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

372 HISTORY OF THE THEORY OF DETERMINANTS

remainders," but its main subject is Cayley's relation of August
1848, and the obtaining of others like it.
Lastly, a chapter is devoted to the bigradient compound determinants which he (Zeipel) drew attention to in his paper of 1858,
and which when of the (2r - ] )th order are equal to
P't-2P and pn -2p r)
respectively. Even to-day these excite a real interest, and a
purely determinantal proof of the identities is much to be
desired.
BRUNO, F. FAA DI (1859).
[THI~ORIE GgNE'RALE DE L'ELIMINAT1ON. Par le Chevalier
Frangois Fah di Bruno.... x ~ 224 pp. Paris.]
In his section (pp. 32-40) dealing with the dialytic eliminant,
Bruno, besides reprinting li,3, R4,4, gives the full expansion of the
discriminant of the equation
ax4 + 4bX3 + 6CX2 + 4dx ~ e = 0
and of the discriminant of
ax5 + 5bX4 + 10cx3 + 10dx2 ~ 5ex + f = 0.
In the printing of the latter discriminant, however, there are at
least seven mistakes. In the next section (pp..40-46) he seeks
to improve on what we have called Cayley's "chain of squares )
by combining the last square with the first, the second from the
end with the second from the beginning, and so on. For
example, his expression for R33, that is to say, for
(a3S3 -d3p3) + (-a2brs2 +cd'2p2q) + {2(-a 2eqS2 + bd2p22r) ~...} +
is
ps2 qrs r3 _prs q2S qr2
qs2 r2s ad2 bcd C3 aed b2d bc2
S3 rs2 bd2 c2d a2(l 3 +3 - I abd I 2 +1
d3 cd2 a2C _ __28____ pqs _
+2 +1 i 2~
a 3 _ _ _ _ _ _ _ _ _ _ a C
+1 1 +a ~3 1 +a 2 ~1
P3 p2q + ab2 ~ 1jq I _ _ _
19q 2 3 b 2C
32 +1
q3 ' + 1 q4 r _ _ _ _ _ _ _ _