1919-20.] Note on Pfaffians with Polynomial Elements. 85 
3.3.5 products which, preparatory to the use of Laplace’s theorem, can be 
recombined in 9 sets of 5, the first and the last set being equal to 
a 2 
a 3 
«5 
. a & 
e 2 
e 3 
e 4 
e 5 
e 6 
bi b 2 
h 
*4 
h 
\ 
fl f 2 
A 
A 
fb 
fb 
. a 2 
« 3 
a 4 
«5 
«6 
c 2 
c 3 
C 4 
C b 
Cq 
■ \ 
^3 
h 
h 
. d 2 
d 3 
d 4 
d 3 
d G 
• C 2 
C 3 
C 4 
C 5 
C 6 
e 2 
e 3 
e 4 
e b 
e 6 
i . d 2 
d 3 
A 
A 
d G 
■ A 
/. 
A 
fb 
fb 
respectively. Lastly, as these two and four others like them are seen to 
vanish, our expression is reduced to 
a 2 
a 3 
«4 
a b 
CIq 
• H 
a 3 
a 4 
a b 
a & 
a 2 
a 3 
«4 
a b 
a 6 
b x b 2 
A 
h 
h 
*e 
• \ 
h 
K 
h 
• 
h 
A 
A 
A 
c 2 
C 3 
C i 
c b 
C 6 
+ 
C 1 C 2 
C 3 
C 4 
c b 
C 6 
+ 
c 2 
C 3 
C 4 
c b 
«6 
• d 2 
d 3 
d 4 
A 
d 6 
d ± d 2 
d 3 
d 5 
A 
d 2 
d 3 
d 5 
d e 
• ^2 
e 3 
«4 
e b 
e 6 
• e 2 
e 3 
e 4 
e b 
e 6 
% 
e 2 
e 3 
e 4 
e b 
e 6 
• fa. 
fa 
A 
fb 
fe 
' fa 
fa 
A 
fb 
fa 
fi 
fa 
f 3 
A 
A 
A 
which manifestly equals 
I a i^2 C 3^4 e 5./6 I * 
When m is greater than 3 the same procedure suffices for proof. 
(6) Just as we had for our groundwork in § 3 Pfaffians of 2-by-4 arrays, 
and in § 5 Pfaffians of 2-by-6 arrays, so we might now proceed with like 
results to use Pfaffians of 2-by-8 arrays, and so on. When the number of 
2-by-2m arrays is less than m the result is 0 : when the number is m the 
result is the 2m-line determinant formed by combining the arrays into 
one : and when the number is greater than m, say z, the result is the sum 
of C z determinants of the (2m) th order. 
(7) It is important to note that both members of the equality may be 
transformed into an aggregate of terms of the form 
| Q'nrfon I • ! ^pdq I • | ^ rf s | • • • 
and that a comparison of the two aggregates is instructive. Taking the 
case of § 5, we observe first that the Pfaffian under consideration being of 
six lines the number of terms in its ordinary expansion is 1 . 3 . 5, and in 
the second place that these being of the type 
{ I a f ) 2 ! + I C 1^2 | + | 2 | } • { | a 8^4 I 4" | c 3^4 | + | e $J *4 | } • { I a f > Q | + I C 5^6 I + | e sf $ | }> 
the total number of terms which are products of two-line determinants is 
1 . 3 . 5 . 3 3 , i.e. 405. On the other hand, the member on the right being a 
six-line determinant, Laplace’s expansion of it as an aggregate of such 
products consists of C 6 , 2 . C 4 , 2 , i.e. 90 terms. The number of terms which 
