Page:American Journal of Sociology Volume 5.djvu/415

 SEMINAR NOTES 4OI

w=^ Vz. Further, in the function determined by the equation

(2 — b')(w — cf=z — a, or w ^^ z -\- \\ • ,

\ s — b

in which a, b, c denote three complex constants, and therefore three points, z^a is a branch-point, at which all three values of the func- tion become equal to w=^c. Moreover, at 3 = ^ all the values of w become infinite. The three functions suffer here an interruption of continuity, and hence it can remain undetermined on which path each is to continue its course, because, when the function makes a spring, it can just as well spring over to one as to the other continuation of its

>^. c

path. As a general rule, those points at which w becomes infinite or discontinuous are branch-points. But there are cases in which points are not branch-points, although at them values of the function are

either equal or discontinuous .Accordingly, the branch-points

are to be looked for only among those points at which interruption of continuity occurs, or at which several values of the function become equal, but whether such points are actual branch-points must still be expressly determined.

"The preceding considerations have shown that, when a variable z, starting from an arbitrary point s„, describes a path to another point z, which leads through a branch-point of the function w, the latter