599 



Finally the value of a becoines: 



261 

 « = ^ + (43) 



The values given by (42) and (43) are inserted into the principal 

 equation (17); retaining the terms of the highest order only, we find: 



1 261 



a = — (44) 



0,710 Z^.K 0.710 Z),'/?' ^ ' 



a attains its maxirtium value if the lower limit D, of the thickness 

 of the vortices is determined by ; 



522 

 A=-^ ....... (45) 



This is much below the value of D given by equation (37). 

 Using (45) we find : 



a = 0,00135 + ... ■ (46) 



and the coefficient of the resistance formula becomes : 



S 1 



Cr=-— =0,00135+ terms of the order - . . (47) 



So this arrangement of the vortices leads to the quadratic law of 

 resistance. 



^ 6. Discussion. 



In paragraph 5 II we have found the value 0,00135, as a higher 

 limit of the coefficient C of the resistance formula using an 

 idealized model of the distribution of the vorticity in a turbulent 

 current. 



If it is possible to calculate C without the use of this special 

 model, using equations (17) and (18) and conditions (6) and (7) only, 

 a still higher limit will probably be found. At the other side if 

 we compare the value of C obtained here to the value given by 

 formula (46), it appears that in the region which is of importance: 

 ft = 10000 to 1000000, the value of C is too high.') 



Hence we may assert that the true resistance is not the highest 

 possible resistance. In order to delermine the true state of affairs, 

 a further condition will be necessary. 



From the result that the value of C appears to be too high, we 

 may deduce that the distribution of the value of — uv over the 

 current is too uniform. Paying attention to the results of measure- 

 ments of the distribution of the velocity over the breadth of the 



') According to Gouette's experiments turbulence sets in at i? — ea. 1900. 



