680 Proceedings of the Koyal Society of Edinburgh. [Sess. 
23. Using also the full set of five results in the case where the columns 
are the multipliers, we have, whatever the x’s may be, 
aq 
x 2 
x 3 
x 4 
o 
II 
rH 
• 
a 2 
a 3 
a b 
— a 2 
• 
h 
h 
- F 
x 2 
CO 
e 
l 
-h 
• 
C 5 
F 
L 3 
-a 4 
" C 4 
■ 
d 5 
— a b 
-h 
- C 5 
— d b 
• 
f 5 
and substituting for the row of xs any row of the determinant, say the 3rd, 
we have, on taking the alternative development of the bipartite, the 
interesting result 
*W F 1 - r 3 r 2- F 2 + ?VV F 3 - W F 4 + W F 5 = °. 
i.e. 
( - a 2 b 3 + a 4 c 4 + a b c b )(b 3 d b - b 4 c b + b b c 4 ) - (a 2 a 3 4 - b 4 c 4 + b b c b )(a 3 d b - a 4 c b + a b c 4 ) 
+ • • • + (a 3 a 5 + b 3 b b - c 4 d b )(a 2 c 4 - a 3 b 4 + a 4 b 3 ) = 0 . 
This result is not greatly altered by substituting an a for the zero of each 
row, because the two terms of r r which vanish separately on account of 
the zeros will in the four cases where p and q are different cancel each 
other when the zeros are replaced by <x’s. In the 5th case where q is the 
same as p there is an additional term, namely, a 2 F p , with the sign + or — 
according asp is odd or even. We thus have the theorem that In the case 
of an odd-ordered shew determinant with univarial diagonal 
W F 1 - W F 2 + W F 3 
= ( - 1 )p U 2 F ' p . (xxviii.) 
Putting p in this equal to 1, 2, ... , n, we have n equations from which 
F 1 , F 2 , . . . , F n may be eliminated, the result being a corroboration of (xxiv.) 
24. If | a n a 22 . . . a nn |, or A say, be an odd-ordered shew determinant 
with univarial diagonal , then 
(A pl , A p2 , . . . , A pn jj Fj , - F 2 , F 3 , - F 4 , . . . ) = ( - l) p-1 Fp . — . (xxx.) 
Cb 
The way to establish this which naturally occurs to one is to substitute 
for the A’s on the left the expressions obtained in § 16 (xxi.) and then per- 
form the simplifications necessary to produce 
( - iy _1 F p {a w_1 + a n ~ z H<pf + 
where Hep, 2 means the sum of the squares of all Pfaffians of degree h. This, 
though not impracticable, is lengthy and troublesome to set forth ; at the 
