A Treatise on the Theory of Screws

^56 THE THEORY OF SCREWS. [416, content, conducted subject to the condition that the intervene between every pair of objects shall equal that between their correspondents. 417. On the Character of a Homographic Transformation which Conserves Intervene. In all investigations of this nature the behaviour of the infinite objects is especially instructive. In the present case it is easily shown that every object 0 infinite before the transformation must be infinite afterwards when moved to O'. For, let X be any object which is not infinite before the transformation, nor afterwards, when it becomes X'. Then, by hypothesis, the intervene OX is equal to O'X'; but OX is infinite, therefore O'X’ must be also infinite, so that either 0' or X' is infinite; but, by hypothesis, X' is finite, therefore 0 must be infinite, so that in a homographic transformation which conserves intervene, each object infinite before the transformation remains infinite afterwards. It follows that in the space representation each point, representing an infinite object, and therefore lying on the infinite quadric 17=0 must, after transformation, be moved to a position which will also lie on the infinite quadric. Hence we obtain the following important result:__ In the space representation of a homographic transformation which con- serves intervene, the infinite quadric U=0 is merely displaced on itself. A homographic transformation of the points in space will not, in general, permit any quadric to remain unchanged. A certain specialization of the constants will be necessary. They must, in fact, satisfy a single condition, for which we shall presently find the expression. Let «J, «2, ®3, Xi be the quadriplanar co-ordinates of a point, and let us transform these to a new tetrahedron of which the vertices shall have as their co-ordinates with respect to the original tetrahedron xf xf xf Xi , X2 , X3 , Xi , ^1 > ^2 } ^3 i tZ?4 , •^1 f *^2 > ^3 , ^4 . If then Xlt X2, X3, Z4 be the four co-ordinates of the point referred to the new tetrahedron x2 = x2 Xi + 2 4- xj"X3 + xf'Xi, x3 = x2 Xi + x2 X2 + x2 "X3 + x2'"'Xi, xs = x3 Xi + x3 X2 + xj"X3 + xj'"Xit Xi = xjXi + + xfX3 + xf" Xt.