

 solution is that it is possible to choose the system of co- 

 ordinates «, /3 so that V 2 ^ is a function of a only, say 



Substitute in (2) and we have 



or 0"fr d n m i v % 



Integrating with respect to ft 



+— r/8^0og/(«))'+P(«). 



