598 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 5 1 



dr 

 Further, if we consider that — is nothing else but the negative of 



the velocity component V n then the equations of motion take the 

 form 



\jr__dv. Vn + v] + iVi + kK (Ia0 



p dr dr r 



= - *Yl . V n - ^ll* - X V n + k V t . . (lb') 



dr r 



To this we add the condition of continuity [as to mass] which 

 requires that [if the density is to remain constant then] as much air 

 flows out of any elementary volume in a unit of time by reason of 

 the horizontal current as flows into it [in the same unit of time] 

 by reason of the vertical current. Now the mass of the horizontal 

 outflow of air for an elementary prism whose altitude is unity, whose 

 base is bounded by two circular arcs of the length dip and by the 

 radii r and r + dr, is equal to 



_^ M V> .drd<p. 

 dr 



on tne other hand by reason of a descending current whose velocity 

 increases by the quantity w within a unit distance, or is itself 

 equal to w at the unit altitude (compare assumption No. 5) the 

 mass of air that flows into the unit volume is equal to pwrdrdcp. 

 Therefore we must have 



d (prVJ 

 — — — = r pw 



dr 



or if p is considered to be constant then 



d(rVJ 



dr 



= — rw (Ic) 



in una equauon w mignt be considered as a known function of r 

 but as already stated in assumption (5), we will make a special 

 hypothesis, i. e., that this quantity has a constant value?;' within 

 the region for which r< R but that outside of this region it has the 



