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

JACOBIANS (BRIOSCHI, 1854)

249

he obtains by solving n +1 sets of equations like. 7f +.... + __A =.
'a f ax+ afi a a f oax
'ax f 'ax fi ax 'af o
-af-ax, -,ax, f+'axl -
aaf -a x -af + ax efn 0
af.~ %+....n
and by using the same sets of equations after row-by-row
multiplication he obtains...ax ax ax,,, ''' a '
af -3af 'f' n ax ax, axil,, '
or, say, SR = 1.
Further, he notes that as a consequence of these two theorems
there results
afj axJ = X 7 'afi,
[73. 1.?f, 'agi -axi
which he might well have generalised by changing the i, ik of
the second factor of both sides into r, s.
In dealing with the "fundamental lemma" his order of
procedure is the reverse of Jacobi's, that is to say, he deduces
the form of 1844 from the original form of 1841. Thus, using
the latter in regard to S, he has
ax'
i=S n 'a
a=fk s'6/ ~- i
i=O
whence, because of R being the reciprocal of S, he obtains
R i=n ax
R+ R. 0;
i=0
so that on substituting
ER _ x ifor -
'=o 'xi af 'af,