Properties of Pfaffians. 
209 
Now in the performance here of the row-by-row multiplication we 
make use of thirty-six results, six of which constitute the set whose 
analogue we know ; and the remaining thirty are those we are in search 
of, namely, 
( 0, A„ A3, A„ A5, Ae I -a„ 0, b„ h,, h^, he) = 0 
( 5 -«3»-^3' 0, C4, C5, Ce) = 0 
0 I 
(-A6,-B6,-C6,-D6,-E 
or, in less instructive form, 
53A3 + h^A^ + &5A5 -f heAe = 0 
- b^A^ + C4A4 + C5 A5 + cgAg = 0 
-C5, -6^5, 0, Be) 
0 
..Ag + 6,B6 + cfie + dj)e = 0 
5-^6 = 
5. On recalling the fact that the vanishing expressions corresponding 
to these in the theory of determinants are viewable as determinants with 
two rows or two columns identical we are led to seek for the analogous 
theorem regarding the nullification of a Pfaffian. It may be stated as 
follows : The value of a Pfaffian is 0 if the elements in the part of the 
ith line which is parallel to a part of the sth line he identical with the 
corresponding elements of the latter, the element common to the two lines 
he 0, and the remaining elements of the ith line differ only in sign from 
the remaining elements of the sth line. For example, the lines being the 
3rd and 5th,* we have 
* The following diagram of the frame-lines of the Pfaffian '| 12 . 23 . 34 . 45 . 56 | will 
help to make this clear : — 
i i \|/ ^ 1 
•12 13 14 15 16-' 
23 24 25 26- 
' 34 35 36- 
' 45 46- 
••56- 
>lst line 
>2nd line 
>3rd line 
>4th line 
>5th line 
6th line 
The 3rd and 5th lines are seen to be parallel at the outset and at the close ; their common 
element is 35; and their remaining elements are 34 and 45. (See my Text-book of 
Determinants, pp. 197-204; Proc. Lond. Math. Soc, xiii., pp. 161-164; and Transac. 
R. Soc. Edinburgh, xl., pp. 49-58). 
