Leathrm — On Two-Dimensional Fluid Motion. 



31 



For a real value % of £, intermediate between and a, the vector angle x 

 of this curve-factor is 



J 



:'?) 



|7r/( K )d K i tan-'!,c/($- k))V"(k) '/k, 





tan-M(g -*)/«)*/(*)&; 



and from this 



p ff = X (Q) = *irf/(a)-/(0)J 



d x = 



2? J (E - «)* 



t /'(k) <fe. 



(46) 



(47) 

 (48) 



On substitution of this value of tf x 



{=« 



i = o 





4 = n k = J 



2f(a-g)*J (£-*)* ' 



4=0 «=o 



Though the subject of the double integration has an infinity for k = 0, ? = 0, 

 this is not of sufficiently high order to preclude change of the order of 

 integration. If this change be effected it appears that 



| = « 



{ = « 



rf 



Y 



4=0 



K = 



K = (7 

 i 



d£ 



4 



£ (« - l)\l - K? 



4=« 



K=0 



--***"*{/(«) -/(0)| =-^«"*. 



This shows that for all curve- factors of this particular class S = 0. 



(49) 



Formula (45; should be compared with Formula (4) of article 6, above. 

 It seems that formula (4) cannot be general, since it gives a curve-factor 

 possessed of this very special property. This casts doubt likewise on the 

 generality of formula (5j, and suggests the possibility that Mr. Levy's method 

 of approximation (paper E) may be less general than it seems. 



