34 E. B. Wilson, 
ologous when and only when their product is a square root of J. 
As the involutory strains of types 0 and m are respectively + / 
and —/, they may be excluded as trivial when referring to the 
product of two. As the product is / when and only when the two roots 
are identical, that case may also be laid aside. In the case of two 
dimensions the only involutory transformation is a¢|@ — |p’. The 
determinant is negative and hence the determinant of the product 
is positive. The product is therefore —J, and it is seen that the 
line through which the reflection takes place in one is the line 
parallel to which it takes place in the other, and vice versa. In 
three dimensions there is a line and a plane entering into the 
characterisation of any involutory transformation, and unless the 
product of two is to be J, it is necessary and sufficient that the 
line of one reflection lie in the plane of the other and vice versa 
if the product is to be commutative. 
Consider next the case of 2 dimensions. Let # denote an in- 
volutory transformation and let 2 by any transformation which is 
commutative with it. Then 
Pia DP Wort, SO 
And 2 = ‘an|@'n + @n=1|@'n—-- - . - 4 e141 — alex 
— 5 — 04] O's 
If (O Carnes (G1, 0an. Senn MtON By Bose se se bns 
2 = By\ a’; + Bo\a’s Sie cateyate Bn|@'n 
and . 2 &2-1= Bal Bn + Ba-11B'n—1 +... + Birtil Bap — Bel Be 
ee he B1| B's. 
If this is to be identical with #, the spaces Ry (fi, Bo,..., Bx) and 
Rx (a, @,..., @k) must coincide, and also the spaces Ry_x (@r41, 
., Bn) and Rn—x (ar4i,..., Qn). Now if it be involutory, the 
transformation between the #’s and a’s in Ay must be involutory ; 
and so must the transformation between the #’s and a’s in Ry_ x. 
If 74, Ya =-* , Ye and yr4y1,.-, , Yn be the fixed elements ofpthe 
involutory transformation ®, it is seen that they all lie in the fixed 
spaces ey and Rn-x <A different way of stating the result is this. 
Let 
pot (OS Glad and! wes en eee z(5) 
1 1 215 
with the spaces Ry, Rn—z and Sj, Sn—i. Suppose A, and S; inter- 
sect in J. Then if the product ¢ # — # @is involutory, FR, inter- 
sects S,-1 in R’;—m and S; intersects Rn, in S‘i-m and the space 
Vi.ti—2 m compounded of R’—m and S’j-m is fixed in the product. 
Furthermore A,-, and Sn»; will have in common a space Tp—4—14+m 
which compounded with 7m gives Vy—r—i42m as a fixed space of 
