A Treatise on the Theory of Screws

436 THE THEORY OF SCREWS. [400 ordinates of the originating objects as constant, and regard the intervene simply as a function of X and p, which we shall denote by /(X, p). The form of this function will be gradually evolved, as we endow it with the attributes we desire it to possess. The first step will be to take a third object on the same range for which the parameter shall be v, where X, p, v are arranged in order of magnitude. Then, as we wish the intervene to possess the property specified in Axiom i,, we have /(X, p) +f(p, v) =f(X, v). By the absence of p from the right-hand side, we conclude that p must disappear identically from the left-hand side. This must be the case whatever X and t> may be. Hence, no term in which p enters can have X as a factor. It follows that f(X,p) must be simply the difference of two parts, one being a function of X, and the other the same function of p. Accordingly, we write, The first step in the determination of the intervene function has thus been taken. But the form of </> is still quite arbitrary. The rank of the objects in a range may be concisely defined by the magnitudes of their corresponding values of p. Three objects are said to be ordered when the corresponding values of p are arranged in ascending or descend- ing magnitude. Let P, Q, Q be three ordered objects, then it is generally impossible that the intervenes PQ and PQ' shall be equal; for, suppose them to be so, then PQ + QQ' = PQ' by Axiom I.; but, by hypothesis, PQ = PQf and hence QQ’ — o. But, from Axiom n., it follows that (Q and Q' being different) this cannot be true, unless in the very peculiar case in which the intervene between every pair of objects on the range is zero. Omitting this exception, to which we shall subsequently return, we see that PQ and PQ' cannot be equal so long as Q and Q' are distinct. We hence draw the important conclusion that there is for each object P but a single object Q, which is at a stated intervene therefrom. Fixing our attention on some definite value S (what value it does not matter) of the intervene, we can, from each object X, have an ordered equi- intervene object p determined. Each X will define one p. Each p will correspond to one X. The values of X with the correlated values of p form two homographic systems. The relation between X and p depends, of course,