304 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
where 
l rs — Ct r \ Q'rl 0's 1 ®r3 ^s4 ®>ri ®s3 
and where therefore 
l rs L 
Using then Cayley’s theorem regarding a determinant which is “gauche 
symetrique,” he concludes that A is expressible as a rational function of 
the V s. This result he might have put in the form Any even-ordered 
determinant is expressible as a Pfaffcan : and at a later date it would have 
been written 
— a il ^22 • • • a 2 m, 2m) ~ J ^13 ^14 • • • • h,2m 
^23 ^24 • • • • 1‘2 , 2m 
^2m— 1 , 2m 
The rest of the paper is occupied with the consideration of the special 
case where 
^12 = ^34 = ^56 = * ' ‘ * = l - 2 ) n - l , 2 m 
and all the other V s vanish. 
Bellavitis (1857). 
[Sposizione elementare della teorica dei determinant!. Memorie . . . 
Istituto Veneto . . . vii. pp. 67-144.] 
For the determinant which Cayley named “ gauche,” Bellavitis introduces 
the term pseudosimmetrico, and for “ gauche symetrique ” he introduces 
emisimmetrico (§41). In the matter of notation also he suggests a change, 
denoting (§ 54) the Pfaffian which is the square root of 
by 
0 
K 
d a 
CL b 
0 
d b 
a 1 
b c 
0 
d c 
a d 
b d 
c d 
0 
Pf. ( 
a, 1 
b c 
,d). 
Nothing else is worth noting save the carefulness of the exposition (§§ 51- 
54, 59). Part of Cayley’s theorem regarding a bordered skew symmetric 
determinant appears as a theorem regarding a non-coaxial primary minor 
of a skew symmetric determinant (§ 59). 
