Lbathkm — On Two-Dimensionul Fluid Motion. 



29 



It is clear that generally the integral in this formula is a continuous function 

 of c and has a definite limit for t > 0, since i -> involves i -i> 0, 

 and £ >" involves £' -^ a. Of course the demonstration contains implicit 

 assumptions as to the continuity of \('i.) and its first and second derivatives, 

 mi both sides of the value £ = a. 

 Now 



\(" + «)- x(«) 



(« - £)* (a + i - £) 



</£ -> f\'(") 



„(«-£)*(« -$ + «) 



-3,4 



4 ' 



2e'x'(«-) 



(39) 



tan- 1 {(a - $) J A"I 



= 2t\'(a)\hTr - tair'(tA0-i, "> 7r\» t * 

 On substitution of this in IT it is seen that (IS) is equivalent to 



jr. f" x(Q - x(«)_ ^ ■ f- x(|)jl^l) (/ , + iKy , {a) + fi() (40) 



J , (« - <0 7 (« -.* + «) I o (« - £) 



where ai is small of the order of e. 



Now if the integral in formula (34) be integrated by parts, it being 

 remembered that \(0) = pir, there results 



- <i\ = — \(") tan" 1 



1 e* 



and therefore, by (40), when i is small 



r 2 i f« \(E) - \(") i 



-8x. = «» -»-»x(«)-- *W_lW d5 _ . 



(41) 



), (42) 



small quantities of the order t- or higher being neglected. Integration by 

 parts within the square brackets leads to 



§Xo = s- 



2p 2 [« d£ 



2[» rfg I 



t Jo («-5)*J 6X 



(«)• 



(43) 



This is the desired improved approximation. It will be convenient to denote 

 the coefficient of i- in this formula by 5. 



A very important inference from equation (4,'!) is that, if the coefficient 

 of s* be zero, - 8\ equals the angle of contingence of the fixed boundary foi 

 an element of arc Ke. Thus in this ease the free stream-line and the fixed 

 boundary osculate at their point of separation. 



l< 



;*i 



