82 E. B. Wilson, 
which correspond to each of the latent roots. It is hardly neces- 
sary at this point to indicate the relation of this result to the 
theory of elementary divisors. Another matter which will be passed 
without examination is the reduction of a real dyadic to a real canon- 
ical form. This is not of importance to the work that follows and 
it was not treated in any detail by Gibbs. All that is essential in 
his treatment of dyadics, as given in his course on multiple alseniee 
has now been set forth. 
Part I].—Some ALGEBRAIC AND GEOMETRIC APPLICATIONS. 
Square Roots of the Idemfactor. 
14. Involutory strains.—lIf a strain represented by # be involutory, 
its square is the identical transformation and analytically 
(66) t= J, (@-D (P+) =0. 
Any dyadic which satisfies this equation may be called a square 
root of the idemfactor. The algebraic theory of these square roots and 
the geometric theory of involutory strains correspond, and each may 
be used to study the other. Equation (66) is clearly of lowest 
degree, and the latent roots are +1 and -—1. As the individual 
factors enter the equation of lowest degree only to the first power, 
the reduction is 
Pig) = L441) = G01 + Gel a@’o +... axl ae, 
Bi—1) = L-1) = aril eg +... + anle’n. 
Hence 
k n 
(67) $= fay—en = 2 a| e i—D a;| a's. 
1 kt 
There are n+ 1 different types of these roots according as # con- 
tains 0, 1,2, ..., m—1, or m negative signs. The first and last are 

1 The relation of involutory strains to the group of unimodular strains 
in the simple case where x =3 has been treated in detail by me in an 
article entitled Oblique reflections and unimodular strains, Transactions 
of the American Mathematical Society, volume 8, pp. 270-298, 1907. 
Reference to the case of three dimensions will be to this article. A 
number of references to the literature of involutory transformations may 
be found there or in my article Involutory transformations in the pro- 
jective group and in its subgroups, The Annals of Mathematics, second 
series, volume 8, pp. 77—86, 1907, where only the most general questions 
are discussed. 
