46 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Bezout’s derived equations (although not in Bezout’s nor Trudi’s notation) 
are 
a bx B + cx 2 + dx + i 
\ 
= 0, 
qx 2 4- ra + s 
ax + b cx 2 + dx + e 
qx 2 + rx + s 
ax 2 + bx + c dx + e 
q rx + s 
= °. 
ax 3 + bx 2 + cx + d e 
qx + r s 
j = 0 
or, in their usual form, 
aq . x 2 + ar . x + as = 
aqx B + (ar + bq)x 2 + (as + br)x + bs = 
arx B + (as + br)x 2 + (bs + cr - dq)x + (cs - eq ) = 
asx B + bsx 2 + (cs - eq)x + (ds - er) = 
0 ^ 
0 
0 
0 ^ 
We then have for the simplified remainders of the 1 st and 0 th degrees the 
determinants 
aq 
ar , 
,x + as 
ar + bq 
(as + br ) . 
, x+bs 
as + br 
(& 
s + cr - dq ) , 
& 
+ 
Ci 
05 
1 
aq 
ar 
as 
aq ar + 
bq 
as + br 
bs 
ar as + 
br 
bs + cr - dq cs - eq 
as bs 
cs — eq 
ds - er 
being only careful to note that both of these contain the irrelevant factor 
a 2 . Trudi, however, does not point out that this factor would not have 
troubled us if we had noted at the outset that for the first two derived 
equations we might have substituted 
qx 2 + rx + s = 0 
qx B + rx 2 + sx = 0, 
thus using Sylvester’s method of derivation for the first m — n equations 
and Bezout’s for the remaining n, as Rosenhain had shown in 1844.* 
The case where B is the derivate of A receives special attention (pp. 152- 
160), the object of course being to show that the quantities D r , U r , V r are 
then expressible in terms of sums of like powers of the roots of the equation 
A = 0. The reason for the possibility of this transformation lies in the fact 
that the coefficients 
h , ^1 ) ^2 ’ • • • • 1 
* Crelle’s Journ xxviii. p. 269. 
