CONFORMAL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 455 



This formula shows that when p < |- the value of R for w = c is zero. It also 

 shows that in the neighbourhood of the point where x = the curve can be made 

 as flat as may be desired by taking A sufficiently small. 



19. It is natural to expect that the parameter a of the transformation (29) will 

 particularly affect the shape of the curve at the extremities of the linear range. 

 In order to study the form of the curve in the immediate neighbourhood of the 

 point w = c it is convenient to put 



-(x-x +iy) exp (ipir) = g+i>,, 



where x is the value of x for w = c ; this makes f and / co-ordinates referred to the 

 point w = c as origin, the axis of being tangential to the curve in the direction 

 of w decreasing. In the expression on the right-hand side of the transformation in 

 is put equal to ce, and an approximation made on the supposition that e is small. 

 This leads to 



d(g+i,,) = Ke 



where 



__ _ 

 X+l J V(2c)'' 



Integration gives 



= 



K 



l-p 



>i 



l+a-p 



sin a- . e 



I+-P 



so that in the immediate neighbourhood of the corner 



(46) 



(47) 



The velocity of slip near the corner is de/d, and consequently is proportional to the 

 {p/(l-p)} ih power of 



20. Motion of a Ship with Pointed Bow and Flat Stern. When the internal 

 boundary in the z plane is symmetrical about a line in the direction of flow, and 



Fig. 5. 



consists on either side of a curve departing from the line of symmetry at an angle p-v 

 and rounding smoothly into a straight line which cuts the line of symmetry at right 

 angles, the configuration has the general character of fig. 5. 



VOL. CCXV. A. 3 P 



