414 Proceedings of Royal Society of Edinburgh. [sess. 
and doing as Richelot here directs, we should first multiply both 
members of the first equation by x 2 - 1 and by x l ~\ then both 
members of the second by x 2 ~ x and by x ^ thus obtaining 
ax 3 + bx 2 + ex = 0 , 
ax 2 + bx + c = 0 , 
ax 3 + fix 2 + yx — 0 , 
ax 2 + (3x + y III 0 , 
and finally eliminate from these four equations a? 3 , x 2 , x 1 , by 
equating to zero the determinant of the system. 
The statement “ Ibi illius linearium,” which seems to 
contradict what we have above said in regard to the absence of 
explanation in Sylvester’s paper, is not literally true. Richelot may 
have meant by it that Sylvester’s result implied that the problem 
had been transformed as stated. 
CAUCHY (1840). 
[Memoire sur l’elimination d’une variable entre deux dquations 
algebriques. Exercises d’ analyse et de pliys. math., i. 
' pp. 385-422.] 
After the appearance of the special papers on this subject by 
Jacobi, Sylvester, and Richelot, a review of the whole matter could 
not but be a desideratum. This was supplied by Cauchy in the 
singularly clear and able memoir which we have now reached. 
After an introduction of four pages there is an account (1) of 
Newton’s method as expounded by Euler in 1748; (2) of Euler 
and Bezout’s method of 1764; (3) of Bezout’s abridged method; 
and (4) of a method * by means of the differences of the roots of 
the equations. 
Euler and Bezout’s method is shown to lead to the same deter- 
minant as Sylvester’s, and the cause is made apparent. Cauchy’s 
says (p. 389) 
11 Supposons, pour fixer les idees, que les fonctions f{x ), F(a?) 
Euler’s, although not called so. 
