( 786 ) 

 or : 



'^JS n" '^14 Mza f^l4 ^2S '^14 -— M23 I f'l4 



^Rl -h ^24 =^ f'si — f'24 •^31 — -^-24 = Msi + f'24 / {'') 



'12 T '-34 /'l3 ^^34 ''12 •''34 ~~ /^ 1 2 1~ /'34 ' 



No^v hü^ve^'e^ 



^^23 + -^'14 

 ^'81 + ^24 

 ^ï'i "t~ ^-3 4 



are the cosines of direction of tlie representant of the system of 

 planes eqniangnhar to the right with P. witli respect to a system 

 of coordinates OX, Y, Z, taken in N,, as that was defined (These 

 Proceedijigs Febr. 1904, page 729). 

 And likewise 



^23 ''14 



''31 '"2 4 



'■12 •^3 4 



are the cosines of direction of the representant of the system of 

 planes eqiiiangnlar to the left with A Avith respect to a system of 

 cooi'dinates OXiYiZi taken in an analogons manner in N/. 



So from the formnlae (u) ensnes that tlie effect of a reflection is 

 what we might call a reciprocal Into'chanije of S,. und Si, i.e. a 

 snchlike interchange that every vector of *S/ takes the place of that 

 vector of S , which has sul)stitnted itself for it. 



Bnt now an arbitrary symmetric transformation of S^ can be 

 replaced by a reflection preceded or followed by a double rotation; 

 which is represented by a recipi'ocal ijiterchange of >S',- and *S'/ 

 preceded or followed by a rotation of S,. and one of >S/; therefore: 



The arbitrary symmetric transformation of S^ is represented by 

 an hitercliaiKje of S,- and S' in arhifracii posi/io/i.s. 



Let ns now consider that for snch an arbitrary interchange of 

 Sr and Si a system of coordinates a of S,. is placed on a system of 

 cooi'dinates ,i of Si whilst that system of coordinates ji of Si itself is 

 placed on a system y of .S', ; then we can rei)lace the interchange 

 by a "reciprocal interchange'' placing a on /^ and (3 on a, followed 

 by a rotation of S,., placing a on y, or also by a rotation of S,, 

 placing a on y, followed by a reciprocal interchange, placing y on /i 

 and /9 on y. 



Conseqnently we have proA'ed : 



"An arbitrary symmetric transformation of aS'^ can be i-eplaced by 

 a reflection preceded or followed by a donble rotation eqniangnlar 



