A Treatise on the Theory of Screws

400] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 437 upon the specific value of 8, but must be such that, when one of the quantities is given, the other shall be determined by a linear equation. It is therefore assumed that X and /z must be related by the equation AXp + BX + Cfi + D = 0, where the ratios of the coefficients A, B, C, D shall depend, to some extent, upon S. If X' and p be a given pair of parameters belonging to objects at the required intervene, then AX' p' + Bx' + Cp + D = 0. by which the disposable coefficients in the homographic equation are reduced to two. The converse of Axiom II., though generally true, is not universally so. It will, of course, generally happen that when two objects coincide their intervene is zero. But on every range two objects can be found, each of which is truly to be regarded as two coincident objects of which the intervene is not zero. Let us, for instance, make X = p in the above equation; then we have JLX2 + (B + O') X + D = 0. This equation has, of course, two roots, each of which points out an object of critical significance on the range. We shall denote these objects by 0 and O'. Each of them consists of a pair of objects which, though actually coincident, have the intervene 8. The fundamental property of 0 and O' is thus demonstrated. Let X be any object on the range; then (Axiom I.) xo + 8 = xo-, and as 8 is not zero, we have XO = infinity. Therefore every object on the range is at an infinite intervene from 0. A similar remark may be made with respect to O'; and hence we learn that the two objects, 0 and O', are at infinity. We assume, in Axiom III., that there are not to be more than two objects on the range at infinity: these are, of course, 0 and 0 . We must, therefore, be conducted to the same two objects at infinity, whatever be the value of the intervene 8, from which we started.* AV© thus see that while ths original coefficients A, B, C, D do undoubtedly contain 8, yet that 8 does not affect the equation AX2 + (J3 + C) X + D = 0. My attention was kindly directed to this point in a letter from Mr F. J. M’Aulay.