A Treatise on the Theory of Screws

416] THE THEORY OF SCREWS IN NON-EUCLIDIAX SPACE. 455 one type. Each object in one system has one correspondent in the other system. But if X be regarded as belonging to the system B, its correspondent in A will not be Y, but some other object, Y . To investigate this correspondence we shall represent the objects by their correlated points in space. We take x„ x2, xs, x, as the co-ordinates of a point corresponding to x, and ylt y2, ys, y4 as the co-ordinates of the point corresponding to y: We are then to have an unique correspondence between x and y, and we proceed to study the conditions necessary if this be complied with. 416. Deduction of the Equations of Transformation. All the points in a plane L, taken as x points, must have as theii correspondents the points also of a plane; for, suppose that the corre- spondents formed a surface of the nth degree, then three planes will have three surfaces of the nth degree as their correspondents, and all their ?i3 intersections regarded as points in the second system will have but the single intersection of the three planes as their correspondent in the first system. But unless n = 1 this does not accord with the assumption that the correspondence is to be universally of the one-to-one type. Hence we see that to a plane of the first system must correspond a plane of the second system, and vice versd. Let the plane in the second system be + ^22/2 + = 0- If we seek its corresponding plane in the first system, we must substitute for 2/1, y2, y«> the corresponding functions of xlt x2, x3, xt. Now, unless these are homogeneous linear expressions, we shall not find that this remains a plane. Hence we see that the relations between xlt x2, xz,xt and ylt y2, ys> yt must be of the following type where (11), (12), &c., are constants: yr = (11) a?! + (12) x2 + (13) x3 + (14) xit y2 = (21) xl + (22) x2 + (23) x3 + (24) xit y3 = (31) xt + (32) x2 + (33) x3 + (34) xit yi = (41) + (42) x2 + (43) x3 + (44) x,. Such are the equations expressing the general homographic transformation of the objects of a content. From the general theory, however, we now proceed to specialize one particular kind of homographic transformation. It is suggested by the notion of a displacement in ordinary space. Ihe displacement of a rigid system is only equivalent to a homographic trans- formation of all its points, conducted under the condition that the distance between every pair of points shall remain unaltered (see p. 2). In our extended conceptions we now study the possible homographic transformations of a