1913-14.] 
Principia Atmospherica. 
89 
Let L be the width of the current, and H its depth ; then the flow over the 
whole section Lx H is LHV ; and by the equation of continuity this must 
be constant as the stream flows northward. 
Now 
LHV= n HLy , 
2(0/0 sin A. 
and Ly is the pressure-difference, A p, between the two sides of the current. 
LHV is constant ; hence, differentiating, we get 
q _ dH dp d sin A 
~H~J 
or 
sin A 
d JLJ± + C otX0X. 
H p 
Now p can only alter by variation of pressure, temperature, or composition ; 
change of pressure is ruled out because the motion is along isobars ; change 
of temperature will be very slight because there is no change of pressure, 
and there are no other causes of any appreciable change of temperature ; 
and change of composition can only occur in consequence of condensation. 
By Law 5, in the absence of change of temperature no change of composi- 
tion will occur. Hence 
dp/p = 0, 
and 
— cot ASA. 
H 
Ln other words, the thickness of the moving layer must increase fractionally 
by the amount cotXSX for the change of latitude <5A. If latitude is ex- 
pressed in degrees and not in circular measure as differentiation supposes, we 
must introduce the factor and thus the formula becomes 
180 
4 ^=-0175 cot A. 
H aX 
Hence, in order that a current may persist over any stretch from south to 
north, it is necessary that the thickness of the moving layer should increase 
fractionally to the extent of '0175 cot \ for every degree of latitude which 
it crosses. 
We have assumed the layer to be unlimited above, and limited below by 
the one-kilometre level. To provide for the additional air by increasing 
the height above the selected base-level would result in altering the 
pressure : that mode of operation is therefore excluded by the condition of 
maintenance of the current as steady. Consequently we must suppose the 
additional thickness to be provided by encroachment upon the lowest 
