210 
Transactio7is of the Boyal Society of South Africa. 
^5 ^4 ^5 he 
I ^ 0. 
With this in front of us it is easy to see that the identities at the close of 
the preceding paragraph can be written in the form 
6. As the vanishing of a determinant which has two rows or two 
columns identical is usually deduced from the fact that a determinant is 
only altered in sign when two of its rows or columns are interchanged, it 
is not unnatural that the theorem formulated in the preceding paragraph 
regarding Pfaffians should be deducible in a similar way. As a matter of 
fact such is the case, the antecedent theorem being as follows : The 
value of a Pfaffian is only altered in sign if the elemeiits in the part of 
the rth line which is parallel to a part of the sth line he interchanged 
ivith the correspondhig elements of the latter, the element common to the 
tivo lines be changed m sign, and the remaining elements of the rth line 
be altered in sign and interchanged with the remaining elements of the 
sth line similarly altered. For example, the 3rd and 6th lines being 
taken, it is seen that the parallel parts of them are 
0 = I . b, b^ b, be 
^3 ^4 ^5 be 
C4 C5 Ce 
d^ de 
C4 Ce 
^3 ^ ^5 ^6 
C4 C5 Ce 
d^ de 
a- 
and 
ae 
be, 
the common element is Ce, and the remaining parts are 
C4 C5 and 
we thus have 
|<X2 a. a^ a^ ae 
^3 ^4 ^5 ^6 
C4 C5 Ce 
d^ de 
^6 
= — I ^2 ae a^ a^ a^ 
be b^ b. b^ 
-de -Ce -Ce 
d, -c, 
-^5 
