559 
of Edinburgh, Session 1885-86. 
The notable point in regard to the earlier portion is, that 
Bezout throws his rule of term-format' on and his rule of signs into 
one. In the ease of finding the resultant of 
a r x + b r y + c f z = 0 (r=l, 2, 3) 
his process consists of four steps, viz, : — 
(1) a. 
(2) a b 
b a 
(3) ab c -a c b +c ab\^b a c + bc a -c b a . 
( 4 ) affa - a Y ef) z + c Y af z - b x a^ + \c 2 a^ - 
The first term of (2) is got from (1) by affixing b , and the second 
is got from the first by advancing the b one place and changing the 
sign. The first term of (3) is got from the first term of (2) by 
affixing c, the second term is got from the first by advancing c a 
place and changing the sign, and the third is got from the second 
by advancing c a place and changing the sign ; the last three 
are got from the second term of (2) in the same way as the first 
three are got from the first term of (2). 
It will thus be seen that while Leibnitz and Cramer direct us to 
find the permutations in any way whatever, and thereafter to fix 
the sign of each in accordance with a rule, Bezout requires the 
permutations to be found by a particular process, and attention 
given to the question of sign throughout all this process, so that 
when the terms have been found their signs have likewise been 
determined. 
Bezout’s contributions to the subject thus are 
(1) A combined rule of term-formation and | 
rule of signs, ) 
(2) The recurrent law of formation of the new functions, (vi.) 
(ii. 2) + (iii. 3) 
VANDERMONDE (1771). 
[Memoire sur l’elimination. Hist, de V Acad. Roy. des Sciences. 
Ann. 1772, 2 e partie (pp. 516-532).] 
This important memoir of Vandermonde and that of Laplace, 
which is dealt with immediately Afterwards, both appear in the 
