A TREATISE ON
THE ANALYTIC GEOMETRY OF
THREE DIMENSIONS
BY THE SAME AUTHOR.
TREATISE ON CONIC SECTIONS.
CONTAINING AN ACCOUNT OF SOME
OF THE MOST IMPORTANT MODERN
ALGEBRAIC AND GEOMETRIC METHODS.
8vo, tax.
A TREATISE ON THE ANALYTIC
GEOMETRY OF THREE DIMENSIONS.
EDITED BY
REGINALD A. P. ROGERS.
Vol. I. 95.
LONGMANS, GREEN AND CO.,
LONDON, NEW YORK, BOMBAY, CALCUTTA, AND MADRAS.
A TREATISE
ON THE
ANALYTIC GEOMETRY
THREE DIMENSIONS
BY
GEORGE SALMON, D.D., D.C.L., LL.D., F.R.S.
LATE PROVOST OF TRINITY COLLEGE, DUBLIN
EDITED BY
REGINALD A. P. ROGERS
FELLOW OF TRINITY COLLEGE, DUBLIN
FIFTH EDITION
VOL. II
NGMANS, GREEN AND CO.
39 PATERNOSTER ROW, LONDON
FOURTH AVENUE & 30r H STREET, NEW YORK
BOMBAY, CALCUTTA, AND MADRAS
HODGES, FIGGIS AND CO., LTD.
104 GRAFTON STREET, DUBLIN
1915
QPt
53
ELECTRONIC VERSION
AVAILABLE
NO.
EDITOR'S PREFACE TO FIFTH EDITION.
VOLUME II.
IN revising this volume I have been fortunate to have
the assistance of Mr. G. R. Webb, Fellow of Trinity
College, Dublin, Miss Hilda P. Hudson, Sc.D.
(Dublin), of Newnham College, Cambridge, and Mr.
Robert Russell, Fellow of Trinity College, Dublin.
The two volumes now form, it is hoped, a concise
and comprehensive survey of tridimensional Eucli
dean Geometry, both algebraic and differential, and
those who wish to specialise further will find ample
references to guide them.
Practically nothing has been omitted from the
fourth edition, but the numbering of the articles has
been altered in some places. New matter is enclosed
in square brackets and the principal additions will
now be described.
Chapter XIII. of the fourth edition has been
divided into three Chapters, XIIL, Xllla and
XHI/y ; XIII& dealing with rectilinear complexes, rec
tilinear congruences and ruled surfaces, contains a
good deal of new matter under the first two headings.
The linear complex is treated analytically (Art. 454^),
Arts. 455#o, which were written by Mr. Russell, deal
with the singular elements of complexes of any order
vi EDITOB'S PREFACE.
and, in particular, with the quadratic complex and
Rummer's quartic surface. I have added some
twentytwo pages on the differential properties of
rectilinear congruences using Kummer's parametric
method (455<?457/). The surfaces and points as
sociated with a congruence are denned. Special
attention is given to normal congruences with their
optical and mechanical characteristics, and the in
vestigation of those normal congruences which are
defined by two directing curves (45 7c), leads naturally
to a discussion of Dupin's cyclides (4576?) which are
the orthogonal surfaces. Arts. 4570 and / contain
a summary of Ribaucour's theorems dealing with the
unique and interesting class of congruences known
as isotropic or " circular," having the characteristic
that the lines of striction of all ruled surfaces of the
congruence lie on a surface. In 462# it is shown how
to apply the parametric method to ruled surfaces.
A few examples of triply orthogonal systems have
been added (486) with a brief account of Lame's
curvilinear coordinates, Lame's equations and the
connection between those and Cayley's differential
equation (486). In 486ft and c, the DupinDarboux
theorem has been proved and generalised so as to
apply to "complexes" of curves. The subject natur
ally leads to an inquiry into normal congruences
of curves. I have explained the parametric method
of dealing with these, and as an illustration, I have
given Ribaucour's beautiful theorem on the deforma
tion of such congruences (486J). Cyclic systems are
touched on in Arts. 4860 and /; 495# and 515# are
also new.
