Lkathkm — On Two- Dimensional Fluid Motion. 



27 



From bhia formula it is seen bhafc, if £"„ have a real value £ greater than ", 



and if G(S, V ) = iT exp ({%,), 



Xo = ^-2tan-><I°^j + 



•«/ 



V I 



it - _ tan 



.,(«.-«)*! 



so that 



f=o 



(« - 5> 



(33) 



{=« 



Xo(«) =1'tt + rf X , and x „ (co ) = 0, 



| = o 

 as was to be expected. 



If Xo( a ) + $Xo :U1( 1 o + e be corresponding values of \ au( l 5o> 



{ = « 



g Xo - - 2p tan- 



2 



7T 



/" 



i 



(34) 



tail" 1 I -. )" dv. 



a j 7T J \a-Kj 



4 = 



When e is very small the first term of this formula can readily be replaced 



by a simpler approximately equivalent expression ; but it is not obviously 



legitimate to substitute je/(a - £)P for tan~ l U/( a ~ £))* under the integral 



sign, since there is a part of the range of integration in which a - £ is very 



small. If, however, e = i; 2 , the theorem of the mean, applied to the subject 



of integration regarded as a function of >j, gives 



tair 1 {„/(« - g)*j = i»(a - £)*/(« - % + »/ s ), 



where i; > i{ > 0. Thus 



vJ f (a - 4)* d x 



»X» 



2^ tan" 1 



a - £ + *' ' 



£ = 



(35) 



where e > t' > 0. The last integral has a definite limit value for e -i> (which 

 involves t' -> 0), and therefore the equation 



- Sxo = 



2par* + 



{ = « 

 r d. 



■ f rt X 

 "J (a -ft* 



(36) 



{ = 



is a valid first approximation to formula (34). Thus £\„ is generally of t In- 

 order of smallness of e*. Of course a higher order of smallness is possible in 

 particular cases. 



If 8s be the corresponding element of arc of the free stream-line, 

 8s = | G(%) I « = -STe- Hence generally 8 x Jm -> co . 



It must be noticed that the above argument has tacitlj assumed the 

 definiteness of d\ld% for £ = a. It may be taken that, throughout the 

 present discussion, the hypothesis is that the sides of the obstacle are 



R.I.A. PROO., VOL. XXXIV., SECT. A. [5J 



