Sylvester s Axisymmetric Unisignant. 201 
Here, as before, the first set of values, five in number, consists of the 
primary minors of in its triangular form ; and the remaining values, 
ten in number, are obtained by multiplying any element of by the 
^corresponding secondary minor, that is to say, by the minor got by 
deleting the two frame-lines to which the element belongs. (HI-) 
5. From the preceding it is incidentally seen that the jpriynary minors 
-of a Sylvester s axisymmetric unisignant are also unisignants though of a 
different kind. (IV.) 
In the case of the primary minors that are coaxial, being of the 
form 
a + {j\ + + i\) + (y, + 73 + 73) + 
have 
16 + 9 + 9 + 16, i.e., 50 
terms ; and the non-coaxial, being of the form 
have 
16 + 6 + 3 i.e., 25 
.terms. 
From this it follows that the number of terms in the adjugate of ^4 
50 25 25 25 
_ 25 50 25 25 
" 25 25 50 25 
25 25 25 50 
= 254.5 = 59 = (53)3, 
which agrees with the fact that the adjugate of is (^4)3 coupled with 
the fact the number of terms in ^4 is 53. 
6. The theorem (IV.) includes a theorem previously known, namely, 
that the sum of the primary minors of /3 is unisignant, and this again 
implies that the axisymmetric bordering of ^ by 0,1,1,1,... gives a 
determinant whose final terms are all negative. It is thus suggested 
to ascertain the character of the final development of the determinant 
■obtained by bordering % axisymmetrically by 0,x,y,z,... 
Taking first the case of which when bordered as indicated becomes 
x y 
X ^2 + 63 —^3 
y -^3 ^3 + 63 
\we see that the latter has all its terms negative, being equal to 
- a^y^ — a^x"" — b^{x + yy. 
