591 



axes of the ellipse have the leiigihs 2b and 2f6, in which expression 

 6 has the value 7» (' ^^ — 1/6) ==: 0,475 ; the smaller one makes the 



angle a/r/t/ - =r « with the direction of the mean motion. The con- 

 ■ s 



jugaled diameters .4 B and C D correspond to the diameters of the 

 circle J„ B^ and C» D^, which make angles of 45° with the direc- 

 tions of the axes of the ellipse. Besides the system of coordinates 

 .r„ //„ used by Lohkntz, the system *•,_(/, along M, B, and M^C, 

 will be introduced. 



From the formulae given by Lokkntz at page 49 we deduce the 

 following expression for the value of no in a point of the vortex, 

 corresponding to the point >i'„ y„ of the circle: 



1 . , 



A/, = — Ml) = — (^'j' — f'i/»') "'' *■*" 2« -\- ix^y^ a>' cos 2« = J 



For the determination "of the mean value of uv along a line 

 parallel to the axis of x, it is necessary to calculate the integral of 

 if„ along a line /' R which is [)arallel to the same axis. This line 

 corresponds to the line f» R^ of the circle; the lengths of these 

 lines are in the constant proportion: 



A,B^ V 2 sin a V 2 



Hence tins integral takes the value: 



^^'^ ^^>j7f==^i'^-.'(l-f') + ^,2/>(l+f=)l- . (24) 

 -i/i 1/7» 

 As has been mentioned already above, tu is a function of 

 /•„=:* A'o' + ,y„' = ^^'r,' + y,' ; this function wUl be taken to be: 



w = c(è"-r.')'/, =r c(6,— ^,'-?/j')V' ') . • . . (25) 



The second terra of the integral vanishes on account of the sym- 

 metry of (u ; the first term gives: 



l/2f(i-6') r 

 ^^' = ~VY^ "J ''•"' '■'' ^^^ '^^ -3/>')'/' 



') In the formulae below everywhere c' occurs; the sign of c is of no im- 

 portance. 



