40 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



The rectangular compoueuts of the velocity of rotation increase in 

 the same ratio as the projections of the portion qs of the axis of rota- 

 tion; hence it follows that the magnitude of the resulting velocity of rota- 

 tion varies for a given particle of liquid in the same ratio as the distance 

 of this particle from its neighbors in the axis of rotation. 



Imagine vortex lines drawn through all points of the circumference 

 of an indefinite small surface. Then will a thread of infinitely small 

 section, which is called the " vortex filament," be thereby cut out of the 

 liquid. The volume of a portion of such a filament included between 

 two given particles of liquid, which volume according to the proposi- 

 tions just proven always remains filled by the same particles, must 

 remain constant during its progressive motion ; therefore its section 

 must vary in the inverse ratio of its length. Hence we can express the 

 last piopusition thus* In a portion of a vortex filament, consisting of 

 the same particles of liquid, the product of the velocity of rotation by the 

 section ever remains constant during its translatory motion. 



From equation (U) it directly follows that — 



Jx + ,~)y "^ ,^z 

 Hence it further follows that— 



=0 (2a) 



II(f+i+f>"^"^=° 



where the integration can be extended over any arbitrary portion ;S^ of 

 the mass of liquid. When we partially integrate this there results: 



/' /'^ dy dz + yy'/; dx dz +^f^fC cix dy=() 



where the integrations are to be extended over the whole surface of 

 the volume of S. If we let doo be an element of this surface and a, fi, 

 y the three angles that the normal to doo drawn outwards makes with 

 the (coordinate axes, then — 



dy J,c=cos a dcsD, dx d2'=cos (ioo, dx dy=cos y doo; 



therefore 



fTi^ cos a-\-r} cos fi+Z cos y)do3 =0, 



or when we let <7 be the resulting velocity of rotation and fi the angle 

 between this velocity and the normal 



/' /'a cos doo = 0, 



where the integration is to be extended over the whole surface of S. 



Let IS be a portion of a vortex filament bounded by two infinitely 

 small planes &?, and co,, perpendicular to the axis of the filament, then 

 will cos (9= + l for one of these planes and cos 8= — l for the other, but 

 cos ^=0 for the whole of the remaining surface of the filament; conse- 

 quently, if (J, and ff^, are the velocities of rotation at oo, and a?/,, respect- 

 ively, the last equation reduces to 



(7, (^, = ff,f 00,, 



