Properties of Pfaffians. 
213 
we see that it may be viewed as an ' extensional ' of the manifest 
identity 
"1 345678 1 = i|34 35 36 37 38 , 
45 46 47 48 
56 57 58 
67 68 
78 
— a fact which makes the analogy with the corresponding theorem in 
determinants still more striking. 
9. On the other hand it has to be noted that although in the case of 
the other known condensation-theorem, Hermite's of 1849, there is also 
an analogue^ the resemblance is not nearly so close, the elements of the 
new Pfaffian being no longer Pfaffians themselves but determinants. 
Thus 
^\aj).c^d^ee\ 
1 
and if the elements of the new Pfaffian here be denoted by 
P., P„ P.; Q„ Q.; 
the next case may be written 
\a^b^c^d^eef,g8\ = 
iP,Q3R,S3T6 
«5 
a. 
ae 
he 
Ce 
a. 
a. 
^5 
a. 
^4 
^5 
a, 
a. 
do 
df 
and similarly for higher orders. 
10. There is apparently no theorem in Pfaffians corresponding to the 
strictly so-called "multiplication-theorem" of determinants. The theorem 
* Hermite, C, " Sur une question relative a la theorie des nombres." Journ. (de 
Liouville) de Math., xiv., pp. 21-30. 
