88 THE MECHANICS OF THE EARTH's ATMOSPHERE. 



tore this line can not twice intersect the boundary surface between two 

 layers, and consequently in the abnormal case this line must necessarily 

 lie between the boundary surface and the horizontal plane located at 

 the pole. Therefore the tangents to the meridional section of the 

 boundary surfaces must intersect the greater arcs on the celestial 

 sphere somewhere between the pole and the equatorial side of the hor- 

 izon. 



The smaller the difference of temperature is relative to the difference 

 of the velocities of rotation so much the nearer does the tangent just 

 referred to approach the pole. 



Moreover at different i^oints of the bounding line of the same two 

 layers there can occur both normal and abnormal inclinations. For 

 since in the expression (see equation -ih) on whose positive or negative 

 value such occurrence depends, the /2 and 6 throughout the extent of 

 each layer are constant, therefore for the same altitude above the earth 

 this value can have a positive value near the equator but a negative 

 value near the poles. Between these the boundary curve must attain 

 a maximum altitude where the quantity under consideration passes 

 from positive through zero to negative. At this iilace also, according 



dr 



to equation (4a), we have j- = 0, therefore r is a limiting value and 



is here a maximum. 



Locution of the strata in the case when the velocity of rotation varies 

 contitmoHsly until the quantity of heat contained. — The considerations 

 hitherto set forth can also be extended to the case where fl is a con- 

 tinuous function of 6, and the value of fl in the atmospheric strata is 

 continually changing. The individual strata are in this case to be con- 

 sidered as indefinitely thin. Equation (4cr) now becomes. 



dr dp 



Or — 2^ 



"[r] ... 

 d{-^ 



4:[^-4f--vO 



In order that the equilibrium may be stable the quantity of con- 

 tained heat (see equation 4/i) must increase in the direction towards the 

 celestial pole. But the layers of similar air are less inclined than the 

 inclination of the polar axis at all places where the quantity 



de ' 



but on the other hand their inclination is steeper where the left-hand 

 side of this inequality is greater than the right. 



