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

88 HISTORY OF THE THEORY OF DETERMINANTS

determinante." "Complete " is used as opposed to " minor," and
"principal minor" for what nowadays we call " coaxial."
Conspicuously frequent use is made of differentiation in the
specification of minors; and it is well to note that, though
the work in this way becomes cumbrous, there is a certain
effectiveness attained by the usage. Thus, A standing for
~( ~ a11a22... a,,,), *Brioschi, like Jacobi, obtains
3A a2 A O2 A
2 + aD2tr +i c + aSMD
__rl. _a a2A a2 ~ar, *aas,
3A,. asl O2A + a.2 O2A + 0
a,1a, Oa + a,3 Da,
and then, by using the multipliers a,1,, arl...,a,,, and adding,
finds
ar,1 a2 Da2ti D,,, a,.3 Daa +,+,,- a,, a2A
a.l a,, a.2?a,13a.2 a., a.3 aarl '3 a3,,-, as 8Th,aan
which is Laplace's expansion-theorem for the case where the
minors of one set are of the second order. The remaining cases,
he says, can be established in the same way.
Again, having proved the multiplication-theorem (row-by-row)
PQ = R,
where
P =,(-~a 11a22... ann), Q = Y_(~t-b11b22... bnn)
R = 1(~-C11 C22... Cnn
he obtains by differentiation with respect to elements of P,
DP DR DRt DR
Q = b+ ORb2s + +. + bns,
Oajs ac,.1 Dc,.2 ac,,,
D2P bXT b,, D2R
cQ (2)
3a,.,3ap.- X by a bys ae,., ac PX
and by twice differentiating with respect to an element of P and
an element of Q,