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UNIVERSITY OF CALIFORNIA PUBLICATIONS
~~ -J~ ill-- -2.
IN
MATHEMATICS
Vol. 1, No. 15, pp. 345-358
November 8, 1923
INVOLUTORY QUARTIC TRANSFORMATIONS
IN
SPACE OF FOUR DIMENSIONS
BY
NINA ALDERTON
I
t
c
v
fi
! RARY
OCT 3 1 1961
UNIVERSITY OF CA11FOS
UNIVERSITY OF CALIFORNIA PRESS
BERKELEY, CALIFORNIA
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MATHEMATICS. Derrick N. Lehmer, Editor. Price per volume, $5.00.
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Vol. 1. 1. On Numbers which Contain no Factors of the Form p(kp + l), by
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2. Constructive Theory of the Unicursal Plane Quartic by Synthetic
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10. On a Birational Transformation Connected with a Pencil of Cubics, by
Arthur Robinson Williams. Pp. 211-222. February, 1920 15
11. Classification of Involutory Cubic Space Transformations, by Frank
Ray Morris. Pp. 223-240. February, 1920 , 25
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15. Involutory Quartic Transformations in Space of Four Dimensions, by
Nina Alderton. Pp. 354-358. November, 1923 25
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a point-to-line Transformation, by Bing Chin Wong. Pp. 371-387.
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UNIVERSITY OF CALIFORNIA PUBLICATIONS
IN
MATHEMATICS
Vol. 1, No. 15, pp. 345-358 November 8, 1923
INVOLUTORY QUARTIC TRANSFORMATIONS
IN SPACE OF FOUR DIMENSIONS
BY
NINA ALDERTON
1. Cayley in his paper "On the Rational Transformation between two Spaces" 1
gives a general discussion of the quadric transformation between two planes and the
cubo-cubic transformation between two spaces. The cubic transformation in space
was further studied by F. R. Morris 2 who gives an analytic treatment of the cases
in which the Jacobian, the sextic curve whose points go into lines by means of this
transformation, breaks up into curves of lower degree, one or more of which are
straight lines. A synthetic treatment of the general case of the cubic space trans
formation is given by D. N. Lehmer 3 in his paper "On Combinations of Involu
tions," and of six of the special cases by Elizabeth J. Easton. 4
2. A discussion of the general involutory quartic transformation in space of four
dimensions has been given by P. H. Schoute 5 in an article, "La Surface de Jacobi
d un systeme lineaire d hyperquadriques Q 3 2 dans Fespace # 4 aquatre dimensions."
The writer was not acquainted with this article when work upon the present paper
was begun and consequently began with a general consideration of the involutory
transformation by means of four hyperquadrics. Schoute makes his transformation
with respect to a pencil of hyperquadrics and defines the transform P of the point
P as being the intersection of the polar spaces of P with respect to the triple infini
tude of hyperquadrics of the pencil. This is equivalent, however, to using four
hyperquadrics, for four independent hyperquadrics determine the pencil. The
present paper gives the discussion of the general involutory quartic transformation
with respect to four hyperquadrics, as originally planned, before going on to a
consideration of the cases in which the Jacobian, now a surface, breaks up into
1 Proc. London Math. Soc., vol. 3 (1869-1873), pp. 127-180.
2 "Classification of Involutory Cubic Space Transformations," Univ. Calif. Publ. Math., vol. 1,
pp. 223-240.
3 Am. Math. Monthly, vol. 18, no. 3 (March 1911). Also Steiner s Ges. Werke, vol. 2, p. 651.
4 Ms., Master s Thesis, "Certain Special Cubo-cubic Space Transformations, " 1917, in Univ.
Calif. Library, Dept. of Mathematics.
5 Archives du Musee Teyler, serie 2, 7, 1900-01.
346 University of California Publications in Mathematics [ VOL - *
surfaces of lower degree including at least one plane. Although it has seemed
advisable to examine the subject analytically as well as synthetically to a consider
able extent, the synthetic treatment only will be presented in most cases in this
paper. .
3. NOTATION
The exponent of the symbol of a surface will be used to denote the infinitude of points on the
surface and the subscript to denote the degree of the surface; thus S 2 3 designates a quadric hyper-
surface.
4. DEFINITIONS
1. All of the 3-spaces through a line form what we shall call an axial pencil of 3-spaces. All
of the 3-spaces through a plane form a plane pencil of 3-spaces. There are o>2 3-spaces in an axial
pencil and in a plane pencil.
2. Harmonic 3-spaces of a plane pencil are four 3-spaces which are cut by any line in four
harmonic points.
3. The polar 3-space of a point P with respect to a hyperquadric is the locus of a point which
is the fourth harmonic to P and the two points in which any line through P cuts the hyperquadric.
