( 722 ) 



unequivocally. (A vector iritJiout length; lateron we shall determine 

 it, likeAvise unequivocally, by a Aector in'tth length, in >S';,). 



It we fartherniore consider the determinant on the coefllcients of 

 /?j, ,?2, ,•?,, ,i^ in (//) it proves to satisfy all the conditions of an 

 ort hogonal transformation . 



We call that transformation with that determinant 

 — «, — (fj + «, + «1 



+ «. - fU — «1 + «2 



— «. + «1 — «4 + «3 



the {-\- f) <^f-transfornialioii: it ai>j)ears iji the relations (H), if the 

 tirst memhei's are arrajiged according to the cosines of direction of 

 the linal position of the I'otating vector. If they are arranged according 

 to tlie cosines of directi(tii of the initial ])Osition the determinant (tf 

 the coefllcients hecomes 



^. ^. /?3 /?4' 



which we shall call the ( — r) /^-transformation. 



(^w'üe analogous to this we have for erpiiangidar double rotations 

 to the left (Oj q., o., (>_,) bilinear homogeneous equations between the 

 cosines of direction of initial and tinal position of a vector, let us say 

 a and i?, \vhich arranged according to the /?'s, give as determinant 

 of the coefficients 



the ( — /) (^transformation. 



We can now state the folloAving: 



If the equianguhir double rotation to the right (.Ti .-r, jr, rrj transfers 

 the vector (a, «., ft, a,) into (i?i ^^ c^^ i?,) then tiie (-f- /•) «-transformation 

 transfers the vector (-Tj nr^ rr^ nr J into (i^j i?., /^j i--?^ 



