419 
1909-10.] The Theory of Persymmetric Determinants. 
It is not noted by the author that having n-\- 2 equations linear and 
homogeneous in the A’s he could at once deduce 
1 
X 
X 2 
( 'o 
a i 
a 2 
«»+i 
a l 
a 2 
H 
^n+2 
i a n 
a n+ 1 
«»+2 • 
. . . 
^2n+l 
The other forms of the equation, however, he gives full attention to. 
Sylvester, J. J. (1853, June). 
[On a theory of the syzygetic relations of two rational integral 
functions, Philos. Trans. Roy. Soc., Loud., cxiiii. pp. 407- 
548 : or Collected Math. Papers , i. pp. 429-586.] 
When dealing in art. 7 with Bezout’s condensed eliminant of two 
equations of the n th degree, Sylvester illustrates by the case of n = 5, that 
is to say, where the equations are 
a 0 x 5 + a Y x 4 + a 2 x 3 + a 3 x 2 + a 4 x + a 5 = 0 ) 
b 0 x 5 + bp 4 + b 2 x 3 + b 3 x 2 + b 4 x+ b 5 = 0 I ’ 
pointing out that the eliminant may be constructed by first forming the 
array 
1 a o b i i 
i «A 1 
a 0 b 3 1 
1 |<A I 
l a 0 b 5 
1 a A \ 
1 a A 1 
1 i 
1 «<A 1 
1 «A 
1 a A 1 
1 «<AH ’ 
1 tt 0 & 5 1 
1 “A 1 
1 a A 
1 Vk 1 
1 a A \ 
1 «A 1 
1 «A 1 
1 «A 
1 a Q b b 1 
1 «A 1 
1 a 2 h 5 1 
1 «A 1 
1 “A 
and then, as it were, superposing 
the array 
| | 
i 1 
1 “A ! 
1 1 
1 «A 1 
1 «A 1 : 
1 aA 1 
1 1 
1 rt :Ai 1 
and next the array 
I a 2 b 3 1 ’ 
In regard to these arrays he says in a footnote (p. 424), “ A square 
arrangement having this kind of symmetry, namely, such as obtains in the 
so-called Pythagorean addition-table as distinguished from that which 
obtains in the multiplication-table, may be universally called persym- 
metric.” This is apparently the first use of the word. 
