Lkathkm — On Two- Dimensional Fluid MoUon. 

 Similarly 



jr i2(a-c))i 



I 



2Ja-J)\i 



a (« + £)/ 



33 



(55) 



The same argument may be applied to this pair of formulae as has been 

 employed in the case of a symmetrical obstacle; when the form of ^ is 

 assigned the simultaneous equations got by equating both coefficients of t^ to 

 zero determine values of a and c corresponding to the How in which the two 

 points of departure of free stream-lines are as far forward as is physically 

 possible. 



20. The approximation to %„(«) - \ {a + s), when t is small and positive 

 may be carried a stage further as follows. 



Since 



d^ _, | a ( a -1) - & ) £(2a + *)$ 



a(a - f + (•) j (,,- _ j»)£ (a _ £ + 6 )' 



and since ^(c -j - y(« +) = (1 - 2]Jj7r, it appears, on integration by parts, 

 that 



\o(a) - X»(« + <0 



COS 



I «(« - c + ) it ' { J -,«.,-) (^ - g»)i (« - ? + g) 



(56) 

 Also, by repeated use of the theorem of the mean, 



H' 



x(5) - x(«) 



(« 2 - sv* (« - £ + e) 



<^ 



(a 



x(g) - x(«) (Z g _ f x( ft + *) ~ %(") rftf 



• (a-g){ .\(«) - xte + 0] + tix(«)- x(S)l 



(«= - r)* (« - ej (o - e + c} 



r>-g) t U'(r)-x> + OMS 



«, 



C.7) 



\"(rn-M £ '-g')^ 



(a«-g>)l( a _ 5 + e) ' 



where (i>!'>§, t > t' > 0, a + e'> £">£', and the integrations are From 

 -ft to ft. The last integral, if \"(?) is assumed continuous, has a definite 

 limit for t -^ ; for if t j> then also t'->0, and if £->" then also £'->«. 

 Hence IV is small of the order of e. 



