( B34) 



— n", jr, ;t, :7r. 



(I) 



7T., -T, jr, — jr^ ^ 

 and we notice that it does not represent a group and does not 

 possess any tlireedimensional properties (it does after composition 

 witli itself, compare for instance tlie theorem of Mr. Jahnke, Jahres- 

 bericlit, 1. c, pap,e 182 : 'Mede endliclie Drelmng im 11^ lasst sich als 

 eine Znsammensetzung ans einer Elementardreluing im B^ mit sich 

 selbst aiiffassen"^) ; ^vhich operation is for the rest bound to a once 

 chosen system of axes). 



We shall now deduce two difterent determinant types likewise 

 determined by a system of cosines of direction .Tj, rr^. jr^, rr^ which 

 do represent a group and liare threedlmensional properties. Those 

 will be the determinants of eqniangidar double rotation to the 

 right and to the left. 



Let us solve the «'s out of the equations {H) (see Proceediiigs of 

 March lO^^S 1904, page 72i); then we have 



a^ — .T, ()\ + jr„ ^?.3 — n^ /?, + jr, ^, 



«, — :t, il, — 7t, /?! + rr, I?, + jr, /?3 , 



, .... (a). 



(h — ^4 i^Z + ^3 /^4 + ^-^ i^l — ^1 i^2 



fU — ^^ ^4 — ^^ i'^3 — -^2 f^2 — ^1 ^1 ' 



Thus the determinant type of the equiangular double rotation to 

 the right is 



^4 ^i —'^^ ^1 



-JTj .T^ rTj n-j 



(H) 



Directly can be verified that this determinant type forms a group. 

 Likewise we deduce for the equiangular double rotation to the left 



(jTi ^2 üT^ ^,), transferring the vector («^ «^ «a «4) hito 0?i /J^ i^s ^4)» the 

 relations : 



«1 = — ^4 i^l - ^, ^^2 + ^, /^3 + ^1 /?4 



«, = .T3 /?, — jr, /?, — jr, /?, + ^, /?4 , 

 «« = — -T, ^1 + iTj /?, — jr, i?3 + ^3 /?4. 



«4 — — -T^l i^l — ^2 i^a — ^3 ^3 — ^4 /?4 



1) "Each finite rotation in S-^ can be legaided as a composition of an elementary 

 rotation in S4, with itself," 



