( 717 ) 



The vectors in llic second |)laiie ol" posilioii are drawn in sneli a 

 wav <Iiii( either 



i""^'"' (1) 



is an e((niangulai' (k)nl>le rolalion (o Ihe i-ii^iit, oi' Ihat snch is llie 

 case with 



\oK-^(JD' 



We sliall sn|)})Ose the tirsl to i>e trne (the reasoning is the same 

 for the second case). Then 



is also an e(iniangidar donhle i-otation to tlie right ; for, the planes 

 (U^y' and (Xrl) can he hronght to coincide with these directions 

 of rotation \\'\{\\ the [)lanes ( J liC and OKI), having the directions 

 of rotation 



on 



OK- 



or 



01) 



So 



C^) 



\ OQ' -> OJJ 

 is an eqniangnlai" donhle rotation to the left. 



If fartherinore OA and ()B are bisectors of the angles N(JFi\nd 

 KOG, and if we have made Z.^X^-^' = /^O^^^^ = Z_H^>^\ = 

 — Z.J''(h\ = z^KOB — /^(rOB, tiicn the pair of planes 



is a pair of planes of rotation as well for Ihe eqniangnlar donhle 

 rotation to the right (1) as for the eqniangnlar donhle rotation to 

 the left (2). So it is the pair of common [)lanes which was looked 

 for of the two systems of })lanes. 



We shall thiidv now that throngh two 

 arbitrary vectors ()A and (JB two planes 

 intersecting each other eqniangnlarly to the 

 left have been laid; we shall now consider 

 more closely the [»osition \vhicli two snch 

 planes have with respect to the plane 

 (JAB and its normal plane. We shall call 

 the indicated eqniangnlar planes to the left 

 H and /i; and indicate OAll by y and its 

 F'g- 2. normal plane by <f. hi tig. '2 the lines 



drawn upwards lie in 6 and those drawn downwards in 7. 



