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

BIGRADIENTS (TRUDI, 1862)

339

Proceeding from the above-noted Cayleyan mode of expressing
the 'simplified remainders,' Trudi puts forward (pp. 145-152)
another mode, each remainder now appearing as a sum of a multiple
of A and a multiple of B; or, in Sylvester's words,* as a syzygetic
function of A and B. For example, the three remainders above
given he considers in the form
a b 1 A + a b. 'B,. p..p1
p q. p q x
a b c d x A + a b d B,.
a b c a b c... p q... p q 1
~ P q ~
p q r.. p q r x
p q r s. p q r s x2
and generally he writes
pr = US.-A + V,..B,
where it is readily seen that, as regards x,
U,. is of the degree — 1, )
V,....... m n —+r-,f
UrA...... + 1,
VrB...... m + nand where, as we know,
pr..... -r.
Observation also shows that the coefficients of the highest powers
of x in U,, V,. are - bD,._,, aoDe._ respectively. By substituting
the new forms for p._-2, p,-_, p,. in Cayley's relation
D_' p-._2 - a,._2a.Q,. '.p.- + Dr_* p,. = 0,
there is obtained
(D2_ * U_2 - a,_2aQ, U._1 + D2,2 ~ U,) A
+ (D2._l-V,. - a2,.Q_, Q,' rV + D -2 ' V,.)B = 0;
* It is worth noting that it was in this connection that the word 'syzygetic'
was first used, the full title of the memoir of 1853 (which clearly had considerable
influence on Trudi) being " On a theory of the syzygetic relations of two rational
integral functions, comprising an application to the theory of Sturm's functions,
and that of the greatest algebraical common measure."