Double Products and Strains in Hyperspace. 25 
coefficients must have equal coefficients, it may be seen that two 
identical polynomials with dyadic coefficients must have identical 
coefficients. Therefore not only may any scalar be put in the place 
of g, but any vector or dyadic may be put in its place without 
disturbing the identity. If # be substituted for g, the righthand 
side vanishes. Hence 
(38) P"—G, "1+ Po, Pr—-2 — ... 4+ (—-1)""1 Gri, s 
+ (—1)" by, J = 0. 
This is the identical equation which any dyadic must satisfy. It is 
sometimes called the Hamilton-Cayley equation. 
The actual equation of the coefficients of the different powers of 
g& gives the dyadic equations 
(39) P Pr_1,¢ = On ii 
@n—1, ¢ + (Pn—2 5 L) e = Pn-1,s vis 
(Bn—2 lve + & (Pn—3 . Lh) A ES eh 
(Po ous eae + & (P © Ln—-2) ¢ = ®y5 I 
(PE Ine + P= Osl 
If the relation (b—g Dn Ig = (@—g 1) (S—v” In~2, ¢) had been used, 
the same identical equation of the matrix would have been found, 
but the equations obtained from comparing coefficients would have 
been 
(39') Pp e To, lain Py (Pn—3 . I) e = DPn_-1, s I, 
Pr—2, ¢ is P s I (@n_-3 . 1) c = P; (Pn—4 = 13) c = Pn_2, s 1; 
Pil alo Bo LADY Typos) 6 1a — Oy ol 
(@S Ins)e + PE I= G I, 
Consider the scalar equation, called the characteristic equation, 
(40) Xn — Bs xnr—-1 + Py H—? — .. ale (—1)n—1 Pn—1,5 X 
+ (—1)" d, = 0, 
and suppose the roots are a, 6, c, ... with the multiplicities /, g, 
r,.... The identical equation (38) may then be factored into 
(38') (b—al)p (@—b 11 (6—cl)\r...= 0. 
As the scalar equation may be regarded as the expansion of (@—x/) p, 
it appears that each of the factors —aJ/, d—b/, d—cT, ... must 
have at least one degree of nullity, that is, there must be at least 
one element a such that (S—a/)a=0 or a=—aza, etc. The 
roots a, b,c, ... of the scalar equation are called the latent roots 
