The condition was therefore shown to be necessary; it remained to prove that it was sufficient. This is obvious at once from the definition of the velocity-potential, which must now be supposed to exist in order that its sufficiency may be proved. If any diameter of the spherical portion be taken as the axis, and any plane through that axis be considered, the velocity of a particle at right angles to that plane can be at once expressed as the rate at which the velocity-potential varies per unit distance along the circle, symmetrical about the axis, on which the particle lies. The integral of the velocity-potential round this circle vanishes, and so the angular momentum for any thin uniform ring of particles about the axis also vanishes, and as the sphere is made up of such rings, the whole angular momentum is zero. Thus the condition is sufficient. Thomson then proves that if the angular momentum thus considered be zero for every portion of the liquid at any one instant, it remains zero at every subsequent instant; that is, no physical action whatsoever could set up angular momentum within the fluid, which, it is to be remembered, is supposed to be frictionless. The proof here given cannot be sketched because it depends on the differential equation of continuity satisfied by the velocity-potential throughout the fluid (the same differential equation, in fact, that is satisfied by the distribution of temperature in a uniform conducting medium in the stationary state), and the consequent expression of this function for any spherical space in the fluid as a series of spherical harmonic functions. To a reader to whom the properties of these functions are known the process can present no difficulty.