44 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



therefore both integrals are equal to zero and the equation (5b) as well 

 as the equations (2) are satisfied. The equations (4) and (5) or (5a) are 

 thus true integrals of the equations (1 4 ) and (2). 



The analogy mentioned in the introduction between the action at a 

 distance of vortex filaments and the electromagnetic action at a dis- 

 tance of conducting wires, which analogy affords a very good means of 

 making visible the form of the vortex motiou, results from this proposi- 

 tion. 



When we substitute in the equation (4) the values of L, M, N, from 

 the equation (5a) and designate by An, Av, Aw the infinitely small 

 portions of the velocities u, v and w in the integral which depend on 

 the material elements da, db, dc and designate their resultant by Ap, 



we obtain 



1 (y-b )t a -{z-c) Va da 



2n r 3 



± («-o)g.-(*-fi)C. da db d 



= 1 ( x -a) th -(y-b)S ada db ^ 

 2n r s 



From these equations it follows that, 



Au(x— a) + Av(y— b) + Aw(z— c)=0, 



that is to say, Ap, the resultant of An, Av, Aw, is at right angles to r. 



Further, 



Z a Au+y a Av+C a Aw=0, 



that is to say, this same resultant, Ap, also makes a right angle with 

 the resulting axis of rotation at the point a, b, c. Finally, 



da db do 



Ap= y/(Auf-\-{Avf + (Awf=-^ — 2~ °" sm y » 



where a is the resultant of [the elementary velocities of rotation] B, a , 

 ?/„, C a , and ?' is the angle between this resultant and r, as determined by 

 the equation, 



ffr cos v=(x-a)£ a +(y-b)i/ a +(s-c)Z a 



Therefore every rotating particle of liquid a causes in every other 

 particle b of the same mass of liquid a velocity that is directed perpen- 

 dicularly to the plane passing through the axis of rotation of the particles 

 a and b. The magnitude of this velocity is directly proportional to the 

 volume of 'A, to its velocity of rotation, and to the sine of the angle between 

 the line ab and the axis of rotation, and inversely proportional to the 

 square of the distance of the two particles. 



The force that an electric current, moviug parallel to the axis of rota- 

 tion at the point a, would exert upon a magnetic particle at b, follows 

 exactly the same law as above. 



The mathematical relationship of both classes of natural phenomena 



