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

BIGRADIENTS (BALTZER, 1864, 1870, 1875) 351
(-l)nbo *A(/3)A(/ 2)... A(/3n) -= am-i a- t... 2
* am am-. *
~ am..... bn b._..... n bn-1 * *..
bn..... n?+'m,
as Hesse in 1858 had shown by direct transformation.
The bigradient form of resultant is also used (~ 11. 7) to show
that when A and B are of the same degree
resultant (A, B +XA) = resultant (A, B).
A fresh proof is given of Jacobi's theorem* that if ~ be a given
function of the (m + n —1)t" degree in x, it is possible to determine two,
functions u, v of the (n — )t", (m- l)th degrees so as to have
uA+vB = So,
where S is Sylvester's bigradient. This consists simply in taking the,
1 +n +mn equations
0 = Cm+n-1 + Cm+n-_X + Cm+n-3 X2 +
A = am + am_ix + am-_22 +
xA = amx + am_-x2 +
x2A = amxc +.
B = bn + bnlx + bn_2x2 +
xB = bnx + bnlx2 +.
and deducing
C0 m+n-1 Cm+n-2 Cm+n-3
A am Cm-1 am-2....
xA am am-................. _ = 0.
B bn bn-x bn-2.
xB. b1...!.+............ *l+n+m

* Crelle's Journ., xv. (1835), p. 108, where however m = n...