679 
1908-9.] Superadjugate and Skew Determinants. 
As has already been seen, there are then associated with the determinant 
n Pfaflians of the degree ^(n—V). These are connected with one another 
by n linear relations, the exact proposition being that If F 1? F 2 , . . . , F n be 
the n primary minors of the quasi- Pfaffian of an odd-ordered zero-axial 
shew determinant, the product of F p — F 2 , F 3 , — F 4 , . . . . by any row of the 
determinant vanishes. (xxvii.) 
The reason for this lies in the fact that the product in question is itself 
expressible as a Pfaffian of the next higher degree, having two frame-lines 
so related as to produce evanescence. Thus, the determinant being 
• 
a 2 
«8 
a 5 
-«2 
• 
^3 
K 
h 
-«3 
^3 
• 
C 4 
C 5 
- a 4 
~ C 4 
• 
( h 
~«5 
~h 
" C 5 
— 
• 
the five signed Pfaffians are 
^3 ^4 ^5 
3 ^3 ^4 ^5 
) | ^2 ^4 ^5 
, | rt 2 ^5 
5 | Cti) 
C 4 C h 
C 4 C 5 
h 
^3 ^5 
h b i 
d 5 
d 5 
f h 
C 5 
C 4 
and the product of them by the row 
~ a 3 5 ~ ^3 J 0, C ' 4 , 6* 5 
is the Pfaffian obtained by prefixing this row to the quasi- Pfaffian of the 
determinant, namely, 
— a. 
which being the square root of 
Clr 
- C. 
— Cr 
• 
^4 
C 5 
<h 
a 3 
h 
h 
h 
C 4 
C 5 
<-h 
3 
jf 
-a 3 
-h 
. 
C A 
C 5 
• 
a 2 
a 3 
« 4 
«5 
~ a 2 
• 
^4 
h 
~ a 3 
-h 
• 
C 4 
C 5 
- a 4 
-h 
~ C 4 
^5 
— a b 
-h 
“ C 5 
-d b 
. 
vanishes because of the equality of the 1st and 4th rows. 
If we take the full set of five relations and eliminate F x , 
— F 4 , F 5 , an already well-known result is obtained. 
-F F 
1 2’ x 3’ 
