416J THE THEORY OF SCREWS IN NON-EUCLIDIAN 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 l} x 2 , x 3 , x 4 as the co-ordinates 

 of a point corresponding to as, and y l} y 2 , y 3 , y 4 as the co-ordinates of the 

 point corresponding to y. We are then to have an unique correspondence 

 between as 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 their 

 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 n 3 

 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 versa. 



Let the plane in the second system be 



AM + A. 2 y, + A 3 y, + A 4 y t = 0. 



If we seek its corresponding plane in the first system, we must substitute 

 for y 1} 7/2, 2/s&amp;gt; 2/4 the corresponding functions of x ly x. 2&amp;gt; x 3 , # 4 . Now, unless 

 these are homogeneous linear expressions, we shall not find that this remains 

 a plane. Hence we see that the relations between X 1) x 2 ,x 3 ,x 4 and y\,y&amp;lt;i&amp;gt;y*,yi 

 must be of the following type where (11), (12), &c., are constants : 



TA = (11K + (12) * 2 + (13) * 3 + (14) * 4 , 



7/ 2 = (21) x, + (22) x z + (23) x s + (24) a&amp;gt; 4 , 



y , = (31 K + (32) x, + (33)^3 + (34) # 4 , 



/4 = (41 X + (42)^ 2 + (43)^ 3 + (44) x 4 . 



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. The 

 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 



