( 723 ) 



and the {- — r) /^-transformation transfers the vector (.Tj jt^ ^r^ jr J 

 into («1 «2 «3 «4). 



Analogous to this : 



If the equiangular double rotation to the left (q^ q^ q^ q^) transfers 

 the vector («j «^ ^3 «4) hito (/?! ^3^ ^^ [i'J, then the (-|- /) «-transformation 

 transfers the vector (Oj q^ q^ q^) into (^j /J^ ^^ /?J 



and the ( — /) /j-transformation transfers the vector (o^ q^ q^ qJ into 

 («1 «2 «a «4)- 



Let us now suppose tliat S^ has first an equiangular double rotation 

 to the right (jt) transferring an arbitrary vector a into /3' ; then an 

 equiangular double rotation to the left (q), transferring /3' into y, then 

 w^e can write : 



rnr =. [(+ r) «] /5' q = [{- I) y^ ^' 



Jt = \[{+r)a-] . [{-1)y-]\q 



where the form between \\ denotes the determinant of transformation 

 having as first row the sum of the products of the terms of the 

 first row of [(-f^)«] with the corresponding ones of respectively the 

 first, second, third and fourth of [( — 1)y'\, whilst the second etc. row 

 in a corresponding manner is deduced out of the second etc. row 

 of [(H-^)«] (all this in the way of forming a product of determinants). 

 If aS'4 has first an equiangular double rotation to the left (p) transferring 

 ft into /3" and then an equiangular double rotation to the right {n'), 

 transferring /?" into y, we ha^'e: 



Q = [(+ «] /?" ^' =- [(- r) y] /3" 

 Jt' = \[{-r)y] . [{+l)a]]Q. 

 But now 



[{+ r) a] . [(- I) y] eee [(- r) y] . [(+ I) a] 

 which can be at once verified, so : 



Thus : 



If S^ is allowed successively an equiangular double rotation to the 



right (ji) and one to the left (q) the order of the two rotations may 



be interchanged. For, in both cases an initial position of a vector a 



gives the same final position y. 



/a ^'\ 

 And if we regard the quadruple a, ^', /5", y, then ( is 



\^ ry 



/« 3" 

 equiangular to the right, according to the rotation (:7r) and ( 



\^ 7 



equiangular to the left according to the rotation (q); by which we 



assuredly once more have proved theorem 1. 



48 

 Proceedings Royal Acad. Amsterdam. Vol. VI. 



