For the rate of change of circulation in a circuit which always consists of the 

 same fluid particles we have 



d r 1 r 



— circ C = - dC-Vp = dS ■ P (8) 



dt Jc p Js 



where P — VPaV( — ), so that P is a vector along the intersection of surfaces of con- 



P 

 stant pressure and constant density. Also v * P — and so by Gauss's theorem P 



defines tubes of constant intensity. Thus we have the famous meteorological theorem 

 of Bjerknes that the rate of change of circulation in C is measured by the number of 

 P tubes which C embraces. 



For plane flow the vector notation leads directly to the complex variable [5]. 



The use of the complex variable in two-dimensional problems has a long history, 

 but it is only in recent years that full advantage has been taken of the methods of 

 function theory as opposed to resolution into equations in x and y. What is beginning 

 now to be more fully realized is that the variables most generally useful are not x, y 

 but the conjugate pair z, Y. For example if & (x, y) is a plane harmonic function, it 

 is the real part of a holomorphic function f{z). The identity 



v (x,y) =K[/(z)+/5)] (9) 



leads to 



f(z) = 2<p(fc, - y 2 iz) - <f(0,0) + iy, (10) 



where y is an arbitrary real constant. 



Again the circle theorem [6] states that the motion of an unbounded liquid 

 whose complex potential is f(z) when disturbed by the circle \z\ — a is governed by 

 the complex potential 



/(z)+/f-\ (11) 



for on the boundary ~z — a 2 /z, so that the boundary is a streamline. That no new 

 singularities are introduced is clear from the fact that of the points z and a 2 /z only 

 one lies inside the circular boundary. 



Combined with conformal mapping the circle theorem enables us to deal with 

 the perturbation produced in disturbing the flow by a cylinder of any cross-section. 



In the same line of thought we know that a stream function can be defined for 

 any two-dimensional flow whether rotational or irrotational. Using the variables z, ~z 

 we can denote this stream function by if/(z, z) and then the velocity is given by 



u - iv = - 2i — . (12) 



dz 



In the case of steady streaming past a fixed cylinder, ty is constant on the boundary 

 and so 



— dz-\ di = 0. . (13) 



dz ds 



The Blasius theorem for the force (X, Y) then gives [7] 



X - iY = jtip I (u - iv)(u + iv)dz = - 2i P I [ — ) dz, (14) 



dz / 



'C JC 



where the integral is taken round the boundary C of the cross-section. But the equation 

 of the boundary is of the form f(z, z) = and so z can be eliminated and Cauchy's 



