404 Proceedings of the Royal Society of Edinburgh. [Sess. 
Hesse himself arrived at in 1843 ; and the resultant coupled with it is 
Euler’s of 1748, which takes the form of a product of differences 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 dealt with. 
The equations being 
a 3 x 3 + a 2 x 2 + a-^x 1 + a 0 = 0 i r <£(a?) = 0 
b 2 x 2 + V 1 + b 0 — 0 J i i f/(x) = 0 , 
Hesse multiplies Sylvester’s eliminant by y 2 -q 2 , the squared difference- 
product of the roots ft, ft of the second equation, with the result 
% a 2 % 
Sq S-, s 2 s 3 s 4 
• CIq Ciy ^2 ^3 
S 1 S 2 S 3 S 4 S 5 
&o b i b 2 ' 
. . 1 . . 
b 0 b 1 b 2 . 
. . • 1 . 
& 0 b 2 
. . . . 1 
<Mh-+aiSi 4- 
1 w ri 
a 0 *i "t ^ 1^2 4* 
Vo + Vl + b 2 s 2 
b 0 S l + b i s 2 + Vs 
V2 + V3 + V 4 
+ «s«s 
+ a 3 s 4 
4 « 3 s 4 
■j - 01/qSz 
u 2 
6, K 
4- cqs 2 4 
cl q s 2 4- a 4 s 3 4- 
Vi 4- V2 4- V3 
V2 + Vs + V 4 
Vs 4 V 4 "t Vs 
where s p = /3 1 p -{-ft 2 ", 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 S, 
S. 
But the determinant on the right is resolvable into 
a 0 + a i/h + a 2p\ + « 3 ft 3 « 0 + a lft + a 2p2 2 + %ft 3 | 
a 0 /3 1 + cqft 2 4- a 2 fi 3 4 - a 3 ft 4 a 0 /3 2 + oqft 2 4- a 2 p 3 4- a 3 ft 4 | » 
6> 0 S 1 
= b 3 
^0^0 4 4 • ■ 
. . +« 3 S 3 
<XqS 4 4 ^1^2 4 • • 
. . + ft 3 S 4 j 
1 fe 2 
A 
^0^1 4 ^1^2 "t • ■ 
, +a 3 s 4 
(XqS 2 4- oqs g 4- . . 
. . + a s s 5 
1 
1 I 
ft 
ft 
that is, into 
and therefore into 
We thus have 
1 1 
ft ft 
1 1 
ft ft 
<Kft) <Kft) 
ft<Kft) ft<£(ft) 
■ ftft) • <Kft) • 
b — ^ 2 3 ^ > (ft) < ^(ft) > 
and, consequently, if cq, a 2 , a 3 be the roots of the first given equation, 
S = & 2 V(ft-ai)(ft-a 2 )(ft-a 3 ) 
(ft - oq)(ft — a 2 )(ft — a s) » 
which is what was to be shown, the co-factor of b 2 3 a 3 2 being Euler’s product 
of differences. 
