24 Proceedings of the Royal Irish Academy. 



The significance of the functional relation between S and £ becomes 

 clearer if the vector angle of dz/d£ on the boundary is denoted by \ ari( l it 

 is noticed that, for £ > <?, 3 = x> w »ile, for £ < c, .7 = \ - tt. Thus along the 

 curved sides of the obstacle d& = <\. and d\ may be substituted for rfc in 

 formula (25). It is of course to be understood that 



I" n 

 means 

 — i 



all sudden change of & or \ in passage through c being accounted for in the 

 fact'u 



f --.or 2 ''. 



\ i.~ the angle which the tangent t" the boundary, drawn in the direction 

 , increasing, makes with the axis of x. Prom the relations (1) and (7) it is 

 Been thai & - a all along the boundary. 



11. The case of symmetrical Bow, as typified in figure 4, lends itself to 

 similar treatment and leads to a result of the same character but simpler 

 form. 



U the geometrical relation be written 



-/: = <.„ . (26) 



and be associated with a ti. -1 • 1 relation »• - *. the problem is to find a 

 enient ai I general form for the function G(X,)> What is required 

 "t i f in this case is that it be a curve-factor "i zero angular rai 2 

 giving the propel corner at Z = U. and such that (i) it is real for £ real 

 and uegativi . b ni'idulus is a constant, Bay K for t real and greater 



than 



Item i>e shown that every ( O satisfying these conditions and regular 

 at infinity can be expn the limit of a product of powers of / k,£, where 



k) 1 - i(Z-a) •> '- ■>>■ 



i • " 7 



Tin- proof of this follows bo closely the method of articles * to 10 that 



only t) utline need be given. An arbitrarily selected value of C in the 



relevant half-plane is denoted by *.„ and its conjugate complex by £„'. 



An auxiliary function .'/'£•? fined by the equation 



(a -:>» ■ - 



. - i, ay 



and it is noticed that the complex conjugate to - - - - »°r 



$ < «. while for g > a it . . intours in th>- relevant half-plane 



