682 Proceedings of the Royal Society of Edinburgh. [Sess. 
( — lp -1 . pp m . F p as one of a group, and then annex a cancelling term 
( — l) p . pp m . ¥ p . Thus, taking the case where n = 7 , p = 2, m — 4, what we 
have to simplify is 
21 4 .F 4 + 23 4 .F 8 - 24 4 .F 4 + 25 4 .F 6 - 26 4 .F e + 27 4 .F 7 , 
where 21 d = y' i \2g/3yj la/3y ] and where generally the cofactor of each F 
is the sum of 10 products of two Pfaffians of the 2nd degree, the first Pfafhan 
of each product having p (i.e. 2) as an umbra, that is to say, being of the 
form [2 a/3y]. As there are only 20 (i.e. 6. 5. 4/1. 2.3) Pfaffians of this form, 
it is clear that each of them must occur thrice. For example, a little 
examination will show that [2134] cannot occur in 21 4 .F X , 23 4 .F 3 , or 
24 4 .F 4 , but occurs in each of the other terms of the given expression, its full 
cofactor being 
[5134]. F 5 - [6134]. F fi + [7131].F 7 . 
Now we know from (xxvii.) that 
[21][5671] - [51][2671] + [61][2571] - [71][2561] - 0, 
from which there follows by “ extension ” 
[21 34][567 1 34] - [5 1 34][267 1 34] + [6 1 34][257 1 34] - [7134][256134] = 0 , 
i.e. - [2134]. F 2 + [5134]. F 5 - [6134]. F 6 + [7134]. F 7 = 0, 
so that the cofactor of [2134] above found is seen to be [2134]F 2 . In like 
manner it can be shown that every other Pfaffian of the form [2a/3y] has 
[2 a/3y] . F 2 as cofactor ; and thus we conclude that 
21 4 .F, + 23 4 .F 3 - 24 4 .F 4 + 25 4 .F 5 - 26 4 .F 6 + 27 4 .F r = 
where a, /3, y are any three of the six integers 1, 3, 4, 5, 6, 7. 
26. From theorem xiv. we know that the left-hand member of (xxx.) is 
equal to the determinant got by substituting F x , — F 2 , F s , — F 4 , . . . in place 
of the p th row of A. Multiplying this determinant row-wise by A, and 
utilising theorem xxvii., we obtain the determinant which, on being divided 
by (-lp- l a , is the determinant got by substituting F 4 , — F 2 , F 3 , — F 4 , . . . 
in place of the p th row of A 2 . There is thus reached the curious theorem 
that If | a n a 22 . . . a nn |, or A say, be an odd-ordered skew determinant 
with univarial diagonal, the determinant obtained by inserting 
F 4 , — F 2 , F 3 , — F 4 , . . . in place of the p Wl row of A m is 
( - :■ • F 
P’ 
(xxxii.) 
This could also be established by proving that the product of 
F 4 , — F 2 , F 3 , — F 4 , . . .by the p* ft row of A m is ( — l) p_1 a m F r , — a theorem 
which has for its first two cases theorem xxviii. and a corollary to theorem 
xxvii. 
