70 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



outside forces like gravity have an influence. But since tliis case 

 occurs only in liquids [i.e., fluids that form drops] that can be regarded 

 as incompressible, therefore (for these) it is not necessary to satisfy 

 equations (3) and (3a). Therefore (for these) the constant n remains 

 arbitrary, and when for thi>5 case this latter constant is so determined 



that — =1, then in equation (la) the intensity of gravity [i. e., the accel- 

 eration, — g) can be added to the left-hand member. 



The boundary condition for a discontinuous surface is that the pres- 

 sure shall be equal on both sides of such a surface, which condition 

 will be satisfied for P when it is so for p. 



As regards the re-action of the fluid against a solid body moving 



in it, the pressure against the unit of area of surface increases as n-r. 



In the same ratio, the frictional forces increase that are proportional to 



du 

 the product of A- £, with the differential quotients such as — , and other 



similar ones. But for corresponding similar portions of the surfaces of 

 the bounding bodies of the forces of pressure and of friction increase 

 as 



n' -' 



The work needed to be done by the immersed bodies to overcome 

 these resistances will therefore for equal intervals of time increase as 

 nq~r. 



In general therefore for compressible fluids [gases] and for heavy 

 cohesive fluids [liquids under gravitation] with free surfaces, if the 

 movement is to be completely and accurately transferred from the first 

 fluid to the other, the three constants n^ q, r are completely determined 

 by the nature of the two fluids. Only in the case of incompressible 

 fluids without free surfaces does one constant remain indeterminate. 



Kow there is a large series of cases where the compressibility not 

 only for cohesive, but also for gaseous fluids, has only an inappreciably 

 small influence. To such cases the following considerations apply: If 

 the constant 7i becomes smaller while r and q remain uncbanged, this 

 indicates that in the second fluid the velocity of sound diminishes pro- 

 portionally with », and similarly for the velocities of the moving mate- 

 rial portions, whereas the linear dimensions increase proportional to 

 the reciprocal of n. For a constant value of r, that is to say, a con- 

 stant density of the second fluid, a diminution of the velocity of sound 

 corresponds to an increased compressibility of the fluid. Therefore 

 with an increased compressibility, the movements remain similar. 

 Hence it follows that when we diminish n, while leaving the compressi- 

 bility of the fluid unchanged, the movements of the fluid themselves 

 change and become similar to those that a more incompressible fluid 

 would execute in a narrower space. Therefore for smaller velocities, 



