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

DETERMINANTS IN GENERAL (CASORATI, 1858) 105
DC
(niL-m+Z -1)
Ders
and making an oversight similar to iesse's of 1853, he does not
divide both sides by n - m +1.
His third theorem,
C l C laDC DC DCDCDC
-ae 'aa,3b.,,b f7a 'a
CDC Dar,~ 2 D2 I'b,2 DCt,.m Dbm'
is more worthy of note. The proof of it depends essentially on
substituting for C in the first factor of each term of the righthand member its equivalent,
DCC DC 'C
Crl -- Cr2 + DCm
Cr a2Dc,.2 + C
in which, it is important to note, the differential-quotients are
necessarily all independent of a,,,, a,.2. The said right-hand
member can then be transformed into
lac bil Dc,.2l ~.. + bmlDC
DC / C DC DCr
Db12 D ~b22 D2.... +bm2
S 'aer aer2C3~
DC (ac DC DC
+ Db12,,1 ~b -_ + + ~b
acri 2nDc,2 mnDac,,
which, if addition be performed columnwise, becomes
aC
0~0~ -t...+-C +0 O+...+0 c,
Dc,.8
because of the fact that the theorem
DC + C lac + + Cm = C when Sr
Dc, 1 D2 aeC,,,D u - r s
holds in reference to the a's and b's as well as to the c's-a fact
which should be noted for other purposes, and which is readily
seen to be justifiable if we view C in its composite form AB and
bear in mind that the operation
a Db+
aa 4 b- + _/ -C