394 THE AMERICAN JOURNAL OF SOCIOLOGY
Observe what is affirmed in this hypothesis. Its significance lies in the question : Does there in actual fact exist between institutions and the constituent desires of which they are the product the relation of function to variable ? And what, indeed, is the nature of that relation ? Recall the definition of function given by Osborne. He says: "When the value of one variable quantity so depends upon that of another that any change in the latter produces a corresponding change in the former, the former is said to be a. function of the latter." Thus
x', log {x^ + i), Vx{x^ i), etc.,
are functions of x. "A function of two or more variables is the rela- tion existing between them such that any change in the one produces a corresponding change in the other." Thus
xy-Vy^ + 2-^.
W-
are functions of.^:, ^, and 2. These illustrations symbolize the func- tional relation of social institutions and the human desires, which is no less intimate and fundamental than are the values of x,y, and 2 to the value of the expression.'
It may be that the human desires named do not embrace all the factors. (In point of fact, they do not.) There may enter in certain constants from the material environment which determine in part the form of the function and the paths of the variables (a, b, c, d, e,/) which we are trying to trace in order that we may have the means fur- nished us of ascertaining in any given case which of the « different values is properly assignable to any special function considered at a special point. For we find that the quantities with which we have to deal are complex, and our functions are multiform. Our analogy, then, to be understood, requires that we have in mind the method of repre- sentation of the imaginary quantities, and for convenience we may quote a paragraph from Durege : "
"A complex variable quantity, z^x-\-iy, depends upon two real variables, *■ and y, which are entirely independent of each other. Hence, for the geometrical representation of a complex quantity, a
' In order to meet objections which might be raised to the use of symbolism sug- gested in this interpretation, it might be well to call attention to the principle which underlies it and makes its use perfectly legitimate, namely, that the operations per- formed affect only the properties belonging to the symbol and not what the symbol indicates.
'See Theory of Functions, pp. 13-15.
