﻿56 Dr. Gr. Green on Fluid Motion 



When the fluid is incompressible, and when a compressible 

 fluid is moving m such a way that ^ — h ^ — \- %- is zero, 



B# oy d~ 

 a solution of a different type obtains. The solution in this 

 case may be written in the form 



u 



I 



=/i{(V-^), (a z x-co x z)}, 



v=j 2 {{a) 1J x-G) x y), (ay z x-co x z)}A. . . (15) 



w =/ 3 { (aye - <o x y) , {co z x - m x z) }, ) 



where f 1} f 2 , f% are arbitrary functions subject only to the 



condition — + ~- + ^- =0. This solution includes as a 

 ooc dy oz 



particular case any motion of rotation of the atmosphere as 

 a solid about the axis of the Earth. 



The solutions which we have above obtained make it clear 

 that the fluid motions relative to rotating axes in which the 

 relative circulation moves with the fluid belong to a very 

 restricted type. A relative motion, for instance, similar to 

 that taking place in a free vortex, does not fulfil the con- 

 ditions required for permanence of the velocity potential, 

 and therefore no steady motion of this type could take place 

 in the atmosphere — as has been assumed to be the case. 



The conditions which we have found to be necessary for 

 the validity of the circulation theorem when the fluid motion 

 is relative to rotating axes, may be obtained in a manner 

 different from that employed above. Taking H, H, Z to 

 denote the components of angular velocity of a fluid ele- 

 ment, and U, V, W to denote components of linear velocity 

 of the element, each referred to fixed axes which coincide at 

 instant t with the instantaneous positions of the moving axes, 

 we may derive the conditions from the equations employed 

 by von Helmholtz in his papers on vortex motion : — 



with two other similar equations. With £, rj, £ to represent 

 the components of relative angular velocity of an element of 

 fluid, referred to the moving axes, we have, 



JJ = u — (o z y + cOyZ; Y = v — co x z + a) z x; W = to — w y x + w x y ; 

 and Dt l)t ~ a ' v W ^' 



