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

BIGRADIENTS (HlESSE, 1858)36

369

HESSE) 0. (1858, October).
[II determinante di Sylvester ed il risultante' di Eulero.- Anneili
dii Mc...... ii. pp. 5-8; or Werlee, 475-480.]
The determinant -referred to is that to which Sylvester was
led in 1840 by his so-called " dialytic" method, and which, as we
have'already seen, Hesse himself arrived at in 1843; and the
resnltant coupled with it. is Euler's of 1748, which takes the
form of a product of difierences of the roots of the two given
equations. Both forms, as well as others, are treated of by
Cauchy in his paper of 1840, already doalt with.
The equations being
a3x 3 ~ a2ce2 + alx + a. = 0 ~ F X
b 2 e+b xi + b0 = 0j or ay (x) = 0,
Hesse multiplies Sylvester's eliminant, by 80832 - 81 2, the squared
difference-product of the 'roots - ~ of the second equation,
with the result
ao al a2 a3 s o 81 82 '83 84
*a0 a a2 Ca 8 82 83 4 8
b0b b2 3 1 2 3 1
bo b0 b1 2.... 1
*. bo b1 b2... 1
a0s0 + ajsj+.. +a,9 a cto8 + a,92 +..+ a,384 'a2 a.
aolsi + al's2 +. + a,3s4 a0s2+ ~alt3 +..+ a3s5, al a2 a3
- bos0 + bis1 + b2s2 b0s1 + bl's2 + b2s3 b2
b0s1~+b1s2 +b2s93 b082+ blts3~b~ b1 b2
b0s2 + bs3 + b2s4 b083 ~ bs4 + b2s3 bo b1 b2
where s3,= /?+3? + 3P and where, therefore, the elements in the
places 31, 32, 41, 42, 51, 52 of the right-hand member all vanish.
There is thus obtained, if we denote Sylvester's eliminant by 5,
8S so -9 b 3 a0s0~+a181 + +.a3s3 a0s + a1s2 +.. a. 4
Si 82 a.81 + als2+.. a3s4 a082+ a183 +.. a.s
M. 1). IL 2 A