Dirichlet's problem is generalized by Bernhard Riemann in 1860 and by Norman R. Lebovitz in modern form in 1965. Let  be the semi-axes of the ellipsoid, which varies with time. Since the ellipsoid is homogeneous, the constancy of mass requires the constancy of the volume of the ellipsoid,
same as the initial volume. Consider an inertial frame  and a rotating frame , with  being the linear transformation such that  and it is clear that  is orthogonal, i.e., . We can define an anti-symmetric matrix with this,
where we can write the dual  of  as  (and ), where  is nothing but the time-dependent rotation of the rotating frame with respect to the inertial frame.
Without loss of generality, let us assume that the inertial frame and the moving frame coincide initially, i.e., . By definition, Dirichlet's problem is looking for a solution which is a linear function of initial condition . Let us assume the following form,
and we define a diagonal matrix  with diagonal elements being the semi-axes of the ellipsoid, then above equation can be written in matrix form as
where . It can shown then that the matrix  transforms the vector  linearly to the same vector at any later time , i.e., . From the definition of , we can realize the vector  represents a unit normal on the surface of the ellipsoid (true only at the boundary) since a fluid element on the surface moves with the surface. Therefore, we see that  transforms one unit vector on the boundary to another unit vector on the boundary, in other words, it is orthogonal, i.e., . In a similar manner as before, we can define another anti-symmetric matrix as
where its dual is defined as  (and ). The problem is one of uniform vorticity  with components given by
The pressure can take only quadratic form, can be seen from the momentum equation (and using the vanishing condition at the surface) given by
where  is the central pressure, so that . Finally, the tensor momentum equation reduces to
where  is the gravitational constant and  is diagonal matrix, whose diagonal elements are given by
The tensor momentum equation and the conservation of mass equation, i.e.,  provides us with ten equations for the ten unknowns,