Page:American Journal of Sociology Volume 5.djvu/410

 396 THE AMERICAN JOURNAL OF SOCIOLOGY

A complex variable may describe very different paths in passing from an initial point 2^ to another point 2;,. The question arises, "whether the curves described by the function w, starting from k'^, which correspond to those described between z^ and 2,, must always end in the same point 7«',, or whether they cannot also end in different points. Now, in the first place, it is clear that, in the case of uniform (one-valued) functions, the final value u\ must be independent of the path taken ; for otherwise the function would be capable of assuming several values for one and the same value of s, which is not possible with uniform functions. This reason, however, does not apply in the case of multiform functions. Such a function has, in fact, several values for the same value of 2, and hence the possibility that different paths may also lead to different points or to different values of the function."

To return to the concrete : Suppose our function w is, say, the institution of property in land. Suppose, upon examination, we find the same point reached in New England, where the institution has developed along the lines familiar in history since time past, and, again, out on the Pacific coast, where entirely at variance with their traditions (and, as experience has shown, at variance also with their welfare, and any code of justice that human ingenuity can devise) the same forms have been forced upon the unwilling and deluded Indians by the Land in Severalty bills of Mr. Dawes. Different paths have led to different values of the function, all of which we must know accu- rately before we may speak with authority concerning the values at given points. To use the symbols again : Let the variable 2 in «/ = 1 2 pass from 1-4 by different paths, and let w start with7X'=: + i, corresponding to 2=1. Let (polar coordinates) 2=;' (cos <^ -+- / sin <^), then ■w=V' r (cos }4 -\-i sin yi <^). Since zt> is to start with the value + i, r^i, and <^:=o; if 2 describe path from 1-4 and does not inclose the origin, then <{> arrives at 4 with value o, while /■=^4.\7V ^2. If 2 goes once around the origin, then at 4, <^= 2 ir and }i  = TT, while again r=4, hence w acquires the value — 2.

The object to be accomplished in introducing the Riemann surface is to convey graphically an idea of the kind of classification which must be made of the different social values (a problem now confronting sociologists) in order that, given any of the human institutions in forms of thought, of personal action, of expression, or of cooperation, we may be able to show corresponding to each, in meaning terms, the variable desires, both in past and present, so that the interdependence