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 THE GIFT OF 
 
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 4553 
 
THE PLANE GEOMETRY OF THE POINT 
 
 IN POINT-SPACE OF FOUR 
 
 DIMENSIONS 
 
 SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE 
 
 OF DOCTOR OF PHILOSOPHY, IN THE FACULTY OF PURE SCIENCE, 
 
 COLUMBIA UNIVERSITY 
 
 BY 
 
 C. J. KEYSER 
 
 Reprinted from the American Journal of Mathematics, 
 Volume XXV, No. 4, pp. 301-330. 
 
 BALTIMORE 
 
 Z-^t £oti (§iittimou fptteg 
 
 The Friedenwald Company 
 
 1963 
 
>•« 
 
 Cornell University 
 Library 
 
 The original of this book is in 
 the Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924032184230 
 
THE PLANE GEOMETRY OF THE POINT 
 
 IN POINT-SPACE OF FOUR 
 
 DIMENSIONS 
 
 SUBMITTED IN PARTIAL FULFILMENT OP THE REQUIREMENTS FOR THE DEGREE 
 
 OF DOCTOR OF PHILOSOPHY, IN THE FACULTY OF PURE SCIENCE, 
 
 COLUMBIA UNIVERSITY 
 
 BY 
 
 C. J. KEYSER 
 
 Reprinted from the American Journal of Mathematics, 
 Volume XXV, No. 4, pp. 301-330. 
 
 BALTIMORE 
 Z^i Boxi i^Aiiimoxt (pre«» 
 
 The Friedenwald Company 
 1903 
 
The Plane Geometry of the Point in Point-Space of 
 
 Four Dimensions. 
 
 By C. J. Keyser. 
 
 I. — Introductory Considerations. 
 
 1. As is well known, the dimensionality (in Riemann's sense) of any given 
 space depends upon the element chosen for its construction ; and in accordance 
 with the Plucker principle of counting constants, any given space may be made 
 to assume any prescribed dimensionality Tc by merely taking for element a con- 
 figuration for whose determination within that space h independent data are 
 necessary and sufficient — a configuration, in other words, whose general analyt- 
 ical representation in the given space involves exactly k parameters. A space 
 being assumed, there are, in general, infinitely many possible choices of element 
 for which the space will have a previously assigned dimensionality. Of such 
 possible choices the great majority would be inexpedient as not leading to inter- 
 esting results. Of all elements, in case of any given space, those are, in general, 
 most practicable which present themselves in pairs of reciprocals, as in the 
 familiar examples of the point and line in the plane, the line and plane in the 
 sheaf, and the point and plane in ordinary space. 
 
 A space that is n-dimensional in points is also n-dimensional in point-spaces 
 of TO — 1 dimensions. It has 2 (n — l) dimensions both in lines and in point- 
 spaces of n — 2 dimensions ; and, in general, its dimensionality is p{n — ^ + 1) 
 if the point-space either of ^ — 1 or of « — p dimensions be taken as element. 
 Not only, however, do the two last mentioned elements furnish the same dimension- 
 ality, which is a necessary though not a sufficient condition for reciprocity, but 
 they are indeed reciprocal elements in w-fold point-space ; for the same system 
 of equations, which on proper interpretation defines one of the elements, admits a 
 second (dual) interpretation defining the other. It thus appears that by taking 
 as elements the various point-spaces of less than n dimensions for the construc- 
 39 
 
2 Ketser : The Plane Geometry of the 
 
 tion of «-fold point-space, there arise n geometries of this space ; or, if we regard 
 two reciprocal theories as but two phases of one geometry, the elements in ques- 
 tion yield — or ^^^ \- 1 geometries according as n is even or odd, the element 
 
 having "~ dimensions being, in case of ?i odd, its own reciprocal, or self-recip- 
 
 rocal. 
 
 Like considerations hold for spaces of n dimensions in other elements than 
 points. It will be convenient, however, and suflBcient to conduct this discussion 
 for space supposed /i-fold in points. 
 
 2. Of such geometries the self-reciprocal, or those arising from the use of 
 self-reciprocal elements, are of special interest as well from the artistic as from 
 the scientific point of view. The precise nature of the distinction in question 
 may be made sufficiently clear by the following considerations. In n-fold space 
 a definite configuration C, including this space itself as a special case, may, in 
 general, be regarded at will as an assemblage of points or of lines or of planes 
 and so on up to w — 1-fold point-spaces. These n assemblages, which may be 
 denoted respectively by ^q, E^, . . . . , ^„_i, the subscripts indicating the point 
 dimensionality of the elements of the corresponding assemblages, are equivalent 
 not only in the assemblage theory sense of the term but also in the logical sense 
 that they serve as so many distinct definitions or conceptions of one and the same 
 configuration. While distinct, they are of course not independent. If, for 
 example, G be supposed to represent a curve of n — 1-ple curvature, Ef^ will 
 naturally be the assemblage of its points, E^ the assemblage of its tangent lines, 
 E^ that of its osculating planes, . . . . , and the ^'s are accordingly to be thought 
 as having a one-parameter dependence, by virtue of which to each element of 
 C belonging to one E, there corresponds in general one and but one element of 
 C belonging to each other ^. Now, under a homographic transformation, the 
 nE's are converted into n other assemblages E'^, E[, . . . . , ^', _i in such a way 
 that any E and the corresponding E' are of the same kind, have, i. e., the same 
 subscript. The ^"s are connected like the E'% and in their turn serve as n dis- 
 tinct definitions of one and the same configuration G', the transformed of G . We 
 may say, then, that each of the indicated definitions or conceptions of a given 
 configuration is preserved in kind under a homographic transformation. Such isi 
 
Point in Point-Space of Four Dimensions. 3 
 
 however, in general, not the case under a dualistic transformation ; for, while the 
 latter converts the n E's into n E''s, of which each serves as a definition of the 
 transformed configuration G' of G, any E and the corresponding E' are, in gen- 
 eral, not of a kind ; if the subscript of the former be 7c, that of the latter will be 
 n — h — 1 , and these cannot be equal unless n be odd, and in this case only for 
 a single value of Jc, namely, k=n — 1:2, n being given. This case excepted, 
 no definition of G is preserved under dualistic transformation'; the point, line, 
 . . . . , conceptions of G pass over respectively into the n — 1-space, n — 2-space, 
 . . . . , conceptions of G' ; but in the case where n is odd and equal (say) to 
 2m + 1 . the assemblage E^ defining G is converted into an assemblage E^ defin- 
 ing G'; the conception is preserved in kind. Now when «=2wi + l, the ele- 
 ments of E^ are self-reciprocal elements of n-fold space, and under no other 
 circumstances are the elements of any E self-reciprocal. We arrive accordingly 
 at this conclusion : The distinction of the self-reciprocal geometries among other 
 geometries is the definitional or conceptual invariance* in case of the former, of all 
 configurations, under hoth the homographic and the dualistic transformations. 
 Because of this property of invariance, one may say that the wj-space conception 
 of configurations in space of 2m 4- 1 dimensions is of higher scientific value, as 
 being more central and penetrating, in more perfect accord with the intimate 
 nature of space itself, than are such conceptions as lose their identity under one 
 or the other of the mentioned modes of transformations — an estimate, moreover, 
 that seems to be justified by the highly artistic analytical iorm which self-recip- 
 rocal theories are, it is well known, capable of assuming. 
 
 3. Point-space of 4 dimensions is also 4-dimensional in ordinary 3-dimensional 
 spaces, or lineoids,f the point and the lineoid being reciprocal elements. It is 
 6-dimensional in lines and in planes, which are also reciprocal elements. This 
 space contains no linear self-reciprocal element and admits of no self-reciprocal 
 construction. Nevertheless there are two self-reciprocal theories of spaces (the 
 point and the lineoid) within 4-fold point space, which, besides their own intrinsic 
 
 * It is interesting that the significance of this property, -which was pointed out and made a principle 
 of procedure in the line geometry of ordinary space by Klein, Koenigs and others, seems not to have 
 been fully appreciated by Pliicker, whose method even in line theory never became quite free from the 
 relatively cumbrous point-plane conception of space. 
 
 •j-Cf. Cole : "On Rotations in Space of Four Dimensions." Amer. Math. Journ., Vol. 12. 
 
