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PEOPERTIES OF PFAFFIANS, 
WITH THEIR ANALOGUES IN DETERMINANTS. 
By Thomas Muir, LL.D. 
(Received May 5, 1914. Read May 20, 1914.) 
1. In the course of a recent inquiry connected with PfafiQans the 
identity 
- 15 !| 2346 1 + 16 !| 2345 1 + 56 '| 1234 1 
= 23!|1456|-24i|13561 + 34'|1256| 
or, in more concrete notation. 
- 1 &3 be 
+ ^6 1 63 
d^ de 
C5 Ce 
+ C4 1 ^2 ^5 ae I 
made its appearance, and had to be verified. It is most readily estabhshed 
by noting that the Pfaffians occurring in it are principal minors of 
'1 123456 1 or ^\aj)^c,d^ee\, 
and that therefore the two members of the identity are by a theorem of 
1903 * each equal to 
^\aJ)^c^d^ee\-\a^h^Ce\. 
There is, however, another way of looking at it which is more instructive 
and which leads with ease to other identities of like kind. 
2. Denoting the said Pfaffian of the third degree by ff, and its 
adjugate by 
ilA^BsC^D^Egl, 
* Trans, of S. African Philos. Soc, xv., pp. 35-41. 
