( 132 ) 



to the vortex element and then farther in the Sp^*~^ of constant azi- 

 muth by a sj'stem of spherical coordinates, of which we take the 

 first axis in the "meridian plane" (see above under § VIII), and in 

 the plane of the vortex element, the second in the meridian plane 

 perpendicular to the first, and the rest arbitrarily ; let us understand 

 meanwhile by (p here the angle of the radius vector with the Spn—2, 

 perpendicular to the vortex element ; let further s be an {n — 3)-dimen- 

 sional element in the n — 3 last coordinates, then this defines for 

 each r and (p an element on the surface of an "— 3gp}^gj.g^ of a size 



dk := ce sink n—^r cos ^^—^(p. 



We then consider a small elementary rectangle in the meridian 

 plane bounded by radii vectores out of the origi-n and circles about 

 the origin and a Spn—i element consisting of the elements dk erected 

 in each point of this small elementary rectangle. Applying to this 

 /Spn—i -element the reduction of an {n — 2)-fold integral along the boundary 

 to a {n — l)-fold integral over the volume according to the definition 

 of second derivative, we find : 



d 



{U cos (f . dr . CE sink "— ^r cos ^—^(p\ dtp — 



d 

 dr 



\U sin (p . sink r d(p . ce si7ih n—3^ (.Qg n— 3^| d^, ^^ 



cosh r 



=z ce sink »— ^r cos "~3«) . sink r dip . dr . — 7— — sm (p. 



sinh "— 'r 



d U cosh r 



(11 — 2) U sinh 1 in — 2) U cosh r =: —r- ^- • 



^ ' dr si?ih''-^r 



dU cosh r 



h (n—2) tanh 1 r . £/ == — — . 



dr smh^-^r 



The solution of this equation is: 



I r 1 



U = . cosh -2(n-2)ir . coth «-3i»' .dir^ — — - 



2»— 3 J {n—2)sinh »-^r 



So we find as planivector potential V of a plane vortex: 



1 



^ I coth n-3Lr .dhr^ F, (r), 



{n—2) sinh "— 2^ 2«— ^ cosh 



directed parallel to that plane vortex. 



Let us now call E the "— 2yector, perpendicular to the plane vortex, 

 the field of which we have examined, and let us also set off" the 

 vector potential V as an "-2vector ; let us then bring in an arbitrary 

 point of space a line vector G ; then the vector V has the property 



