50 E. B. Whlson, 
Moreover it appears that the choice of o is arbitrary except that it 
shall not lie in any of the invariant spaces of m—1 dimensions. 
The result is stated for 2 dimensions because the proof is the same 
as for four. In particular these scalar invariants may be chosen so 
as to make the Hamilton-Cayley equation of the product a recip- 
rocal equation with distinct roots, and hence the product is resoluble 
into two reflections. Geometrically this means that a reflection of 
the first type may be found in co”! ways such that the product 
of the reflection and any collineation (or any unimodular proper 
or improper strain) which has distinct roots is resoluble into two 
reflections. In other words, 2 in this case is always resoluble into 
three reflections, or analytically may be regarded as the product 
of three square roots of the i1demfactor. 
18. Some special cases of the product.—The theorem which has 
just been established for the general case where the roots of 2 
are distinct may be extended. It is clear that @ may be such 
that the product 2 (J—2 o|o) cannot have arbitrary scalar invariants: 
for if 2 were a reflection the product would have to have such 
invariants as to make a reciprocal equation. It may, however, be 
shown that: If the Hamilton-Cayley equation of a dyadic 2 is the 
equation of lowest degree, the choice of a reflection /—2 6|6 may 
be made in co”-! ways so that the scalar invariants of the product 
Q ({—2 o|o) are arbitrary with the exception of the determinant 
which is —@2n. It should be noted that the condition that the 
Hamilton-Cayley equation be identical with the equation of lowest 
degree is equivalent to saying that in the canonical form (64) to 
which the dyadic may be reduced all the shearing terms correspond- 
ing to equal roots must be present. If 2 — 4 the possible cases are 
(98) 2Q=aca '. bs 2a ee 2a ea : 
arm ho +a PB ny +a eel 
ee YY see es 219 ee i 
+4568 +458 + @éee"? 
D4 ae f 
; 2 The vertical bar has 
ae ; 
; been omitted for brev- 
sh CUM aes 
Ba ee 
As the proof of theorem in general involves great detail, and at 
the same time general reasoning on tolerably varied and involved 
formulas, it will be well to carry the computation through in these 
cases, after which the general cases will offer no particular difficulty. 
The expressions for 3@) and ZF) in each of the cases above are 
