1919-20.] Note on Pfaffians with Polynomial Elements. 
83 
X. — Note on Pfaffians. with Polynomial Elements. 
By Sir Thomas Muir, F.R.S. 
(MS. received January 28, 1920. Read March 1, 1920.) 
(1) So far back as March 1855 Brioschi in effect formulated the theorem 
that any even-ordered determinant is expressible as a Pfaffian ; for example, 
in the case of the fourth order he gave the equality 
KVAt = I a, 2 ) (i,3) (i,4) 
(2,3) (2,4) 
(3, 4) 
where ( h , k) stands for 
a hhk ~ bh a k + C hdjc ~ d h C k , 
this expression being obtained by multiplying the h th column 
a h > > c h j d k 
of the given determinant by the k ih column 
b]e , — a k > d k , - C k 
of an equivalent determinant. 
(2) It is manifest, however, that we may think of ( h , k) as arising in 
a totally different manner, namely, through the addition of two deter- 
minants of the second order 
I a nbk | > I c hdk | j 
and it is thus suggested to ascertain the consequences of such a change 
in the point of view. 
(3) The Pfafftan whose every element is the sum of the corresponding 
elements of the identically vanishing Pfaffians 
| ! 1 1 %^3 1 1 1 
| | C l^2 1 1 C 1^3 1 1 *1^4 1 
| 1 e if 2 1 1 e i f 3 1 1 e i/4 1 
\ a A\ 1 «2 & 4 1 
> 1 *2^3 1 1 *2^4 1 
j 1 * 2/3 1 1 e 2 f 4 1 
1 a 3 h i \ 
1 C 3^4 1 
l e 3/4| 
is expressible as a sum of 4<-line determinants, namely, the sum 
I | + 1 af) 2 e 3 / 4 1 4- j e 1 d 2 e 3 f 4 ^ | + . . . , 
where, if the number of vanishing Pfaffians be m, the number of deter- 
minants is Jm(m — 1). 
The reason for this is that the Pfaffian 
I I a A I + 1 I + . . . | aff 3 I + 1 cff 3 I + . . . I affi | + | i + . . . 
I a 2p3 I + I *2^3 | + • • • | I + I *2^4 I + • • * 
I I "I" I *3^4 I + * ‘ ‘ 
