212 
Transactions of the Boyal Society of South Africa. 
6 -3 
-6-22 
-21 
-168+132 + 18 
= -18. 
8. With its help also there may be deduced a condensation-theorem 
or Pfaffians similar to that for determinants, — that is to say, a theorem 
expressing a Pfaffian of the nth order by means of one of the (w — l)th 
order. 
Thus, taking \\aj)^c^d^e(fjg^\ and performing in succession the- 
operations 
a2line3 — a3line2, a^me^ — a^ine^, ^aline^ — a^inQ^, ... 
we change into 0 all the elements of the 1st line except a^, and thence- 
derive the result 
^\aj)^c^d^e(f^g^\ = - j'l^^^&aCj ^\aj)^c^\ ^\aJ)^Ce\ ^\aj)^c^\ ^\aj)^c^\ 
^\aj)^d^\ 'la^Msl 'I'^sM?! 'kaMsl 
^\aj)^ee\ '1^2^567! ^\aj)^e^\ , 
The case of the preceding order, 
^\aj)^c^\ '1^2^3 061 , 
^\aj)^ce\ 
is not so interesting, because the Pfaffian on the right is then a minor of 
the adjugate of the Pfaffian on the left, and the identity is thus otherwise^ 
known. 
If we write the theorem in the umbral notation, for example. 
^\aj)^c^d^ee\ 
0-2 
\\ajj^c,\ 
i| 123456781. (12)- = 'I i|1234| !|12351 '|1236| 
1112451 '112461 
