82 CONSTRUCTION OF ISOBARIC CHARTS 



as E^i, Efei, Eij, there are always 50 solenoids between each pair of adjacent curved 

 lines. 



Significayice of n[';-maps. — The dynamic significance of the charts of Y\\\ results 

 from a second principle enunciated by Bjerknes.* If the velocity of the air be in- 

 dicated by u, and the density of the air by p, then u = pu expresses the amount of the 

 so-called specific quantity of motion of the air. The tangential component Ui of this 

 quantity, when integrated along a closed curve, we call the " moment-circulation " 

 of that curve. By moment-solenoid we designate the tubular figure formed in the 

 atmosphere by surfaces of equal density (isodense surfaces) and by level surfaces of 

 gravity, when these surfaces are constructed for each unit difference of density and of 

 gravity potential, respectively. If we further assume that the barometric formula for 

 static conditions also holds true under dynamic conditions, then Bjerknes' second 

 theorem may be stated somewhat as follows : 



The increase, during a unit of time, of the moment-circulation of a closed curve con- 

 sisting of particles of air in the atmosphere, is equal to the number of moment-solenoids en- 

 closed by that curve. We designate the moment-circulation of the closed curve by G 

 and the number of enclosed moment-solenoids by A ; then this theorem is expressed 

 by the formula 



^ = A = Jp.IV (29) 



where integration is to be carried out along the whole closed curve. Now the num- 

 bers and positions of the moment-solenoids are readily determined from the n ['--maps. 

 To demonsti'ate this let the closed atmospheric curve be composed of two lines ab and 

 ab, lying in the level-surfaces of gravity V = Vq and V = l\ respectively, and of the 

 two verticals aa and bb, joining the ends of these two lines. Then pd V, and fpd V, each 

 equal zero along the lines ab in the level surfaces and it only becomes necessary to 

 carry out the integration along the verticals aa and bb. 

 Thus we obtain 



By the aid of the static barometric formula 



dp= -pdV 



the above integrals may be transformed. The integration of this formula along the 

 vertical aa gives 



*V. Bjerknes — "On the formation of oiroulatory movements and vortices in friotionless fluids." — Chiisiiania, 

 Videnskabssehkabels Skriften, 1898, No. 5. 



