148 
Proceedings of Royal Society of Edinburgh. [sess. 
(7) Both the general identity, however, and the special cases 
acquire new significance if we make use of a recently discovered 
theorem regarding Pfaffians in order to alter the form of the right- 
hand side of the identity. 
This theorem, in so far as it concerns the present subject, may 
be described as giving an expansion of a special Pfaffian in the 
form of a series of terms, each of which is the product of a deter- 
minant and an element of the Pfaffian, the specialty of the Pfaffian 
being that the elements in the places where n - 1 of the 2 n frame- 
lines intersect are zeros. 
Thus the Pfaffian 
| « 4 a 5 a Q 
& 3 b A b 5 b 6 
c A c 5 c 6 
d 5 d 6 
e 6 > 
which is of the 3rd order, and has a zero at the place (12) where 
two of the frame-lines intersect, is equal to 
— | af> A 1 6 6 + | af> b K - | af>Q | d^ — | a A b 5 1 c 6 + af> Q | c 5 — | 1 c 4 , 
where the first factors of the terms are the six Tdeterminants 
formed from 
and the second factors are the remaining non-zero elements 
Similarly the Pfaffian 
d 5 d Q 
ct^ ccq a 7 cl g 
\ \ h \ 
C 4 C 5 C 6 C 7 C 8 
dr, d 6 d 7 d s 
e 6 e 7 e 8 
A A 
98 » 
which is of the 4 th order, and has a zero at the places (12), (13)* 
(23), where three of the frame-lines intersect, is equal to 
