( 721 ) 



system of equiangular planes of rotation to the right with direction 

 and the angle of rotation. 



For all those [»lajics of i-otation 



have the same values. These three values ft^, n^, «t^, besides the 

 cosinus of the angle of rotation a^, we can take as determining- 

 quantities of the equiangular double rotatioji to the right. A rotation 

 <^ 2 rr is uneciuixocallv determined by that (foi-, whcthor the angle 



of rotation is -^^, ^^lHch is left undecided by the value of the cosinus, 



follows from the sign of the a^, a.^, a^). 



Suppose an arbitrary \'ector a to be transferred by the rotation 

 into (?, then it holds good for each pair of vectors u^i that: 



^f, t^3 — <'a 1^.. + (^l ^A — «4 t^l ~ l'^' • «1 



«a 1^1 — 'fl i^-i + (^2 t^4 — ^'4 ti — J<^ • «, 



«1 <■?. — «. lA + «3 t^. — «4 (^^ = ^^' • «Z 



t^i i"?i + «.. 1^. + 'f. t^.. + 'r, i^4 = ^'4' 



If however we consider that K = -\- \/s(/i:' ih, if i> is tlie angle of 

 rotation, then K proves to be a constant for all paii's of vectors so 

 that we may regard K.a^, K.k.., K-<(-^ and a^ as (hMerniining quan- 

 tities of the double rotation which we shall call .ti, :it.,, rr^, rr^; and 

 we shall write the relations: 



— «4 .^1 — «. ^'1 + «2 t^3 + «1 1^4 = ''«^1 



f^Z i^l — ^^4 1^2 — «1 t^3 + «. l"?4 = ^-Z , 



^<. i'^l + «1 i^. — ^^4 /^.3 + ^^n 1^4 = -"^S 



«1 i^l + «. t?. + «3 i^Z + «4 i*4 = ^T^4 



in which we have at the same time arranged the first members 

 according to i^i^ i^o' i^3H^4- We now perceive: 



= /iMi+2X0) + «4\ 



= 1. 



So we caji regard .t^, -r„, .Tj. :t^ as cosines of dii-ectiou of a ^'ector 

 ihi'onuh (/ in S^ and we can represent an e(piiangular double rotation 

 U» the right by a vector through (f in >\ , which determines it 



