Chaplygin dipole

this post is the Chaplygin dipole, it's an interesting issue.

I understood the steps till (2.4) included, Can someone explain me please the next steps in other words please? any help appreciated!

In 1903 Chaplygin published another remarkable paper (Chaplygin 1903) devoted to the motion associated with a compact vorticity distribution in a two-dimensional unbounded inviscid flow. In the introduction of that paper he gave a precise formulation of the problem:
Consider an unbounded mass of incompressible fluid in which the motion is parallel to the OXY plane; let the motion outside some circular cylinder be irrotational, the velocity being equal to zero at infinity. The question is to find a distribution of vortex lines inside the cylinder that gives rise to a uniformly translating vortex column with a continuous velocity distribution and with a positive pressure all around.

As a first example of the solution Chaplygin considered in detail a case of rectilinear motion of a circular vortex of radius 'a' with a constant translation velocity v_0. By superimposing on the whole fluid fluid a uniform velocity -v_0 he obtained a stationary problem of a steady vortex cylinder placed in a potential flow with uniform velocity at infinity. By choosing the polar coordinate system (r,theata) , with the origin at the centre of the cylinder, the stream function 'psi_1' for the potential flow around the cylinder is written as: