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

CIRCULANTS (BERTRAND, 1850). 40

405

BERTRAND, J. (1850).
[TRAIT9~ EL~~MENTAIRE D'ALG1hBRE, avec un grand nombre
d'exercices: par Joseph Bertrand. 407 pp. Paris..]
On page 25 the student is asked to prove that if
A = be' + Wb + aa',
B = ab'+bW'+ cc',
C = ac' + ea' + bb,
then
A+B+C = (a+b+c)(a'+b'+c,'),
A2 +B2+C2 - AB-AC-BC = (a 2+b 2+C2 -ab-ac-bc)
a2+ b'2+ '2- a'b'-a'c' _b
A3+ B + C3 - 3ABC- (a +3+ V+c5- 3abc)
*(a'53 + b' + c'3 - 3a'b'c').
These results may possibly have been got by the multiplication
of two circulants of the third order, but as against this it
has to be noted that determinants are not referred to in the
book.
SPOTTISWOODE, W. (1853).
[Elementary theorems relating to determinants. Rewritten and
much enlarged by the author. Crelle's Jowrn., li. pp. 209 -'271, 328-381.]
In the section (~ xi.) which did not appear in the first edition,
and which bears the title "MNis-cellaneous -instances of determinants," the following is given (p. 375), being the fourth of the
said instances:
"Let 1, 1 2..,i, be the n + 1 roots of the equation
Xnl+1 -1= 0,
then, whatever be the values of A, Al, AD.. A21
A Al... All
A1A2... A (A +-A, +. + An)(A +iiA, +.. + inA,,)..(A +inA, +... n n
AA1.