1917-18.] Note on the Construction of an Orthogonant. 151 
In other words, P is such that its square is elementally a multiple of its 
duplicant ; and therefore by § 8 it follows that 
Pn-J M P\2 Vis P\+ 
P-21 P22 ~~ ^ ^ P 2 3 P‘2+ 
P 3 I P32 P33 ~ 2^ P34 
P+l P+2 P+3 P+ 
is an orthogonant. 
(12) In similar fashion to this an orthogonant can be obtained from 
the quadric portion of the matrix of the adjugate of any unit-axial skew 
determinant whatever. The case of the third order was intentionally 
passed over by us, because the orthogonant thence obtained merges in the 
three-line orthogonant of § 6. We shall presently consider the case of the 
fifth order, it being necessary to have an odd-ordered case separately 
investigated, since the occurrence of a zero-axial skew determinant in a 
piece of work is almost certain to bring about two more or less variant 
types of result. 
(13) Meanwhile it is seen to be of some consequence to be able to tell 
what the desired portion of the said matrix is without having to calculate 
the whole of the elements of the adjugate. The following theorem gives 
what is wanted : the determinant of the quadric portion of the matrix of 
the adjugate of any unit-axial shew determinant is 
S — r 1 r l 
~ r \ r 2 
~ r + V 3 
r i r 2 
S - ?\,r, 
r 2 V 3 
- r i V 3 
~ V 2 r 3 
8 — r 3 r 3 
where r h r k is the product of the 1C and h fh rows of the given determinant 
when made zero-axial, and S is the sum of the squares of the elements on 
one side of the diagonal. The practical rule involved in this directs the 
given determinant when made zero-axial to be multiplied by itself, S to be 
subtracted from each diagonal element of the product, and then the signs of 
all the elements to be changed. For example, if the determinant of the 
5 th order be 
1 
a 2 
a 3 
a 4 
a 5 
— a 2 
1 
$3 
ft 
ft 
— a.. 
— @3 
1 
y 4 
75 
— a 4 
— (3 4 
-y+ 
1 
“ a 5 
"ft 
-75 
-85 
1 
the first column of the quadric portion of the matrix of its adjugate is 
