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



If x be, as usual, the angle between the axis of z real and the tangent at any point 

 of the curve corresponding to w real, the sense of the tangent being that of tv 

 increasing, the external angle of the polygon whose limit is to be the curve may be 

 taken to be d x , so that, if Q be the value of w at the corner, the Schwarzian factor is 

 (^fl)-''*/". The parameter 9 varies continuously along the curve, and the curve- 



factor 



lira IL(w-0)- d * lr 



can generally be replaced by the expression 



the integral extending over the whole curve. 



It is to be noticed that in this expression there may be an infinity in the subject of 

 integration at a point corresponding to a real value of ir. This infinity, being less 

 powerful than an infinity of the type (ir 0}~\ does not interfere with absolute 

 convergence of the integral. Formal proof of this statement is considered unnecessary 

 here, being of a character readily suggested by familiar treatments of convergence 

 tests. It may be illustrated, in the hydro-dynamical application, by the fact that, 

 while a convex angle in the boundary gives rise to an infinite velocity, a convex 

 smooth curve does not. 



But, while there is no divergence of the integral, the formula (113) is nevertheless 

 indefinite in the absence of any specified functional relation between the variables 6 

 and x- The Schwara transformation takes account only of the angles of the 

 configuration to which it is applied, and leaves the adjustments of all lengths to 

 be dealt with after transformation by the assigning of suitable values to the 

 various parameters associated with in in the factors. It is therefore not surprising 

 that a transformation containing only Schwarzian factors and limits of products 

 of such factors should ensure only that the directional and angular aspects of 

 the configuration are properly dealt with, leaving all settlement of correct linear 

 dimensions to be adjusted subsequently by the assigning of suitable values to isolated 

 parameters and suitable functional relations to parameters which vary continuously. 



It is therefore proper to assume a functional relation between x an d , say 

 X = vf(6), with the reservation that the nature of the function f is determined 

 by the configuration which is being dealt with, and depends not merely on the curve 

 to which ^4! corresponds but on the whole angular and linear configuration of the 

 prescribed boundary. 



45. Before attempting to. formulate the condition which f must satisfy in order 

 that ^ 41 in a transformation of suitable type may represent a curve of assigned form, 

 it is convenient to note the usefulness of an alternative method of applying formula 

 (113) which consists in assigning such arbitrary forms to f as lead to simple 



