208 
Transactions of the Boyal Society of South Africa. 
they make their appearance when we try to establish the relation 
between a Pfaffian and its adjugate. 
From the theory of determinants we know that the square of this 
adjugate is 
A. 
A3 
A. 
A3 
Ae 
-A. 
B3 
B. 
B5 
Be 
-A3 
-B3 
C4 
06 
-A, 
r>5 
-A3 
D5 
-K 
-B, 
-0, - 
-E, 
. A3 
d6 
e6 
where 
and where, generally, the cofactor of any element of the original Pfaffian 
^\a2b^c^d^e(,\ is got by deleting the two frame-lines in which the element 
is situated and prefixing the sign - when the sum of the numbers of the 
lines is even. Multiplying this determinant row-wise by the determinant 
which is the square of ^{a^b^c^d^e^] itself, namely, by 
we obtain 
a^ 
a, 
«4 
a, 
ae 
-a^ 
be 
-a. 
ce 
-a^ 
-K 
de 
-a, 
ee 
-de 
ff 
-be 
-de 
-ee 
in other words, we have the equality 
(adjug. oiffY.jf^ = 
from which it follows that 
adjug. oiff = p- 
and, generally, that 
