4 8 2 DE. J. G. LEATHEM ON SOME APPLICATIONS OF 



hitherto considered; but in connexion with the problem typified in fig. 3, the 

 problem namely of obtaining shapes of doubly-pointed ships whose hydrodynamical 

 effect in longitudinal motion can be exactly specified, something can be done with the 

 more general transformation. 



It is easy to verify that the transformation for a semi-circle and the productions of 



a diameter, namely 



dz = G (w 2 -c 2 }-^ {w+ (w 2 -c s ) l/2 } dw, 

 is equivalent to 



d* _ A?- dw /, 27 \ 



*"(5C^K !l 



and that the transformation for a semi-ellipse, (Cc cosh a, Cc sinh a), and the 

 productions of its principal diameter, namely 



dz = C (nf-cY 1 ' 1 {w sinh + (w'-c 2 ) 1 '- cosh a] dw, 

 is equivalent to 



an intermediate variable f being for convenience introduced in each case. 



In these forms it is to be noticed : (i) that the factors in the denominators on the 

 left-hand side do not lead to zeros of dz/dw or to corners because the points in the 

 z plane where they have zeros are not in the relevant region, and (ii) that the only 

 factors left on the right-hand side are the Schwarzian or corner factors. Thus the 

 effect of the (z, f) transformation is to straighten out the curved part of the boundary 

 without alteration of the corner-angles, and the effect of the (, w] transformation is 

 to smooth out the corners. 



In both cases the (z, c) transformation is of the Schwarzian type, making the axis 

 of z real correspond to a rectilineal open polygon in the f plane, but these polygons 

 do not constitute the boundaries of the regions which are relevant to the present 

 problem. A straight line joining points in the first and last arms of the open 

 polygon in the plane screens off all the original corners from the relevant region, as 

 it were short-circuiting a part of the boundary including all the corners ; and the 

 corresponding line in the z plane is a curved line which screens off from the relevant 

 region all those points on the real axis which corresponded to the corners of the 

 Schwarzian transformation. 



This aspect of the (z, f) transformation suggests generalisation by the introduction 

 of any number of corners on the broken line in the plane which corresponds to the 

 part of the axis of z real which is screened off from the relevant region. 



The desired configuration in the z plane being as shown in fig. 17, the values pv, 

 q-ff of the marked angles being prescribed, and the values c, -c being assigned to z at 



