442 



DE. J. G. LEATHEM ON SOME APPLICATIONS OF 



a r . For real values of w intermediate between a r and a r+l the vector angle of *& 

 must change continuously with w. 



(ii.) ^ must not be zero or infinite for any definite value of w which is real, or for 

 any definite complex value of w corresponding to a point on the relevant (positive) 

 side of the axis of w real. Such zeros or infinities would destroy the conformal 

 character of the transformation, and would, if occurring on the boundary, interfere 

 with the prescribed arrangement of corners. The word " definite " is here used so 

 that the restriction may be understood not to apply to w infinite. Certain 

 singularities at definite points might be capable of interesting interpretations in the 

 application to hydrodynamics, but it seems best at present to exclude them. 



(iii.) *$ must not have any definite branch points on the relevant side of the axis 

 of w real. It may however have branch points on the axis of w real, since this axis 

 serves as a barrier preventing such circulation round a branch point in it as would 

 make the function many- valued. 



(iv.) The form of ^ for w infinite must conform to conditions depending on the 

 nature of the particular problem to which the transformation is to be applied. 



5. Linear and Angular Ranges. Tlie range on the axis of w real corresponding 

 to values of w for which the vector angle of ^ is variable may be called the " linear 

 range" of the curve-factor. The difference between the vector angles at the 

 extremities of the linear range may be culled the "angular range" of the curve- 

 factor, this being reckoned as positive when the greater angle corresponds to the 

 smaller value of w. 



In the case of a simple curve-factor the linear range consists of the real values of 

 w between those typified by ,. and r+1) and the angular range is the angle between 

 the tangents at the extremities of the corresponding curved side of the polygon, the 

 standard case being that of convexity towards the relevant region. 



A curve-factor may of course have its angular range zero ; in that case there will 

 be an inflexion on the corresponding curved side of the polygon. 



It is readily seen that if ^ be a curve-factor, and if n be a constant, then #* also 

 is a curve-factor, having the same linear range as ^, but its angular range equal to 

 n times that of ^. It is important to notice that this statement is valid even if n is 

 negative, for as ^ has no zeros or infinities in the relevant region (except possibly 

 for w infinite), the raising of ^ to a negative power does not introduce any fresh 

 infinities or zeros. But if 9 has a positive angular range and if n is negative the 

 angular range of V* is negative. Thus, if * gives a curve which is everywhere 

 convex to the relevant region, positive powers of give curves having the same 

 characteristic, but negative powers of ff give curves which are concave to the 

 relevant region. 



In connexion with requirement (iv.) above, the form of # for w infinite is 

 sometimes important. If, for w^oo, V + ku, where k is a constant, it will be 

 convenient to speak of m as the "order at infinity" of the curve-factor. Clearly 



