( 721 ) 
Mathematics. “The equation of order nine representing the locus 
of the principal axes of a pencil of quadratic surfaces. By 
Mr. K. Bers. (Communicated by Prof. J. CARDINAAL). 
1. In These Proceedings of Jan. 28» 1905 appears a communication 
by Prof. CARDINAAL: “On the equations by which the locus of the 
principal axes of a pencil of quadratic surfaces is determined.” 
2. Prof. CARDINAAL deduces three non-homogeneous equations of 
order two between two variable parameters 4 and /, and tries to 
arrive at the equation of the demanded surface by elimination of 
these parameters. The result obtained by him (8) seems to be an 
equation of order 12. This is incongruent with the result arrived at 
geometrically, which made an equation of order nine to be expected. 
This incongrueney is attributed to factors, which the equation arrived 
at may contain, but these factors are not indicated. 
3. The method of elimination described in my paper ‘Théorie 
générale de Vélimination” (Verhandelingen, Vol. VI, n°. 7) gives the 
means to set aside this incongruency and to determine in reality the 
equation sought for by Prof. Carpinaan. 
To this end we can start from his equations (5) after having made 
them homogeneous with respect to the variable parameters, which 
may be done by assuming the equation (1) of the pencil of surfaces 
in the form: 
pA +AB=0. 
If now we develop the equations (5), they assume the following form : 
(a,,4,+4,,4,+0,,A,)u’"+(a,,3,+0,,B,+a,,B,+b,,A,+6,,A,+6,,4,)Aut\ 
4(b,,B,+b,,B,+b,,B,)2+A,pktA ak=0, 
(a,,A,+a,,4,+0,,A4,)u’+(a,,B,+0,,B,+a,,B,4b,,A,+b,,A,+),,A,)at 
+b,,B,+b,,B,+b,,B,)2+Auk+B,dk=0 , (a) 
(a,,4,+4,,A,+4,,4,)u°4(a,,B,+4,,B,+0,,B,4),,A,+b,,A,+0,,A,)aet+ 
+(b,,B,+-b,,B,.+b,,B,)+A,uk+Bak=—0. 
The coefficients of these equations are linear functions of the 
variable coordinates 7, y and z. To simplify we can introduce the 
following notations : 
ERO Ne ne ARL 
Pe = Taal, a. Cre S Ta Ae : 
Pr dn Ait ass + a,,A. , 
Q, = 4,,B, + 4,,B, + 4,,B, + b,A, + beds + b,,A,, 
Q, =4,,B, + 4,,B, + 4,,B, + b,,A, + bad, + bad, 
Q, Url Sn Belg ar Op dap a ba, dt brada 6,4, , 
R, = },,B, + 6,,B, + 6,,B, , 
i Dees Gielen SS ban 
R, = 6,,B, + 6,,B, + 4,,B, , 
