( B.S7 ) 



In nil analogous way we (loduce from llic rolalions (//) for an 

 equiangular doul)le rotation to the left the following rclalions between 

 the coefficients of i)Osition of a plane before and after the rotation : 



^•23 + ^4 =.«23 -f f*14 

 ^•31 + ^-M =f*31 + M-24 

 ^•12 + ^-34 ~/<12 + (hi 



-j- 2(.X,.T, + .T, Jt J(fl 1 ., — fig J 



''n^— '^•34=2G-T3.Tj4-jr,;rJ(M.,3— Mi.,) + 2(^,rr,— .-Ti^^KHaj— ft,J^- 

 -f(-■T3'+-T4'— -^i'— -'T^2')(f'i2— ftsJ- 

 As nowJS' (;..,, -|- ;.jj- = 1 and 21' (-^-.a-^.i,)" = 1 and the determinant 



2(rT,.T,-jr,.T,) .Tr/ + .T/-.T,^-rr/ 2(.t,.t,+^^.t,) (IF) 



represents the general congruent threedimensional transformation 

 about a fixed point expressed in the four parameters of homogeneity, 

 we can regard the motion grouj) with the determinant type (II) 

 as a congruent motion group of the twodimensional Euclidean 

 star of the (^.6c+'^-a4)'s and the motion group with the determinant 

 t^'ite (III) as a congruent motion group of the twodimensional 

 Euclidean star of the (P.^e — P.a4)'s; namely according lo the determinant 

 type (IV) about an axis Avith cosines of direction 



^7 ^TTo Jr, 



v\-7ty i/i-v' j/ï^^'' 



over an angle equal to 2 arc cos jr^. 



Let us call the *S, of the {hc-{-^-a^y^ '^the representing space to the 

 right" or the Sr of S^ and the S^ of the (Aic—^4)'s the "representing 

 space to the left" of the SiOïS^, then we find that to two equiangular 

 double rotations to the right (leftj (.t/ n-,' n-g' .t:/) and (.t/' jt," ^j" jt/') 

 of S^ whose angles of rotation are arc cos rr/ and at-c cos rr/' and whose 

 Systems of planes of rotation make an angle ^^ ith each other equal to 



arc cos ' ^ ' L .,„"__ ^ ' ' "Jl^L , (see Proceedhigs, March, 1904, page 724) 



correspond two rotations oï Sr{Si) over angles 2 arc cos. t\ and 2 arc cos jr J', 



whose axes make an angle equal to arc cos 



v\-:t:' . v\ 



^. 



