90 
Proceedings of the Royal Society of Edinburgh. [Sess. 
kilometre : that region is already supposed to be occupied by an extension 
of the current which is disturbed by surface friction ; hence, unless there is 
a continual flow-off of air from below the one-kilometre level, the steady 
state cannot be maintained. 
The south-to-north current implies a high pressure on the eastern side 
and a low pressure on the western side, and near the surface there is a 
component of flow from high to low across the isobars. Hence we may 
suppose a case in which the northward-flowing current is maintained steady 
by the flow-off from east to west in the surface layer. We proceed to 
calculate the amount of this east-to-west current which will suffice to draw 
off the increase of the current above 1 kilometre. 
We suppose, for the purpose of calculation, that the east-to-west com- 
ponent is uniform over the lowest half kilometre of the western section. 
The fractional increase of thickness in the upper layer has been shown to 
be ’0175 cot X for each degree of advance northward. The increase of the 
thickness is the same over each elementary layer of height into which the 
whole thickness can be divided ; consequently the air to be removed is the 
fraction -0175 cot X of the transverse vertical section at every level. If 
the removal is confined to the lowest half kilometre, which contains a 
fraction of the atmosphere approximately one-twentieth of the whole, it 
follows that a fraction 20 x ‘0175 cot X of the lowest half -kilometre layer 
has to be removed for each degree of advance northward. 
For each metre of advance northward, therefore, a fraction ^ X 0175 cot X 
llTlxlO 3 
of the lowest half-kilometre layer has to be removed ; and, similarly, for 
each metre per second of the wind velocity from south to north a fraction 
20 x -0175 cot X i t i 7 
— — — — — must be removed every second. 
-L JL _L I X JL v/ 
Suppose that the breadth of the advancing current which is supposed 
to be maintained steady is L kilometres, the westerly flow at the western 
end of the lowest half kilometre must carry away air at the rate of 
20 x 0175 cot X x l kilometres per second, or there must be a cross com- 
UlTxlO 3 
ponent of wind there amounting to 
20 x *0175 cot X 
HIT 
xL metres per second. 
If the cross wind be referred to the width of a current expressed in 
degrees of longitude at the latitude X, and if l be the width of the current 
in degrees, we get 
L = 111T cos XL 
Whence it follows that in order to maintain a south-to-north current of V 
