594 



vortices, the thickness D of which lies between the limits D and 

 D -\- dD, and the lower tangents of which (i.e. the tangent at the 

 point D ill fig. 2) are enclosed between tiie limits y = ï and 

 y=^^-\-d'^, then we may say that ail of them are lying between 

 the same lines parallel to the axis of x, and by what has been 

 remarked above all of them will give proportional contributions to 



the field of uv-values. As the integral — I dx uv extended over a 



section FR of a single vortex has been calculated in (26) and 

 (27), we may write the contribution of the whole class: 



b (Z>, 5) ii'(\--riY dD di = bffi^i) dD d%. 

 In this expression: ij = (y — §)//), and the factor b {D,§) dD d^ 

 represents the product of the number of these vortices contained in 

 a strip of unit length parallel to the axis of x, their mean intensity 

 (i.e. the mean of c'), and the otiier factors which are contained in 

 the letter A of formula (27). If the function 6 (Z>, §) is given, the 

 distribution of iw can be calculated. 



It is not necessary to know the value of the quantity ?" at every 

 point of the current, its integral only over the whole breadth being 

 wanted, which integral can be found as the sum of the integrals 

 of ?* over all vortices contained in a strip of the full breadth, and 

 of unit length. With the aid of formula (30) we find as the 

 contribution of the considered class of vortices: 



rr '294 rr 



I I dxdy §' = —— \\ dxdy uv 



294 r Cy-^\ 



= -l>dDd,jdy„(^-^y 



f+'D . • . (31) 



63Ö ]T 



A simplification further arises from the fact that the second and 

 third equations (16) which determine r and x are homogeneous as 

 regards to the intensity of the vortices. In using these equations it 

 is allowed to multiply h with an arbitrary factor. The true value 

 of (J is found from the principal equation (17). It would be possible 

 to calculate the true value of b afterwards, but this is of no use. 



The problem put in paragraph 3 : to make a as great as possible, 

 obliges us to search for a function b [D, i.) which gives a value of 

 — uv as nearly constant as possible. Two rather simple types of 

 functions will be discussed. 



I. We will begin with an investigation of what can be reached 

 if all vortices have the same thickness D. In that case in order to 