4 Keyser : The Plane Geometry of the 
 
 interest, are of the greatest importance in building up as well the point-lineoid 
 as the line-plane geometry of 4-fold space itself. Just as any lineoid of this space 
 is 3-dimensional in points and in planes and 4-dimensional in lines, so any point 
 of the same space is 3- dimensional in lineoids and in lines and 4-dimensional in 
 planes ; and just as the line geometry (the Pliicker theory) of a lineoid, regarded 
 as a space of lines, is a self-reciprocal geometry, so the plane geometry of a point, 
 regarded as a space or plenum of planes containing it, is a self- reciprocal theory. 
 With the evidently possible parallelization of these coordinate self-reciprocal 
 theories with the point-lineoid geometry of 4-space, we are not here concerned. 
 Our interest lies in the theories as such, in their relations with one another and 
 in the completely correlative roles they play particularly in the development of 
 the line-plane geometry of 4-space. The line geometry of the lineoid has been 
 often treated and is familiar enough, at least in its elements. On the other hand, 
 the plane geometry of the point (in 4-space) has not, so far as we are aware, 
 been systematically developed.* This paper undertakes to construct so much of 
 this theory as in connection with the other will be of immediate service in 
 investigating the line-plane geometry of 4-space, to which subject this article is 
 intended as a preliminary contribution. 
 
 II. — Homogeneous Coordinates of the Plane. 
 
 4. We enter here directly upon the subject proper of this paper : the plane 
 theory of the point in 4-space. The space with which we have to deal is the 
 point regarded as the assemblage of all the lineoids, planes and lines of 4-space 
 that pass through (or contain) it. As the plane is to be taken as element, the 
 point will be for us primarily a space of planes. This hypersheaf will be sup- 
 posed given once for all, and, except where the contrary is indicated, all lines, 
 planes and lineoids considered will be supposed to belong to it. 
 
 Let the assumed point be given by the four lineoids 
 
 ^1 = 0, J-2=0, -43=0, A=0, 
 where A = af + Xaf^X^ (* = 1 , 2 , 3 , 4) . 
 
 The assemblage of generating lineoids may be represented by the equation 
 
 Xi Ai -|- aig J-a + 0:3 ^3 + X4 JL4 = 0, 
 
 *In the cited article by Cole several theorems of the present paper are established. 
 
Point in Point- Space of Four Dimensions. 5 
 
 where the x^ are parameters. Each system of values of Xc defines a lineoid and 
 to each lineoid corresponds a imique system of values of the ratios cci : icj : Xg : 0:4 
 A linear equation 
 
 ^1X1 + ^2^2 + ^3^3 + ^i^i=(> = '^^i^i (1) 
 
 where the ^i are supposed given, will define a line ^< as an envelope of lineoids. 
 On the other hand if the ^, be regarded as variable and the Xi as given, the same 
 equation will define a lineoid cCj as locus of lines, a bundle. Accordingly a pair 
 of equations 
 
 will represent a plane (^j, ^j) as an envelope of lineoids, while a pair 
 
 will represent a plane (ajj, x'^ as a locus of lines, a ^t pencil. 
 
 We will employ Xi and ^^ respectively as associated homogeneous lineoid 
 and line coordinates.* The configuration of common reference will be that 
 composed of the four lineoids A^, A^, A3, A^, the six planes A-^A^, A^A^, A^A^, 
 A^A^, A^Ai, AsAi, and the four lines A^A^A^, A^A^A^, A^A^A^, A^A^A^^. The 
 coordinate lineoids will be represented in line coordinates by f ^ := , ^2=0, 
 ^3 =: , ^4 = 0, and in lineoid coordinates by Xg = a^a = 0-4 ^ , ajj = ccg = 0:4 = , 
 a;j = Xa = 0:4 = , ajj = Xg = Xg = ; the fundamental lines will be represented by 
 Xi = 0, Xg = 0, Xg = 0, X4 = , or -by ^2 = £3 = ^4 = 0, ^1 = ^3 = ^4 = 0, ^1 = ^a 
 = ^41=0, ^1=^2 = 03= 0; while the planes of reference will be given by Xi = X2=0, 
 Xi = X3 = 0, Xi = X4 = 0, Xg = X3 = 0, Xa = X4 = 0, xg = X4 = 0, or by ^g = ^4 = 0, 
 ^3 = ^4 = 0, ^a = ^3= 0, ^, = ^4=0, ^1= ^3= 0, ^1 =:^a = 0. The equation (1) 
 also signifies that the line ^i and the lineoid Xi are united in position, i. e., that 
 the line lies in the lineoid and the lineoid contains the line. 
 
 5. As already indicated, any plane whatever may appear in either of two 
 aspects : as a hcus of its lines, i. e., as a Jlat pencil, or as an envelope of its 
 generating lineoids, i. e., the lineoids containing the plane. These dual concep- 
 tions of the plane correspond precisely, in the order named, to the two Pliicker 
 
 *Such systems might legitimately have been assumed immediately by virt.ie of the projective 
 correspondence, already noticed, between the elements of the hypersheaf under investigation and the 
 elements of the lineoid, according to which the lines, planes and lineoids of the one assemblage corre- 
 spond respectively in a one-to-one way to the planes, lines and points of the other. 
 
Keyser : The Plane Geometry of the 
 
 (3) 
 
 conceptions of the line (in ordinary space), namely, axis and ray {Axe, Strahl). 
 The plane, being geometrically determined as a flat pencil by any two of its 
 lines, is determined analytically by two sets of line coordinates; while, being 
 geometrically determined as an envelope by any two of its (generating) lineoids, 
 it is determined analytically by two systems of lineoid coordinates. As explained 
 below, the first two sets when combined will furnish one system of homogeneous 
 plane coordinates and the second two sets similarly combined will yield a second 
 system. 
 
 Consider any two lines ^^ and jj^ (i = 1 , 2, 3, 4). These determine a plane 
 n, which is equally determined by any two lines of the pencil 
 
 X{-klx, + iiY,iX,) = 0, (i = l, 2,3,4), (2) 
 
 and in particular by any two of the four special lines obtained by equating 
 successively to zero the coefficients of ccj, x^, x^, x^ in (2), 
 
 (^i Vz — ii >7i) ^i + (^1 % — ^3 m) 3^3 + (^1 m — ii ni) Xi—0, 
 
 — {ki V'z — ^2 57i) a;i + (^2 m — ^3 Vz) ^i + (^2 Vi — ^4 ^z) ^i = 0, 
 
 — (^1 '73 — ^3 Vi) ^1 — (^a m — ^3 ^z) ^i + (^3 '74 — 04 'Is) ^i = 0, 
 
 — (^1 Vi — ^4 m) ^1 — (^3 >74 — ^4 ^Iz) a^a — (^3 '74 — ^3 '74) ^3 = 0- 
 
 These are the four lines of the pencil (2) that lie one in each of the fundamental 
 lineoids. The ratios of the coefficients of any two of the lines furnish the four 
 constants upon which the position of n depends The choice of two of the lines 
 would, however, be arbitrary, and we may avoid such choice, while at the same 
 time securing symmetry, by retaining for coordinates of it the entire six coeffi- 
 cients of equations (3). These, again, may be replaced by an arbitrary multiple 
 of them, since only their ratios are material. We will accordingly have for 
 coordinates of the plane regarded as a flat pencil the six quantities 
 
 9Pik = byik — hyii- (4) 
 
 On expanding the identically vanishing determinant 
 
 ^1 h h L 
 
 >7i >78 
 
 m yiz 
 
 m 
 
 574 
 
 ^3 
 
 ^4 
 
 ^73 
 
 >74 
 
 = A, 
 
 in terms of quadratic minors, we find that the six coordinates are connected 
 
Point in Point-Space of Four Dimensions, 7 
 
 by the identity 
 
 io) (p) = = 2)12 ^34 + _Pj3 p^^ + Pj, ^j3 , (5) 
 
 showing, as ought to be the case, that the 5 ratios of the 6 ^'s are equivalent to 
 but 4 independents. It can be readily shown that any six quantities ^^j, satisfying 
 the identity (5) and being such that pj„ = — p,,., serve to determine a plane (or 
 pencil), and that if ^[ and >?; be any two lines of the pencil, p^^ : p[^ — k, a constant. 
 
 5. To find a corresponding st/sfem of coordinates for the plane conceived as an 
 envelope ofUneoids, suppose it given by two lineoids Xi and y^ Of the lineoids 
 of the pencil 
 
 2 {7^x, £, + fiyi ^,) = , (i = 1, 2, 3, 4) (6) 
 
 of generators of 7t, the following 
 
 ' ?12 ^2 + qi3 Es + 9'U 04 = , 
 qzi 01 + ?23 03 + ?24 04 = , 
 qsi 01 + g'32 08 + ^34 04 = . 
 $'4101 + ^42 03 + ^43 03 = 0, 
 
 ^here <5qiu = x^y, - x,y, (8) 
 
 are the four generators of which each contains one and but one of the fundamental 
 lines. 
 
