450 DR. J. G. LEATHEM ON SOME APPLICATIONS OF 



15. Longitudinal Motion of a Ship with Curved Sides terminating in a Pointed 

 Bow and Stern. -The problem now to be considered is that of the disturbance 

 produced in a liquid stream, which if undisturbed would be uniform, by a stationary 

 object which is bounded by two curves and is symmetrical about a line parallel to the 

 stream. The curves meet at one end at an angle 2p-7r and at the other end at an 

 angle 2qir, p and q being each less than a half. The configuration in the z plane is of 

 the general character indicated in fig. 3. 



As there is a line of symmetry which is obviously a stream line, only half the 

 configuration need be dealt with, and a transformation is required which will make 

 the boundary shown by a thick line in the z diagram correspond to the axis of w real, 

 and conformally represent the region to the left of the arrows. 



The angles in the boundary must be taken accouiit of by Schwarzian factors in the 

 transformation ; if c and c be the values assigned to 10 at the corners, these factors 

 are (iv c}~ p and (; + c)~ ? . The curved side must be represented by a curve-factor <@ 

 of linear range c to c. 



Let the angular range of a transformation be defined as the angle between the 

 limiting directions of the vector dz for w real and -> + oo and for w real and -^ oo , 



-c 



Fig. 3. 



measured in the positive sense from the former to the latter. In the present 

 instance the angular range of the transformation is clearly zero. But, if y be the 

 angular range of the curve-factor, the angular range of the transformation is made 

 up of -p-TT contributed by one Schwarzian factor, -q-w contributed by the other, and 

 y contributed by the curve-factor. Hence y- ( p + q) * = 0, and the curve-factor 

 must have an angular range of (p + q) TT. This can be secured by taking the (p + q) th 

 power of a curve-factor of angular range TT. Thus the type of transformation is 



dz = C(w-c)-p(w + c^-"^ ( P^dw, ....... (28) 



where ^ has the linear range c to -c, and an angular range TT. 



Of the curve-factors so far considered the most general is #,. The use of this, in a 

 slightly modified form, leads to the transformation 



w , , 



where X > 0, c > k > - c, and it has been arranged that | dz/dw \ + V" 1 for w -> + oo . 



