38 £. B. Wilson, 
can be resolved into two reflections, the transformation in #-dimen- 
sional space may be so resolved by merely combining the elements 
through which the transformation takes place in the two spaces R, 
S and the elements along which it takes place. Thus the question 
has been reduced to the same question for a fewer number of 
dimensions provided that there are two independent fixed spaces 
Rand S in 2. 
There remains to consider only the cases where there is just one 
; : 1 5) de Sas Bye 
pair of reciprocal roots a, 7 OF one root which is either +1 or —1. 
The first of these arises when z is even and the type of both re- 
; ie Grae 
flections is = with the z consequents of the two reflections and the 
2 
m antecedents each independent. However, if there is only one 
pair of reciprocal roots, the dyadic may be written in the form 
(76) Q=aay|a'y, @ag|@o+... +4 Gn] O'n + a! Bi|B'1 +a! Bol Bo’ 
Tee 
= aco + a—! Bn | B'n 
Bical 23 
+ ay| a's ia li@igi te. tr > 21] B4-4> Balla 
where it must be assumed that the shearing terms which occur in the 
second row are equal in number for both roots and are similarly 
distributed. Moreover it may be assumed that none of them are 
lacking, namely, that their number is z—-2: for otherwise the rea- 
soning just given for different pairs of roots would apply. The 
transformation may be written in oblique coordinates as 
(76’) G4 = O24, 5 Sha Oy es, Ke hn en 
2) 2 a 
\ fl F 1 ; 1 
Ve = Vy Dae 9, 3 Yun=—-Yn + In 
a a aA Sy ins 
This transformation leaves a quadratic form invariant. _For con- 
sider the terms 
(77) Ay myi+ Ag X21 + Alyy Ky Ae ahs + An Xn 1 
ea 
+ Ay X13 tery aon el CPA La ds i anes An x Xn Mii sin 
yo oe 
+ Aim Vin + Aon X2Vn +Asn X3Yn . +A.n Xsyn .«- (0) 
+Ajin X1Vn + Aon X2Vn +As,n X3Vu Otc. een 
+ Ai,n NVn + Aon X2Vn O 3 Reon ae 5 
prot fer oerL roa et 
+Ai,n Vn O O Se Na eee! 6c 5 Ot 
2 2 
