The kinematical relations above explained now lead to the conclusion that in calculating the effect of extraneous forces in an infinitely short time t we may take moments about an axis passing through the instantaneous position of G exactly as if G were fixed; moreover, the result will be the same whether in this process we employ the true velocities of the particles or merely their velocities relative to G.
The step he took is really nothing more than the kinematical principle of the composition of linear velocities, but expressed in terms of the algebraic imaginary.
For applications of the hodograph to the solution of kinematical problems see Mechanics.
This is sometimes discussed as a separate theory but for our present purposes it is more convenient to introduc kinematical motions as they are required.
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 aw is constant for all time, and the same for every cross-section of the vortex filament.
Obviously the number of such geometrical or kinematical definitions is infinite.
The determination of the O's and x's is a kinematical problem, solved as yet only for a few cases, such as those discussed above.