4. The simplex of reference in 4-space is a figure bounded by five 3-spaces. The five 3-spaces
intersect two at a time in ten planes, three at a time in ten lines, and four at a time in five points
which are the vertices of the simplex.
5. PRELIMINARY THEOREMS
1. In a plane pencil (4, 1) of 3-spaces, if four planes are cut by one line In four
harmonic points they are cut by every line in four harmonic points.
Proof. Upon any line p take four harmonic points. These points together with
the plane of the plane-pencil determine four harmonic 3-spaces; for, cut across the
plane-pencil by any 3-space through p. We then have an axial pencil of planes
and we know that planes corresponding to four harmonic points of the line are
four harmonic planes which are cut by any line in four harmonic points. Since
this is true in any 3-space through p, we see that any line cuts the four 3-spaces
corresponding to four harmonic points of the line in four harmonic points. Hence
as a point P moves along p, the polar 3-spaces of P with respect to four hyper-
quadrics will form four projective plane-pencils. Similarly, as P moves over a plane,
the polar 3-spaces of P will form four projective axial pencils.
2. There are o 2 lines cutting four planes in 4-space. Proof: Call the planes
en, a2, a 3 , a 4 . If we pass a 3-space through ai, it will cut a 2 , a 3 , and a 4 in three lines.
There will be o lines in the 3-space cutting these three lines, and these lines will also
cut ai, since they lie in the same 3-space with ai. Hence in every 3-space through
0,1, there will be < lines cutting ai, a 2 , a,3 and 0.4. But there are < 3-spaces about
ai (4, 1) and consequently < 2 lines cutting ai, a 2 , a 3 and a 4 .
1023] Alderton: Involutory Quartic Transformations 347
CASE I
GENERAL TRANSFORMATION BY MEANS OF FOUR
HYPERQUADRICS
6. We may set up an involutory one-to-one correspondence between the points
of 4-space by means of four arbitrarily chosen hyperquadrics. To a point P corres
ponds a point P 1 , the intersection of the four polar 3-spaces of P with respect to the
four hyperquadrics. Since the polar 3-spaces of P must all pass through P and
since four 3-spaces can intersect, in general, in only one point, the point P also
corresponds to the point P and we have an involutory, one-to-one correspondence.
7. Thus, in general, to a point P will correspond a point P , but there are certain
points to which correspond a whole line of points. The locus of such a point is the
Jacobian. We shall show that
Theorem I. The locus of all points whose transform is a line is a surface of the
tenth degree in 4-space, J 2 w. (2, 3). Proof: Let the four hyperquadrics be
The polar 3-spaces of a point P with respect to A, B, C and D are then
i+x sB, =
z 3 C 3 + z 4 C 4 + x 6 C& =
x lDi+x zDz+x sDz+x tDt+x tDs =
Ordinarily these four 3-spaces will intersect in a point. If, however, the equations
are linearly dependent they will intersect in a line. The condition for this is that
the matrix of the coefficients be of rank three; i.e. that all of the four-rowed deter
minants of the matrix vanish. Hence from the matrix
A,
A,
A,
A,
A,
B,
Bz
B 3
B,
B,
C,
C
C 3
C 4
C,
D l
D 2
D 3
D,
D,
we shall have five four-rowed determinants equal to zero. Each of these equations
represents a quartic hypersurface. The Jacobian is the locus of points lying on all
five of these hyperquartics. The hyperquartic we get by omitting the fourth
column and the one we get by omitting the fifth column will intersect in a surface
of degree sixteen, S 2 ^. But they have in common the matrix formed of the first
three columns which represents a surface of the sixth degree through which the
other hyperquartics do not pass. Hence the Jacobian, the surface through which
all five hyperquartics pass, is a J Z IQ.
348
University of California Publications in Mathematics
[VOL. 1
8. Theorem II. The lines which are the transforms of the points of / 2 i form a
ruled hypersurface, j 3 ^.
Proof: If we denote by X = 0, 7 = 0, Z = 0, W = 0, and 7 = the hyperquartics
of the matrix of 7 which we get by omitting the first column, then the second, etc.,
we know that the equation of the Jacobian hypersurface, which is made up of the
lines which are the transforms of points of J 2 10 , may be found by equating to zero
the determinant of the partial derivatives of X, Y, Z, W and V with respect to
#1, 2, 3, 4 and # 5 ; thus
1 ~~~
x l
Y l
W\
Vi
Y 2
Z 2
W,
V,
W
X,
F 4
Z,
W,
F 4
X,
F 5
wl
F 5
=
Each element of this determinant is of degree three in x i} x 2 , x 3 , 4 and 5 and hence
the equation represents a hypersurface of degree fifteen which we shall designate
asjV
9. Thus we see that the points of 4-space go by this transformation into other
points with the exception of points on a / 2 i whose points transform into the lines
of a ruled J 3 15.
