( 724) 



Let us fartliermore suppose (0) and (t) to be two equiangular 

 double rotations to the right, transferring a given vector s the for- 

 mer into ?, the latter into t]. Then 



The same orthogonal transformation transferring into S, transfers 

 T into t], so that the angle between S and v is the angle between the 

 vectors o and r independent of the initial position b. As a special 

 case theorem 3 is included in this, for the case that the two double 

 rotations take place about an angle i :t ; for then the angle between 

 S and 7? is the angle of the two planes of rotation through ?, proving 

 to be independent of f. 



We have still to mention that theorem i in the second form is 

 entirely included in the applications of the biquaternions on S^ as 

 given by Dr. W. A. Wijthoff in his dissertation : "De Biquaternion 

 als bewerking in de ruimte van vier afmetingen" (the biquaternion 

 as an operation in fourdimensional space). For an equiangular double 

 rotation to the right is represented by Q . f ^ + ^ s (P- 127) where 

 Q represents a certain quaternion with norm unity. 



This applied to an arbitrary double vector 



changes it into 



so it leaves the isosceles part of that double vector to the left 

 unchanged and so also the equiangular system of planes to the left to 

 which it belongs. This holds good for an arbitrary double vector, 

 so particularly for a plane. 



Finally theorem 3 can be proved as follows : 



If (fi and ip are the acute angles of position of two planes, then 

 if we represent the coefficients of position respectively by ?.'s and ft's : 



cos <p COS tp = ;.23 fi,3+;.3, !hl-\-K, l-ll,+^-14 ^^14 +-^-34 f^ï44-^-J4 f-^84 = 

 = ^ 0^^3 + ^-14) (."i,3+f*:4) — ^ (Kz /^14+^-14 f^Ss)- 



For intersecting planes with angle of position (p: 



COS(p=::S (;.,3+-?.14) (f^33+ftl4) = ^ i^,Z—^',4) (f^33-^ti4)- 



So for two intersecting planes, belonging to two definite equiangular 

 systems to the right or two to the left 



^•'33 + ^'l4='^-"23 + '^-"l4 and fi',3+l^'l4=/^"23+ft"l4 



resp. -^-'ss— ■^'n^'^-'^s— -^-"m and fi'j3— ft'i4=ft"23— ft"i4 and cos (p' z= cos cp" . 



We shall now resume our geometrical reasoning dropped after 

 theorem 3. Let us take through a definite vector OW in S^ but 

 not movable with S^ and let us represent each system of planes equi- 



