ELEMENTARY PRINCIPLES OF KINEMATICS OP CONTINUOUS MEDIA. 29 



when mercury expels water. The specific momentum will be solenoidal within 

 each homogeneous part of the fluid column, but non-solenoidal at the surface of 

 of discontinuity separating water and mercury. Instead of a discontinuous system 

 like this, we could also have considered a fluid system with continuously varying 

 density, for instance, a column of water with continuously varying salinity. Even 

 in this case we would have a solenoidal velocity-field and non-solenoidal field of 

 specific momentum, the solenoidal condition being violated by this vector not 

 only at a certain surface of discontinuity, but at every point where density showed 

 variations in space. 



Let us now, on the other hand, consider motions in a compressible medium, 

 atmospheric air. Setting aside the insignificant influence of humidity, we know that 

 the density of the air depends upon temperature and pressure. Therefore, if the 

 fields of temperature and of pressure are maintained stationary in space, the field 

 of mass will also be stationary, and the specific momentum will be a solenoidal 

 vector. Let us then consider a tube having its lower end near sea-level and its upper 

 end in the region of cirrus. If one ton of air enters the tube per second at its lower 

 end, one ton per second must leave it at its upper end. But measuring by volumes, 

 we find that one ton of air has at sea-level a volume of about iooo cubic meters, 

 and at the height of cirrus a volume of about 3000 cubic meters. There is a volume- 

 outflow from the closed volume limited by the walls and the cross-sections of the 

 tube equal to 2000 cubic meters per second. This volume-outflow is equal to the 

 velocity of expansion of the column of air which is contained in the tube. This 

 expansion is due to the motion up toward lower pressures. Reversing the direc- 

 tion of the motion, we get a corresponding inflow, equal to the contraction per 

 second which the column of air will have in virtue of its descending motion. 



117. The Fields of Motion in Atmosphere and Hydrosphere. We can now 

 take up the discussion of the chances of arriving at a satisfactory diagnosis of 

 atmospheric or hydrospheric motions. The great incompleteness of the observations 

 of air-motions is that they give only the horizontal components, and no information 

 on the vertical components. The same has also hitherto been the case with all 

 observations of sea-motions. The conditions for a satisfactory diagnosis will then 

 be that we should be able to derive the unknown vertical components from the 

 observed horizontal components. This will be possible if the motions can be con- 

 sidered solenoidal, and the question will be if we can suppose this to be the case with 

 sufficient approximation for the purpose of the kinematic diagnosis. 



In the case of the hydrosphere there is no doubt. We can put out of consider- 

 ation both the slight compressibility of the sea-water and the slight changes in the 

 field of mass following local changes of temperature, salinity, and pressure. Doing 

 so, we find that both the field of velocity and the field of specific momentum will 

 fulfil the solenoidal condition. Using this condition for deriving the not-observed 

 vertical components from the observed horizontal ones, we shall obtain an accuracy 

 depending entirely upon that of the observations; for the errors introduced by 

 neglecting compressibility and heterogeneity will be insignificant. 



