IGO THK MECHANICS OF THE EARTH's ATMOSPHERE. 



It we assume 



F{x+ iij)=\og (.r+ iy) = (pJt-i'/- 

 and if we substitute 



.T;=r cos ^ ; y = r siu ^ 



tbeu follows 



f/j=\ogr, il- = H. 



lu this case the isobars consist of coucentric circles. The paths of the 

 wind are logarithmic s[)irals having the equation 



tt—'-, los:>'=constant. 



T. STEADY SYSTEMS OF WINDS. 



It is certainly at present g'enerally assumed in meteorology that the 

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

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

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

 which a current of air ascends whose velocity in tlie 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 determine all the quantities coming 

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



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

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

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

 region satisfy the partial differential equation Acp=0, and in the inner 

 region must satisfy the equation J(^=— c; therefore* 



9 



= -,~Jr(ff\ogp (22) 



In this p indicates the distance of the element of the surface da 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 tt that covers the 



region of the ascending current of air. The function qj itself, as also 

 its first dilT'irential quotient, varies continuously throughout the whole 

 l^lane up to the boundaries ot the outer and inner regions. 



*See G. Kirchhoff, Vorlesungen ilber Mechanik, 187(i, p. 195. 



