04 THE MECHANICS OF THE EARTH's ATMOSPHERE. 



locity if iu tbe expression for r we substitute ri, g, Xi c, z, m, iu place 



of T, J, f/, Z, C, Wi. 



In this process « and Z7 remain unchanged and we obtain, 



WTj+Wirii/=0 (S) 



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

 caused by the vortex filament nii whose coordinates are r/ and c, we 

 find: ' _ _ 



If now we call ?c, the velocity at the locality of Wi parallel to the 

 axis of z^ 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: 



„ ., ,, ., 2mmi , -.^ ^ , 



1m icx- — 2 rih n\ (j~— m rxz—rii 1 r i</c= ---- f(fx U. . . (8a.) 



Sums similar to (Sj and (Srt) can lie found for any number of vortex 

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

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

 all the other rings are r„ and ?c,„ 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 o„ 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 '"^iitl z. 

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

 rings, be ^'„. By forming and adding the equations (8) and (8«) corre- 

 sponding to each pair of vortex rings, there results 



2 [m,, Pn r,.J=0. 

 2 [2 w„ »•„ p-,, — m„ r„ p„ /\,J = 2 [m„ p, >/•„]. 



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

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

 ?/■ only those parts of these quantities that are due to the i)resence of 

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

 rings keeping the space continuously filled, then v becomes the poten- 

 tial function 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 Erdmaijnefismus in the liesultate dts mar/netischen 

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

 vol. u. 



