Note on Double Alternants, 
185 
There is a difference of forra in the result when the given determinant 
is of even order. Thus, in the case of the fourth order, while we come as 
before to the equation 
1 
-1 
-1 
1-1 -A3 
-1 
1 1 
U As 
— /24 fdA 
1 
/34 
and partition the determinant on the right into the sum of a five-line and 
a four-line zero-axial determinant, it is not the latter but the former that 
vanishes, giving 
d;.- = i/.. /.3 /.4 
/34 
(VIII.2) 
13. Underlying these results we have evidently the general theorem 
that If all the elements of a zero-axial skezo determinant he increased by 1, 
the restdting determinant is an exact square, ivhatever the order may be ; 
the reason being that where the order is even the value of the determinant 
is unaltered by the change, and where the order is odd the new deter- 
minant is expressible as a zero-axial skew determinant of the next higher 
order. (IX.) 
13 
