The first equation tells that  is just function of , the function is defined as
Different authors defines the function differently, for example, Landau defines the function with a factor . But following Whitham, Rosenhead the  momentum equation becomes
Now letting
the  and  momentum equations reduce to
and substituting this into the previous equation(to eliminate pressure) results in
Multiplying by  and integrating once,
where  are constants to be determined from the boundary conditions. The above equation can be re-written conveniently with three other constants  as roots of a cubic polynomial, with only two constants being arbitrary, the third constant is always obtained from other two because sum of the roots is .
The boundary conditions reduce to
where  is the corresponding Reynolds number. The solution can be expressed  in terms of elliptic functions. For convergent flow , the solution exists for all , but for the divergent flow , the solution exists only for a particular range of .