PROBLEMS AND SOLUTIONS. 
593 
[485 
485] PROBLEMS AND SOLUTIONS. 593 
at observe 
not given) 
i a given 
4. A construction has been given by Aronhold (Berl. Monatsber., July, 1864) by 
which, taking any 7 given lines as double tangents of a quartic curve, the remaining 
21 double tangents can be constructed, and which, when the seven given lines are real, 
leads to a system of 28 real double tangents; but wishing to construct the figure of 
the 28 real double tangents, it occurred to me that the easier manner might be to- 
ts 0, k, to 
construct Pliicker’s curve UV — k 2 = 0, as the envelope of the conic 0U + 0 _1 V + 2k = 0, 
and then to draw the tangents of this curve: the construction is, however, practically 
one of considerable difficulty, and I have not yet accomplished it. 
milar and 
centre, the 
o see that 
[Yol. ix. p. 87.] 
2451. (Proposed by Professor Cayley.)—If A, B, G, I) are the intersections of a 
conic by a circle, then the autipoints of A, B, and the antipoints of G, D, lie on a 
confocal conic. 
the conic 
g through 
N.B. If AB, A'B' intersect at right angles in a point 0 in such wise that 
OA' = OB' = i . OA = i . OB {where i = V(— 1) as usual}, then A', B' are the antipoints of 
A, B, and conversely. 
may con- 
i of these 
d~ x V=0\ 
points in 
= 0, V = 0. 
n also be 
V = 0, at 
have thus 
g at each 
2k = 0 is 
-0-^=0, 
[Yol. ix. pp. 101—108.] 
2590. (Proposed by Professor Cayley.)—It is required to verify Professor Rummer’s 
theorem that “if a quartic surface is such that every section by a plane through a 
certain fixed point is a pair of conics, the surface is a pair of quadric surfaces (except 
only in the case where it is a quartic cone having its vertex at the fixed point).” 
Solution by the Proposer. 
! envelope 
a curve 
f showing 
[n fact, if 
e implicit 
f positive 
its of the 
3es which 
erefore of 
i of each 
iach such 
four real 
l, =28. 
The theorem may be more generally stated as follows; if a surface is such that 
every section through a certain fixed point (is or) includes a proper conic, then the 
surface (is or) includes a proper quadric surface. In order to the demonstration, I 
premise the following Lemma: If a surface is such that every section through a 
certain fixed line includes a conic, then the line meets each of these conics in the same 
two points. 
In fact, if the line meet the surface in any n points, then it is clear that each 
of the conics will meet the line in some two of these n points; and as the plane of 
the section passes continuously from any one to any other position, the two points of 
intersection with the conic cannot pass abruptly from being some two to being some 
other two of the n points, that is, they are always the same two points. 
Consider now a surface which is such that every section through a fixed point 
0 includes a conic; and consider three lines xx, yy', zz meeting in the point 0. Let 
the conics in the planes yz, zx, xy be A, B, G respectively; then since the conics 
c. VII. 75