152 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



which the expressions for the velocity of the air at the earth's surface 

 and for the pressure follow different laws. But they have not at- 

 tempted so to deduce the expressions for the velocity and for the pres- 

 sure in these two regions from one common principle, that these veloci- 

 ties and pressures merge into each other continuously at the boundary. 

 In the computation of numerical examples tbey have sought to help 

 over this difficulty by not applying their formula to the region in the 

 neighborhood of the boundary, but have here by interpolation intro- 

 duced numerical values passably good, but therefore certainly rather 

 arbitrary. Above all however it is a serious matter that according 

 to their theory the direction of the wind at the boundary suddenly 

 varies through a definite angle. The want of continuity here spoken 

 of can originate either in the assumptions adopted as a basis or in the 

 execution of the computation. I have arrived at the conviction that 

 the latter is the case. 



I have therefore started from the same assumptions as Guhlberg and 

 Mohn ; these are given in the following Section (n) and I add only 

 thereto the following principle, about which there can be no doubt: 

 " The pressure of the air, as also the velocity of the air and its direction, 

 ought to experience only continuous variations throughout the whole region 

 under consideration.''' 



By the application of this fundamental principle the theory of cyclones, 

 even in the case of circular isobars, deviates not a little from the theory 

 established by Guldberg and Mohn. 



II. ASSUMPTIONS THAT ARE THE BASIS OF THE PRESENT TREATMENT. 



The following assumptions form the foundation of my mathematical 

 development : 



(a) The portion of the earth's surface coming into consideration is 

 assumed to be a plane. A constant average value will be assumed for 

 the geographical latitude of this region. 



(6) The air will be treated as an incompressible fluid. 



(c) The investigation here carried out refers only to a stratum of air 

 of moderate height above the earth's surface. The latter surface ex- 

 erts a retarding influence on the movements of the air that can be con- 

 sidered as a force opposed to the movement and proportional to the 

 velocity. 



('/) The currents of air at the earth's surface are ordinarily directed 

 toward a center or flow away from the neighborhood of such a center. 

 Such currents can not be imagined without the existence of a vertical 

 current in their neighborhood. If, therefore, we in general confine our- 

 selves to the consideration of horizontal currents, still the consideration 

 of vertical motion is not to be avoided for the neighborhood of such a 

 center. We have, therefore, to distinguish between regions of pure 

 horizontal motion and regions with vertical motion. As to the latter, 



