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Mathematics. — “The equations by which the locus of the principal 
aves of a pencil of quadratic surfaces is determined” by 
Mr. CARDINAAL. 
1. The communication following here can be regarded as a con- 
tinuation of the preceding one included in the Proceedings of Nov. 26 
1904. It contains the analytical treatment of the problem, of which 
a geometrical treatment is given there. It ought to have been con- 
ducive to the finding of a surface of order nine; this has not been 
effected on account of the calculations becoming too extensive; 
however, the form of the final equation has been found. 
2. In the first place the equation must be found of the cone of 
axes of the concentric pencil of quadratic cones, at the same time 
director cone of the locus of the axes of the pencil of surfaces. To 
this end we regard the intersection of the two cones, determining 
the pencil of cones, with the plane at infinity P, and besides the 
isotropic circle situated in this plane; then we have the three 
equations in rectangular cartesian coordinates : 
A=a,, 2 + 4,,y° Haeg 2° + 2a,, vy + 2a,, 22 + 2a,, yz = 0, 
B=b,, 2? + by + bea 2 + 2b,, vy + 2b,, wz + 2b,, ye = 0, 
C=27 fy? -- 2? — 0: 
Out of these equations we find that of the cone of axes in the 
same way as we determine the Jacobian curve of a net of conics: 
| 
| 
| A, B, Gi | 
| A, 18 Gi | ==), 
LAS op SBE SCA 
| 
where A,, A,, A,, ete. are the derivatives of A with respect to z, y, 2. 
So the equation of the cone becomes 
OF Ant Hay Has bur Hbiny + bn 
by, v Ae bs 4) = bys 
bie + bn 4 + ban 
co) 
a 
n 
a 
> 
| 
ly Myf Gy, @ + As, Yi > Gs, 
| ; 
RS a,,% + Aya Y + Ass 
a 
x 
Without harming the generality we can always assume that the 
principal axes of one of the cones coincide with the axes of coordinates ; 
from this ensues that we may put 6,,=06,,=6,,=0, by which 
the equation of the cone is simplified. 
3. After having found the equation of this cone we can pass to 
the formation of the set of equations, by means of which is found 
the equation of the locus of the axes, 
