484 



DK. J. G. LEATHEM ON SOME APPLICATIONS OF 



(expressed now in terms of z) into the left side of the transformation. For example 

 if one factor & of linear range a r to a r+l and angular range yir were introduced, the 

 transformation would be of the type 



with the condition 



.y = p + q. 



(132) 

 (133) 



A practical inconvenience of the present method of constructing transformations 

 applicable to the ship problem lies in the fact that, while the condition (129) or 

 alternatively (133) secures that the angles PTT and q-rr shall have any prescribed sum, 

 it does not suffice to secure prescribed separate values for these angles. When this 

 is desired there is a further condition to be satisfied in the form of a relation between 

 the parameters c, a } , a 2 , ..., c. What is wanted is that the vector angle of the 

 complex f (c) f ( c) shall be pir, and this is equivalent to 



pc 



Vector angle of f(z)dz = p-jr. 



Generally the fulfilment of this condition cannot be arranged for without previous 

 evaluation of the indefinite integral of f(z). Thus the problem of integration, which 

 must inevitably be faced in any case in the detailed interpretation of a transformation, 

 presents itself here at an earlier stage. 



50. The method of the previous article may be further modified by associating 

 with dz such Schwarzian factors, powers of zc, as shall remove the corners at c 

 and make the first and last portions of the boundary productions of the straight line 

 that corresponds to the curve in the z plane. In this case there are no Schwarzian 

 factors to be associated with dw, so f and w become identical save for a constant 



to- 



(nn 



multiplier, and the configuration in the w plane is as shown in fig. 19, the marked 

 angles being called rmr and mr. The transformation is of the form 



~ , 



= C dw. 



In comparing the z and w configurations as represented by figs. 17 and 19, it is to 

 be noted that corresponding angles at c, though not equal, are proportional. Hence 



