50 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



all motions take place in planes that are perpendicular to the axis of 

 z and are precisely the same in all such planes. 

 Therefore we put 



_ yu __ dv_ = ?p = ?y = 0t 



W ~~ dz dz ' dz ~ A- 

 Then equations (2) reduce to 



c n o or - <> u dv 



£=0,77=0,2:-—-^, 



the equations (3) become 



^ = 

 6t 



Therefore the vortex threads, in so far as they have constant sectional 



areas, have also constant velocities of rotation. 



The equations (4) reduce to, 



In this I have put P = in accord with the remark in Sect. ill. Therefore 

 the equation of the streamline is N = constant. 



In this case N is the potential function of infinitely long lines ; this 

 function itself is infinitely large, but its differential coefficients are 

 finite. Let a and b be the coordinates of a vortex filament the area of 

 whose cross-section is da db, then is 



<)iV Qdadb x—a 

 '^x~ n r 2 



t W Cdadb y—b 



v ~~ d% 7t r 2 



Hence it follows that the resultant velocity q is perpendicular to the r 

 drawn perpendicular to the vortex filament and its value is 



Cdadb 



If within a liquid mass of indefinite extent in the direction x aud y we 



have many vortex filaments whose coordinates are respectively x x , y x ; 



o? 2 , y 2 , etc., while the products of rotatory velocity by the sectional 



area are for each distinguished by m : , m 2 , etc., and if we form the 



sums, 



U = mi Ux + m 2 u 2 -f m 3 1*3, etc., 



V = m x Vi + m 2 v 2 + m 3 v 3 , etc., 



then these sums are each equal to zero, because that part of each sum 

 that is due to the action of the second vortex filament on the first is 

 counterbalanced by the action of the first vortex filament on the sec- 

 ond. That is to say, the two effects are, respectively, 



w? 2 Xi—x 2 . wii x 2 —Xi 



m x • • — =— and m 2 • — • — 5— > 



1 71 r 2 ' n r 2 



