For applications of the hodograph to the solution of kinematical problems see Mechanics.
So far these theorems on vortex motion are kinematical; but introducing the equations of motion of § 22, Du + dQ =o, Dv+dQ =o, Dw + dQ dt dx dt dy dt dz and taking dx, dy, dz in the direction of u, v, w, and dx: dy: dz=u: v: w, (udx + vdy + wdz) = Du dx +u 1+..
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 terms of 0 may be determined one at a time, and this problem is purely kinematical; thus to determine 4)1, the component U alone is taken to exist, and then 1, m, n, denoting the direction cosines of the normal of the surface drawn into the exterior liquid, the function 01 must be determined to satisfy the conditions v 2 0 1 = o, throughout the liquid; (ii.) ' = -1, the gradient of 0 down the normal at the surface of the moving solid; 1 =0, over a fixed boundary, or at infinity; similarly for 02 and 03.
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 argument may be shortly put as follows: As the Nature which is the object of mechanics and all natural sciences is not natural substances, but phenomena and ideas; as mass is not substance, and force is not cause; as activity is not in the physical but in the psychical world; as the laws of Nature are not facts but teleological conceptions, and Nature is teleological, as well as not mechanical but kinematical; as the category of causality is to be referred to " conation "; as, in short, " mind is active and matter inert," what then?
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.
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.
We have also the kinematical relation x=a.