 The six coefficients q^c are connected by the quadratic identity 
 
 ia{q) = Q = ^,3 qsi + qis q^ + qu fe - (9) 
 
 and for reasons precisely analogous to those given for the ^'s, any six quantities 
 g-j,, satisfying the identity (9) and being such that g',i=: — qid, suffice to determine 
 a plane uniquely, and may be taken for homogeneous coordinates of the same 
 regarded as enveloped by lineoids. 
 
 7. Inasmuch as the configurations with which we shall be concerned are, 
 most of them, to be conceived as assemblages of planes, and since the latter are 
 self- reciprocal elements (cf. §1), there is, in general, no advantage, but often 
 rather a disadvantage, in observing the distinction between the two aspects of 
 the plane, as flat pencil of lines and as envelope of lineoids, in which alone the 
 
8 Keyser : The Plane Geometry of the 
 
 diiFerence between the two systems* piu and g^^ originates. And in fact it is easy 
 to show that these systems, while they differ in the sense indicated, are as coor- 
 dinates, as data fixing the position of a plane, identical function for function, a 
 proportionality factor being of course excepted. To effect this identification it is 
 suflBcient to find the condition that pi^ and g^^j shall determine one and the same 
 plane. Suppose the plane determined by pi,, to be that represented by equations 
 (3). If qiu give the same plane, then the latter must lie in each of the lineoids 
 Xi, yi, which requires 
 
 PizXi+PiiXs+Pu^i = ^A (10) 
 
 Vi%y%-\- PizVz + Puyi = 0,1 
 
 Puyz+pisys+Puyi 
 Pnyi+Pzsys+Payi 
 
 '4 = 0,3 
 
 P31^l+P3Z^Z+P3i^i = 0A (12) 
 
 Pziyi-^-PwVi +i'34 2^4 = 0,i 
 
 and a third pair, not needed. 
 From (10) we derive 
 
 P\i ■ q.3i = Pi3 ■ qa = Pii •• ?23. 
 
 from (11) Pi2-q3i=Pz3-qii=Pi2-qi3> 
 
 and from ( 1 2) p^g : q^^ = p^^ : qu = Psi ■ qn • 
 
 On combining we have 
 
 Pii ■ qu = Pu ■ qa = Pii ■ ?23 = P3i ■ ?i2 = Piz ■ qn = Pzs : ?i4 . (l 3) f 
 
 which shows that if the quantities pi,,, taken in any order, as 12, 13, 14, 34, 42, 
 23, determine a plane as a flat pencil, the same quantities, taken in the equally 
 general corresponding order 34, 42, 23, 12, 13, 14, determine the same plane as 
 an envelope of lineoids. 
 
 8. In general, two planes have no intersection, i. e., no common line. The 
 condition that they shall have a line in common, or, what is tantamount, shall 
 
 * There are, of course, many equivalent systems of coordinates for the plane in the hypersheaf as 
 there are for the line in ordinary space. Indeed, in the appendix to his very first paper on the latter 
 subject Pliicker presents no less than eight distinct systems. Cf. Pliicker, " On a New Geometry of 
 Space." Phil. Trans, of the R. Soc. of London, Vol. 156. Also Wiss. Abh. 
 
 fOf. Pliicker : " Neue Geometric des Raumes," p. 4. 
 
Point in Point-Space of Four Dimensions. 9 
 
 lie in a same lineoid, will, like the corresponding condition* for the intersection 
 of two lines in the Plucker line theory, assume the following four forms, if the 
 distinction of locus and envelope be observed. According as the two planes 
 7t and 7t! are regarded (a) both as flat pencils, {h) n aa a pencil and n' as an 
 envelope, (c) n as an envelope and n' as a pencil, {d) both as envelopes, the forms 
 in question will be 
 
 PnPL + P13P42 + ^14^33 + PuP^ + P^p[i + PisP'u = , (a) 
 
 2pifc ■q'ik = 0, 2^;, . y,., = , (6), (c) 
 
 qi2 qL + qn q'a + qu ■ qk + qm ■ q'12 + q^ • q'u + ?23 • q'u = . [d) 
 
 By virtue, however, of (13) we may write for coordinates of the plane 
 simply six quantities r^i, such that r,.^= — r^^ and that a{r)=zO, and then 
 disregard the distinction of locus and envelope. The condition that two planes 
 
 { r,j V2 + risrg + r^^Vi = 0, 
 
 1 —^nVi + r^vs + r^^v, = 0, 
 
 ( Kz vz + r(3 vs + r'li ?i = , 
 ( —^12^1 + r'i^fz + 'r'u^i = 0, 
 shall lie in a same lineoid, or intersect in a line, is, then. 
 
 
 
 »*13 n* 
 
 '^33 
 
 r^s 
 
 '12 
 
 ^•is 
 
 
 ''23 *"24 
 
 = 0, 
 
 from which by help of the conditions, r.^ = — r^ and o (r) = 0, we readily find 
 
 »-12 ^34 + »-13 »-42 + ''14 r'z3 + ^•34 ^12 + ^42 ''la + »*23 ^'4 = . 
 
 Writing the left member of this polar form 
 
 we have the fundamental-proposition : The necessary and sufficient condition that 
 
 * Cf ■ Cayley : " On the Six Coordinates of a Line," Collected Papers, Vol. VII. Klein : "Einleitung 
 in die hohere Geometrie," Vol. 1, p. 168. 
 
 fCf. Pasch: "Zur Theorie der linearen Complexe," Crelle, Vol. 75, p. 11. Also, Koenigs : "La 
 geometrie regie," Annales de La Faculte des Sciences de Toulouse, Vol. Ill, p. 9. 
 
 40 
 
10 Keyser : The Plane Geometry of the 
 
 two planes j-j^ and Ta shall have a line in common, or lie in a same lineoid, is that 
 the polar form, w (r, r') with respect to these planes, shall vanish. 
 
 9. The coordinates r^^^ admit of generalization. We know from the theory* 
 of forms that the new variables Vi in the transformation 
 
 where the modulus is not zero, are connected by a homogeneous quadratic 
 identity £1 (r) =: 0, where £L {v) is the transformed of u{r). Moreover, the polar 
 form CO (r, ?•') of o (?■) is converted by the same transformation into the polar 
 form £i [v, v') of II [v) . It is well known that o (r) regarded as a quadratic form 
 has a non-vanishing discriminant and that it is possible to find a linear trans- 
 formation which will convert this form into any quadratic form £l {v) whose 
 determinant does not vanish. Accordingly we may employ for homogeneous 
 plane coordinates any six variables v^ connected by the quadratic relation 
 £L{y)^^0, where £l{v) has a non-zero discriminant. Hence the condition that 
 the planes Vi and r'i shall intersect in a line, or lie in one lineoid, is that the 
 polar form £i {v, v') shall vanish. 
 
 10. It is now perfectly clear, it was indeed a priori evident, that the theory 
 here in process of construction and the line theory of ordinary space, while they 
 are geometrically distinct, disparate in fact, may be made to assume one and the 
 same analytical aspect. Accordingly three courses lie open. The theories being 
 coordinate in rank and being correlative auxiliary instruments for the construc- 
 tion of the line-plane geometry of 4-space, the ideal would seem to be to develop 
 them as such, side by side. On the other hand, as the line theory already exists 
 in a score of presentations, one might be content to derive the plane theory from 
 it by translation, by merely replacing the old system of ideas by the new. 
 Again, as neither doctrine can claim logical priority as against the other, it 
 appears to be desirable to present the new doctrine 07ice on its own account, the 
 old having been often so presented, and not as a secondary discipline derived 
 from another. The first course is rejected as being too long ; the second is 
 scarcely shorter and offers, besides, a false perspective. The third recommends 
 itself as a compromise, and accordingly we shall continue, as we have begun, to 
 construct the theory in question, as self-justified, upon its own foundations, — a 
 course which will allow occasional pauses to note correlative propositions in the 
 corresponding line geometry. 
 
 * Of. Klein : Op. cit., pp. 190, 191. 
 
Point in Point- Space of Four Dimensions. 11 
 
 III. — Systems of Planes. — The Linear Complex of Planes. 
 
 11. We pass to the study of systems of planes. Of such systems there are 
 five sorts as follows : (a) the Aparameter system, which is composed of all the 
 planes of the point, or hypersheaf under investigation, and which may be 
 regarded as the locus of a single plane it of the system, n being subject to no 
 condition ; (h) the 3-parameter system, or complex, which is defined by imposing 
 one condition upon the 4-parameter system ; (c) the 2-para.meter system, or con- 
 gruence, the assemblage defined by a pair of conditions upon the planes of the 
 hypersheaf; (d) the 1-parameter system, or configuration or plane series, an 
 assemblage defined by a 3-fold condition ; (e) the zero-parameter system, always 
 a finite assemblage, defined by a set oi four conditions upon the parameters of 
 system (a). A plane will be said to have 4, 3, 2, 1, or degrees of freedom or 
 indetermination according as it is regarded as belonging to a 4-, 3-, 2-, 1-, or 
 0-parameter system. 
 
