597 



conditions can he found, if all vortices are put against tiie walls. 

 If this be done, it is of course unnecessary to use the variable § 

 introduced in tlie beginning of this pai'agraph, as the positions of 

 all vortices are fixed. Only a <iet,erniination of the function B[D) 

 is wanted. The following form of this function gives the right 

 distribution of ity-values: 



1. the class of vortices whose thicknesses lie between the limits 



D and D-\-dD have a total intensity proportional to _6f/Z)= 2 — ; 



these vortices are divided into two equal groups, each of them 

 situated along one of the walls; 



2. besides the vortices mentioned under 1), there is a number of 

 vortices of thickness D^i, which have the total intensity '/, (in 

 same unit as used above). 



With this determination of B{D), the value of — uv appears to 

 be. if Z),<,v<l-0.: 



1 1 



V 1 — 1/ 



1 1 



J V J V * 



+ JY'(.V)- 



1 (40) 



i-y 



1 



"~28Ö 



The first term represents the contribution of the vortices lying 

 along the wall y = 0; of these vortices only those are of importance 

 for which Z) ]> //. The second term represents the contribution of 

 the vortices situated at the other side; here only those for which 

 D^l — y are of importance. The third term represents the contri- 

 bution of the group of vortices whose thickness D is equal to 1 '). 



') If we should take the quantity B proportional to D~", with n < 1, the 



integral / ^'^ di/ would take a smaller value, but now the first term of equation 



(40) which gives the contribution of the vortices situated against the wall y = 0^ 

 would become: 



1 1 



J'^v(^^ = y''"'jdvri'+"0-r}y (for y>D, 



y y 



If y becomes small, this expression approaches to zero. Only if n = 1 it ap- 

 proaches to a value independent of y, which is necessary in order that a constant 

 value of — uv at all points outside of the boundary layer may be obtained. 



