102 



Mi' Brill, On the Generalization of [Feb. 24, 



Hence, by means of the first of equations (12), we have 



a Um = ax + hy + gz. 

 From equation (14) we obtain 



a ( Fm) 2 = ax 2 + by 2 + cz 2 + 2fyz + 2gzx + 2hxy. 

 If we introduce a second matrix of the given form, viz. 

 m ' — x + py + qz' ; 



then, with the aid of equation (13), we readily deduce 

 aLmm = axx + byy' 4- czz +f(yz' + y'z) 



+ g izx' + z'x) + h (xy + x'y). 



4. We now come to the establishment of a connection of our 

 analysis with geometry. We will suppose in the first place that 

 our axes of coordinates constitute a right-handed screw system. 

 Next, we will suppose that we have a set of three points whose 

 coordinates are (x 1} y 1} z x ), (x 2 , y 2 , z. 2 ), (x 3 , y 3 , z s ), and also a second 

 set whose coordinates are (a?/, y-[, zl\ {x%, y 2 ', z 2 '), (x 3 ', y 3 , z s '). We 

 will consider these two triads of points as determining two planes ; 

 and will suppose the origin to be so placed that the cyclical order 

 of the subscripts given above corresponds, in each instance, to a 

 right-handed rotation about that normal to the plane of the triad 

 which is drawn away from the origin. Further, we will write 



2X = 



and will suppose that X', Y', Z' denote quantities formed in a 

 similar manner from the coordinates of the points constituting the 

 second triad. 



We will now write 



m =X + pY + qZ, 

 m'=X'+pY'+qZ'; 



and, then, by means of the results obtained in the last article, we have 

 a ( Wm) 2 = aX 2 + b Y 2 + cZ 2 + 2/YZ + 2gZX + 2hXY, 

 a ( Wm') 2 = aX' 2 +bY' 2 + cZ' 2 + 2fY'Z' + 2gZX' + 2hX'Y', 

 aLmm' = aXX'+ bYY'+cZZ' +f(YZ' + Y'Z) 



+ g (ZX' + Z'X) + h(XY' + X'Y). 