10. It seems at first thought as though there might be points in 4-space whose
polar 3-spaces meet in planes, but this is not true in general. The condition for this
would be that the matrix of the coefficients of the four polar 3-spaces of 7 be of
rank two. The forty cubic hypersurfaces obtained by setting each three-rowed
determinant equal to zero would all have to pass through the points which trans
form into planes. Taking the first three rows of the matrix,
AI A 2 A s A 4 A b \
A,
B 2
C 2
C 3 C 4
the three cubic hypersurfaces represented by columns 1, 2, 3; 1, 2, 4; and 1, 2, 5
intersect in a cubic surface and a curve. The other seven cubic hypersurfaces
represented by this matrix will not pass through the cubic surface but will pass
through the curve. Similarly, taking the matrix
C 1
B 2
C 2
D,
C 3
D*
D
we find that the ten cubic hypersurfaces whose equations are the different three-
rowed determinants of the matrix set equal to zero pass through another curve.
Two curves do not, in general, intersect in 4-space and therefore the twenty cubic
hypersurfaces so far examined have no points in common and hence the forty have
none. Consequently there are no points whose transforms are planes when the
hyperquadrics are unrelated.
1923] Alderton: Involutory Quart ic Transformations 349
11. Theorem III. If a point P moves along a line p, its corresponding point P
moves along a quartic curve in 4-space.
Proof: As the point P moves along the line p, its polar 3-spaces with respect to
the four hyperquadrics revolves about four planes forming what we shall call plane
pencils of 3-spaces. These four plane pencils are projective pencils for there is a
one-to-one correspondence between the points of p and the 3-spaces of the four plane
pencils, and to four harmonic points of the line correspond four harmonic 3-spaces
of the pencils. (2, 4, and Theorem I, 5.) The locus of the intersection of corres
ponding 3-spaces of the four projective plane pencils will be the transform of the
line p. If we cut across the plane pencils by a 3-space, we have four projective axial
pencils of planes in a 3-space. Four and only four sets of corresponding planes of
these pencils meet in points, (Reye, Geometrie der Lage, vol. 2, XII, p. 93), so there
are four points of the locus in every 3-space and hence the transform of the line p
is a quartic curve in 4-space. The single infinitude of points of the curve corresponds
to the single infinitude of points of the line p.
12. Theorem IV. If the point P moves over a 3-space, the corresponding point
P moves over a quartic hypersurface.
Proof : If the point P moves over a 3-space, the polar 3-spaces of P with respect
to the four hyperquadrics revolve about four points forming what we shall call
four points of 3-spaces. There will be a triple infinitude of points P corresponding
to the triple infinitude of points P and hence points P lie on a hyper-surface. In
order to find the degree of this hypersurface, cut across it by a line. Transforming,
the line goes into a quartic curve, as we have just seen, and the hypersurface back
into the 3-space, giving off also the J 3 i- . The hypersurface must give off the J 3 i 5
when transformed, for it contains J 2 W , since the 3-space of which it is the transform
cuts all of the lines of J 3 i 5 . The points of the hypersurface which do not transform
into lines but into points will go back into the points of the 3-space on account of
the involutory relation between the points under this transformation. But the
quartic curve cuts the 3-space in four points. Therefore the line cuts the hyper
surface in four points and it is a quartic hypersurface.
13. Theorem V. If P moves over a plane, its corresponding point P moves over
an S 6 2 .
Proof: The plane may be considered as the intersection of two 3-spaces, RI and
R<I. When we transform, these two 3-spaces go into two quartic hypersurfaces
which intersect in a surface S 2 W . But we have seen (12) that every quartic hyper
surface which is the transform of a 3-space must pass through J 2 w. Hence J 2 io is
a part of S 2 W and the remaining part is an $ 6 2 which is then the transform of the
plane.
14. Theorem VI. The multiplicity of lines of J 3 i& through points of J 2 lo and the
multiplicity of points of J 2 W on lines of J 3 i& is four.
Proof: If p is a line which is the transform of a point P of J 2 10 , then the polar
3-spaces of all points of p f will pass through P. Ordinarily the four polar 3-spaces
350 University of California Publications in Mathematics [VOL. 1
of points of p r intersect in only one point and this must be the point P. Since P is
always a common point of four polar 3-spaces of points of p , in order for p to
transform into a quartic curve it must go into four lines passing through P. But
P is any point of J 2 i Q and hence the Jacobian must be a surface of J 3 ]5 of multi
plicity four. The points of p all go into the point P except four points whose trans
forms are the four lines through P. Hence there are four points of </ 2 i on every
line of J 3 is, or the lines of J 3 i 5 cut / 2 i in four points.
