( 72Ó ) 



equiangular to the ri.üht: in other words tlie fiiuil position would 

 have been ohtaiuat)le out of the initial position by a single equiangular 

 double rotation to the right: with whieli is proved: 



Theorriii 2. E(iniangnlar doulde rotations of the same kind form a 

 group. 



Let us suppose given two equiangular systems to the right and two 

 vectors OA and OB through each of winch we bring the planes 

 behmging to l)Oth systems; then the equiangnlar double rotation to 

 the left, transferring (I A into OB. Avill transfer at the same tiine 

 the angle of position formed in OA into the one formed in ^>7i, thus : 



Theoirm 'A. Two equiangular systems to the right form in each 

 vector the same angle, Avhich can be called the angle of the two 

 systems. 



The obtaiiied results we shall verify by deducing analytically 

 theorems 1 and o, avhich deduction will also throw some more light. 



Suppose a rectangular system of coordinates to be given in such 

 a way that 



OX,, OX, 



is equiangular to the right. The same then holds good for 



nX,. n\J \j:>X,. OX, 



A vector a through O we can determine by its four cosines of 



direction ('i- '^,. ((,.- <(,. 



A |>lane })assijig through the vectors a and ^i with direction 

 of rotation from c. to ,1 is determined by its six coeflicients of 

 position (i. e. projections of a vector unity) ?..^^, X^,, ).,.^. X, ,,?.„,, z.^,, 

 which ai-e dctined by the following relations, if we represent «.,/?3—«st^3 

 by i._,3 : 



We must take the positive sign iji the denominator, for ).„„ imist 

 be posirixc, when the projection of « on ''-'A.^A'., to that of ji on ^7A„A'3 

 rotates ihrouuh an angle less than .t in the same way as ^^A'.^ to 

 OXg-, and in that same case $35 gives us a positive value. If we 

 now represent that positive root of the denominator by K, theji 



/,,= -^,:;,„ = -^. etc. 

 An etpiiangular double rotation to the right can be given by the 



