589 



according to (18) C approxiiriutes to the same constant value, and 

 thus according to (1) tlie quadratic law of resistance is obtained. 



This reasoning is in many respects vagne, and it does not admit 

 of a determination of the values of c, and c,. It only shows that 

 tlie particles of fluid with high values of the vorlicity l^t must be 

 concenlrated along the walls. To get a more definite result it is 

 necessary to develop a piclure of the structure of the turbulent 

 motion. Two ways may be followed : we may try to analyze the 

 possible motions into a sum of elementary functions (gonionietrical 

 or others) in a manner analogous to a series of Foukieh ; or we may 

 imagine the motion to be built up from an assembly of individual 

 vortices (\ oriex filaments with their axes perpendicular to the plane 

 of x-y), distributed in some way or other throughout the fluid. In 

 the calculation of the critical value of R (i.e. the value at which 

 the turbulence occuis for the first time) analogous methods have 

 iieen used: REVNOi.ns, Orr and other writers have directed their 

 attention to disturbances which are propagated in a periodic way 

 through the whole fluid; Lokentz at the other hand has studied the 

 disturbance caused by a single vortex '). 



The statistical treatment of such an assembly of elementary motions 

 is very difficult on account of the circumstance that every elementary 

 motion is damped by the action of the viscosity. At the other side 

 the mutual actions between the elementary motions (brought forth 

 by the quadratic terms in the equations of hydrodynamics) and the 

 influence of ihe mean motion continually generate new motions. 

 From the formula given by Lorentz it follows that types of motion 



for which I J dx dji uv is negative, are intensified by the action of 



the mean motion. Hence a mean stationary state can exist, in which 

 every elementary motion changes continually its intensity and its 

 phase (or its position, if it is an individual vortex), but in which 

 every one of these motions has a constant mean intensity. It is 

 obvious that for the greater part, if not exclusive, these will be 



types of motion for which I I dx dy uv <^0. 



The statistical problem will not be attacked here. On the contrary 

 a simple type of turbulent motion will be studied in the following 

 paragraphs, built up from an as<semltly of elliptic vortices, all of 

 them having Ihe same configuration, but having different dimensions. 



1) 0. Reynolds, I.e. p. 570; 



W. Mc. F. Ork, Proc. Roy. Irish Acad. 27, p. 124-128, 1907; 

 H. A. Lorentz, I.e. p. 48. 



