36 EE. B. Wilson, 
second type. In fact, when 2 = 2, a dyadic which satisfies an 
equation of the type x? + ax—1=0 is .not in general resoluble 
into two reflections. The case of the fourth type must be examined 
more in detail. 
Consider the product of two reflections in an even number of 
dimensions and let the determinant of the product be negative. It 
is evident that the two reflections cannot be the same. In fact if 
the types are & and /, it is necessary and sufficient that & + / be 
odd in order,that the determinant of the product be negative. The 
product may then be written in the form 
k l 
One? Sy 87> al = 2 Sasa). Eee 
1 1 
The spaces fj, Bo, .--, Bk, G4, G3, .--, a of the consequents are 
therefore together greater than 7 and must intersect. The trans- 
formation of vectors in this space of intersection is either identical 
or is such as to reserve the direction of each vector without in- 
troducing any other change. In the former case +1 and in the 
latter case —1 is a root of the scalar equation. On substituting 
either of these values in the fourth equation of (71), if is seen that 
Qn = 0. In other words it appears from special considerations 
a 
that the equation of the fourth type is also reciprocal if 2 is the 
product of two reflections. As there is this additional condition in 
this case, the question might arise whether there were not also 
additional conditions in other cases. 
This question may be phrased as follows: Given any reciprocal 
equation 
dete eed 
(72) Hn—G, KH" + dg xP-2 — Nee a6 PR: mae os) 
of degree v, can a dyadic 2 be found such that 
1 1 
(73) Qs = QQ, Qo s— Qo, eee aoe s— aia qi QH=1, so ale = 
a2, N; 
Qn — 7 1 
and such that 2 may be written as the product of two square roots 
of /? Suppose that the roots of (72) with their respective multip- 
licities are 
115 Fi WY SAV ayTa yg SVs), 1a les le oo 3 e ee e 
The dyadic which has these roots may be written as 
(74) 2=7, |e +7 a9] @'9 +... 7% Cm| Om + 7o Om +1| Om +1 
= Vy Copy 329 Grong aagiciae «ae 
aa) PANS to Paleo oo snl ae Bma|B’m + 72} Bmi+1 |B'm: +1 
ae Bmi+2| P'mi+2 arith ee 

