( 722 ) 
by which the equations («) pass into the following : 
Py + Qa + RA + Ak + Bak =0, 
Po? + Q,ap +R, + Auk + Bak = 0.) EN 
Py? + Q,4u + R,2 + Ayu + Bak =0, | 
4. Which condition now must exist between the coefficients of 
these equations if they are to allow of a mutual system of roots? 
The answer is that no condition is demanded for this. These equations 
are namely satisfied independent of the value of the coefficients by 
the system of roots: 
A= 05 wi 0 e arbitrary. 
The result arrived at by applying the method indicated in § 118 
of my paper. “Théorie générale de l’élimination” agrees with this. 
According to this method we should have to find for the resultant 
the quotient of two determinants successively of order 15 and of 
order 3. In the case under consideration where we have 
a0) AO ande RAN 
we always obtain, in whatener way we choose the determinants, 
as quotient a quantity which is identically zero. 
So the above-mentioned equation (8) can be nothing else but an 
identity. 
5. This result having been fixed it is no longer difficult to answer 
the question how, to obtain the equation of the demanded locus. 
To this end we must express the condition that the equations (4) are 
satisfied by a second system of roots. 
The condition in demand is, that all determinants are equal to zero 
contained in the assemblant (85) appearing in § 118 of the already 
mentioned paper. Applied to the equations (4) it gives but one 
equation, namely 
P, P P, 
| 2 8 
| Q, Fi Q, LE Q; P, 
RP aR eRaRe oh Ps a 
A, Q, A, Q, A, Q, | 
B, R, Q, B, R, Q, B, R, Q, |—=0, 
| R, R, Ri} 
| BS A, il | 
Bel Ba Be 
B, B, B, 
this being the equation of the demanded locus. It is of order nine 
agreeing to the geometrical researches of Prof, CARDINAAL. 
