l6o DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



simply by forming the field of divergence of the specific momentum. Then we could 

 multiply the field of ^ by a suitable interval of time dt, and add it to the field of 



ct 



density at the time /. We should then get the field of density at the time t+dt. 



This method could be formally carried out if we had sufficiently complete and 

 exact observations of specific momentum V. But as we have to form the divergence 

 in space, we need observations not only of the horizontal, but also of the vertical 

 component of specific momentum. Therefore, as long as we can get an idea of the 

 vertical motions only in the indirect way by making a diagnostic use of the equation 



cD 



of continuity, supposing simply the field of density to be stationary in space, = o, 



every prognostic use of the equation of continuity in this directway will be excluded. 

 But we could also think of a prognosis of a more summary character, which 

 would also be of great value if it could be carried out practically. We shall return 

 to the equation of continuity in the integral form 



W -TT -/"* 



ct u 



and apply it to a vertical cylinder going from the ground to the limit of the atmos- 

 phere, or at least to a height in which the density of the air is so low that it can 

 cause only an insignificant mass-transport. It will then be sufficient to integrate 

 the horizontal specific momentum over the cylindrical surface, and our ignorance 

 of the vertical motion will cause no difficulty. 



Now the ground carries the weight of the mass of air M in this cylinder. <r 

 being the area of the base and p the pressure, we have pa = Mg, g representing an 

 average value of acceleration of gravity. Multiplying equation (a) by g, introducing 

 pa instead of gM, and remembering that the cylinder is stationary and therefore 

 its base a constant, we see that the equation can be written 



ib) -f = z-fvj* 



Therefore, if we know horizontal specific momentum V sufficiently well up to suffi- 

 cient heights, we should be able by this equation to forecast the change of pressure 

 at the ground. Evidently this would be of high practical value. 



The question whether this will succeed will depend on the 'degree of complete- 

 ness and of accuracy required in the knowledge of V, or of the corresponding velocity 

 v. In order to estimate it, we can express the vertical dimension by pressure and 

 at the same time substitute velocity for specific momentum. Thus we have first 

 da = dz ds, dz being a vertical and ds a horizontal element of line. Then we can 

 express dz approximately by pressure, writing dz = o.i a dp, where z is measured 

 in meters and p in the m. t. s. unit of pressure, centibar. Thus 



J Vda =jj Vdzds = |ji',(-o.i dp) ds 



