54 THE MECHANICS OF THE EARTHS ATMOSPHERE. 



locity if iu the expression for r we substitute n, g, Xi c > z > m i in P lace 



of r, x, 9, z, c i w i- , a 



In this process n and U remain unchanged and we obtain, 



mrx-\- / )»irig=0 (8) 



If now we determine the value of the velocity w parallel to the axis, 



caused by the vortex filament »*i whose coordinates are g and e, we 



find : n o o 



,m, lg jj.mi l~dU * (z-c) 2 +g 2 -X 2 



w * = l^x + ^v gx U Sx (g+W+(*-rcf 



If now we call w, the velocity at the locality of m, parallel to the 

 axis of 2, which is caused by the vortex ring m whose coordinates are 

 z and j, then in order to determine this, we only need to execute the 

 interchange of appropriate coordinates and masses as above shown. 

 Thus we find : 



2mwx 2 -^miUhg 2 — , mTX!S—m l T 1 gc=—-^^JgxU. . . (8a) 



Sums similar to (8) and (8a) can be found for any number of vortex 

 rings. For the nth of these rings I designate the product a dg dc by 

 m n ; the components of the velocity that is communicated to this ring by 

 all the other rings are r n and u\, in which however I provisionally omit 

 the velocities that every vortex ring can communicate to itself. Fur- 

 ther I call the radius of this ring p n and its distance from a surface 

 perpendicular to the axis A, which two latter quantities agree with x 

 and z as to direction, but, as belonging to this particular ring, they are 

 functions of the time and not independent variables as are x aQ d %• 

 Finally let the value of ?/<, in so far as it depends on the other vortex 

 rings, be tp n . By forming and adding the equations (8) and (8a) corre- 

 sponding to each pair of vortex rings, there results 



2 [m n p n r n J=0. 

 2 [2 m H w n f? n —m n t n p n l n } = 2 [m n p v ip n ]. 



So long as we have in these sums only a finite number of separate 

 and infinitely slender vortex rings, we must understand by w, r, and 

 if) only those parts of these quantities that are due to the presence of 

 the other rings. But when we imagine an infinite number of such 

 rings keeping the space continuously filled, then y> becomes the poten- 

 tial fuuction of a continuous mass, w and r become partial differential 

 coefficients of this potential function, and it is known* that both for 

 such functions and for their differential coefficients, the portions of the 

 function that depend upon the presence of matter within an infinitely 

 small space surrounding a point for which the function is determined 

 are infinitely small with respect to those portions that depend on finite 

 masses at finite distances. 



* See Gauss, Allgemeine Theorie des Erdmagnetismiis in the Resaltate des magnetischen 

 Vereins im Jahre, 1839, page 7, or the translation iu Taylor's Scientific Memoirs, 

 vol. II. 



