450 
Transactions of the Boyal Society of South Africa. 
This is virtually the same as saying that If the sum oj every row, except 
one, of a determinant be 0, the co-factors of the elements of the excepted 
row are identical. Consequently we can assert that Any determinant which 
has the sum of every roio equal to zero has in the case of every row the same 
co-factor for every element of the row ; and from this it follows at once 
that Any axisymmetric determinant luhich has the sum of every roio equal 
to zero has the co-factors of all the elements identical — which is Borchardt's 
theorem. 
5. It is very doubtful, however, whether it is desirable to view this 
common co-factor as a determinant at all ; for when it is of the nth. order 
it is a function of only ^n{n + 1) elements and yet contains {71 -f terms 
and all of them positive. Thus 
«2 + ^3 + ^4 
— 0^ 
a. + 6. + c. 
(^4 + ^4 + I, 
which is a function of the six quantities a^, a^, a^, b^, b^, c^, is equal to 
a.a^a^ + (^a3{^4 + ^4) + ^20^4(^3 + ^4) + ^3^4(^3 + ^4) 
+ {a, + ^'3 + ^4) (^3^4 + ^3^4 + ^4^4)- 
Further, although it is of the third order, there are not two but three 
other ways of writing it like this ; and, generally, n -\- 1 ways in all in the 
case of the nth. order — a property unnatural to a determinant. The reason, 
of course, is that these n + 1 ways correspond to the co-axial primary 
minors of the determinant of the {n + l)th order whose primary minors are 
all equal. 
6. Much more convenience results from arranging the ^n{n + 1) 
elements in semi-quadrate form, that is to say, in the manner in which 
the elements of a pfaffian are arranged ; and if after doing this we bracket 
them in some specially distinctive way, we shall at the same time secure 
a convenient notation. Thus, we shall have 
a. a/ 
c 
instead of the determinant in the preceding paragraph, and 
I a^ ^3 
b. 
instead of 
'^2 + ^3 - ^3 
- 63 ^3 + 
