A Treatise on the Theory of Screws

403] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 443 Take any infinite object 0. Construct a series of ranges in the extent, each containing 0. Each of these ranges will have another infinite object, 01, 02, 03, &c. The values x2, x3, which define 0lt 02, 03, &c., must fulfil some general condition, which we may express thus: /(«!, x2, x3) = 0. Form a range through 0x and 02. There must be two infinite objects on this range, and of course all other objects thereon will be defined by a linear equation L — 0 in x2, x3. Every object satisfying the condition /(«1; x2, xsj — 0 is infinite, and there- fore all the values of x1, x2, x3 common to the two equations L = 0 and /(«!, x2, x3~) = 0 must denote infinite objects. But we have already seen that there are only two infinite objects on one range; therefore there can be only two systems of values common to the two equations. In other words, /(«i, x2, xs) must be an algebraical function of the second degree. There can be no infinite object except those so conditioned; for, suppose that $ were one, then any range through $ would have two objects in common with f, and thus there would be three infinite objects on one range, which is contrary to Axiom ill. Hence we deduce the following important result:— All the infinite objects in an extent lie on a range of the second degree. We thus see that every range in an extent will have two objects in common with the infinite range of the second order. These are, of course, the two infinite objects on the range. 403. On the Periodic Term in the Complete Expression of the Intervene. We have found for the intervene the general expression H (log X - log /i). We may, however, write instead of X, (cos 2m7t + i sin 2k7f) X, where n is any integer; but this equals e2i,wX; hence, log X = Zimr + log X; and, consequently, the intervene is indeterminate to the extent of any number of integral multiples of The expression just written is the intervene between any object and the same object, if we proceed round the entire circumference of the range. We may call it, in brief, the circuit of the range.