Li'-.Ai'HEM — Periodic Conformal Curve-Factors and Corner- Factors. 41 



w plane, and has no branch-points on the lines i^ = + 3 A (save possibly at the 

 corners of the strip), so -6^ is a curve-factor in ^ whose range of curvilinearity 

 does not extend outside the range from - c to + c. 



If the range of curvilinearity covers only part of the range from - c to + c, 

 the curve in the z plane corresponding, in the transformation dz = (Jdv:, to 

 »// = will have one or more straight portions, without loss of smoothness at 

 points between the points w = ± |A where curved and straight portions 

 meet. In such cases 6^ is not a simple curve-factor. 



It is to be noted that Q is periodic in w, of linear period 2A. Therefore 

 any function of Q, defined so as to be single-valued over any region of the 

 d plane, is, when expressed as a function of w, periodic of linear period 2A 

 within that region. So, in particular, if the plane of Q be cut along the real 

 axis from = - c to 6 = c, any curve-factor in d whose branchings are all in 

 tliis cut, and which is single-valued in the cut plane, is periodic in lo with 

 linear period 2A. But what is required of 6^ is periodicity of linear period A, 

 so that not all curve-factors in 6 satisfy the requirement. 



Thus the attempt to generalize has led to the following verbal formula for 

 a periodic curve-factor :— Any curve-factor in the variable 6 = c sin(Tr«'/A) 

 which is periodic in w with linear period A, and has the linear range - c to c 

 or any range within that range. 



As regards the required periodicity, it is to be noticed that the addition 

 of A to w changes 6 into - 9, so that (9 must be a function of tt whose value is 

 unaltered by change of the sign of 0. But, seeing that the variables dealt 

 with are complex, and that there may be branch-points or a continuous 

 distribution of branching along the range from = - c to = c, the effect 

 upon ff ol & change in the sign of 6 cannot be estimated by a mere glance at 

 the functional formula, but must be studied more closely. 



If the positive half-plane of tu be divided up into a continuous series of 

 semi-infinite strips of breadth A, one of which is the strip from j> = - \\ 

 to (j> = ^A, the transformation (14) represents only alternate strips of the 

 series upon the positive half-plane of d. The other strips, including those 

 immediately adjoining the above specified strip, are conformally represented 

 on that half-plane of for which the imaginary part is negative. The addition 

 of A to vj involves a passage from a point in one strip to the corresponding 

 point in the next strip ; but this passage must be along a path which does 

 not cross the axis i/; = 0, and therefore does cross the boundary between the 

 strips. The corresponding change is from a value to a value - 6 ; but in the 

 6 plane the passage is not along any arbitrarily selected path, it must be 

 along a path which does not cut the part of the real axis between 6 = - c and 

 d = c, but crosses the real axis somewhere outside that range. 



