412 Proceedings of Royal Society of Edinburgh. [sess. 
from which we have to deduce an equation involving no power of x 
higher than the 2nd. To do so we employ, as just stated, exactly 
the same method as was used in obtaining the “leading theorem ” 
of the preceding paper. That is to say, we form multipliers 
a b 
. a 
a 
a 
a p 
> 
a b 
effect the multiplications, and add, the result being 
. a b 
. a c 
. a d 
a b c 
X 2 + 
a b d 
x + 
a b 
a P y 
a ft S 
a p e 
0 . (liv. 2) 
This is what Sylvester’s third rule would give. His second rule is 
simply a case of the third, viz., where r= 1 ; and his first rule is 
another case, viz., where r = 0. Had he followed the order of his 
former paper, he would have called the third rule his “ leading 
theorem,” and given the others as corollaries from it. 
RICHELOT (May 1840). 
[Nota ad theoriam eliminationis pertinens. Crelle’s Journal , xxi. 
pp. 226-234.] 
Just as Jacobi (1835) brought determinants to bear on Bezout’s 
abridged method of eliminating x from two equations of the n tYl 
degree, so did his fellow-professor Richelot, in treating of the other 
method of elimination, Euler’s and Bezout’s, discovered in the same 
year (1764). Euler’s method, it will be remembered, consists in 
transforming the problem into the simpler one of eliminating a set 
of unknowns from a sufficient number of linear equations; and 
Richelot in a few lines (p. 227) points out that this may, of course, 
be done by equating to zero the determinant of the system of equa- 
tions. An investigation connected therewith occupies the main 
portion of the paper. 
Sylvester’s method (1840) is described in passing, and the 
principle at the basis of it given. We have just seen that, when 
originally made known by the author, it was merely in the form of 
