Sylvester's and other Unisignants. 
191 
to which, as it is readily seen to have all its coaxial minors unisignant, 
the theorem quoted at the end of § 3 may be profitably applied, in order 
to obtain a unisignant of greater generality than C^, namely, 
a. 
^134 
-h 
4 
Co 
d,._ 
a. 
-c, 
d. 
or, say. 
The final development of this contains, in addition to the 24 terms of 
the 57 terms involved in 
boCodo.a,^^ + SCo(^o.o^34&i34 + ^do{a^h^^c,^ + b^a,_^c,^ + c,a^J)^^) 
The variant forms of 
6 are 
^' which include the forms of 
^2 
«4 
^2 
a. 
^134 
^1340 
K 
-^^034 
C124 
^1240 
"4 
^024 
Co\ 
C02 
<^I23 
^1230 
-^023 
-do3 
— 6^02 
dj_ 
(IX.) 
used in 
and two others are 
^034 
0134 
ft. 
C0X24 
"•'0123 ) 
(X. 
^234 
" Coi 
-do. 
C04 
dr.^ 
a. 
^04 
dr^t 
bo, 
C02 
d. 
Ct - Co 
d,-d. 
^34 
^24 
^23 
^0 
^X3 
^•14 
-Co 
<?I2 
^13 
d,2 
-do 
these latter leading easily to the theorem that if the vacant places ii 
^^234 ^2 
■c. 
c. 
dr 
'234 
-c. 
d. 
be filled up ivith any positive elements ivhatever, the restdtiiig determinants 
are unisignant.'^' (XI.) 
9. Taking now C3, and treating it as was treated in § 6, we find 
that the form of it corresponding to (V.) is 
a^ 
a. 
«4 
a, 
K 
-^45 
^45 
Ci 
C25 
-C24 
-^23 
^34 
^24 
-623 
-^345 
^245 
* From summing these two unisignants we reach a result already given in a paper on 
Boole's Unisignant " in the Proc. R. S. Edinb , xxxi., pp. 448-455. : 
