Sylvester's and other Unisignants. 
193 
11. Taking in the form used in § 9 and developing it according to 
products of the diagonal elements we obtain an alternative proof of its 
unisignancy. For the result of the development is 
— ^c^d^e^ 
6I 
-b 
345 
-^245 
a. 
+ 
L 
— 
'^245 
'^^235 
(?35 
-r?25 
— 
+ terms free of suffix 1, 
and in this it is readily seen that the sum of the terms free of the suffix 1 
is equal to 
?r<^2345 X determinant like ; 
that the cofactor of is a determinant like ; and, if we care to say so, 
that the cofactor of d^e^^ is a determinant like ^3 and the cofactor of c^d^e^ 
a determinant like : in other words, that all the determinants which 
appear in the development are unisignants. 
12. Eeturning now to Sylvester's unisignant we have to note the 
curious theorem that if it be bordered horizontally and vertically luith the 
eleinents 0, 1, 1, 1,... the resulting determinant has in its final development 
nothing but negative terms. 
This is easily verified for the cases of S2 and S3, the number of terms 
in the one case being 6 and in the other 48. Supposing the verification 
has been made, let us examine the case of S4, namely, 
i'0234 
d. 
0134 
1 
— a^ 
-d. 
^4 
dor, 
Taking the development according to products of ^o, b^, Cq, do we see that 
the cofactor of ^o^o'^o is - 1 ; that the cofactor of Codo is a bordered S2, and 
therefore by hypothesis a negative unisignant : that the cofactor of d^ is a 
bordered S3, and therefore for the same reason a negative unisignant : and 
that the aggregate of the terms independent of the suffix 0 is the negative 
sum of the signed primary minors of 
'^234 
-«2 
- a. 
-a. 
- b^ 
^.34 
-63 
-K 
- C2 
^^X24 
-^4 
-d. 
-d. 
d. 
and therefore is equal to 4 times the negative sum of the primary coaxial 
