~ 
Double Products and Strains in Hyperspace. 15 
square roots of the idemfactor, two square roots are not generally 
homologous. It is possible to define logarithms and exponentials 
of dyadics, and to show that these are homologous with the original 
dyadic; but this does not appear to be very useful. 
The system of homologous dyadics which is most useful is that 
which consists of a given dyadic, the idemfactor, and the dyadics 
derivable by means of rational operations on these two. For in- 
stance, let a dyadic @ satisfy an equation, with scalar coefficients, 
of the type 
(16) @P + a, PP-1 +... + ap-1 B+ Gy I= 0, 
and consider the scalar equation 
BP Opa! sp) 1 at ap — 0: 
The roots of this equation may be found and the equation factored 
into 
(x---7,) (x—1rg) .. . (X%—1p) = 0. 
So likewise the equation involving # may be factored into 
(16') (b—1, 1) (@—1ry 1)... (P—1ry [) = 0. 
Again, two polynomials in @: 
['(d) = oP + a, Ge-1 +... tap 1 S+ayl 
and A (d) = 6" + qa, OM-1 +... + am-1 G+ Aml 
may be divided according to the usual algorithm. If 4 is of lower 
degree than I’, the result of the division may be written 
(17) [(d)= B(#) A(#)+ PC) 
where the remainder P(#) is a polynomial of degree less than m7. 
Thus the Euclidean algorithm for the highest common factor may 
be applied to two such polynomials. 
Any dyadic # may be shown to satisfy a polynomial of degree 
not greater than *. To see this, let @ be expressed as a block 
of nm? terms in the form (9), where the antecedents and conse- 
quents are chosen as reciprocal sets. The higher powers of # are 
likewise expressible in terms of the same mu? dyads and certain 
combinations of the coefficients c;;. Consider the system of equa- 
tions, 77+ 1 in number, formed by the first 2? powers of # and 
the idemfactor. From these equations the m? dyads may be elimi- 
nated as if they where ordinary variables in 7? + 1 linear equations 
in #? unknowns. The result is obviously an equation of the form 
(18) €, 6 + 6, GV. toy. + cm [=0. 
As a matter of fact, it will be shown in article 11 that any dyadic & 
satisfies an equation of degree »—the Hamilton-Cayley equation— 

1 See, for instance, The Scientific Papers, volume 2, pp. 78-84. Some 
simple differential equations are also solved in these pages. 
