text: It states that if a motion determined by  is admissible under the conditions of Dirichlets problem, then the motion determined by the transpose  of  is also admissible.  In other words, the theorem can be stated as for any state of motions that preserves a ellipsoidal figure, there is an adjoint state of motions that preserves the same ellipsoidal figure'.
  By taking transpose of the tensor momentum equation, one sees that the role of  and  are interchanged. If there is solution for , then for the same , there exists another solution with the role of  and  interchanged. But interchanging  and  is equivalent to replacing  by . The following relations confirms the previous statement.
  where, further
  The typical configuration of this theorem is the Jacobi ellipsoid and its adjoint is called as Dedekind ellipsoid, in other words, both ellipsoid have same shape, but their internal fluid motions are different.