Sylvester s Axisymmetric Unisignant. 
199 
3. By reason of the permutability of the frame-lines we have 
^3 K 
\ ^3 ^4 
and we thus see that the function is invariant to the simultaneously 
performed circular substitutions 
and can be expressed by 
or 
o O 
Similarly it is found that the function of the 4th degree is invariant to the 
simultaneous circular substitutions 
and so on for the 5th and higher degrees, the first substitution in each 
case being composed of the elements lying on the hypotenuse of the 
triangular array together with the element situated at the right angle 
of the same, and the remaining substitutions being similarly obtained 
from lines parallel to the hypotenuse. (II.) 
4. Since the adjugate of a determinant is an integral power of the 
same, it follows that the adjugate of a unisignant can have only positive 
terms in its final development. It consequently becomes of interest to 
ascertain whether such an adjugate can be represented in a form which 
attests unisignancy. For the particular case of the unisignant ^ the 
answer is in the affirmative, and a very interesting theorem is connected 
therewith. 
