(4)
so that the component spin is
| ½ ( | dv | − | du | ) = ζ, |
| dx | dy |
(5)
in the previous notation of § 24; so also for the other two components ξ and η.
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, ξ, η, ζ, 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 ωα is constant where α is the cross-section of the filament and ω 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 | = 0, | Dv | + | dQ | = 0, | Dw | + | dQ | = 0, |
| dt | dx | dt | dy | dt | dz |
(6)