480 DE. J. G. LEATHEM ON SOME APPLICATIONS OF 



It seems simplest to deal with the curvature of the curve. When dz is put equal 

 to dsexp(ix), it appears that, for w real and between c and c, 



)-f( c )} 



(118) 



the assumption pir = vf(o) being equivalent to the previously implied assumption 

 that the curve-factor of the type ^ 41 has no latent Schwarzian singularity at either 

 end of the range, so that the transformation dz = <$ 4l div would give a boundary 

 without corners. It appears also that 



ds = (c\ 



so that 



- P f (6) \og\w-6\d9\dw, . . (119) 



J -c 



, expj-f /'(0)log w-0 de\ 



i (120) 



Now on the prescribed curve ds/dx is a known function of x, say E,(X/TT); hence/ 

 must satisfy 



expj- f f'(e)log\w-8 dO\ 

 U - *- 



which can be regarded as an integral equation inf. 



A. less complicated-looking form of the condition can be got by taking logarithms 

 of both sides of (121) and differentiating with respect to w. In carrying out this 

 operation it is useful to note that the differentiation of the definite integral is effected 

 by differentiation under the sign of integration and taking the Cauchy principal 

 value (indicated by P.V.) of the resulting integral ; proof of this is omitted as the 

 result is probably well known. The resulting expression of the condition which / 

 must satisfy is 



_ 



f (w) c-w 



(IB) 



47. Transformation for any Prescribed Boundary. Instead of confining the 

 application of the method of 44 to a single curved portion of the boundary of 

 the region which is to be represented conformally on the half-plane of w, it is 

 legitimate to deal with the whole boundary in a single formula, namely 



dz = exp\--\log(w-0)dx\dw, (123) 



I TT J J 



where 6 ranges from +00 to -co, and there is an unknown and possibly 

 discontinuous, but definite, functional relation between and X - A straight part of 

 the boundary, along which d x is zero, makes no contribution to the formula. A 



