585 
of Edinburgh, Session 1885-86. 
which we at once recognise as the simplest case of the theorem con- 
necting (as we now say) the discriminant of any quantic with the 
discriminant of the result of transforming the quantic by a linear 
substitution. 
Putting now in compact form all the identities obtained from the 
three preceding memoirs of Lagrange, we have — 
(1) (xy'z" + yz'x" + zx'y" — xz'y" — yx'z " — zy'x") 2 
= aa'a" + 2 bb'b" - ab 2 - a'b' 2 - a"b " 2 , (xvii.) 
where a — x 2 + y 2 + z 2 , a' = 
(2) $ 2 + y] 2 + £? = a' a" - b 2 , where £ = y'z" - z'y", rj — . . . . (xviii.) 
(3) y'£-zrj = bx r -a'x”. (xix.) 
(4) £A = ax + P"x' + fix", where a = a' a" - b 2 , /3" = 
and A = xy'z" + yz'x" + zx'y" - xz'y" - yx'z" - zy’x". (xvii. 2) 
(5) X = Ax , where X = ?/£" - £rj" . (xx.) 
(6) (xy'z" + yz'x" + zx'y" - xz'y " - yx'z" - zy'x") 2 
= w + + t£i" - w - - tvT. (xxi.) 
(7) PK - Q 2 = (pr - q 2 ) (M n - Xm) 2 , (xxii.) 
if p(Ms + Xcc) 2 + + Xa;) (ms + nx) + r(ms + nx) 2 
= Ps 2 + 2Q sx + Rx 2 identically. 
EEZOUT (1779). 
[Theorie Generale des Equations Algebriques, §§ 195-223, 
pp. 171-187; §§ 252-270, pp. 208-223. Paris.] 
In his extensive treatise on algebraical equations B4zout was 
hound, as a matter of course, to take up the question of elimination ; 
and, as he had dealt with the subject in a separate memoir in 1764, 
one might not unreasonably expect to find the treatise giving 
merely a reproduction of the contents of the memoir in a form 
suited to a didactic work. Such, however, is far from being the 
case. He merely mentions the necessary references to the work of 
Cramer, himself, Vandermonde, and Laplace; and then adds — 
“Mais lorsqu’il a 4te question d’appliquer ces differentes 
methodes au probleme de Telimination, envisage dans toute 
