V. A. Manevich

A GENERALIZED CAYLEY PROBLEM AND RELATED QUESTIONS

(Presented by Academician P. S. Aleksandrov on 16 XII 1961)

In 1873 A. Cayley posed the following problem: Given two tetrahedra \(ABCD\) and \(A'B'C'D'\). Find the locus of points \(P\) and \(P'\) such that the pencils \(P(PA, PB, PC, PD, \ldots)\) and \(P'(P'A', P'B', P'C', P'D', \ldots)\) are congruent \((^1)\).

Many eminent geometers worked on the solution of this problem (see \((^2)\), p. 428). In 1909 R. Sturm showed that the solution of A. Cayley’s problem is connected with the construction of spheroidal collineations \((^3)\), but this construction was not obtained.

In the present paper two theorems are proved and two problems are solved, from which, as consequences, follow the construction of a spheroidal collineation and the solution of some other problems.

Theorem 1. Given two triples of points \(A, B, C\) and \(A', B', C'\), and two conic sections \(\alpha^2\) and \(\alpha'^2\), not lying in the planes \(ABC\) and \(A'B'C'\). Then in space there is established a point correspondence in which to each point \(M \notin \operatorname{pl} ABC\) there correspond 8 points \(M'\) such that the pencil

\[ M(A, B, C, \alpha^2, \ldots)\ \overline{\wedge}\ M'(A', B', C', \alpha'^2, \ldots). \]

The basic idea of the proof is as follows:

\[ \alpha'^2 \cap \operatorname{pl} A'B'C' \equiv (D', E'). \]

In the plane \(ABC\) a collineation \(\Gamma : D \to E\) is established such that for any pair of corresponding points \(D, E\) we have

\[ (ABCDE \ldots)\ \overline{\wedge}\ (A'B'C'D'E' \ldots). \]

To each point \(M\) of space there corresponds the curve

\[ \alpha_1^2 \equiv \text{cone }(M\alpha^2) \cap \operatorname{pl} ABC. \]

In the plane \(A'B'C'\) there exist 4 curves \(\alpha_1'^2\) passing through \(D', E'\) and satisfying the condition

\[ (A, B, C, \alpha_1^2, \ldots)\ \overline{\wedge}\ (A', B', C', \alpha_1'^2, \ldots). \]

There exist two cones of the 2nd order passing through \(\alpha'^2\) and \(\alpha_1'^2\), whose vertices \(M_1'\) and \(M_2'\) are the desired ones.

Analogous arguments make it possible to solve the following problem:

Problem 1. Find the locus of points \(M'\) and \(M\) such that the pencils \(\{M\}\) and \(\{M'\}\), projecting respectively the figures \([A, B, C, D, \alpha^2]\) and \((A', B', C', D', \alpha'^2)\), where \(ABCD\) and \(A'B'C'D'\) are plane quadrilaterals, are collinear. The desired loci are curves of the 2nd order. If \(\alpha^2 \equiv \alpha'^2\) is the absolute curve of space, we obtain Möbius’ theorem on bringing two collinear fields into perspective position and the locus of centers of perspectivity.

Theorem 2. Given a tetrahedron \(ABCD\), a plane quadrilateral \(A_1B_1C_1D_1\), and two curves of the 2nd order \(\alpha^2\) and \(\alpha_1^2\). Then there is a spatial-

an curve of the 16th order, to each point \(M\) of which there corresponds a point \(M'\) such that the bundles \(\{M\}\) and \(\{M'\}\), projecting the figures \((ABCDa^2)\) and \((A_1B_1C_1D_1a_1^2)\), are collinear.

The idea of the proof is as follows: \(a_1^2 \cap \operatorname{pl} A_1B_1C_1 \equiv (F_1,G_1)\). Let \(F\) be an arbitrary point of the plane \(ABC\). In the collineation \(A_1B_1C_1F_1 \to ABCF\), the points \(G_1\) and \(D_1\) pass respectively into \(G\) and \(D'\). We shall seek such points \(M\) for which \(MD \supset D'\), and \(MG\) and \(MF\) intersect \(a^2\). To each line \(DD'\) there corresponds a cone of the 2nd order \((Ga^2)\), intersecting \(DD'\) in two points lying on the surface \(\varphi^4\) of the 4th order. On the other hand, to the line \(DD'\) there corresponds a cone of the 2nd order \((Fa^2)\); this correspondence generates a surface \(f^4\) of the 4th order, \((\varphi^4 \cap f^4)\) is the required curve.

In the case when \(a^2 \equiv a_1^2\) is the absolute curve of space, we obtain a solution of the first problem of N. F. Chetverukhin \((^4)\) and the Pohlke—Schwarz theorem.

Problem 2. Two tetrahedra \(ABCD\), \(A'B'C'D'\) and two conic sections \(a^2\) and \(a'^2\) are given. Construct a collineation in which the given tetrahedra would be corresponding, and to some surfaces of the 2nd order passing through \(a^2\) there would correspond surfaces of the 2nd order passing through \(a'^2\).

The idea of the solution is as follows: we seek a plane \(\alpha_1\), corresponding to the plane \(\alpha \supset a^2\), such that the conic section \(a_1^2\), corresponding to \(a^2\) in the collineation
\[ (A,B,C,D,\alpha,\ldots)\to(A',B',C',D',\alpha_1\ldots), \]
intersects \(a'^2\) in two points. The cone \((D'a'^2)\cap \operatorname{pl} A'B'C' \equiv a_1'^2\), \(\alpha_1 \equiv \alpha'\cap \operatorname{pl} A'B'C'\) (\(\alpha'\) is the plane of the curve \(a'^2\)); \(\operatorname{pl} ABC\cap \alpha \equiv a\); \(DA\), \(DB\), \(DC\) intersect \(\operatorname{pl}\alpha\) respectively at the points \(A_1\), \(B_1\), \(C_1\).

Problem 2 is reduced to the following:

Find such a line \(a' \subset \operatorname{pl} A'B'C'\) for which the collineation
\[ (A_1,B_1,C_1,a_1,\ldots)\to(A',B',C',a',\ldots) \tag{1} \]
takes the conic section \(a^2\) into a conic section \(\varphi^2\) intersecting \(a_1'^2\) at points \(K,L\) such that \(KL\cap a' \equiv M \subset a_1\).

It has been shown that for each point \(M \subset a_1\) there exist 12 lines \(a'\) satisfying condition (1). Thus, there exists \(\infty^1\) collineations satisfying the requirement of Problem 2.

In the case when \(a^2 \equiv a'^2\) is the absolute curve of space, we obtain spheroidal collineations, thereby resolving the question of bringing two given tetrahedra into perspective position.

Moscow Institute
of Railway Transport Engineers

Received
8 XII 1961

CITED LITERATURE

¹ A. Cayley, Proc. London Math. Soc., 4, 396 (1873). ² Encyclopädie d. mathematischen Wissenschaften, 3, H. 3—4, 1907. ³ R. Sturm, Die Lehre von den Geometrischen Verwandtschaften, 3, Leipzig u. Berlin, 1909, p. 211. ⁴ N. F. Chetverukhin, Tr. Gruz. polytechn. inst., No. 2 (63), 5 (1959).