ON DOUBLE POLYADICS — THE LINEAR MATRIX EQUATION. 381 



«n^i 2 + 2 aio.Vi.ro + «22^2 2 (82) 



and a variable vector may be written 



x = Xl i + .r 2 j (83) 



but this vector is regarded as merely formal. We have identically 



[anii + a 12 (ij + ji) + aoojj]: xx = a n Xi 2 + 2ai2.ri.T2 + a 2 2.T 2 2 (84) 



so that any binary quadric is the double dot product of a dyadic into 

 the dyad xx. We now make up the new system, orthogonal with 

 respect to star product. 



V li+ jj V "~ jj TP - lj + jl (XZ\ 



Xi— — , J! 2 . , * 3— — 7=" \po) 



v+2 V— 2 V— 2 



and also write 



an + a 2 2 a 2 2 — a u ,— , oa ^ 



x = — , y = -=-, z = ai2V — 2 (86) 



V+2 V— 2 



when we shall have identically 



anii + an (ij + Ji) + «2 2 jj = zFi + ?/F 2 + ^F 3 = <p (87) 



The system Fi, F 2 , F3 is now taken as a set of rectangular unit vectors 

 in ordinary space. Thus any binary quadric corresponds to a vector 

 drawn from a stipulated origin in space. As a single but important 

 example of very numerous correspondences which arise, we have 



fy = & + y- + z 2 = 2 facto ~ a 12 2 ) (88) 



This brief outline renders scant justice to the very beautiful system of 

 Waelsch, from which it differs in slight details, owing to difference in 

 definitions and point of view. It differs in particular by identification 

 of the dyadic with the desired vector through the star product and the 

 relation (87). The essential thing is the correspondence set up be- 

 tween geometry and invariant theory by relations of the type (88). 



More generally, if we take the ordinary dyadics of Gibbs, making 

 K = 1, N = 3, with unit vectors i, j, k, it is easy to verify that the 

 system of nine dyadics given by 



F1V2 =ii+jj, F2V^4-kk-2ii, F 3 v / ^4 = 2jj-kk, F 4 V^2 = 



_ ij+ji 



F 5 V-2 = jk+kj, F 6 V / -2 = ki+ik, F7V2 = ij-ji, F 8 \ / 2 = jk-kj, 



F 9 V2 = ki-ik 



