A Treatise on the Theory of Screws

406] THE THEORY OF SCREWS JN NON-EUCLIDIAN SPACE. 447 or H' /u\H I 4tH' . • 4Æ' \ &c. &c. &c. We see that the correspondence between X and p, cannot be of the homo- graphic character, unless H = H’. The necessity for this condition may be otherwise demonstrated by considering the subject in the following manner:— The intervene between any two objects on one range is, of course, ambiguous, to the extent of any integral number of the circuits on that range. Let C and C be the circuits, and let 8 be an intervene between two objects on the second range. If we try to determine two objects, a and X, on the first range that shall have an intervene 3, we must also have another object X', such that its intervene from a is 8 + O'. Similarly, there must be another object X" with the intervene 8 + 2(7', &c. It is therefore impossible to have a single object at the intervene 8 + mC from a, unless it happened that G = C, or that H = H'. Thus, again, are we led to the conclusion that ranges cannot be equally graduated unless their circuits are the same. The circuits on every range in the content being now taken to be equal, we can assume for the circuit any value we please. There are great advan- tages in so choosing our units that the circuit shall be tt ; but we have as its expression, 'ZiHrr; whence we deduce 2iH = 1. 406. On the Infinite Objects in the Content. Certain objects in the content are infinite, and it is proposed to determine the conditions imposed on xlt x2, x3, when they indicate one of these. If an object 0 be infinite, then every range through that object will have one other infinite object. Let these be 02, &c. These several objects will conform with the condition, x2, æ3, «4) = 0. Every infinite object in the content must satisfy this equation; and, conversely, every object so circumstanced is infinite.