 12. Two or more planes having a line in common may be called colUnear ; 
 two or more planes contained in a same lineoid may be called collineoidal. An 
 assemblage of planes that are all of them at once collinear and collineoidal is 
 an ordinary axal pencil of planes. We will, however, call such a pencil a fiat 
 axal pencil, reserving the name axal pencil for the totality of planes containing 
 a line. Denote by v/ and v'/ any two collinear, or collineoidal, planes and 
 
 consider the expression 
 
 Vi = 7i,iVi + "k^Vi . (the /I's arbitrary) 
 
 We have by hypothesis 
 
 cD.{v') =0, 
 
 \£i{v") =0, 
 
 lil{v',v") = 0. 
 
 Also, by identity 
 
 £i{v) = a (;ii v' + A.3 v") = a {v') ;i? + n {v") ;ii + 2fi (y , v") -k^ \ , 
 
 whence H (v) =: , 
 
 i e. the quantities v, are the coordinates of a plane for all values of Xj and \. 
 If 7t- be any plane whatever having a lineoid in common with each of the planes 
 
 v[ and v'i, 
 
 n(v', 7t) = 0, n(r", 7t) = 0, 
 
 and therefore 
 
 fl (r, 7t) = n iy', Tt)-k^-\-£i {v'\ 7t) ^2 = , 
 
1 2 Keyser : The Plane Geometry of the 
 
 i. e.,the planes Vi are coUineoidal with 7t< and they consequently contain the 
 common line of r,' and vi'. It is likewise plain that the planes r< are all contained 
 in the common lineoid of vl and v'i'. The planes Vi are therefore all found in the 
 flat axal pencil {v[,v'i'). Is the converse true? Is every plane of the pencil 
 one of the planes Vil Suppose v'J' to be an arbitrarily chosen plane of the 
 pencil and let ni be any plane coUineoidal with vi" but not with any other plane 
 of the pencil. Let v["' be that one of the planes Vi for which 
 
 n (v, n') = a {v', 7t') ^, + a (v", n') ?.^ = o. 
 
 The planes vi" and vi'" are identical. We see, therefore, that the planes Vt con- 
 stitute the flat axal pencil {vi, vi'). Accordingly, any two coUineoidal planes 
 vi and v'i' determine a flat axal pencil and the coordinates of the planes of the 2>encil 
 
 are of the form 
 
 Vi = Aj vi + ;i2 v'i'. 
 
 This last is identical with the form giving in ordinary space the line coordinates 
 of the lines of a flat pencil determined by two concurrent lines. 
 
 13. As a plane of a flat axal pencil has one degree of freedom and that of 
 a complex three degrees, a plane that belongs to both will have zero degrees 
 of freedom, being subject to four conditions. The number of planes of a given 
 complex that belong to an arbitrary flat axal pencil is, therefore, finitft. This 
 number will be called the degree of the given complex. 
 
 The assemblage of planes having a line in common — the axal pencil proper — 
 and the assemblage of planes contained in a lineoid — the ordinary bundle of 
 planes — are the analogues respectively of the sheaf and the plane of lines in 
 ordinary space. The two assemblages in question, i. e., the axal pencil and the 
 bundle, being each bi-dimensional, may with propriety receive a common name. 
 Following a suggestion of Koenigs, we will adopt for such common designation 
 the term hyperpencil of planes. 
 
 14. We may now prove that a hyperpencil of planes is completely determined 
 by any three planes vi, v'i, v'i" , such that each is coUineoidal [or collinear) with each 
 of the other two, and that all and only the planes of the hyperpencil are given hy 
 coordinates of the form 
 
 Vi = \v'^^-7,^v'i' -{--K^v'i". 
 
Point in Point-Space of Four Dimensions. 13 
 
 Two cases may arise. The three given planes may determine three distinct lines 
 and one lineoid containing them or three distinct lineoids and one line contained 
 in them. In the former case the planes are the faces of an ordinary trieder and 
 the hyperpencil will be a bundle. We will conduct the argument for the second 
 case, for which the hyperpencil will be an axal pencil, the proof being identical 
 in form for both cases. By hypothesis, we have 
 
 n {v') = 0, n {v") = , n {v<") = o , 
 
 n {v", v"') = , a [v'", v') = o, n {v', v") = o , 
 
 from which it follows that 
 
 n (v) = n (;li y + 1^ v" + ^s v'") 
 
 = D. {v') 2?, + a {v") ^l + D. {v'") %\ + 2n {v'\ T'"')A ^3 
 
 + 2D {v'", v') 2,3 2,1 + 211 {v', v") Ai ;i2 = . 
 
 Hence for every system of values of the ratios 2,i: 2,^:2,3, the six quantities 
 Vi determine a plane. Now let 7ti be an arbitrary plane collineoidal (or collinear) 
 with each of the given planes vi, vi', vi". Then 
 
 £l{7t) = 0, n(r', 7t) = 0, n{v",n) = 0, fl(v'", 7t) = 0. 
 
 Consequently 
 
 £l{v, n) = Q. {2ii v< + \ v" + %3 v'") = fl {v', 7t) ;ii + n {v'<, 7t) ;ia + a (^"', n) %3 = 0, 
 
 i. e., every Vi is collineoidal with every Tt; and hence contains the line {v',v", v'"). 
 Conversely, every plane vi" containing this line is one of the planes Vi. For let 
 ni and 7t-' be any two planes each collineoidal with but not belonging to the asal 
 pencil. Only one plane v"' is collineoidal with each of the planes n- and n" . 
 We prove that one of the planes Vi is so collineoidal, whence it follows that this 
 Vi is identical with vi"'. The proof consists in showing that 
 
 fl(^,7t') = 0, Sl{v,n") = 0. 
 Now 
 
 £l{y, «') =£i{v', 7t')2.i + Cl{v", 7t>)2., + £i{v<", 7t')^3. 
 
 a iy, 7t") = n {v', 7t") Ai + n {v", n") t^^ + d. (v'", «") ^s> 
 
 which may both be made to vanish by a proper choice of values of the ratios 
 of the ;i's. Hence the planes v, constitute the planes of the axal pencil deter- 
 mined by the collinear planes vi, vi', vi". In like manner, if vi, v'/, vi" be the 
 
14 Kbyseb: The Plane Geometry of the 
 
 faces of an ordinary trieder, they determine a bundle whose planes are given by 
 the formula 
 
 15. A plane that is required to belong at once to a complex and a hyper- 
 pencil, being subject to three conditions, has one degree of freedom. The locus 
 of such a plane is, therefore, a "configuration." We will name it a cone Cj, or 
 a cone G^ of the complex according as the hyperpencil is an axal pencil or a 
 bundle. Ci and G^ correspond precisely and respectively to the curve and the 
 cone of a complex of lines in ordinary space. Just as the curve has all its 
 lines in a bi-dimensional point manifold, the plane, so Cj has its planes in a 
 bi-dimensional lineoid manifold, the line ; and just as the (line) cone has all its 
 lines joined by a point while their points require for their representation a 3-fold 
 manifold of points, oi-dinary space (a lineoid), so Cg has all its planes in a lineoid 
 while their (generating) lineoids require for their construction a 3-fold manifold 
 of lineoids, a point; and so on. Every line has its G^ and every lineoid its Cj 
 of any given complex. A flat axal pencil will be said to belong to a given 
 hyperpencil when the latter contains the planes of the former. The degi-ee of 
 a Cj or a G^ will signify the number of planes common to the cone and an 
 arbitrary flat axal pencil belonging to the hyperpencil to which the cone belongs. 
 It should be noted that as the notions, locus and envelope, of the Pliicker geometry 
 correspond respectively to envelope and locvs in the present theory, so also the 
 notions of order and class in the former doctrine correspond to those of class 
 and order in the latter. Thus the curve of a line complex is an envelope of lines, 
 but its correlate, G^ of a plane complex, is a hcus oi planes. The degree of Cj 
 will be called the order of this cone, and the degree of G^, will be called its elms. 
 We have immediately the proposition : The degree of a complex is equal to the order 
 of any of its G^s and to the class of any of its G^s. 
 
