458 Proceedings of Royal Society of Edinburgh. [july is. 
Similarly by taking the three next identities before obtained, 
which for shortness we may write in modern notation, 
«/ / 
\ab'c"\d - 
- \ab'd"\c 
+ 1 ac d \b — 
| bcd"\a — 
0, 
\ab'c"\d' - 
- \ab'd!'\d 
+ \acd"\b' - 
\bc'd"\a' = 
0, 
| ab'c"\d" - 
- \ab'd"\c" 
+ \acd"\b" - 
\bc'd"\a" = 
0, 
there is 
deduced in regard to the quantities 
a, b, 
c, d, e 
) 
a ', If 
c', d\ e 
/ 
a", b\ 
cf d", e 
the identities 
\ab'e" | 
. \dd | 
- \ab'd! 
j.|ce' | + 
\ac'd n \ . | be f | 
- \bc'd"\ 
. | ae | 
= 0, 
\ab'c"\ 
. \de" | 
— | ab'd! 
j.|ce"| + 
\add"\.\be” \ 
- \bcd"\ 
. | ae" | 
= 0, 
\ab'c" | 
. \d'e" 
| - | ab'd 
"|.|cV'| + 
\ac’d"\.\b'e"\ 
\ - \bc'd"\ 
. \ae"\ 
= 0. 
Finally these last three identities are taken, both sides of the first 
multiplied by /", both sides of the second by-/', both sides of 
the third by f and then by addition there is obtained in regard to 
the quantities 
a , b , 
a', b', 
af bf 
the identity 
\ab'e'\.\def"\ - \ab'd''\.\ce'f"\ + \acd"\.\bef"\ - \bcd"\.\ae'f"\ = 0. 
c, d, e, f 
f it t /•/ 
^ 5 ^ 5 ^ 5 a f 
// iff ft aft 
c , d , e , / 
The subject of what may appropriately be called vanishing aggre- 
gates of determinant-products is not pursued farther, the concluding 
paragraph being 
“ (223) En voila assez pour faire connoitre la route qu’on doit 
tenir, pour trouver ces sortes des theoremes. On voit qu’il y 
a une infinite d’autres combinaisons a faire, et qui donneront 
chacune de nouvelles fonctions, qui seront zero par elles-memes : 
mais cela est facile a trouver actuellement.”* 
* It is very curious to observe, in passing, that although Bezout does not 
obtain all his vanishing aggregates directly by means of the principle which 
he so carefully states at the commencement, nevertheless every one of them 
can be so obtained. He does not extend the principle beyond the case where 
only one of the original equations is repeated. If, however, we take the 
equations 
ax + by +cz + die = 0 , 
a'x + b'y + c'z + d'tv = 0 , 
