( 835 ) 



fVoin \vliic'Ii ensues the detenrünaiit type for equiangular double 

 rotation to tlie left : 



— jr, — rr, rr, .t^ 



(III) 



and for tliis too the {)foperty of a group can he verified. 



If we call (I') the determinant type formed by interchange of the 

 rows and colunuis of (I), we can remark : 



If we reverse the signs in the bottom row of type (II) type (I') 

 appears. 



If we reverse tJie signs in tiie last column of type (III) type (I') 

 appears. 



If we ask oursehes wiiether each arbitrary congruent transformation 

 can be replaced by tiie succession of a transformation (III) and a 

 transformation (II) the answei' must be afiirmative ; for we shall have 

 but to take those transformations (III) and (II) belonging to the 

 transformations (I') whicli when successively applied transfer the 

 given initial position into the given linal one. (For those two ways 

 only the intermediate positions will (Htfer in as far as tliey will l)e 

 each otiier's retlection with regard to their A^'^-axis.) 



This is the algebraic proof of 1". 



At the same time it lias become evident that the meaning of 

 the type (I') is an arl)itrary e(iuiangular double rotation to the 

 right preceded by a retlection accor(bng to tiie AVii^ij^ (tiiat is the 

 AV^^xis of tiie initial position) or an arbitrary equiangular double 

 rotation to the left followed by a reflection according to the A>axis 

 (that is the AV^ixis of the final position), and that the meaning of 

 the ty[)e (1) is an arbitrary e({uiangular double rotation to the right 

 followed by a reflection according to the AVti^xis or an arbitrary 

 eqniangular double rotation to the left preceded by a reflection according 

 to the AV^xis. 



Thus according to a preceding communication made in this meeting 

 of the Academy (see page 785) it has been proved that the types (I) 

 and (I') represent the most general symmetric transformation of S^, 

 of which the determinant type has been simplified only by jiarticular 

 choice of the system of coordinates. 



We shall now give a i)roof for 2". 



Out of the relations {a) for equiangular double rotation to the 



55* 



