588 
Proceedings of the Poyal Society 
Je change successivement x en a', y en V, z enc', t en cl', et 
observant la regie des signes, j’ai pour seconde ligne 
(ab' - a'b)zt - (ad - a'c)yt + (ad! - a!d)yz 
-f (bd - b’c)xt - ( bd ' - b'd)xz + (cd! - dd)xy . 
Je change successivement x en a ", y en b ", z en c", t en d", et 
observant la rkgle des signes j’ai pour troisieme ligne 
[(ab' - a'b)d' - (ad - a!d)b" + (bd - b'd)a''~\t 
- [(ab' - a'b)d " - (ad! - a! d)b" + (bd' - b' d) a' 'Y 
+ [(ad - a'd)d" - (ad' - a'd)c" + (cd' - dd)a"]y 
- [(bd - b'd)d" - (bd' - b'd)c" + (cd' - dd)b"]x . 
D’ou (198) je tire 
_ - [(be' - b'd)d!' - (bd' - b'd)c" + (cd' - dd)b "] 
(ab 1 - a'b)c" - (ad - a'c)b" + (bd - b'c)a" ’ 
_ + [(ad - a'c)d" - (ad' - a'd)c" + (cd' - 
(ab' - a' b)c" - (ad -a'd)b" + (bd - b'c)a"Y 
- [(ab' - a'b)d" - (ad' - a'd)b" + (bd' - b'd)a" ] „ 
(ab' - a'b)c" - (ad - a'c)b" + (bd - b'c)a" 
Among the other examples are included (1) one in which the 
coefficients in the set of equations are given in figures ; (2) one in 
which some of the coefficients are zero ; (3) one showing the sim- 
plification possible when the value of only one unknown is wanted ; 
(4) one showing the signification of the vanishing of one of the 
“ lignes” ; (5) one showing the signification of the absence of one of 
the unknowms from the last “ligne”; and (6) one or two concerned 
with the allied problem of elimination. 
B6zout nowhere gives any reason for his rule ; it is used through- 
out as a pure rule-of-thumb : its effectiveness being manifest, he 
leaves on the reader the full burden of its arbitrariness. The unreal 
product xyzt at the very outset must have been a sore puzzle to 
students, and none the less so because of the certainty which many 
of them must have felt that a real entity underlay it. 
To throw light upon the process, let us compare the above solu- 
tion of a set of three linear equations with the following solution, 
which from one point of view may be looked upon as an improve- 
ment on the ordinary determinantal modes of solution as presented 
to modern readers. 