 16. A complex of first degree is said to be linear. A G^ of such a complex 
 is of order 1, it is a flat axal pencil, to be viewed as a lineoid of collinear planes ; 
 while a G^ of the linear complex, being of class 1, is also a flat axal pencil, to be 
 viewed, however, as a line of collineoidal planes : the lineoid Cj is a locus of the 
 planes of the flat axal pencil ; the line G^ is an envelope of the planes of the flat 
 axal pencil : the line and the lineoid are thus but reciprocal phases of one con- 
 
Point in Point- Space of Four Dimensions. 15 
 
 figuration, just as in line geometry the flat pencil is regarded now as a point and 
 again as a plane. Given an arbitrary linear complex of planes. Of these 
 there pass through any line whatever a single infinity of planes all contained 
 in a lineoid and constituting a flat axal pencil; the lineoid so determined 
 will be called the polar lineoid of the given line. Reciprocally every lineoid 
 contains a single infinity of the planes of the given complex and these, too, are 
 collinear, constituting a flat axal pencil ; the axis, or line so determined, will be 
 called the polar line of the given lineoid. Accordingly with respect to any linear 
 plane complex, every lineoid has a polar line, and every line has a polar lineoid. 
 Every lineoid or line is united in position with its polar line or lineoid. 
 
 17. These propositions, showing the distribution of the planes of a linear 
 complex, are of such fundamental importance as to justify their separate estab- 
 lishment by analytical means. As a preliminary we will show that a linear 
 plane complex is representable by an equation of first degree in Vi, and con- 
 versely, that every such equation defines such a complex. 
 
 Let the equation 
 
 F{v;) = 
 
 represent a linear complex of planes. The identity 
 
 D.{v) = 
 
 is, of course, supposed given. Denote by vi and v^,' any two collineoidal planes. 
 We have seen that the coordinates of the planes of the flat axal pencil determined 
 by v- and vi' are 
 
 Vi = :^ivi + "k^vi'. 
 
 The condition that one of these planes shall belong to B' is 
 
 Since by definition of F, only one plane of the pencil belongs to F, the last 
 equation must be linear in "k^-.^, and is, therefore, of the form 
 
 ^CiVi = 0. 
 
 The converse is obviously correct. 
 
 Now let 7t denote any plane whatever and let {x-^, x^, x^, ccj and {y^, y^, y^, y^ 
 
16 Keyser: The Plane Geometry of the 
 
 be any two generating lineoids of n. Then for coordinates of n we may take 
 
 r vi =: 031 ?/3 — xs 2/] , v^ — x^yz — 'x^yi, 
 -j ^2 =: xi^/g — a-3 2/i, r'5 = 3322/4 — x^ y^, 
 (vs = X:^yi — Xiyi, Vf, = x^y^ — Xiy^. 
 
 The condition that n shall belong to the complex 
 
 XciVi = 0, 
 may, therefore, be written 
 
 (cj Vz + (i%yz + cs yd ^1 + (— Ci Vi + Ciy3 + C5 2/4) a;^ 
 
 + (— C2?/i — C42/2 + Cey^) 033 + (— C32/1 — C52/2 — Ce^/s) a^i = 0- 
 
 This equation, if the y's be regarded as fixed and the cc's as variable represents 
 a straight line, and as the equation is satisfied by Xj^ = yi, X2 = y2, 333 = 3/3, 
 x^ = yi, this line lies in the lineoid y. It thus appears that the ^Zawes 7t 0/ a 
 given lineoid y that belong to a given linear complex envelope a line, the polar of a 
 given lineoid. They constitute a flat axal pencil within y . 
 
 In like manner, if 7t be supposed given by two of its lines ^ and ri, reasoning 
 analogous to the foregoing will show that the planes of a given line y that belong 
 to a given linear complex have for locus a lineoid, polar of the given line. 
 
 The flat axal pencils which are thus determined, one for each line and one 
 for each lineoid, by any given linear complex, may be called the pencils of the 
 complex. 
 
 Denote by I any line and by L any lineoid containing Z, and consider L' and 
 V , the polars respectively of 7 and L with respect to a given linear plane com- 
 plex G . The plane {L, L') being contained in L' and containing the 
 polar I of L', belongs to G, and, therefore, as it is contained in L, it contains I'. 
 Consequently, I' lies in L'. Hence, the polars of a line and lineoid united in 
 position are themselves united in position. 
 
 Let 7t be any plane. Every generating lineoid i of 7t is united in position 
 with every generating line 7 of 7t. Hence, every polar line I' of the X's is united 
 in position with every polar lineoid L of the ?'s Hence, the L'^s and the 7"s 
 generate one and the same plane 7t'. Two planes n and 71' thus related will be 
 called conjugate planes with respect to the given complex. Tico conjugate planes are 
 such that either of them is the locus (or envelope) of the polar lines (or lineoids) of the 
 generating lineoids (or lines) of the other. 
 
Point in Puint-Space of Four Dimensions. 17 
 
 18. It thus appears that a linear complex of planes serves as a dualistic 
 transformation establishing a unique and reciprocal correspondence* between 
 lines and lineoids, and between planes and planes. In this correspondence each 
 plane of the complex corresponds to itself; for obviously, if n belongs to the 
 complex, 7t and its conjugate 71' coincide, i. e., every plane of the given complex is 
 self-conjugate or self-polar with respect to that complex. On the other hand, no 
 other plane is self-conjugate. In fact, if two conjugates 7t and n' are not planes 
 of the complex, they are not collineoidal, for suppose them contained in a lineoid 
 L ; the polar line of L lies in both n and rt!, and hence these planes belong to 
 the complex and consequently coincide. Therefore, two conjugate planes coincide 
 and so ielong to the complex or else they are non-collinear and so do not belong to the 
 complex. This proposition is a corollary to the following : If two conjugates, n^ 
 and n[, are each collineoidal with the plane n, the latter belongs to the complex. To 
 prove this proposition, denote by Ly the lineoid determined by Ttj and n, and by 
 L[ that determined by ni and n ; the polar line l^ of L^ lies in n[, and hence in 
 L[, and the polar line l[ of L[ lies in n^ and hence in Zj ; therefore, li and ll are 
 both lines of 7t, the common plane of Zj and L[ ; hence 7t belongs to the com- 
 plex. 
 
 Let 7t, any plane of the complex, be collineoidal with a plane Ttj. If i be 
 the lineoid containing n and Tti, the polar line I of L lies in n, and as L con- 
 tains 7ti, I also lies in n[, the conjugate of 7t[ ; hence, if a plane of a complex is 
 collineoidal with any other plane, it is aho collineoidal with the conjugate of the latter. 
 
 19. The foregoing and additional properties of conjugate planes may be 
 investigated analytically as follows : The condition 
 
 „ , , . - ^ an (v') 
 £1 (v', v) = =Z ^\ Vt 
 
 that the planes v- and lu shall be collineoidal (or collinear) will assume the form 
 
 r'iVi + v'^Vi + ^6^3 + ViVi+ viv^ + rive = 
 on taking Q. (v) to be of the form 
 
 • Exceptions to the one-to one character of this correspondence will he noted at a later stage. 
 41 
 
18 Keyser: The Plane Geometry of the 
 
 The condition, being linear in v'i and Vu shows that the assemblage of planes of 
 which each is collineoidal with a given plane is a linear complex. Such a linear 
 complex will be called a special complex. The condition that the complex 
 
 'Zc.Vi = 
 
 shall be a special complex is thus seen to be 
 
 Q, (c) = C1C4 + c^c^ + c^Cft = , 
 
 which is identical with the condition in the Pliicker geometry that every line 
 of a line complex shall have a point in common with a given line. Employing 
 Klein's terminology for the line theory, we will call the quadratic form £1 (c) the 
 invariant of the complex. In case of a special complex the plane which is col- 
 lineoidal with each plane of the complex will be called the director plane or direc- 
 trix of the complex, this plane being the analogue of the directrix of the special 
 line complex of the line theory. 
 
 Let XciXi = (1) 
 
 be an arbitrary chosen complex, and denote by vi any given plane. The latter 
 is director plane of the special complex 
 
 a (v', r) == . (2) 
 
 The planes common to (1) and (2) are identical with the planes of the flat 
 axal pencils (lines) that are polar to the generating lineoids of r,' with respect 
 to (1). Denote by v'/ the plane common to any two of these pencils. The planes 
 common to (1) and the special complex 
 
 D. {v", v) = (3) 
 
 are identical with the planes of the polar flat axal pencils (lines) of the generating 
 lineoids of Vt'. The plane vi" common to any two of these pencils is identical 
 with v't; for if Zj and L.^ be the lineoid.s whose polar lines (pencils) give r-', then, 
 as the polar line I of any lineoid L of r-' must lie in both L^ and ig, it follows 
 that v't is the locus of such polar lines I. We have accordingly the proposition : 
 Any pair 0/ complexes G and C, of which one of them as C is special, determines 
 a third special complex G" such that the assemblage of planes common to G and C' 
 is identical with the assemblage common to G and G". 
 
