26 EE. B. Wilson, 
of the dyadic. There is at least one element fixed except as to 
magnitude for each latent root of @. 
If any element is a fixed element (except for magnitude will be 
understood) corresponding to the latent root a, no product of the 
form 
(41) Il (®) = (@—61)1 (@—c I)... 
can annihilate the element. For by direct substitution it is seen that 
IT a = a (a—6b)2 (a—c)... 
In like manner, if P is an element which satisfies the equation 
(6—a IP 8 =0 but does not satisfy the equation (6—a J)P—1 B =0, 
then (#/—a /)P—! B is a fixed element corresponding to the root a, and 
no product of the above type (41) can annihilate it. It follows, 
therefore, that factors of the type (@—aJ)?, (6—c /)4, ... are entirely 
independent in their nullities, and the product of such terms has 
the same nullity as the sum of the nullities of the factors. It ap- 
pears also that the equation of lowest degree must contain each 
of the factors of the type /—al, /—d/, ... at least once, or there 
would be some elements which would not be annihilated by the 
product. The equation of least degree may therefore be written as 
(42) A(®) = (S—a 1)v’ (@—b 1) (S—cI\"’... = 0, 
PERG EG FEI OS 
where none of the exponents vanish and the degree of the equa- 
tion is not greater than 7. 
12. The reduction of a dyadic.—With his usual desire for general 
hypotheses, Gibbs made no use of the Hamilton-Cayley equation 
and the resulting fact that the degree of the equation of least 
degree is not greater than 2 when he came to reduce the dyadic 
to a canonical form. He based his work on the existence of an 
equation of least degree as proved in (16) of article 6. Suppose 
this equation were factored by the methods indicated in that ar- 
ticle. Let the equation be 
(43) A(&) = (b—al)P (6-6 14 (@—c Tl)... , 
and let a fen tae dg ee ay 
Further let 
(44) @-al= ¥, 6—b/=¥#-+ (a—b) I, 6—cl=P-+ (a—o)l.... 
Then 
(45) ($—bJ)1(@--cly ...=AIAB¥PACH-...4 Aes 
where 
A = (a—b)1 (a—c)"... = 0. 
Divide 47/+-BY+ Cy?+...4+ HPm-v into J by the ordinary 
