252 Proceedings of the Royal Irinh Academy. 



XVI. — KoTEs oif THE Kinematics and Dynamics of a Rigid System in 

 Elliptic Space. By Eobekt S. Ball, LL.D., F.R.S. 



[Eead, June 9, 1884.] 

 If we write the quadriplanar transformation 



2/1 = (11) 0-1 + (12) ;r3 + (13) :r3 + (14) ^4, 

 ^2 =(21)^1 + (22) ;r2 + (23) a-3 + (24) 3^4, 

 y3 = (31):ri + (32)^2+ (33):P3 + (34)a;*, 

 ^4 = (41 ) 0^1 + (42) X., + (43) x^ + (44) x\ 



we have then the general homographic transformation of a point in 

 space. This is too general to correspond with the displacement of a 

 rigid system in elliptic space, but if we further specialize the transfor- 

 mation by the assumption that it is to be orthogonal, then it will cor- 

 respond with the most general displacement of the rigid system in 

 elliptic space. 



If it be orthogonal, then we have the further conditions 



x, = {\\)y, (21)^3 (31)y3 (41)^4, 

 x, = (12)yi (22) y3 (32)^3 (42)^4, 

 ^3 = (13)yi (23)2/3 (33)2/3 (43)2/4, 

 ;r4 = (14)yi (24)2/a (34)^3 (44)2/,. 



From these we deduce at once the conditions 



xy^ + x^' + Xi' + Xi- = 2/i^ + V'i + ys" + Vi, 



and consequently 



Q, = x-c + x<^ + x^ + :r4- = 



is the equation to the absolute. 



We thus have the general conditions of movement in elliptic space 

 exhibited in a symmetrical manner. 



The absolute is only one member of a family of quadric surfaces, 

 each of which possesses the property that a point thereon before the 

 displacement remains thereon after the displacement. Write the 

 expression 



