420 Proceedings of Royal Society of Edinburgh. [sess. 
lines dealing with his introductory example will suffice to establish 
the fact. He says : — 
“ Let it he required to eliminate x from the equations 
X 2 +px + q = 0 , 
x 2 +p'x + q' = 0 . 
Multiply each of the proposed equations by x, and you obtain 
x s +px 2 + qx = 0 , 
x s -\-p’x 2 + fx=0. 
These two combined with the two given equations make a 
system of four equations containing three quantities to be 
eliminated, viz., x, x 2 , x s ; and they are of the first degree with 
respect to each of these quantities. We may, therefore, elimin- 
ate x, x 2 , x s by the rules for equations of the first degree. 
The result is .... ” 
He enunciates a general rule, and then takes up the analogous 
subject in Differential Equations, where successive differentiation 
takes the place of successive multiplication by x. In a postscript 
he acknowledges Sylvester’s priority which the editor had pointed 
out to him. He knew nothing of determinants. 
CAUCHY (March 8, 1841). 
[Hote sur la formation des fonctions alternees qui servent a 
resoudre le probleme de l’elimination. Comytes Rendus .... 
Paris , xii. pp. 414-426; or CEuvres Completes * P Augustin 
Cauchy , l re Ser., vi. pp. 87-99.] 
Recalling the fact that the final equation, resulting from the 
elimination of several unknowns from a set of linear equations, has 
for its first member “ une fonction alternee,” and pointing out the 
further fact that the same holds good in regard to the elimination 
of one unknown from two equations of any degree, “puisque les 
methodes de Bezout et d’Euler reduisent ce denier probleme au 
premier,” Cauchy affirms the importance of being able easily to 
write out the full expansion of such functions. There can be little 
