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 



as an immediate consequence of the formula? 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=0 be a function of x 1} x 2 , x 3 , # 4 . Let #/, x 2 , x 3 , x 4 be a system 

 of variables cogredient with x lt x. 2 , x 3 , # 4 , and let us substitute in U the ex 

 pressions x l + kxi, # 2 + kx 2 , &c., for x lt x 2 . The value of U then becomes 



where 



. _ , d , d , d , d 



LA = Xi j -J- X.2 i ~ ~\~ Xn ~^j -p X z 



If U be changed into V, a function of y, by the formulae of transformation, 

 we have, of course, 



U-Yt 



but since y l is a linear function of x l} &c., i.e. 



y 1 = (11) x, + (12) x 2 + (13) x 3 + (14) x., 



it follows that if we change x^ into x 1 + kx 1 , &c., we simply change y^ into 

 y l + ky l . Hence we deduce, that if U be transformed by writing x 1 + kx 1 , 

 &c., for x, then V will be similarly transformed by writing y^ + %/ for y, 

 and, of course, as the original U 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 



=y l , 

 l dy, 



Hence the intervene between two objects before displacement remains 

 unaltered by that operation ; for 



,dU 

 * dx +&C 



rx _ (Mil 



and by what we have just proved, this expression will remain unaltered if 

 y be interchanged with x. 



