Double Products and Strains in Hyperspace. 29 
valid for incomplete dyadics. If the dyadic # be arranged in matric- 
ular form with no terms beneath the main diagonal, the existence 
of the Hamilton-Cayley equation is evident. Finally it may be noted 
that the reduction yields the same form for all of the transformed 
dyadics of # For if # satisfies the equation : 
A(@) = "+ A, S14 ...+ Am-1 © + An l= 0, 
then 
(54) A(VSP—-1) — (PSP-1)m+ 4,(Chb P-1)m-1-,,, Am POE! 
ota Am vp 
= Por P+ 4, Por! G+... +Am_; PbP-1 
+ Am I 
= P A(b) P-1 = 0, 
It is obvious that this proof could have been given just as well in 
article 6, and that in particular the equation of least degree is 
identical for all the transformed dyadics ##&#—1, The fact that 
the scalar invariants of #@ @ #@—1, as shown in article 10, are identical 
with those of # shows that the Hamilton-Cayley equations are the 
same in both cases. The remaining steps to fill in for the purpose 
of establishing the identity of the reduction of # and ¥ # #—1 are 
too obvious to need detail. 
13. The canonical form of a dyadic.—The equation of least degree 
gives the relation 
(54) (b—al)p Ig =0 or (b—aly Ig? = 0 or (Pa—al,)r = 0 
For brevity let 
(55) e—-gia— 7. 
The further classification and reduction of dyadics therefore depends 
on the classification and reduction of nilpotent dyadics. Consider 
the successive powers . 
(56) ZIPS... GER PO: 
These have increasing nullities, but the change of nullity between 
two successive powers never increases. This may be expressed as 
(57) De == 0, Ly ig, 0, 5 Dep ey tg 1 9s 
Lith: +. + bh, 9 0. 
where, by the theorem at the end of article 5, 
(58) p= hi =e »Shp_o S hp-1 
and R-+- hy + hot... + Rp—2+ Rp--1 = m, 
if m be the multiplicity of the root a, where it is understood that 
the next higher double powers of each dyadic must vanish. The 
subscripts therefore denote the number of independent dyads in the 
dyadics. It remains to show that, with the aid of these relations, 


