ROTATIONS IN HYPERSPACE. C59 



we will express ^ in terms of planes (2-vectors). This is done by 

 means of the double product of Gibbs.^° If $ = Z A-,/, is a dyadic 

 which transforms 1-vectors into l-vectors then ^^x^ = I ^ (/li^^'y) 

 (/,x/,) represents the same transformation expressed in plane coordi- 

 nates, that is a transformation which transforms planes into planes. 

 To show this let r and s be two 1-vectors. Then the plane r^s is 

 transformed by $ into (r-^)x{s-^) which can be written 



r'xs' = (r-$)x(5.<|)) = i[(r-$)x(5-<J)) - (5-<J>)x(r-$)] 



= hmr-k,)h]x[^is-kj)l,] - p(^-A-,)/,]x[2(r-A-,)/,]} 

 = ^{Z[{r-ki)(s-ki) - (s-kMr-kj)](I,-I,) 

 = iZ[(rx5)-(A-,x/,)](/,x/.) 



If $ is the dyadic used in (12) 



(14) $ = viiikoki — kik-z) + vhik^kz — A-3A-4). 

 Then 



(15) ^ = ^4>^$ = 77i-rki2kn + mo'^ksiksi + ?ni7»2(W>4 + ^'24^-13 



— A'i4A'23 — k-zskii). 



If 



P = 2a, J kij 

 be any complex 2-vector 



p.-qf = vii-a]oki2 + mivioankis — viimoaozku — mimodnk-is + viiviiCiy^hn 



+ luo-aziksi 



and if P-^ = XP we see that this is satisfied by the planes A-i2 and ^-34 

 and by the complexes 



ai3(A'i3 + A-24) + (iiiiku — k-a); ciaikis — ku) + ai4(^"i4 + ^-23). 



These complexes satisfy the condition for all values of 013 and 054. 

 The invariant planes will then be A-i2 and ^.-34 and the planes belonging 

 to either of these pencils of complexes. These last named planes are 

 obtained from the values of 013 : Ou satisfying either of the relations 



[ai3(A-i3 + A-24) + Onikii — A-23)]x[ai3(A-i3 + kii) + 014(^14 — ^23)] = 

 [ai3(A-13 — k24) + fll4(A-14 + A-23)]x[ai3(A-i3 — A-24) + ai4(A-i4 + A-23)] = 



10 See Phillips and Moore, Dyadics occurring in point space of three dimen- 

 sions. These Proceedings, vol. 5.3. 