15. Summary. The principal facts which have been established concerning
the general involutory quartic transformation in 4-space are: that lines go into
quartic curves, planes into surfaces of the sixth degree, and 3-spaces into quartic
hypersurf aces ; also, that the locus of points whose transforms are lines is a surface
of the tenth degree, and the lines which are the transforms of these points form a
hypersurface of the fifteenth degree.
CASE II
ONE FUNDAMENTAL HYPERQUADRIC IS A SPACE-PAIR
16. (a) In Case I the four hyperquadrics of the transformation were perfectly
general. We shall now consider the case in which one of the hyperquadrics A is a
space-pair whose 3-spaces intersect in a plane ai. It is evident that we may still
set up a one-to-one correspondence between the points of 4-space, for ordinarily to
a point P will correspond a point P , the intersection of polar 3-spaces of P with
respect to B, C and D and the 3-space conjugate with respect to the 3-spaces of A to
that determined by P and the plane ai.
17. The polar 3-space with respect to A of a point P on cu is indeterminate and
hence to such a point corresponds the line of intersection of the polar 3-spaces with
respect to B, C and D. Hence ai is a part of / 2 io and
Theorem VII. The Jacobian is a plane and an *S 2 9 when one of the fundamental
hyperquadrics is a space-pair.
18. Theorem VIII. The Jacobian hypersurface is an $ 3 3 and an $ 3 i 2 when one of
the fundamental hyperquadrics is a space-pair.
Proof: The transforms of points of a\ are lines of J 3 lb . As P moves over ai, the
polar 3-spaces of P revolve about three lines forming three projective axial pencils.
(1, 5). Now three projective axial pencils of 3-spaces intersect in a hypersurface
of the third degree, for if we cut across them by any 3-space we have three points
of planes intersecting in a cubic surface. Hence J 3 15 breaks down into a hyper
surface of the third degree and one of the twelfth degree.
19. (b) If the two 3-spaces of A coincide; i.e. if A is composed of two coincident
3-spaces we cannot set up an involutory relation between points of 4-space, for
the polar 3-space of any point will be A itself and the points of 4-space will trans
form into those of a 3-space.
1923] Alderton: Involutory Quartic Transformations 351
CASE III
TWO FUNDAMENTAL HYPERQUADRICS ARE SPACE-PAIRS
20. (a) Suppose A and B are the two space-pairs and ai and a 2 , the planes of
intersection of the pairs of 3-spaces of A and B respectively, have only a point in
common. It is evident that we may still set up an involutory one-to-one corres
pondence between the points of 4-space. The planes ai and a 2 are part of 7 2 i and
we have the theorem
Theorem IX. The Jacobian is composed of two planes and an $ 8 2 when two of
the fundamental hyperquadrics are space-pairs.
21. The planes a t and a 2 both go into cubic hypersurfaces of J 3 i 5 . Hence
Theorem X. The Jacobian hypersurface is composed of two $Ys and an iS 3 9 when
two of the fundamental hyperquadrics are space-pairs.
22. There is a plane lying on both of the cubic hypersurfaces and hence forming
part of their intersection; namely the transform of the point of intersection of the
two planes ai and a 2 . This point transforms into a plane since its polar 3-spaces
with respect to A and B are indeterminate, and its transform is then the intersection
of its polar 3-spaces with respect to C and D.
23. A line I cutting either ai or a 2 will transform into a line and a cubic curve.
Suppose it cuts ai. The line Zi is the transform of the point where the given line I
cuts ai. To get the transform of the rest of I, allow the point P to move along I.
The point P and the plane ai always determine the same 3-space and hence all
points of I have the same polar 3-space with respect to A. The polar 3-spaces of
points of I with respect to B, C and D form the three protective plane pencils
of 3-spaces and these intersect in lines of a ruled /S 2 3 , for if we cut across them by a
3-space we have three projective axial pencils whose corresponding planes intersect
in points of a cubic curve. The polar 3-space of points of I with respect to A will
cut this $ 3 2 in a twisted cubic. Hence I transforms into a line and a twisted cubic
lying in the polar 3-space with respect to A of points of /. Similarly, a line cutting
a 2 will transform into a line and a twisted cubic lying in the polar 3-space with
respect to B of points of the line.
24. A line I cutting both ai and a 2 will transform into two lines li and h and a
conic C 2 lying in the plane of intersection of the two polar 3-spaces of points of I
with respect to A and B.
