Lkathkm — On Two-Dimensional Fluid Motion. .'39 



Though there is no loss of generality in dealing with an arbitrarily selected 

 value of a, it is not permissible for the present purpose to keep the value of 

 c fixed. So c must be eliminated. This gives 



— a —a 



If, for one selected value of £, a small increment S\ were made in d\, the 

 increment of i\ . I, would be /(£)(?%', where 



= 2p»r 



« - c a + £\j fa + c a - £\i ) 



a + c a 



5 ; + U~ c - f 77ij I" (72) 



Hence if two simultaneous increments of d%, namely 8,^' for f = £,, and 

 82X' Ior ? = ?2> ' Je suen that ^ey leave the equality (71) still true, it is 

 necessary that 



/(f,)8^+/(g,y8.x'-0. (73) 



Now, the two terms in /(£) have a product independent of (J, and are equal 

 when £ = c Hence /(§) has its minimum value for £ = c, and increases as 

 | varies from c towards either a or - a. And so, if a > £, > |, > c, or 

 c > 5. > & > - «, tlien /(&) >/(?,), and therefore, by (73), | 8, X ' I > I &*' I ■ 

 Thus if <S,x' is positive S.%' is negative, and S,\' + S«\ is positive, so that 

 there is a net increase in the whole range of ^'. 



This means that any change in the functional form of \ which has the 

 effect of bringing curvature nearer to the prow c increases the negative range 

 of y, and so makes the free stream-lines less divergent at their points of 

 departure. The bearing of this principle on the resistance in a case of actual 

 (as distinguished from theoretical) How is the same as when the flow is 

 symmetrical. 



K.I.A. PROC, VOL. XXXIV. SECT. A, [71 



