46 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



rents, instead of potential functions which can not exist here, there occur 

 other common functions such as are expressed in the equations (4), (5), 

 and {5a). On the other hand, for simple progressive movements of 

 liquids and for the magnetic forces, just as for gravitation, for electric 

 attractions and for the steady flow of electricity and heat, we have to 

 do with single-valued potential functions. 



The integrals of the hydro-dynamic equations, for which a single-val- 

 ued velocity potential exists, we can call integrals of the first class. Those 

 on the other hand for which there are rotations in one portion of the 

 liquid particles, and correspondingly a many-valued velocity potential 

 for the non-rotating particles we call integrals of the second class. It 

 can happen that in the latter case only such portions of the space are 

 to be considered in the problem as contain no rotatory particles of 

 liquid, e. g., in the case of the movements of liquid in a ring-shaped 

 vessel, where a vortex filament can be imagined traversing the axis of 

 the vessel, and where notwithstanding this the problem belongs to those 

 that can be resolved by means of the assumption of a velocity potential. 

 In the hydro-dynamic integrals of the first class the velocities of the 

 liquid particles have the same direction as, and are proportional to the 

 forces that would be produced by a certain distribution of the magnetic 

 masses outside of the liquid acting on a magnetic particle at the loca- 

 tion of the particle of liquid. 



In the hydro dynamic integrals of the second class the velocities of 

 the liquid particles have the same direction as, and are proportional 

 to forces acting on the magnetic particle such as would be produced 

 by a closed electric current flowing through the vortex filament and 

 having a density proportional to the velocity of rotation of this fila- 

 ment, combined with the action of magnetic masses entirely outside the 

 liquid. The electric currents within the liquid would How forward with 

 the respective vortex filaments, and must retain a constant intensity. 

 The adopted distribution of magnetic masses outside of the liquid or on 

 its surface must be so defined that the boundary conditions are satisfied. 

 Every magnetic mass can also, as is well known, be replaced by electric 

 currents. Therefore instead of introducing into the values u, v, and w, 

 the potential function P of an exterior mass 1c, we can obtain an equally 

 general solution if we give to the quantities £, ?/, and £ external to the 

 fluid or even only on its surface, such arbitrary values that only closed 

 current filaments arise, and then extend the integration of the equa- 

 tions (oa) over the whole region for which £, //, and C differ from zero. 



IV. VORTEX SHEETS AND THE ENERGY OF THE VORTEX FILAMENTS. 



In the hydro-dynamic integrals of the first class it suffices, as I have 

 already shown, to kuow the movements of the surface; the movement 

 in the interior is then entirely determined. For the integrals of the 

 second class, on the other hand, the movements of the vortex filaments 