25. Other lines which do not transform into quartic curves are those lying on
ai (or a 2 ). A line lying on ai (or a 2 ) and not passing through PI, the intersection
of ai and a 2 , becomes an *S 3 2 , the locus of intersections of corresponding 3-spaces
of three projective plane pencils of 3-spaces. If the line passes through this point
Pi, also, the cubic surface breaks up into a plane, the transform of PI, and an
$ 2 2 lying in the polar 3-space with respect to B (or A) of points of the line.
352 University of California Publications in Mathematics [T L - 1
26. The surface S<? which is the transform of a plane has two lines lying on it,
the transforms of the points of intersection of the plane with ai and a 2 . If these two
points coincide at PI, the >S 6 2 breaks down into a plane which is the transform of
PI and an $ 2 5 . Further degeneration of the $ 6 2 occurs when the plane intersects
ai (or a 2 ) and both ai and a 2 in lines.
27. A 3-space containing ai (or a 2 ) will transform into a ruled cubic hypersurface
which is the transform of a x (or a 2 ) and the 3-space which is the polar of points of
the 3-space with respect to A (or B).
(b) Now suppose A and B are so related that 0,1 and a 2 intersect in a line L 3 ;
i.e. that all of the 3-spaces of A and B pass through L 3 . Then we have the theorem
Theorem XL The Jacobian is composed of three planes and a S 2 7 when space-
pairs A and B have a line in common.
Proof: The points of L 3 transform into the planes of a planed hyperquadric
which are the intersections of two plane pencils of polar 3-spaces of points of L 3
with respect to C and D. Hence the plane <n (or a 2 ) will transform into the planes
of a planed hyperquadric and o 2 lines of the 3-space which is the polar of points of
ai (or a 2 ) with respect to B (or A). Hence ai and a 2 are still parts of J 2 10 and have a
line lying on them whose points transform into planes. Now ai and a 2 determine a
3-space since they have a line in common. Points of this 3-space will have a single
polar 3-space with respect to A and a single polar 3-space with respect to B and these
two 3-spaces will intersect in a plane /8 3 passing through L 3 . The polar 3-space of
points of j8 3 with respect to both A and B will be the 3-space determined by ai and
a 2 , and this 3-space will cut the polar 3-spaces with respect to C and D in lines.
Hence /3 3 is also a part of J 2 w, as are the planes ai and a 2 .
28. Theorem XII. The Jacobian hypersurface is composed of a planed hyper
quadric counted twice, three 3-spaces, and an $ 8 3 when the four 3-spaces of A and
B have a line in common.
Proof: The plane ai (or a 2 ) goes into a hyperquadric which is the transform of
L 3 and the 3-space which is the polar 3-space with respect to B (or A) of points
of ai (or a 2 ). The plane /3 3 goes into the 3-space determined by ai and a 2 . Hence
three 3-spaces and a hyperquadric counted twice will be part of the J 3 i& . It should
be noted that a line on J 2 io whose transform is counted twice is a triple line on / 2 io;
in this case the three planes <n, a 2 , and /3 3 pass through L 3 .
29. Again, lines which have a special position with respect to ai and a 2 will not
transform as usual into quartic curves. Any line cutting L 3 will transform into a
plane and a conic lying in the plane of intersection of the two polar 3-spaces of points
of the line with respect to A and B. The transform of a plane cutting L 3 or passing
through L 3 will be a degenerate 6 2 . The 3-space determined by ai and a 2 will go
into the planed hj^perquadric and the two 3-spaces which are the remainder of the
transforms of ai and a 2 . All points of the 3-space not on ai or a 2 go into points of /3 3 .
30. (c) Suppose one of the 3-spaces of B (or A) passes through the plane of
intersection of the 3-spaces of A (or B). If A and B are so related that one of the
1923] Alderton: Involutory Quartic Transformations 353
3-spaces, R 3 of B, passes through ai, then ai and a 2 determine R s and must intersect
in a line L 3 as in (b). The plane /3 3 will now coincide with ai, the intersection of the
conjugate 3-space of R 3 with respect to A and R 3 itself, since it is self-conjugate with
respect to B. Hence
Theorem XIII. The Jacobian is composed of a single plane, a double plane, and
an $ 7 2 when one of the 3-spaces of B (or A) passes through ai (or a 2 ).
31. Theorem XIV. The Jacobian hypersurface is composed of a planed hyper-
quadric counted twice, a single 3-space, a double 3-space and an $ 8 3 . The single
3-space is the conjugate with respect to A of R s and the double one is Rz itself.
32. (d) If ai and a 2 coincide in ai, the points of 4-space go into points of ai and
we no longer have a one-to-one correspondence.
