THE DYADICS IN THREE DIMENSIONS. 423 



i. e. any quadratic complex consists of the lines thus transformed by 

 some two-two dyadic. In fact g consists of all lines p satisfying the 

 equation 



[^p-^p] = li^c-'^p] = [p'^p] = 0. 



We have just shown that <J> can be determined such that '^ is any self- 

 conjugate two-two dyadic. Now any homogeneous quadratic equa- 

 tion in the Pliicker coordinates of a line p can be written in the form 



[p ^ p] = 



where ^ is a self-conjugate two-two dyadic. Hence any quadratic com- 

 plex is the complex g of some tioo-two dyadic $. 



Suppose next, that $ is an anti-self-conjugate two-two dyadic. 

 Let qi, 92- .. -96 be six linearly independent complexes. Then $ can 

 be written 



$ = ^aikqiqk- 

 Since 



$ = - $c 

 ^aikqiqk = — ^aikqkqi 

 whence 



ttifc = — aki. 



The dj'adic can therefore be written 



$ = 012(9192 — 9291) + aniqiqs — qsqi) + a.u{qiqi — g4?i) + • • . 

 Suppose one of the coefficients a a-, say au, is not zero. Let 



X14P4 = 012^2 + 01393 + 01494 + O1595 + 01696, 



Xi4 being an arbitrarily assigned number which is zero if the right side 

 of the equation is zero. By using this equation eliminate 94 from $ 

 and so reduce it to the form 



^ = Xi4(9iP4 — 2^491) + 



In this form 91 appears only in the term 91^4 — 94^1- The others con- 

 tain 92, 93, 94, 95, 96- Similarly, if 92 occurs in more than one of the 

 combinations qi qk — qkqi a new complex can be introduced such that 

 92 will appear in only one combination. Finally 93 can be treated in 

 the same way. The dyadic will then have the form 



^ = y^uiqiPi — Piqi) + y^ihiq-iPb — P592) + ^ssiqsPe — p^qs) + • . 



