φ = Uφ1 + Vφ2 + Wφ3 + Pχ1 + Qχ2 + Rχ3,
(1)
where the φ’s and χ’s are functions of x, y, z, depending on the shape of the body; interpreted dynamically, C − ρφ represents the impulsive pressure required to stop the motion, or C + ρφ to start it again from rest.
The terms of φ may be determined one at a time, and this problem is purely kinematical; thus to determine φ1, the component U alone is taken to exist, and then l, m, n, denoting the direction cosines of the normal of the surface drawn into the exterior liquid, the function φ1 must be determined to satisfy the conditions
(i.) ∇2φ1 = 0. throughout the liquid;
(ii.) dφ1/dυ = −l, the gradient of φ down the normal at the surface of the moving solid;
(iii.) dφ1/dυ = 0, over a fixed boundary, or at infinity;
similarly for φ2 and φ3.
To determine χ1 the angular velocity P alone is introduced, and the conditions to be satisfied are
(i.) ∇2χ1 = 0, throughout the liquid;