ON THE ORDER OF CERTAIN SYSTEMS OF ALGEBRAICAL EQUATIONS. 
but to justify this, it must be shown first that these equations reduce themselves to 
two independent equations; and next that system is really one of the fourth order. 
We may remark in the first place, that if 
u, = arf + 4 b^i] + Qcrfrf + 4>d£rf + erf, 
is a perfect square, the coefficients will be proportional to those of • 0) 
Thus the conditions requisite in order that u may be a perfect square, are given by 
the system 
2 (ac — b 2 ) : {ad — be) : ae + 2bd — 3c 2 : be — cd : 2 (ce — d 2 ) 
= a : b : 3c : d : e, 
or these equations are equivalent to two independent equations only (this may be 
easily verified a posteriori); and by writing 3 J in the form 
e {ac — b 2 ) — 2d {ad — bc) + c {ae + 2bd — 3c 2 ) — 2b {be — cd) + a{ce — d 2 ), 
the remaining equations of the complete system (3) are immediately deduced; thus the 
latter system contains only two independent equations. (The preceding reasoning shows 
that the system (3) expresses the conditions in order that the equations 
arf + 3b£ 2 7] + 3c%rf + drf = 0, b£ 3 + 3c£ 2 ?7 + 3d£rf + erf = 0, 
may have a pair of unequal roots in common: we have already seen that the equa 
tions / = 0, J = 0 represent the conditions in order that these two equations may have 
a pair of equal roots in common.) Finally, to verify a posteriori the fact of the 
system (3) being one only of the fourth order, i^ae may, as Mr Salmon has done in 
the memoir above referred to, represent the system by the two equations 
that is, by 
a {ce — d?) — e {ac — b 2 ) = 0, e {ad — be) — 2b {ce — d 2 ) = 0, 
ad 2 — eb 2 = 0, 2bd 2 — 3bce + ade = 0. 
These equations contain the extraneous system (a = 0, b = 0) and the extraneous 
systems {b = 0, d = 0) and (d — 0, e = 0), each of which last, as Mr Salmon has remarked 
from geometrical considerations, counts double, or the system is one of the 4 th order only. 
More generally whatever be the order of u, if u contain a square factor, this square factor may easily 
, , . dhi cP u 
be shown to occur in 
at; 1 ar) 
KB5S3