 The director planes of G' and G" are evidently conjugates with respect to G. 
 In order, therefore, that v- and t,'' shall be conjugate planes with respect to (1), 
 
Point in Point- Space of Four Dimensions. 19 
 
 it is necessary and suflScient that 
 
 2c,r, = \il iy', v) + \D. {v", v) , (4) 
 
 whence Cy = Vi + ^,v'^, c, = 7,^v[ + Vi"- ) 
 
 C2 = Vb + Vb', C5 = ;Iit;2' + V2'. h (5) 
 
 for some value of the ratio \:\; or, writing the complexes in the form 
 
 s3§(^).i^ = „, s?^).;=:o, X?^..' = 0, (6) 
 
 the condition may be written 
 
 ^^ = Vi' + Vi', (» = 1, 2, 6). (7) 
 
 By the aid of the condition 
 
 a(3nif)_v') = o («) 
 
 that the v'i shall be coordinates of a plane, we readily find 
 
 \ = Q.{c):Xciv[, (9) 
 
 whence the coordinates of the conjugate v'' oiv[ are given by 
 
 ri(c)::;^0, :ZcM=f=0, (a) 
 
 n(c):^o, ^cM=o, (b) 
 
 n(c)=0, SCir;:^0, (c) 
 
 D.{c) = 0, XCivi = 0. (d) 
 
 In (a), which is the general case, the complex (1) is non-special and the 
 plane vi does not belong to (1). From the symmetry of (7), in respect to r/ and 
 Vi, it appears that v-' does not belong to (1). Hence, 0/ two conjugates with respect 
 to a compkx either hoth belong or neither belongs to the complex. Equation (10) 
 shows that under (a) to two planes vi there correspond two planes vi' and recip- 
 rocally (cf §18). The equation 
 
 £liv',v") = (11) 
 
 Four cases may arise : 
 
20 Keyser : The Plane Geometry of the 
 
 signifies indifferently that rf belongs to (2) or that vi belongs to (3) ; the direc- 
 tor plane of a special complex may be considered as belonging to that complex ; 
 hence, if (11) be true, both r[ and v'J belong to both (2) and (3) and hence also 
 to (1), but this is contrary to (a). Hence, ttvo conjugates that do not belong to the 
 given complex are non-collineoidal. 
 
 In (b), vi belongs to (1), which is non-special ; ^j = oo , and we have from (7) 
 
 ;ii : ;i2 = — vi' : vi, 
 
 which shows that every plane of (1) is self-conjugate. 
 In (c) the complex (1) is special, ;ij = 0, and 
 
 hence the conjugate of any v[ with respect to a special complex not containing vi is 
 the director plane of the complex. 
 
 In case (d), X^ is indeterminate ; the meaning is that the conjugate of any 
 v'i with respect to a special complex containing v[ is indeterminate, i. e., may be 
 indifferently taken to be either vi itself, as in (b), or the director plane, as in (c). 
 One and the same line I is polar to all the lineoids of a given Vi, and I is the 
 intersection of Vi and the director plane. 
 
 The results under the four cases may be summarized thus : Given a complex 
 
 G and let n stand for plane ; it is self-conjugate or not so according as it belongs or 
 
 does not belong to G ; if tl^ and n^ be any two planes, their conjugates are distinct 
 
 or not according as G is non-special or special. In case of G special, the director 
 
 plane is conjugate vnth all planes, itself included. 
 
 IV. — Linear Gongruences of Planes, and Pencils of Gomplexes. 
 
 20. A two-parameter system of planes, i. e., a system in which a plane has 
 two degrees of freedom, has been named congruence (§11). It follows that the 
 assemblage of planes that are common to two complexes is a congruence. The 
 two-dimensional assemblage of planes constituting an axal pencil is a congruence. 
 In like manner, the manifold of planes contained in a lineoid constitute a con- 
 gruence, i. e., a bundle of planes is a congruence. A congruence being given, the 
 number of planes it has in common with an arbitrary axal pencil will be called 
 its order, while the number it has in common with an arbitrary bundle will be 
 
Point in Point- Space of Four Dimensions. 21 
 
 called its class. In other words, the terms order and class of a congruence 
 signify respectively the number of planes common to the congruence and an 
 arbitrary line and the number common to the congruence and an arbitrary 
 lineoid. The notions order and class of a plane congruence are seen to correspond 
 respectively to class and order of a line congruence in the Pliicker theory, the 
 order of a line congruence being the number of its lines that go through an 
 arbitrary point, and its class the number that lie in an arbitrary plane. A bundle 
 of planes is of class one and order zero, while an axal pencil is of class zero and 
 order one, just as in the Pliicker theory a plane of lines is of class one and order 
 zero, while a sheaf of lines is of order one and class zero. 
 
 21. We shall be chiefly concerned in this chapter with such congruences as 
 are definable by two linear complexes of planes. Such congruences may be 
 themselves called linear. We have seen that, given a linear complex, an arbi- 
 trary lineoid contains a flat axal pencil of planes belonging to the complex. It 
 follows that an arbitrary lineoid contains one and but one plane of a given linear 
 congruence. Reciprocally, an arbitrary line is contained in one and but one 
 plane of the given congruence. It thus appears that a congruence composed of 
 the planes common to two linear complexes is of order one and class one. 
 
 The assemblage of complexes represented by the equation 
 
 :K%ciVi + ^'tc'o^i = 2 (;ic, + 7Jc[) Vi = 0, (1) 
 
 the c's being supposed given, and the ;i's being parameters, will be called a pencil 
 of complexes. The given complexes Ci and c[, which determines it, may be called 
 ih& fundamental complexes of the pencil. 
 
 If 7^-^ : /L{ and 1^ : ^Ig be any two complexes of the pencil (1), this last is iden- 
 tical with the pencil 
 
 2 [((3;ii + (o%) c, + {6%[ + 6%) cQ Vi = Q, 
 
 the (3's being parameters, for, in order to identify any given complex of either 
 pencil with one of the other, we need only the relation 
 
 ;ii : ;i^ = (67,^ + <6% : 67.[ + <o%, 
 
 which, it is plain, can always be found. Hence, the pencil of complexes determined 
 hy any two complexes of a given pencil is identical with the given pencil. 
 
22 Ketser: The Plane Geometry of the 
 
 22. It is obvious that the congruence defined by any two compleaies c^ and 
 c'i is common to all the complexes of the pencil determined by c,- and c- , and from 
 the foregoing theorem it follows that the congruence may be regarded at will as 
 defined by any pair whatever of the complexes of the pencil. A pencil of com- 
 plexes and the congruence determined by a pair of the complexes may be said to 
 correspond. In this way, to every pencil there corresponds a congruence, and 
 conversely. 
 
 The condiiion that a complex of the pencil determined by the complexes 
 Ci and c'i shall be special is 
 
 fl {7.C + ^'c') = fl (c) A^ + n (c') X'' + 2a (c, c') 7^'X! = 0. (3) 
 
 Two cases are to be considered according as the roots of this equation, regarded 
 as an equation in "k : Ti', are 
 
 (a) determinate 
 or (b) indeterminate. 
 
 Under (a), the general case, there fall two sub-cases : 
 
 (a') A{c,c')=t=0, 
 (a") A{c,c') = 0, 
 
 where A{c,c') = Sl (c) £1 (c') — Q.^{c,c'). 
 
 In (a) equation (3) has two and but two roots. These, which may be real 
 or conjugate imaginaries, are definite and distinct. The pencil accordingly 
 contains two definite and distinct special complexes. The congruence corresponding 
 to the pencil consists of the planes that are at the same time coUineoidal with 
 the two director planes of the special complexes. These director planes may 
 therefore be called the director planes of the congruence. 
 
 Denote by vl and v'i the director planes in question. The special complexes 
 
 of the pencil 
 
 2 {U, + -k'ci) v, = (4) 
 
 will be given by the equations 
 
 £l{vi,v) = 0, £i(y<\v) = 0. (5) 
 
 As the director plane of a special complex is, with respect to that complex, 
 conjugate to all planes whatever, the planes v[ and v[' are conjugate with respect 
 
Point in Point-Space of Four Dimensions. 23 
 
 to both complexes (5). These two planes are conjugate (§19, Eq. 7) with 
 respect to all and only the complexes of the pencil 
 
 2 (pw{ + pS") Vi = (6) 
 
 where 
 
 j w[, w'^, w^, wi, wi wii = vi, vi, vi, v{, v'i, v's. 
 
 in" Hi" — i/' I," 
 
 Therefore (6) contains the complexes (5), and as these are also contained in (4), 
 it follows that (4) and (6) are identical. Hence, the two director planes of a linear 
 congruence are conjugate with respect to every complex of the corresponding pencil. 
 Also if two planes are conjugate with respect to each of two complexes, they are the 
 director planes of the common congruence of those complexes. 
 
