84 
Proceedings of the Boyal Society of Edinburgh. [Sess. 
is expressible as a sum of m 2 Pfaffians, of which m are the given vanishing 
Pfaffians, and the remaining m 2 — m combine in pairs to form the 
equivalents of %(m 2 — m) determinants, thus 
l b 2 | | | I a i b 4 
c 2 d s\ 
I C 2 d 4: ! 
I c 3 rf 4 | 
+ | | c^d 2 | | e^d 3 | | | 
I a 2 b 3 I I a 2 b i I 
I « 3 & 4 I 
I &i^ 2 C 3^4 I * 
(4) The case of this theorem where m is 2 is essentially identical with 
Brioschi’s theorem. The vanishing- Pfaffians which are taken as the founda- 
tion of it are those whose elements are the determinants of the arrays 
a l 
« 2 
a 3 
a i 
c l 
C 2 
C 3 
C 4 
K 
h 
h 
’ 1^1 
d 2 
d 3 
d 4 
so that, for shortness’ sake, we may appropriately speak of 
| 1 «A 1 1 a \ b 3 \ 1 i 
as the Pfaffian of 1 a i a 2 a 3 a 4 
1 a 2^3 1 1 a '2^4 1 
II b l b 2 b 3 b \ ! 
1 tt A 1 
(5) The Pfaffian whose every element is the sum of the corresponding 
elements of the Pfaffians of the arrays 
a i a 2 ‘ ' 
• «6 
C 1 c 2 * ' 
. c 6 
e i «2 ' 
. . e 6 j 
} || 9i 92 • • 
• • 
V». • ■ 
•• \ 
5 
d\ d 2 . . 
. cf 6 
fl / 2 ' 
••/ei 
|i h x h 2 . . 
•• 'h 
is expressible as a sum of 6-line determinants, namely, the sum 
I a i b 2 C 3 d & ft I + \ a A C 3 d i9A ! + I a A e 3 f A9hh I + I C l d 2 e J$A I + . ... . 
where , if the number of vanishing Pfaffians be m, the number of deter- 
minants is Jm(m — l)(m — 2). 
Taking the case where m is 3, and using repeatedly the corresponding 
case of the preceding theorem, we obtain in the first place the given Pfaffian 
I \a x b 2 | + | c x d 2 I + | erfA 2|aA| 2\a 1 b i \ 2 | «A | 2 | a A | 
2 | a 2 b 3 | % | | 5 | a 2 b b | 2 | | 
2 I a 3 & 4 | 2 | a 3 b 5 | ^ | a 3 & 6 | 
2 | aj) 5 | 2 | af>Q | 
2 | a 5 b 6 | 
= { ! a l b 2 I + I I + I e l/ 2 I } * { I ! + I a -A e A I + I C 3 d 4?5 /« I } 
— { | a 1 b 3 1 + | c^d 3 | + J e 1 f 3 | } . { | a^b^cplQ | + | <^^b^ b f 3 | + j efL^e^f , | } 
+ { | CL-fb^ | + I C^d^ I + | e 4 / 4 I } . { I tt 2^3 C 5^ 6 I "h I a 2 b 3 e 5f6 I I C 2 d 3 e, 3 ^ 6 I } 
- { I I + I C l d 5 I + I e if5 I } * {I «2 6 3 C 4^6 bt I a 2 b 3 e J(i I + I C 2 d 3 e A I } 
+ { | «A | + | c x d & | + | ej/ 6 | } . { | 
a 2 b 3 C 4: d 5 i \^2 b 3 e 4 fh 1 + 1 C 2 d 3 e A 5 I } * 
Next, if the five multiplications here indicated hb performed, there result 
