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448                      ON THE  DYNAMICS  OF  REVOLVING FLUIDS                       [413
For the present purpose we assume symmetry with respect to the axis of z, so that u, v, w, and P (assumed to be single-valued) are independent of 6. So simplified, the equations become
du       du    i>2       du _    dP                             ,^
dt       dr     r        dz        dr
dv       dv    uv       dv
dw       dw .     dw       dP dt        dr
of which the second may be written
'd .     d   .     d
signifying that (rv) may be considered to move with the fluid, in accordance with Kelvin's general theorem respecting "circulation." If r0> v0, be the initial values of r, v, for any particle of the fluid, the value of v at any future time when the particle is at a distance r from the axis is given by rv  r0v0.
Respecting the motion expressed by it, w, we see that it is the same as might take place with v = 0, that is when the whole motion is in planes passing through the axis, provided that we introduce a force along r equal to v2/r. We have here the familiar idea of " centrifugal force," and the conclusion might have been arrived at immediately, at any rate in the case where there is no (u, w) motion.
It will be well to consider this case (u  0, <w = 0) more in detail. The third equation (8) shows that P is then independent of z, that is a function of r (and t) only. It follows from the first equation (6) that v also is a function
of r only, and P = j^dr/r.    Accordingly by (2)
ldr.    .....................(10)
If V, the potential of impressed forces, is independent of z, so also will be p and p, but not otherwise. For example, if gravity (g*) act parallel to z (measured downwards),
gravity and centrifugal force contributing independently.    In (11) p will be constant if the fluid is incompressible.    For gases following Boyle's law
a2 (log p, or log p) = C +-gz + jWr/r................(12)
Jbe capable of raising the pitch. But we must remember that we are here dealing with a vibration such that the phase at both ends of the minor axis is the opposite of that at the centre. A parallel case which admits of complete calculation is that of the rectangle regarded as a deformed square, and vibrating in the gravest symmetrical modef. It is easily shown that a departure from the square form raises the pitch. Of course, the one-dimensional vibration parallel to the longer side has its pitch depressed.