88 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Proposition 1 . — The Conditions necessary to maintain a Steady 
Atmospheric Current. 
The conditions which must be complied with if a steady current is to 
be persistently maintained must satisfy the first law, the law of relation of 
motion to pressure. 
The law prescribes that the velocity V is related to the pressure 
gradient y, density p, latitude X, and the angular velocity of the earth’s 
rotation w, by the equation 
F=y/(2a>p sin X). 
Provided that the latitude X remains constant during the persistence of the 
current, this condition presents no difficulty ; the flow will always be de- 
termined by the distance apart of the isobars, but the auxiliary condition 
that the current shall not change its latitude implies that the isobars are 
parallel to the circles of latitude. Hence we may infer that, neglecting a 
very small correction for curvature, a circulation round the pole along 
isobars parallel to the circles of latitude is a “ steady ” circulation which 
will be persistently maintained. The only forces which will interfere with 
it are frictional forces due to the relative motion of adjacent layers of air, 
and, except in the immediate neighbourhood of the ground where friction 
is aided by turbulent motion, these are extremely small. Hence a west-to- 
east circulation or an east-to-west circulation in the upper air, once steady 
will remain so, unless it is disturbed by changes of pressure- distribution. 
But, on the contrary, when the air movement is from south to north 
or from north to south, or has any component which gives a motion across 
the circles of latitude, a change in sin X has to be dealt with. 
Motion from South to North. 
We propose to deal first with a current moving from south to north. 
We shall suppose the current to be uniform over the section from the one- 
kilometre level upivards. We leave out the lowest kilometre because we 
know that it is disturbed by quasi-frictional forces at the surface. 
In this case the value of sin X is increasing, and therefore greater pressure- 
difference is required to get the same quantity of air through the same 
section. But the pressure-difference is limited by the isobars, which are by 
hypothesis supposed steady. Any convergence of the isobars themselves 
provides its own remedy, because the gradient velocity is inversely pro- 
portional to the distance. We have, therefore, only to deal with the 
change in sin X in the formula 
F=y/(2 mp sin X). 
