1917-18.] Note on the Construction of an Orthogonant. 147 
accounts is its generality, it being always carefully pointed out that the 
resulting expression contains the full number of arbitrary parameters. 
There has, however, been a tendency to accept this as meaning that 
Cayley’s orthogonant of any order includes all possible orthogonants of 
that order, and that therefore if a theorem be proved to hold for Cayley’s 
orthogonant it must hold for orthogonants in general. Such a theorem as 
this last is Brioschi’s of 1854 concerning the latent roots, which originally 
was stated and proved in regard to Cayley’s form only. Another pertinent 
example is connected with Stieltes’ projected theorem concerning the 
hypothetical vanishing of the determinant of the sum of two orthogonal 
matrices, Netto’s proof of it being dependent on the assumption that 
Cayley’s orthogonant could represent a product-orthogonant. In regard to 
this assumption some little discussion took place at the time, and Kronecker 
being referred to gave it as his opinion that as a representation of a positive 
unit orthogonant Cayley’s is not perfectly general. 
In 1890 Kronecker gave the subject much attention, finding in the 
first place that Cayley’s result had to have a condition attached to it 
which necessarily narrowed its scope ; in the second place, that it could 
not represent an axisymmetric orthogonant unless in an approximate 
way by proceeding to a limit; and, thirdly, that nevertheless any defect 
which it had was not of essential importance and could be remedied by 
multiplying it by another of the same kind. 
A less general statement was made in 1892 by Metzler, who affirmed 
that an orthogonant having both + 1 and — 1 for latent roots could not 
be represented by Cayley’s. The result, however, of his investigation was 
to show that Cayley’s three-line orthogonant will include the special form 
in question if the signs of the second and third rows be altered. 
In 1893 H. Taber, taking Ivronecker’s result as his starting-point, 
namely, that Cayley’s includes every positive-unit orthogonant except 
those that have — 1 for a latent root, reached in that year and the 
following a number of interesting results, all more or less bearing on the 
defect and the remedying of it. 
(4) In all the effective part of the work of these writers the pure 
theory of determinants is not adhered to, Kronecker making use throughout 
of a method dependent on the properties of so-called “ modulus-systems,” 
and Taber employing the algebra of Cayleyan matrices. It is thus seen 
that there is room for additional consideration of the subject, and for 
contributions, however slight, to the elucidation of the general question. 
(5) A first gain in clearness is attained from fixing our attention on the 
fact that Cayley’s orthogonant has the peculiarity that its diagonal elements 
