A Treatise on the Theory of Screws

424] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 469 and the similar one with y and y', instead of x and x'. The denominators are clearly equal, and we have only to notice that x^' + + x3x3 + = y^yi + y2y2 + y3y3 + yty< ; as an immediate consequence of the formulae connecting the orthogonal transformation. 424. Application of the Theory of Emanants. We can demonstrate the same proposition in another manner by revert- ing to the general case. Let U=Q be a function of xlt x2, xs, xt. Let x,', x2, x3, be a system of variables cogredient with xlt x2, x3, xit and let us substitute in U the ex- pressions «j 4- kxi, x2 + kx2, &c., for xlt x2. The value of U then becomes ü + k^U+. A2tf + &C., 1 . Li where . d , d , d , d A = a?1 dx*^ dx2 + Xi dx3 + Xi dx,‘ If U be changed into V, a function of y, by the formulæ of transformation, we have, of course, U= V-, but since yT is a linear function of xit &c., i.e. y^ = (11) + (12) x2 + (13) x3 + (14) x^ it follows that if we change x-, into x1 + W, &c., we simply change yT into jfj + kyi. Hence we deduce, that if U be transformed by writing + kxi, &c, for x, then V will be similarly transformed by writing y. + fey/ for y, and, of course, as the original Ü and V 'were equal, so will the transformed U and V be equal. It further follows that as k is arbitrary, the several coefficients will also be equal, and thus we have ,dU dXy ,dU ~ yi dyi Hence the intervene between two objects before displacement remains unaltered by that operation; for cos 8 = ,dU , „ Xi +&C. ”OT" and by what we have just proved, this expression will remain unaltered if y be interchanged with x.