The theory of determinants in the historical order of development, by Sir Thomas Muir.

ORTHOGONANTS (BARDELLI, 1876)

297

Ai is equal to 0 or the square of a Pfaffian according as n is odd or
even. It is equally readily seen that any determinant containing
some rows taken from A, and the others from A can in the same way,
namely, by multiplication by A, be shown to be expressible as a
determinant of lower order whose elements are p's. Thus, taking
A = a(32y3d4 1,
and the determinant of mixed origin to be
a1 (x, a3 a4
/ Y,Y, or M, say,
~1 62 63 64
we obtain
MA = 1 P13 *
P* 21 P23 P21 P23 2
= Pl3P23 -P31 P33 P 31 *
P* 41 P43 1
When the rows taken from A, correspond with those not taken from
A, the resulting determinant is zero-axial and skew: thus
a a2 a3 a 1 p2 p13
/3 2 3 4 A p22 P23 P P23 )2.
l 7Y 73 4. 4P32 P33. P32
Sl 62 3, 84 P42 P43 1
Next taking the equations which define a column of p's, namely,
all as1 +122... +.. + aCt Pls
a21 asl + a22 as2 + + (.,, a2, -,,, P2,
a*21a1 -i- a pa22e tly t-.... c27i' an,
a7,1 as1 + -H a2 a>-1 —... + -H a c,,,a,, = pi,,
Bardelli, apparently without knowing of Jacobi's general theorem
of 1831 (Hist., i. pp. 438-439), deduces
aClrls+, 2+ a22p2,. + a,.nrns, = a;.
Pl +Ps+ * * +P2 = a + a2 + '2* * +'2
p +p~+.. -H+p = >Ha + +a... -+ a2.
Going a step further, however, he arrives at the result
pirP,,S +-P,2P,2 - ~ +P -Ps, = -^,1 al 1 +aa.2a -2.+.. - + ar,,,,,
- ' - -.~~2. qrn as