A Treatise on the Theory of Screws

422] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 467 seen the necessary characteristic of the homographic transformation which preserves intervene. The infinite quadric which the transformation fails to derange can be written at once, for we have ®i2 + + «s2 + = y-i + y2 + y3 + y? = 0. It is also easily seen that the expression «13/1 + + <«3y3 + W = 0 is unchanged by the orthogonal transformation. We thus have the following quadric, which remains unaltered :— ®i [(11)®! + (12)«2 + (13)®3 + (14)«4] + x.2 [(21)«j + (22)«a + (23)«3 + (24)a?4] +«3[(31)®1+(32)aj2+(33)a;3+(34)a;4] +x4 [(41)«1+(42)®2+(43)a?3 + (44)«4] = 0; or, writing it otherwise, (11 + (22)«/ + (33>,8 + (44>? + [(12) + (21)] xxx2 + [(13) + (31)] x,x, + [(14) + (41)] x1Xi + [(23)4- (32)] x2x3 + [(24)+(42) ] + [(43)4- (34)] = 0. If this be denoted by U, and x? + x2 + a1? + «'42 by ft, then, more generally, Ü — hil is unaltered by the transformation. We now investigate the intervene 3, through which every object on Ü — AH = 0 is conveyed by the transformation. If we substitute x1 + Xyx &c. for x^ &c. in the infinite quadric we have n + 2xor + x2n = o, and, accordingly, the intervene 3, through which an object is conveyed by the orthogonal transformation is defined by the equation n U cos 3 = y, ; hence the locus of objects moved through the intervene 3 is simply U — fl cos 3 = 0. 422. Proof that U and fl have Four Common Generators. The equation in p has four roots of the type , 1 „1 p, P , — • P P These correspond to the vertices of the tetrahedron (fig. 47). Symmetry shows that the conjugate polars as distinguished from the generators will be the ray joining the vertices corresponding to 1 ' — ana p , p 30—2