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



transformation might have no discontinuity of direction in the full linear range there 

 would be discontinuities in the analytical form of the equation to the curve at 

 the exlremities of the inner range. 



This illustrates the fact that a curve-factor, which, for its total linear range 

 of curvilinearity, is not necessarily simple, may have branch-points not only at 

 the extremities of the linear range but also at points within the range. If the 

 general character of the types already chiefly considered is to be maintained in such 

 a multiple curve-factor, the net effect of proceeding along the real axis of w with a 

 suitable detour round each branch-point must be simply a change of sign when the 

 whole range has been traversed. Thus the formula 



^, i = w-k+(u--a)"(w-c 1 )^(w-c 2 )^...(w-bY,. . . . (138A) 



where a > C; > c., > ... > c r > k > c r+1 > ... > b, gives a curve-factor provided all the 



indices are positive, and 



1 ......... (139) 



The function has no real zeros, and there can be no imaginary zeros in the relevant 

 region unless the angular range is different from IT. On writing down the expression 

 for the tangent of the vector angle x in any sub-range c, to c, +l it can be verified 

 that, except in the case of s = r, infinities of tan x, if they occur at all, must occur in 

 pairs which are not separated by a zero. Thus all the sub-ranges have zero angular 

 ranges, except one whose angular range is TT, and so the total angular range is IT. 



The complete curve corresponding, in any transformation, to a factor ^ 44 m consists 

 of two undulatory curves of zero angular range extending from a to c r and from c r+1 

 to b, joined by a curve of angular range m-n- corresponding to the interval c r to c r+l . 



Of course no one of the c's may be equal to k, as if wk were a factor of ^ 44 there 

 would be a corner at w = L 



53. The multiple character of the curve-factor ^44 arises from the discrete 

 distribution along the linear range of branch-points of definite order. This feature 

 can be eliminated by substituting a continuous distribution of branch-points, each 

 factor having an infinitesimal index except the factors at the ends of the range. 

 Thus to the factor w 9 is assigned the index f(9)dd, and the limit is taken for 

 vanishing of dd in each case. The form thus suggested is 



. . . (140) 



So long as f(Q) is free from infinities in the linear range, the second term of this 

 expression has no zeros in that range, and it will be supposed that f(Q) is thus 

 restricted. An infinity of f(9) at 6 = k might introduce a power of w k as a factor 

 iii ^ and so introduce a corner; infinities of f(d) at other points of the range, if 

 their effect were to introduce factors of the type (w c) n into the second term, would 

 leave ^ still a curve-factor but not a simple curve-factor. 



