﻿dV gx 



3V gy 



av 



a« ~ r ' 



ay R' 



S* 



58 Dr. G. Green on Fluid Motion 



TsY' 



corresponding to apparent gravity, so that — ^ — at point 



(# = 0, y = 0, £ = 0) is the value of —g at the reference 

 point 0. In applying the above equations to motion of the 

 atmosphere, we may take 



av av av t . 



in the immediate neighbourhood of in the region within 

 which the value of apparent gravity may be regarded as 

 constant in direction and amount. If we neglect a change 

 o£ direction of one degree in g, our equations (18) to (21) 

 would then represent conditions of motion within a radius 

 of about seventy miles from point O. In order to render our 

 equations suitable to represent circulations of air of diameter 

 exceeding, say, 150 miles, we might employ the approximate 

 values 



, . . (22) 



wherein we neglect the variation of g with height. 



If we exclude certain cases of motion relating specially to 

 the tides, very few solutions of the above equations have been 

 recorded. In order to make ourselves familiar with the 

 types of fluid motion possible in the atmosphere it is of 

 interest to examine all solutions which can be obtained 

 having a bearing on meteorological problems. We, accord- 

 ingly, take first the steady rotational motions of incom- 

 pressible fluid under the force of gravity alone. 



We may take the boundary condition w = Q to apply at 

 the surface of the Earth. A simple rotation of all the fluid 

 about a vertical axis through 0, with a uniform angular 

 velocity o>, would be represented by 



u=—(oy; v=+(Die; io = 0. . . (23) 



With these values in equations (18) to (20), it would be 

 impossible to satisfy (19) and (20) simultaneously ; but 

 the motion represented by 



u= — co(y — /3z) ; v = cox\ io = 0, . . (24) 



fulfils all the conditions contained in our equations, pro- 

 vided (3 is given by 



o_ 212 cos j> m . 



