Since the circulation round any triangular area of given aspect is the sum of the circulation round the projections of the area on the coordinate planes, the composition of the components of spin, [xi], [eta], [zeta], is according to the vector law. Hence in any infinitesimal part of the fluid the circulation is zero round every small plane curve passing through the vortex line; and consequently the circulation round any curve drawn on the surface of a vortex filament is zero. If at any two points of a vortex line the cross-section ABC, A´B´C´ is drawn of the vortex filament, joined by the vortex line AA´, then, since the flow in AA´ is taken in opposite directions in the complete circuit ABC AA´B´C´ A´A, the resultant flow in AA´ cancels, and the circulation in ABC, A´B´C´ is the same; this is expressed by saying that at all points of a vortex filament [omega][alpha] is constant where [alpha] is the cross-section of the filament and [omega] the resultant spin (W. K. Clifford, _Kinematic_, book iii.). So far these theorems on vortex motion are kinematical; but introducing the equations of motion of § 22, Du dQ Dv dQ Dw dQ -- + -- = 0, -- + -- = 0, -- + -- = 0, (6) dt dx dt dy dt dz _ / Q = | dp/[rho] + V, (7) _/ and taking dx, dy, dz in the direction of u, v, w, and dx : dy : dz = u : v : w, D / \ Du D dx -- (u dx + v dy + w dz ) = -- dx + u ---- + ... = -dQ + ½dq², (8) dt \ / dt dt and integrating round a closed curve _ D / -- | (u dx + v dy + w dz) = 0, (9) dt _/ and the circulation in any circuit composed of the same fluid particles is constant; and if the motion is differential irrotational and due to a velocity function, the circulation is zero round all reconcilable paths. Interpreted dynamically the normal pressure of the surrounding fluid on a tube cannot create any circulation in the tube. The circulation being always zero round a small plane curve passing through the axis of spin in vortical motion, it follows conversely that a vortex filament is composed always of the same fluid particles; and since the circulation round a cross-section of a vortex filament is constant, not changing with the time, it follows from the previous kinematical theorem that [alpha][omega] is constant for all time, and the same for every cross-section of the vortex filament. A vortex filament must close on itself, or end on a bounding surface, as seen when the tip of a spoon is drawn through the surface of water.