CASE IV
THREE FUNDAMENTAL HYPERQUADRICS ARE SPACE-PAIRS
33. (a) Suppose A, B and C are space-pairs and their planes, ai, a 2 , and a 3
respectively, have only points in common.
Theorem XV. The Jacobian is composed of three planes and an $ 7 2 when three
of the fundamental hyperquadrics are space-pairs. The planes ai, a 2 and a 3 which
form part of the J 2 w intersect two at a time in three points PI, P 2 and P 3 . These
points transform into planes. There are two such points on each of the three planes.
34. Theorem XVI. The Jacobian hypersurface is composed of three hypercubics
and an $ 6 3 when three of the fundamental hyperquadrics are space-pairs.
The three hypercubics are transforms of the planes ai, a 2 and a 3 .
35. If ai and a 2 intersect in P 3 , a 2 and a 3 in PI, and a 3 and ai in P 2 , then a line
such as the one joining PI with any point of ai transforms into a plane and two lines.
A plane through two of the points PI, P 2 , P 3 , say PI and P 2 , will go into two planes
which are the transforms of PI and P 2 , another plane which is the transform of the
other points of the line PI P 2 , and a cubic surface lying in the 3-space conjugate to
that determined by the plane to be transformed and a 3 . The plane PI, P 2 , P 3 , goes
into six planes.
36. (b) Suppose two of the three planes ai, a 2 and a 3 , say ai and a 2 , intersect
in a line L 3 .
Theorem XVII. The Jacobian is composed of four planes and an $ 2 6 when two
of the three fundamental space-pairs A, B and C have a line in common.
The four planes are the planes ai, a 2 and a 3 and the plane j8 3 which is the inter
section of the two polar 3-spaces of points of the 3-space determined by ai and a 2
with respect to A and B.
354 University of California Publications in Mathematics [VOL. i
37. Theorem XVIII. The Jacobian hypersurface is composed of three 3-spaces,
a planed hyperquadric counted twice, a hypercubic and an S$ 3 when two of the
three fundamental space-pairs A, B, C have a line in common.
38. (c) Suppose the three fundamental space-pairs A, B, C are so related that
a 3-space R s of B passes through ai. This implies that a! and a 2 have a line L 3 in
common.
Theorem XIX. The Jacobian is composed of two planes, a double plane, and an
$ 6 2 when three of the fundamental hyperquadrics are space-pairs and a 3-space of
one of them passes through the plane of another.
This is true since two of the four planes of Theorem XVII now coincide in ai.
(Compare Theorem XIII.)
39. Theorem XX. The Jacobian hypersurface is composed of one single 3-space,
one double 3-space, a planed hyperquadric counted ibwice, a hypercubic and an
$ 5 3 when three of the fundamental hyperquadrics are space-pairs and a 3-space
of one of them passes through a plane of another.
40. (d) The three fundamental space-pairs may be so related that A and B
have a line L 3 in common and B and C have a line LI in common. In this case
Theorem XXI. The Jacobian is composed of six planes and an $ 4 2 when A and
B have a line in common and B and C have a line in common.
The six planes are ai, a 2 , as, the plane /3 3 which is the intersection of the two polar
3-spaces of points of the 3-space R f determined by ai and a 2 with respect to A and
B, the plane pi which is the intersection of the two polar 3-spaces of points of the
3-space R" determined by a 2 and a 3 with respect to B and C, and the plane /3 2 which
is the intersection of the polar 3-space of points of R with respect to A and the
polar 3-space of points of R" with respect to C.
41. The lines L 3 and LI intersect since they both lie in a 2 and this point P which
lies on each of the three planes ai, a 2 and a 3 transforms into a 3-space. It should be
noted that the planes |8i, /3 2 and /3 3 also pass through P and hence a point on <7 2 i
whose transform is a 3-space of j 3 ^ counted three times has six sheets of the surface
passing through it. The 3-space into which all points of ai except those lying on L 3
transform is the 3-space determined by a 2 and a 3 , and similarly for a 3 . This may be
shown by taking A, B and C as space-pairs through three planes of the simplex
of reference. (4, 4.) The 3-spaces determined by ai and a 2 and by a 2 and a 3 are
then two of the five faces of the simplex. Hence in this case
Theorem XXII. The Jacobian hypersurface is composed of one triple 3-space,
four double 3-spaces and an $ 4 3 when the three fundamental space-pairs A, B and
C are so related that A and B have a line in common and B and C have a line in
common. Two of the double 3-spaces are of course the 3-spaces determined by
ai and a 2 and by a 2 and a 3 .
42. Any line lying in a 2 and not passing through P transforms into three planes
since it cuts both LI and L 3 . If it passes through P it transforms into a 3-space
and a plane.
