676 Proceedings of the Royal Society of Edinburgh. [Sess. 
In the “ umbral ” notation the former of these is 
[12fel2], 
and the latter 
-fel23]|>123]. 
The rather striking result to which after some difficulty we are finally 
led is that If | a n a 22 . . . a nn | be a skew determinant with univarial 
diagonal, then 
-V/ = a n -‘.[pq\ - + a n ~ i . 2 |>ga/?][a/3] 
- a”- 5 . 2 y][i a M +••••» 
where a, ft, y, . . . are any of the integers 1, 2, 3, . . . , n other than p 
and q. (xxi.) 
Thus, when n — 5 we have 
A 23 = « 3 [23] - a 2 ([21][31] + [24][34] + [25][35]) 
+ a([2314][14] + [231 5][ 1 5] + [2345][45]) 
- [2145][3145] ; 
so that if | a n a 22 . . . a 55 | be given in the form 
X 
a. 2 
CO 
3 
a 4 
- a 2 
X 
h 
-h 
X 
C 4 
-a 4 
~ C 4 
X 
~ a 5 
-h 
~ C 5 
— d 5 
we know that 
B 3 - x 3 .b 3 - x 2 (a 2 r/ 3 + b 4 c 4 + b 5 c 5 ) 
+ x / a 4 | a 2 a 3 ct 4 
+ a 5 | a 2 a 3 a F) 
+ d b \b 3 b 4 b b 
h K 
^3 ^5 
C 4 C 5 
1 
C 5 
d 5 
a., a 4 a b 
b 
d 
\ h 
a 3 a 4 a 5 
C 4 C 5 
d c 
Further, it deserves to be noted that an alternative for the coefficient of 
in (xxi.) is r p r q ; also that when n is odd there appear in the develop- 
ed 
n — 3 
ment n Pfaffians of the highest possible degree, and that when n is even 
there is only one Pfaffian of highest degree. The latter unique Pfaffian may 
be conveniently denoted by F, and the former set of n by F 1? F 2 , . . . , F w , 
wliere F,- is got by deleting the r th frame-line of 
