178 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



to be correct iu the inotiou of liquids.* But I would not hereby assert 

 that the same uuraerical coefficient is to be used as is given by the 

 laboratory experiments on the internal friction of the air made under 

 the exclusion of all attendant disturbing circumstances. More likely 

 is it that along with the greater horizontal currents there will arise 

 small vertical currents of a local nature which will increase the 

 friction. The air can either be held fast at the earth's surface or glide 

 with more or less resistance. This fact, as is well known, is exi)ressed 

 in the boundary equations of condition by a number, the coefficient of 

 slip, whose value may lie between zero and infinity. 



(5.) The density of the air must be considered as dependent upon 

 the temperature, since the effective cause of the currents results from 

 this. But I have not objected to use, as the equation of continuity, 

 that simpler expression that ol)tains for incompressible liquids. The 

 error introduced hereby can be eliminated if, at places where the 

 density is less than the average, one increases to a corresponding extent 

 the velocity found for that locality, but considers the velocity as dimin- 

 ished at locations where the density exceeds the average. 



(6) A hydro-dynamic problem is only perfectly defi^nite when the fluid 

 occupies a definite space, and its behavior is known for all limiting- 

 boundary surfaces, 1 have therefore assumed that the atmosphere is 

 bounded both by the earth's surface and by a second spherical surface 

 concentric therewith. The distance of the two spherical surfaces, which 

 I will briefly designate as the height of the atmosphere, can remain un- 

 determined. But this is quite small in comparison with the earth's 

 radius. The above assumption just made however, only expresses the 

 idea that for a given altitude above the earth's surface the radial or 

 vertical currents are very small, or rather that when they are present 

 they exert an inappreciably small influence on the remaining motions. 

 This is certainly the case, since at very large altitudes the density is 

 very small. Since moreover it is assumed that the air can glide without 

 resistance on the upper spherical surface, therefore in my opinion no 

 limitation of the motions of the atmosphere, contradictory to the real 

 phenomena, results from the introduction of such an upper boundary 

 surface. 



III. 



The following notation will be used for the principal equations of the 

 problem. The position of a point in the atmosphere is determined by 

 the rectangular coordinates x, y, z. The center of the earth is the origin 

 of coordinates and the earth's axis in the direction of the North Pole is 

 the positive axis of z. The positive directions of the two other axes are 

 to be so chosen that the axis of y as seen from the Xorth Pole must be 

 turned through an angle of 90° in the direction of the motion of the 

 hands of a watch in order to be made to coincide with the axis of x. 



*[The term liiction as here used therefore includes vi.scosity and slip, but excludes 

 the resistance due to wave motion and to vortex motion and all the resistances 

 implied in turbulent How of lluids. — C. J.] 