 In sub-case (a) the two roots of (3) are equal, the two director planes coin- 
 cide, are one plane. This plane is to be counted twice, once as belonging to the 
 one and once as belonging to the other, of the two (coincident) special com- 
 plexes. This double plane {v[ = Vi) is, therefore, a plane of the congruence. All 
 planes of the congruence are collineoidal with r/, but the converse is not true, 
 for while a plane of a congruence has two, a plane merely required to be colline- 
 oidal with a given plane has three, degrees of freedom. The precise way in 
 which a plane having three degrees of freedom as specified, loses one degree on 
 being required to belong to a linear congruence having the given plane v'l for 
 double directrix, may be seen as follows : If n belongs to the congruence, the 
 lineoid {n, v'^, since it contains two planes of each complex of the pencil, is, in 
 regard to each complex, the polar lineoid of the line {it, v[). Of the oo^ planes 
 containing this line, only oo^ planes, viz., those contained in the lineoid (n, v'^, 
 belong to the congruence. If, therefore, a bilinear relation be set up between 
 the generating lines and the generating lineoids of any plane v'i, then the assem- 
 blage of planes obtained by taking all and only flat axal pencils that are contained 
 in the lineoids corresponding to the axes (lines) is a linear congruence having v[ 
 as double directrix. 
 
 The necessary and suflBcient conditions for the occurrence of case (b) are 
 
 n(c) = o, n(c') = o, n(c,c') = o. 
 
 The first two of these equations indicate that the fundamental complexes Cj and 
 c- are special, and the third equation signifies that the director planes of Cj and 
 c- are collineoidal. The indetermination of "^-.X shows that the pencil is com- 
 
24 Keyser : TTie Plane Geometry of the 
 
 posed of special complexes. The corresponding congruence has for director 
 planes any pair of an infinity of planes, the director planes of the pencil, all of 
 which belong to the congruence. We may readily prove that these director planes 
 constitute aflat axal pencil. For if 
 
 2 (XCi + TJd) Vi — 
 
 be any complex of the pencil of complexes and if r^' be the corresponding direc- 
 tor plane, then 
 
 ,_ dn{^'c + i'c') 
 
 oci del 
 
 which shows that as /L : ^' varies through all real values, the director plane 
 vi generates a flat axal pencil p, namely, that determined by the director planes 
 of the complexes C; and Cj. The congruence consists of all the planes of which 
 each is collineoidal with each plane of^. It is accordingly composed of two 
 hyperpencils, one of each kind : the bundle contained in the lineoid contain- 
 ing the planes oi p, and the axal pencil having for its axis the axis of p. 
 Each of these components is of itself a congruence, the former of order zero and 
 class one, the latter of order one and class zero, showing what should be the case 
 that the congruence under consideration is of order one and class one. 
 
 23. The propositions of line geometry that correspond to the foregoing may 
 be briefly stated as follows : 
 
 Case (a), sub-case (a'). The linear congruence has two distinct non-inter- 
 secting directrices (lines). The assemblage of lines joining the latter is the con- 
 gruence. The directrices are conjugate lines with respect to every line complex 
 of the pencil of complexes defining the congruence. 
 
 Case (a), sub-case (a"). The congruence has one (double) directrix. If a 
 bilinear relation be established between the points and the planes of any given 
 line I, this line is the double directrix of the congruence obtained by taking all 
 and only the lines of such flat pencils as lie each in the plane that corresponds 
 to the vertex (a point of I) of the pencil. 
 
 Case (b). The congruence has an infinity of directrices, which belong to it 
 and which constitute a flat pencil of lines. The congruence is composed of two 
 congruences : the assemblage of lines of the plane of the pencil and the sheaf 
 
Point in Point-Space of Four Dimensions. 25 
 
 of lines containing the vertex of the pencil, the former component being of order 
 zero and class one, while the latter is of order one and class zero. 
 
 24. The geometric properties of the discriminant A of 11 {^c + ^'d) show 
 that A is an invariant both under a reversible linear transformation of the 
 variables v and under such a transformation of the %^s, ; for the coincidence or 
 non-coincidence of the director planes of a congruence is independent alike of 
 the coordinates employed and of the particular choice of a pair out of the 
 corresponding pencil of complexes for fundamental complexes. We will merely 
 state without proof that, \i M^ be the modulus of the ■^-transformation and Ai be 
 the new discriminant, and if if^ be the modulus of the ;i-transformation and Ag 
 the resulting discriminant, then 
 
 (Ai = if|A, 
 
 The ^-transformation being merely equivalent to replacing the fundamental 
 complexes c^ and c[ by a new pair as \ c^ -\- A{ Cj and \ Cj + X^ c[ , we may write 
 
 Aa = A (Ai c -f- ;i{ c', 7.^0 + K c') = {\ K — \ ^^0^ • ^• 
 
 As the vanishing or non-vanishing of A is not a mark of any particular pair of 
 complexes of the pencil but characterizes the pencil as such, we may name A(cic') 
 the discriminant of the "pencil "kc^ -\- 7Jc[ or of the corresponding congruence. A. 
 pencil of complexes being given, it contains two distinct or two coincident special 
 complexes, and the corresponding congruence has two distinct or two coincident 
 director planes, according as the discriminant vanishes or does not vanish. 
 
 V. — Projective Transformations hy Means of Complexes. 
 Orthogonal Complexes : Involution. 
 
 25. We have seen that a linear complex of planes serves as a means of 
 dualistic transformation according to which lines or lineoids correspond to lineoids 
 or lines, and planes correspond to planes. If c^ be any linear complex and if 
 7t and n' be any pair of conjugate planes with respect to c^, then the generating 
 lines or lineoids of 7t or 7t' are projectively related through c^ to the generating 
 lineoids or lines of 7t' or n, any line or lineoid being with respect to c^ the polar 
 of the corresponding lineoid or line. If ?i, 4> k, k be any four lines of 7t or n', 
 
 42 
 
26 Ketser : The Plane Geometry of the 
 
 and Li, L^, ig, Z4 be the corresponding lineoids (of n' or n), the anharmonic 
 ratio of the Z's is equal to that of the L's, i. e., 
 
 ill I2 Ig li) = (A Lz L3 Li) . 
 
 If 7t belong to c^, 7t is self-conjugate, and if generated by I it will at the 
 same time be generated by L, the (polar) correspondent of I. The plane being 
 at once a locus of lines (a pencil of lines) and an envelope of lineoids (a pencil 
 of lineoids), we have the proposition : Any linear complex of planes estahlishes a 
 projective correspondence between the lines and the lineoids of each of its planes. 
 
 If now we suppose n to be common to the two complexes Cj and c[, and if 
 to the lineoids ij, L.^, L3, L^ of n there correspond with reference to Cj, the lines 
 ?i) 4' hi h ^"^^ with reference to c[ the lines Zj, Zg, Z3, Z^, then, as 
 
 (A Lz Lg Li) = (?! Z2 I3 li) , 
 
 (A Lfs L3 Li) = {l[ I2 13 1'i) , 
 
 we have iJikkh) = {VA^li)- 
 
 Accordingly, if we regard n as two superposed pencils, viz., of lines I (associated 
 with Ci) and of (the same) lines I' (associated with c^'), it is seen that these pencils 
 are brought into projective relation by means of the complexes C; and c,'. 
 Reciprocally, n may be conceived as two superposed pencils of (its generating) 
 lineoids, L and L', and these, too, are projectively related through the complexes 
 in question. Given either of the line (or lineoid) pencils, the other pencil (of 
 the same kind) is obtainable from the given one by means of a linear line (or 
 lineoid) transformation of n. Hence, the assemblage of two given complexes plays 
 the role of a definite linear transformation at the same time of the lines and the 
 lineoids of any given plane of the corresponding congruence. 
 As 7t is common to all the complexes of the pencil 
 
 2 (;ic, + ;i'cO Vi = , (7) 
 
 it follows that the lines and lineoids of n are transformed by every pair, 7J •.'X=-'ki, 
 ;\,' : ;\, = ^2, of these complexes. The equation of the transformation will assume 
 its simplest form when referred to its foci. What then are these ? Denote by 
 7t<j and n'g, the director planes of the congruence, and let 7^ and V^ be the lines and 
 L^ and L'^ the lineoids determined by 7t and the director planes. As n^ and n'^ 
 are conjugates with respect to both \ and h^^ld and L'^ are each the other's 
 polar in regard to both Fs. Similarly, l'^ and L^. Hence l^^ and Z^ are the foci 
 
Point in Paint-Space of Four Dimensions. 27 
 
 of the line transformation, and La and i^ are the foci of the lineoid transforma- 
 tion, eflFected on n by the pair Jc^, k^ of complexes. Taking 7^ and l^ as base 
 elements of homogeneous coordinates, z = xi:x^, of the lines of the pencil deter- 
 mined by la and l'^ , the line transformation assumes the form 
 
 In like manner, if z be interpreted as coordinates of the lineoids of the pencil 
 lineoids referred to L^ and L'^ as base elements, the lineoid transformation takes 
 the form 
 
 z' = p'z. 
 