1923 ] Alderton: Involutory Quartic Transformations 355
43. (e) Let one of the 3-spaces R 3 of B pass through ai while a 2 and a 3 still have
a line LI in common.
Theorem XXIII. The Jacobian is composed of four single planes, a double plane
and an $ 4 2 when one of the 3-spaces of B passes through ai and B and C have a line
in common.
This is due to the coincidence of two of the planes of Theorem XXI, the double
plane being the plane ai.
44. Theorem XXIV. The Jacobian hypersurface is composed of two triple
3-spaces, two double 3-spaces, a single 3-space and an 4 3 when one of the 3-spaces
of B passes through ai and B and C have a line in common.
The 3-space R 3 is one of the triple 3-spaces and the plane determined by a 2 and
as is the single 3-space.
45. (f) Suppose one of the 3-spaces R s of B passes through ai and one of the
3-spaces R- of C passes through a 2 . Then
Theorem XXV. The Jacobian is composed of two single planes, two double
planes, and an $ 4 2 when one of the 3-spaces of B passes through ai and one of the
3-spaces of C passes through a 2 . The two double planes are ai and a 2 while a 3 is one
of the single planes.
46. Theorem XXVI. The Jacobian hypersurface is composed of a triple 3-space,
four double 3-spaces, and an $ 4 3 when one of the 3-spaces of B passes through ai,
and one of the 3-spaces of C passes through <x 2 .
The 3-spaces R s and R 5 are double 3-spaces.
47. (g) Suppose one of the 3-spaces, R 3 , of B passes through ai and the other,
Ri, passes through a 3 .
Theorem XXVII. The Jacobian is composed of two single planes, two double
planes and an $ 4 2 when one 3-space of B passes through ai and the other through a 3 .
The planes ai and as are double planes while a 2 is now a single plane.
48. Theorem XXVIII. The Jacobian hypersurface is composed of a triple 3-space,
four double 3-spaces, and an $ 4 3 when one 3-space of B passes through ai and the
other through a 3 . R 3 and R 4 are double 3-spaces.
49. (h) Let the planes ai, a 2 , and a 3 intersect in lines two at a time.
Theorem XXIX. The Jacobian is composed of six planes and an $ 4 2 when the
planes ai, a 2 and a 3 intersect two at a time in lines.
Call the intersection of ai and a 2 L 3 , the intersection of a 2 and a 3 LI, and the
intersection of a 3 and ai L 2 . The planes ai, a 2 and a 3 now lie in a 3-space R and
intersect in a point P so the lines LI, L 2 and L 3 are concurrent. The 3-space R will
have conjugate 3-spaces with respect to A, B and C which will intersect two at a
time in planes j8 3 , (3i and 182 which together with planes ai, a 2 and a 3 are the six
planes of the <7 2 io-
356 University of California Publications in Mathematics t v L - l
50. Theorem XXX. The Jacobian hypersurface is composed of one triple 3-space,
four double 3-spaces and an $ 4 3 when the planes ai, a 2 and a 3 intersect two at a time
in lines. This is true since the plane ai transforms into lines of /3i and the three
3-spaces which are the transforms of L 2 and L 3 , a 2 into lines of j8 2 and the three 3-
spaces which are the transforms of LI and L 3 , and a 3 into lines of j8 3 and the three
3-spaces which are the transforms of LI and L 2 .
51. The transform of points of R with respect to A, B and C will be three 3-spaces
intersecting in a line L 4 . Hence the planes #1, jS 2 and /3 3 all transform into R, the
transform of points of L 4 , which is then one of the double 3-spaces of f^. (28.)
Points of this line L 4 will transform into planes of R. Hence there are four lines,
LI, L 2 , L 3 and L 4 , whose transforms are 3-spaces of planes.
52. Any line in ai, a 2 or a 3 will transform into three planes if it does not pass
through Pj for it will cut two of the lines LI, L 2 and L 3 . If it does pass through P
it will transform into a 3-space and a plane through one of the lines LI, L 2 or L 3 .
The planes (3i, (3 2 and /3 3 all pass through L 4 and hence any line in 0i, /3 2 or /3 3 will
transform into two planes since it cuts L 4 and either LI, L 2 or L 3 .
53. (i) Suppose one of the 3-spaces R s of B in (k) passes through ai. The planes
ai, a 2 and a 3 now all lie in R 3 and hence R 3 passes through a 3 . Hence part of / 2 i
is ai taken twice, a 3 taken twice, a 2 and |8 2 or
Theorem XXXI. The Jacobian is composed of two double planes, two single
planes, and an $ 4 2 when a 3-space R s of B passes through ai and a 3 .
54. Theorem XXXII. The Jacobian hypersurface is composed of one triple
3-space, four double 3-spaces, and an $ 4 3 when a 3-space R s of B passes through
ai and a 2 .
