For applications of the hodograph to the solution of kinematical problems see Mechanics.
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.
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.
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.
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.