593 



Tlie integral of the square of the vorticity iV, = | j dxdyZ,'' 



extended over tlie same area becomes according to the formula 

 given by Lohentz : 



b 



iV.=iL,3+2,.+30j'*.v(^-;;) 



42 







(29) 



From (28) and (29) we deduce: 



TV, _ K5 (3 4 2f' + 3f«)( l+ e') 1 

 ^""T e'(l-f') ï»' 



or, introducing the "thickness" D of the voitex (cf. fig. 2), so that: 



4f 



we get : 



A', 3 4- If- -y 36* 1 294 



-^=r30— =: .... (30) 



This fraction surpasses only by a small amount its minimum 

 value, calculated by Lohentz : 



2(3+2e' + 3f*) 1 288 ') 



14,68 



6(1-8*) I>' X>' 



^ 5. Distribution of the vortices over the fluid. 



It has already been remarked in ^ 1 and 3 that our object in 

 this paragraph is not to analyse the true distribution of the vorticity 

 of the fluid, but that we will construct an idea! case only, a "model", 

 which atfords us an admissible image of the behaviour of the 

 quantities uv and S'. This model is obtained by distributing a number 

 of elliptic vortices, of the type studied in the foregoing paragraph, 

 over the mean current U [y). In doing this we do not want to pay 

 any attention to the abscissae of tlie centra of the vortices, if only 

 their mean distribution along lines parallel to the axis of x be 

 uniform. Positively and negatively rolatiiig vortices are distributed 

 uniformly through each other. If two or more vortices may happen 

 to overlap, they may as well strengthen as enfeeble their respeclive 

 fields; hence in calculating the mean values uv and ?' it is un- 

 necessary to lake account of these overlappings, and the contribu- 

 tions of the different vortices may be simply summed. 



If for a moment we direct our attention to a special class of 



1) Gomp. a remark made by Loeentz, I.e. p. 54/55. The function defined by 

 eq. (25) above fulfils the condition : du./ds = for s — 1 (s = rjb). 



39 



Proceedings Royal Acad. Amsterdam. Vol. XXVI. 



