( 719 ) 



OF' -^ O A' 



OII-^OK 

 and tliis beiiio: of tlie same kind as 



OF' -> OA 



OA -^ OF 

 wliich in its turn can be made to coincide with 



OC' -^ OA' 



OA -» OC 

 hy a single rotation about the pkne OAA', the kind of equiangukir 

 intersection is the same as the kind of the double rotation 



OC'-^OA' 



OA -» OC 

 which is to the left according to the data. 



So the plane (}BG is identical with the ph^ne ^i, for through OB 

 only o]ie plane equiangular to the left with « can pass. 



If we now introduce the notation "I j equiangular to the right" 



indicating if abed denote four vectors through 0, Ihat the planes 

 {(/b) and (cd) are equiangular to the right and that the same double 

 equiangular rotation to tl»e right transferring a into b, also transfers 

 c into (/, then 



'OA, OB^ 



OF, OG^ 



is equiangular to the right and the corresponding equiangular double 

 rotation to the I'ight transfers a into /?. In other words we have 

 proved the 



Theorem 'J. Iff j is equiangular to the right, then ( j is equian- 

 gular to the left; oi- in other words though less significant: 



By an equiangular double rotation to the right any plane passes 

 into one equiangular to it to the left. 



If we suppose three vectors ^//vt! (whose position of course determines 



the position of S^) to have come after some equiangular double rotations 



7 . Cah\ 



to the right into the position <k/,\heu ( I is equiangular to the left 



and ( j equiangular to the left; so I j equiangular to the right 



and I.I equiangular to the right, so tinally 



/ad\ 



