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1919]                                    THE  TRAVELLING  CYCLONE                                        657
and thus from (7)
= const. -2ttCfy-ia?(#6-*2/«*+2fi>)»,   ............... (11)
where, as usual, R2 = y" + (x — Ut)z. May llth.
The completion of this paper was interrupted by illness.
The two-dimensional solution requires a ceiling, as well as a floor, to take the pressure. In the absence of a ceiling we must introduce gravity, and since in the supposed motion no part of the fluid is vertically accelerated, the third equation of motion gives simply
-£-•= const. — gz.
Thus (10) is altered merely by the addition of the term —gz.
I had supposed too that the solution would remain substantially unaltered even though p were variable as a function of p. But these conclusions seem to be at variance with those put forward by Dr Jeffreys in the January No. of the Philosophical Magazine. I am not able to pursue the comparison at present.
June 25^, 1919.
[The following note was contributed by Sir Joseph Larrnor; and was appended to the paper as originally published.
This paper was left incomplete on Lord Rayleigh's decease on June 30. It may therefore be permissible to direct attention to its main conclusion from another aspect, by way of paraphrase. Two questions are involved. If a vortical system can persist at rest, in an atmosphere rotating with the Earth, can it also persist, slightly modified, with a translatory velocity U? And if so, how will the distribution of pressure in it be modified ? The equations of fluid motion relative to the ground are (1) and (2) ; in them the last terms Da/Dt and Dv/Dt express the components of relative acceleration, and these are clearly the centrifugal accelerations — £-X, - t?y in the relative orbits assumed to be circular, as found analytically lower down. On substituting these values, the equations give for Bp an exact differential form which is integrated in (7); therefore a modified motion is possible, and the first question is answered in the affirmative, in agreement so far with fact*. The displacement of the pressure-system due to the progressive motion is
* The conditions of stability for flow of liquid with varying vorticity had been considered in a series of papers, for which reference maybe made to the section Hydrodynamics of the catalogue appended to this volume.
K. VI.                      '                                                                                                         42readers to follow from the facts detailed. I trust we may hear still more from him.