1909-10.] The Theory of Persymmetric Determinants. 
417 
a 0 
a Y 
a 2 
a z 
" i 
a 2 
a 3 
« 4 
a 2 
a z 
a 5 
«3 
«4 
«5 
a 6 
where v — '72ep l 2 p 2 2 p s 2 p^I and I is the quadratic invariant s 4 — s 2 2 of 
x 4 + s^y + s 2 x 2 y 2 + s s xy 3 -f s A y 4 or (x + X Y y){x + X 2 y)(x + \^y)(x + X A y) . 
The fact that the coefficients of y 3 , v 2 , v 1 , v° are invariants of the octavic is 
insisted on, and generalisations effected for functions of the degree 4m 
and the degree 4m + 2. 
Further, it is pointed out that when the said even-degreed functions after 
transformation are without the last (or unique) term, — that is to say, are in 
Sylvester’s phraseology “ meio-catalectic,” — the last of the series of invari- 
ants must vanish : for example, the condition that (a 0 , oq , . . . , a 6 $ x , y ) & 
may be expressible as the sum of three sixth powers is 
= 0 
This, of course, may be proved independently, but is seen to be a conclusion 
from putting e = 0 in the f oregoing. 
Sylvester, J. J. (1852, April). 
[On the principles of the calculus of forms. Cambridge and Dub. 
Math. Journ., vii. pp. 62-97, 179-217 : or Collected Math. Papers, 
i. pp. 284-327, 328-363.] 
Here the same subjects and the same special determinants are dealt 
with as in the preceding ; and the determinant whose vanishing has been 
seen to be the condition for “ meio-catalecticism ” is denominated (p. 62) 
the catalecticant * of the even-degreed function in question, while the 
determinant whose resolution into linear factors furnishes Sylvester’s 
canonical form of an odd-degreed function is called the canonizant f of 
the said function. As the former is an invariant of its function, so the 
latter is a covariant. 
Bruno, Faa de (1852, May). 
[Demonstration d’un theoreme relatif a la, reduction des fonctions 
homogenes a deux lettres a leur forme canonique. Journ. (de 
Liouville) de Math. . . . xvii. pp. 193-201.] 
The subject of the whole of this paper is simply the solution of the set 
of equations dealt with in Sylvester’s paper of 1851 (October). The process 
* “ Meicatalecticizant,” Sylvester truly says, would have been the more correct word, 
but even he took alarm sometimes. 
t The name would have been equally appropriate for the determinants of the preceding, 
paper which have v in their diagonal. 
