160 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



If we assume 



F{x+iy)=\og (x+iy) = (p+i>p 



and if we substitute 



x=rcos 6-, y=r sin 6 



then follows 



cpz=:logr, ip=6. 



In this case the isobars consist of concentric circles. The paths of the 

 wind are logarithmic spirals having the equation 



H— log >=constaut. 



V. STEADY SYSTEMS OF WINDS. 



It is certainly at present generally assumed in meteorology that the 

 winds at the earth's surface owe their origin and maintenance to ver- 

 tical currents of air that are limited to definite regions. Let us assume 

 that there is given such a region having auy arbitrary boundary above 

 which a current of air ascends whose velocity in the neighborhood of the 

 earth's surface is determined by the constant (c). By this assumption 

 the whole system of winds dependent thereon, as well as the distribu- 

 tion of pressure, is determined for the whole region. It is therefore 

 the province of mathematics to determiue all the quantities coming 

 into consideration both for the inner and also for outer region. 



To this end the functions cp and to are to be propely determined. The 

 first of these is found without further difficulty from well-known theorems 

 in the theory of the poteutial. Since these functions must in the outer 

 region satisfy the partial differential equation z/rp=0, and in the inner 

 region must satisfy the equation J<p=^c; therefore* 



V^-^fdelogp ,...,,„,. ..(23) 



In thisp indicates the distance of the element of the surface d a from 

 the point x, y. The integral is to be extended over the whole of the 

 given inner region. Therefore the velocity potential is the logarithmic 



potential of a layer of matter having the density — c/2 n that covers the 



region of the ascending current of air. The functiou cp itself, as also 

 its first differential quotient, varies continuously throughout the whole 

 plane up to the boundaries of the outer and inner regions. 



*§ep Q. Kjj-chhoff, VorJesungen iiber Uechanilc, 1876, p. 195, 