 26. In case of the line transformation, the characteristic constant p is the 
 anharmonic ratio formed by any pair z and z' of corresponding lines of n with the 
 foci la and I'a. Writing 
 
 o^^^logp, 
 
 a will be the (generalized) angle of the lines z and z'. If the foci be the isotropic 
 lines of the pencil in question, a will be the ordinary angle of z and z'. In like 
 manner a', where 
 
 c^'=^logp' 
 
 fs the angle between any pair z and z' of corresponding lineoids. 
 
 42. Since the transformation is determined completely by the given com- 
 plexes ki and Jc^, it must be possible to express p and p' in terms of the parame- 
 ters ki, kz and the coefficients of the fundamental complexes c^ and c/. We pro- 
 ceed to determine such expressions for p. 
 
 Let L be any lineoid of 7t. With respect to each complex of the pencil (7), 
 L has a polar line I. By virtue of this one-to-one correspondence, if ^i, k[, k^, k^ 
 be any four of the complexes and Zj, l^, 4. h ^^^ corresponding polar lines of L , 
 we shall have 
 
 ('1 'l 4 '2) ~~ ("'1 "'I "'S "'i) • 
 
 If, now, we replace ki and k^ respectively by s^ and s^, where ;i' : A =Si, A,' : ;i = s^ 
 are the special complexes of (7), li and li will be replaced by the foci la and l'^ 
 and we shall obtain, since l^ and l^ are any pair of correspondents, 
 
 p = {l^hl,l'a) = {S^iSiK,S,). 
 
28 Keyser : The Plane Geometry of the 
 
 In precisely like manner, it may be shown that 
 
 whence P = p'' °^ = ^'• 
 
 Hence, The line and lineoid transformation effected hy two given complexes hi, h.^ 
 on any plane 7t of their common congruence are algebraically identical. The trans- 
 formation remains unchanged if n be supposed to generate the congruence. The 
 angle of two corresponding lines of a given n is constant. The same is true of the 
 angle of two corresponding lineoids. 
 
 27. The angle of any pair of lines of one plane is equal to that of any pair 
 of any other plane of the congruence. The two pairs are, however, not congruent 
 (in the ordinary sense). The equality is merely arithmetic, not geometric. The 
 systems of measurement in the one plane and in the other are not the same. 
 The foci (the Cayleyan absolute) in the one plane are not congruent with the 
 foci in case of the other. Like remarks apply to the lineoid transformations, and 
 to the equation a = a'. 
 
 28. There is a simple infinity, qo^, of linear transformations having the same 
 assigned foci. On the other hand, there are oo^ pairs of complexes defining one 
 and the same congruence and each pair gives a line (lineoid) transformation of 
 the planes. It is, therefore, to be expected that, if one pair gives a transforma- 
 tion, there are 00^ pairs giving the same transformation. That such is the case 
 appears in the equation 
 
 P — - [hlSitC^S^), 
 
 which, being linear in k^ and in Jc^, is satisfied by 00^ of pairs of values of the ^'s 
 This statement is independent of the value of p, and as the foci in case of any 
 plane are the same for all pairs of k's, we have the proposition: The lines {lineoids) 
 of every plane of a congrue)ice are transformed by co^ distinct transformations by 
 pairs of complexes of the corresponding pencil of complexes, eaoh transformation is 
 effected by co^ pairs, and all transformations of any given platie have the same foci. 
 Holding p fixed, the above equation may be regarded as a linear transforma- 
 tion of the pencil (7) of complexes. The foci of the transformation are the special 
 complexes Sj and s^, for on writing 
 
 _ { h — Si ) {^^2 — ^a) 
 P- (si-h){s.,-Jcij' 
 
Point in Point- ):ijpace of Four Dimensions. 29 
 
 it is seen that h^ and k^ coincide when and only when one of them coincides with 
 Sj or Sg (the s's are here supposed distinct). The constant p is, then, the anhar- 
 monic ratio formed by any pair ki and k^ of complexes with the pair of special 
 complexes Sj and s.^. We may therefore call 
 
 <x = ^.logp, 
 
 the angle of tlie complexes k^ and k^. Hence the proposition : The angle of any 
 pair of complexes is equal to the angle between any tico corresponding lines (lineoids) 
 in the transformation effected by the given complexes upon any plane of their common 
 congruence. 
 
 The case where p = — 1 and a-= -- is of special interest. In this case the 
 
 complexes k^ ank k^ may be said to be orthogonal, or, since the pair k^, \ is har- 
 monic to the pair s^, Sg, the complexes k^ and k^ may be said to be in involution 
 with respect to the two special complexes. The condition for such orthogonality 
 or involution is 
 
 2^1 k^ — (si + SzW + ^2) + 2si §2 = , (8) 
 
 which shows that every complex of a pencil of complexes is in involution udth one 
 and [in general) only one other complex of the pencil. 
 
 It is plain also that any pair of corresponding lines {lineoids) of a plane trans- 
 formed hy two orthogonal complexes is harmonic to the pair of foci in that plane. 
 
 If, in equation (8), we let k^^ 0, then k^ ■= ^SiS^ : (si + s^, and this will be 
 
 00 when and only when s^-\- s^=.0, or, since the s's are the roots of equation 
 
 (3), the condition is 
 
 XI (c, c') = 0. 
 
 The values /fci = , /fcg = 00 correspond to the fundamental complexes c^ and c/ of 
 the pencil (7). The vanishing of fl(c, c') is, therefore, the condition that the 
 complexes Cj and c^ shall be orlliogonal. This function of the coefficients c is that 
 which in the Pliicker geometry, Klein has called the simultaneous invariant of 
 the complexes c^ and cl. 
 
 29. Suppose that the plane v- belongs to c{, and that v'^ is the conjugate of 
 v'i with respect to c^, then (IV, Eq. 7), 
 
 g '^ ' = ^1 V[ + Aj < , 
 
30 Keyser : The Plane Oeometry of the Point, etc. 
 
 whence ^ d£i (c) , -i v^/ / _i- n v ' // 
 
 2 g'- ' o! = A,i Zc[ v'i + 7^ ZCi v[\ 
 
 i.e., lQ.(c,c') = :^^'Zc[v['. 
 
 Since, by hypothesis, 
 
 M< = . 
 
 Accordingly, if vi' belongs to ci , the simultaneous invariant vanishes. Therefore, 
 if two complexes are such that one of them contains a plane conjugate to one of its 
 -planes with respect to the other complex, the complexes are orthogonal. 
 
 It is, moreover, seen that, \iv[' is in c|, then v'l" the conjugate of any other 
 plane v'^' of c- is also in c[, i. e., the planes of c/ fall into pairs of conjugates with 
 respect to Cj, and, as the relation is a reciprocal one, the planes of c^ fall into 
 pairs of conjugates with respect to Cj. Two complexes thus related may be said 
 to be each its own conjugate, or self-conjugate, or self-polar, with respect to the 
 other. This property of self-conjugateness characterizes both complexes of every 
 orthogonal pair and might he taTcen as the defining property of orthogonality. 
 
 30. The notion of orthogonality has as yet been attached only to pairs of 
 non-special complexes. Nevertheless if (following Klein and Koenigs in the line 
 theory) we agree to say that two complexes are orthogonal always when their 
 simultaneous invariant is zero, then it readily follows: (1) That every special 
 complex is orthogonal to all complexes cont lining its director plane, and conversely ; 
 (2) that the necessary and sufficient condition for the orthogonality of two special 
 complexes is that their director planes be collineoidal. 
 
 Consideration of the net 
 
 ;io 2Ci V, + \ Xc( Vi + X^Sc't' Vi=0, 
 
 and the web /loSCjVi + + 2.sXci"vi = 0, 
 
 of complexes, which correspond respectively to the one-parameter system (con- 
 figuration, plane-series) and the zero-parameter system of planes, is reserved. 
 
 Columbia University, Aug. 10, 1901 
 
arY991 ^'""" ""'^or^^V Library 
 ^IlllillftfiiiSaf.J,';^,,,,?' 'lie point in point 
 
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