54 E. B. Wilson, 
that therefore the method adopted in the last two articles for 
showing that such a strain may be resolved into three reflections 
must break down in some of the special cases (for instance when 
the given strain is itself a reflection), the theorem that any strain 
is resoluble into three properly chosen reflections is strongly 
suggested. If 2 be a strain and @ a reflection, the necessary and 
sufficient condition that 2 @ be resoluble into two reflections is seen 
from article 15 to be that the Hamilton-Cayley equation of Q@ 
shall be reciprocal, and that the invariant numbers which corre- 
spond to a pair of reciprocal roots of the scalar equation shall be 
equal. The first part of this condition is not hard to state and, in 
the simplest cases, to examine. The last part of the condition ap- 
parently requires very detailed consideration. ; 
For the present purposes the fourfold division of the problem, 
according as z is odd or even and &, is + 1 or —1, may be some- 
what abridged by the use of (70). If X—@@, the conditions be- 
come 
(102) Xis=Xpy, k< EG), Xns=0 
with the supplementary condition necessary aie when z is even 
and the determinant of the product Xn — — 1, and with the further 
condition that the invariant numbers which correspond to any pair 
of reciprocal roots of X must be equal. These are the necessary 
and sufficient conditions that 2 be resoluble into three reflections. 
In case @ happens to be of type 1, these conditions reduce to 
(102’) O12 oe) (O, 2) 120, 
o [BO —B-Y— 4 (Q5—Qj,) 1]a=0, 1< hk < E(2), 
oz) — 
andi # 1s even and X,,— 1, 22s I\o—0 
where 3(-" has been written as an BUBLevien for the &th in- 
variant dyadic Z associated with 2-1. Thus there are E : equa- 
tions to be satisfied in case z is odd, or in case m is even and 
: n 
Xn=—1; but if ” is even and X,—-+-1 there are only E\ 5) — 
equations to be fulfilled. In all cases they must be satisfied sub- 
ject to the restriction oo=|=0. If & were of type 2, or higher up 
n 
to type £ it the conditions which would be analogous to (402’) 
might be expressed in terms of the invariant dyadics of higher class 
referred to in article 17 but not investigated. 

