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

LESS COMMON SPECIAL FORMS (HORNER, 1865) 477

that if each term of | ablc2 be multiplied by the corresponding term
of I delf2 1, then similarly, by the corresponding term of
- de2f1i, I dle2f, -| dlef2 1, I d2efi, - Id2elf1,
the aggregate of the 36 terms thus obtained is the product desired.
He also, without reference to Cayley (1851), gives the companion
theorem
a b c ~* e f = I ctd bl el C2f | - ad' 1)bc2~ *Cf/I +..

al bi1 C di el fi
a2 1b c2 d2 e2 f2
and thus is able to indicate how the product of three or more
determinants can be expressed.
In his illustrative examples Horner is unfortunate, the expression
obtained for I ablc2 12 being quite incorrect, and the expression for
a0blc2 12 being much more readily got from
1 1 1 1I2 l a a2
a b c i than from 1 b b2
a bs2 (2 1 c c..,

HAMMOND, J. (1879).
[Question 6001. Educ. Times, xxxii. p. 179; lii. p. 338: solution by
T. Muir, lxv. p. 139, or Math.from Educ. Times, (2) xxii. pp. 49-50.]
The proposition of which proof is sought is that in the final
expansion of the product
(aCC11+ ca122 +.. + axXn) (a21x1 + a22x2 +.. + a2nln)
* * * (anlXlX+ ( n2X2+ ~ + CXnnXn)
the coefficients of x0Z, xln-x2,...
are all expressible as permanents, or, as Hammond rather unhappily
calls them, 'alternate determinants.' He instances the first two
coefficients, namely,

+
a I.. t21
a31
I.,

+ +
a,,ll
a11...(
~.!. ~o
* * a,/,1I e.:.... a2
(C12
C21 * ~* * (* 22
a21 a31 * * * 32... oC.2..... (;~n2.