596 



(if tlie terms of tiie highest order only are written down). This 

 formula gives a maximum value for a if the thickness D of the 

 vortices is determined hy : 



which gives: 



V R 



(37) 



(38) 



The coefficient C of the resistance formula (1) now becomes, 



according to (J 8) : 



5 0,027 1 



C= = —-=-4- terms of the order - . . (39) 



C diminishes proportional!}' lo — =:; hence we do not obtain the 



quadratic law of resistance, but the resistance appears to be propor- 

 tional to the l^-power of the velocity. This does not conform to 

 the result of paragraph 3. In the kilter paragraph, however, it was 

 assumed that the most intensive vorticity was concentrated in the 

 'neighlioiirhood of the walls only, whereas in tlie model considered 

 above it is distributed uniformly over the whole breadth. If all 

 vortices have the same dimensions, it is not possible to distribute 

 them otherwise, without disturbing the field of «I'-values. Hence we 

 must try to obtain a better result by using vortices of different 

 dimensions. 



II. If we take vortices of different dimensions, say with thick- 

 nesses ranging from D =: 1 to a lower limit Z), (to be determined 

 later on), the thickness of the boundary layers in the most favourable 

 case will be of the same order of magnitude as D,. The same 

 applies to the quantity t. If now the contribution of the vortices of 



thickness D to the integral I ?' % becomes asymptotically propor- 

 tional to -—- for small values of D, the value of this integral will 



1 

 become of the order of magnitude of — . In this case we shall be 



in the circumstances considered in the deduction of equations (J 9) 

 and (20). Paying attention to equation (31), it is necessary that 



/• 1 



B := I bdl shall be proportional to — for small values of D. 



Now it appears that a distribution of vortices fulfilling these 