The 3-space R 3 is one of the double 3-spaces.
55. (j) If L 2 coincides with LI, then L 3 also coincides with LI. In this case all of
the points of 4-space go into points of LI and we no longer have a one-to-one corres
pondence.
CASE V
THE FOUR FUNDAMENTAL HYPERQUADRICS ARE SPACE-PAIRS
56. (a) Let A, B, C and D be four space-pairs any two of which have only a
point in common. The planes ai, a 2 , a 3 , and a 4 of A, B, C and D respectively are
planes of the J 2 IQ . Hence
Theorem XXXIII. The Jacobian is composed of four planes and an $ 6 2 when
the four fundamental hyperquadrics are space-pairs intersecting in pairs in points.
57. Theorem XXXIV. The Jacobian hypersurface is composed of four hyper-
cubics and an iS 3 3 when the four fundamental hyperquadrics are space-pairs inter
secting in pairs in points.
1923] Alderton: Involutory Quartic Transformations 357
58. There are 2 lines cutting ai, a 2 , a 3 and a 4 . (2, 5.) Any one of these lines
will transform into four lines which are rulings of the J 3 u. It cuts J 2 io four times
so the point into which the remainder of the line transforms is the point through
which the four lines pass. Hence the four lines are concurrent. Lines cutting only
three of the planes ai, a 2 , a 3 and a 4 also transform into four lines but these lines
are not concurrent.
59. (b) Suppose the 3-spaces of A and B have a line L 3 in common.
Theorem XXXV. The Jacobian is composed of five planes and an $ 5 2 when two
of the four fundamental space-pairs have a line in common.
The planes are ai, a 2 , a 3 and a 4 and the plane /3 3 which transforms into lines of
the 3-space determined by ai and a 2 .
60. Theorem XXXVI. The Jacobian hypersurface is composed of three 3-spaces,
a planed hyperquadric counted twice, two cubic hypersurfaces and an $ 2 3 .
61. (c) Suppose a 3-space R 3 of B passes through 0,1.
Theorem XXXVII. The Jacobian is composed of three single planes, a double
plane, and an S b 2 when a 3-space of B passes through en.
The double plane is the plane ai.
62. Theorem XXXVIII. The Jacobian hypersurface is composed of a single
3-space, a double 3-space, a planed hyperquadric counted twice, and an S 2 3 , when
a 3-space of B passes through ai. The double 3-space is R s .
63. (d) Suppose A and B have a line L 3 in common and C and D a line L 2 in
common.
Theorem XXXIX. The Jacobian is composed of six planes and an $ 4 2 when A
and B have a line in common and C and D have a line in common.
The planes are ai, 0,2, a 3 , a 4 and also /3 3 and (3 2 which transform into the 3-spaces
determined by ai and a 2 and by a 3 and a 4 respectively.
64. Theorem XL. The Jacobian hypersurface is composed of two hyperquadrics
counted twice and seven 3-spaces when A and B have a line in common and C and
D have a line in common.
65. (e) If the lines L 2 and L 3 of (d) intersect in a point P, then every point in
4-space transforms into the point P and we no longer have a one-to-one corres
pondence.
66. (f) Suppose the space-pairs A, B, and C are so related that A and B have a
line L 3 in common and B and C have a line LI in common.
Theorem XLI. The Jacobian is composed of seven planes and an $ 4 2 when
A and B have a line in common and B and C have a line in common.
The planes are a x , a 2 , a 3 and o, 4 and the planes (3i, j9 2 and /3 3 of Theorem XXI.
67. Theorem XLI I. The Jacobian hypersurface is composed of a triple 3-space,
four double 3-spaces, a single 3-space, and a hypercubic.
358 University of California Publications in Mathematics [VOL. 1
68. (g) Suppose the space-pairs A and B have a line L 3 in common, B and C a
line LI in common, and C and Z) a line L 2 in common.
Theorem XLIII. The Jacobian breaks down into the ten planes of the simplex
of reference (4, 4) when the four fundamental hyperquadrics are space-pairs and
three pairs of them intersect in lines. This is more easily seen analytically. Take
as the fundamental space-pairs
A=x\-bx 2 2 =
5 = * 2 i-c* 2 3 =
The Jacobian turns out to be, in addition to the planes en, a 2 , a 3 and a 4 , the planes
< 2 \ * \ 3 s * \ and \
69. Theorem XLIV. The Jacobian hypersurface breaks down into the five faces
of the simplex of reference counted three times when the four fundamental hyper-
quadrics are space-pairs and the three pairs of them intersect in lines.
The equation of the Jacobian hypersurface turns out to be
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