( 821 ) 

 2 (jk) (lm) = O, we shall continually be able to break up each ofthese 



quantities into two |>;irls (if)' and (//')", so thai 2(kl)' (jm)'-=.2 (kl)" (yw)"=0. 



ï i 



Tr is easy to see that this decomposition can be done in go 4 ways. 



For each decomposition holds good : 



2(kl)'(jm)"=:2(kl)(jm) , (4) 



for : 



2 (kl)' (jm)" = 2 (M)> \(jm) - (jm)'\ = 2 (kl)' (jm), 



t i i 



likewise : 



2 (kl)' (jm)" = 2 (kl)" (jm), 



t i 



from which by addition : 



2 (kl)' (jm)" = 2 (kl) (jm). 

 i i 



Giving a geometrical interpretation we regard a homogeneous 



system of 10 arbitrary quantities a$ and a$ as the coordinates of 



a system a of cc 4 lines, in pairs a system « of <x 4 planes, in pairs 



a', a" conjugated by the relations : a', a" conjugated by the relations: 



a'ij + a"ij =<hj . . . (5) a'ij + «"$ = a {j . . . . (5') 



All these lines lie in one Sp t B All these planes pass through 

 having as coordinates : one point X, having as coordinates: 



ii = 2 ajd (ij m .... (6) a>t = 2 au «ƒ,„ .... (6') 



We now annul the homogeneousness of the p-, n-, a- and «-co- 

 ordinates. 



This causes those elements to assume vector-nature and makes 

 them interpretable respectively as force, as rotation, as dynam and 

 as double-rotation. The equations (5), (5') determine the reduction 

 of the vectors a and a on the conjugate pairs of lines and of planes 

 of the systems a and « under consideration and not yet partaking 

 vector-nature, whose structure now becomes revealed. 



II. In connection with the meaning given in /; of the equation 

 2 py Ttij = we interpret 



2 «ijpij = .... (7) 2oj|jrj = 0, .... (7') 



as the condition that a line p cuts as the condition that a plane n 

 a pair of conjugated planes of cuts a pair of conjugated lines 

 system a. of system a. 



This gives us a very fair survey of the structure of the linear 

 complex of lines and planes. The reduction of the equation of the 

 complex of planes to its diametral space is now easy to do ; likewise 