EDITOR'S PREFACE. vii
Chapters XV. and XVI. dealing with cubic and
quartic surfaces have been revised and enlarged by
Mr. G. R. Webb who has made several interesting
additions. In 522^, b, c, further details are given
on singular points of cubic surfaces with some
account of Segre's method of analysing higher sin
gularities ; 527# contains a geometrical proof of the
uniqueness of the canonical form of a cubic. In 5'J7/>
the reader is introduced to the focal surface of the
congruence of lines joining corresponding lines on the
Hessian, this being the analogue of the Cayleyan of a
plane cubic. In 536# various proofs are given of
Schlafli's theorem, independently of the theory of cubic
surfaces, and in 537 the connection is exhibited
between the twentyseven right lines on a cubic
surface and the twentyeight bitangents of a plane
quartic.
In Chapter XVI. Mr. Webb has added a full
bibliography of work done on the quartic surface
(Art. 545), with articles on Steiner's quartic (554a),
on the relation between Kummer's cones and
the sixteen right lines on a quartic with nodal conic
(559#), on cyclides with nodes (567), on Weddle's
surface, symmetroids, and Kummer's quartic surface
(572# to 573c), and on Holm's investigation of the
maximum number of ovals that a quartic can possess
(574). There are also some minor additions.
Chapter XVII. of the Fourth Edition has been
subdivided into two Chapters, XVII. and XVII0.
This portion has been revised by Miss Hilda P.
Hudson, who has also added six articles on
the interesting subject of Cremona transformations
viii EDITOR'S PREFACE.
(Arts. 587/). The sections on the contact of lines
and planes with surfaces (Arts. 588608) have been re
arranged in a more logical order with minor additions.
The principal changes are in Cayley's " addition on the
theory of reciprocal surfaces " (Arts. 620630). One
is sorry to lose anything so picturesque as the
"points of an unexplained singularity," but the
phrase is no longer justified. This section has been
rewritten with very kind assistance from Professor
Zeuthen to whom Miss Hudson and the editor offer
their grateful acknowledgments.
REGINALD A. P. ROGERS.
TRINITY COLLEGE, DUBLIN,
October, 1914.
CONTENTS OF VOLUME II.
PAGE
PREFACE v
CHAPTER XIII.
PARTIAL DIFFERENTIAL EQUATIONS OF FAMILIES OF SURFACES.
GENERAL CONCEPTION OF A FAMILY OF SURFACES 1
Equations involving two parameters and one arbitrary function . . 2
Cylindrical surfaces 4
Conical surfaces or cones 5
Conoidal surfaces, the right conoid, the cylindroid 7, 8
CHAPTER XIII (a).
COMPLEXES, CONGRUENCES, RULED SURFACES.
The Rectilinear Complex for triply infinite system of lines ... 87
Order of an algebraic complex 37
The Rectilinear Congruence for doubly infinite system of lines . . 87
Its rays are bitangents of a surface 87
Order, class, and orderclass of algebraic congruence .... 88
Ruled surfaces of congruence 38
The Ruled Surface for singly infinite system of lines .... 88
SECTION I. RECTILINEAR COMPLEXES.
THE LINEAR COMPLEX 39
Its simplest form, geometrical construction 40
Curves of the linear complex ... 42
Definition of the QUADRATIC COMPLEX, and of its equatorial, complex, and
singular surfaces '
SINGULAR POINTS, PLANES, AND SURFACES OF COMPLEXES OF ANY DEGREE 48
APPLICATION TO THE QUADRATIC COMPLEX 45
The singular surface has sixteen nodes and sixteen planes of contact . 46
Its equation 47
Its planes of contact and nodal points *8
It is a general Rummer's sixteennodal quai tic 50
Conjugate lines ; the conjugate complex .
ix
X CONTENTS.
PAGBi
Cosingular complexes 52
Double tangent lines of singular surface . . ... . .52
The principal linear complexes 53
The rays common to two complexes form a congruence. Deduction of
focal points and planes 55
Application to quadratic complex. Its focal surface .... 56
SECTION II. RECTILINEAR CONGRUENCES.
The congruence of rays meeting two fixed right lines . . . .57
Two methods of dealing with differential properties of congruences . 58
The limit points and principal planes 59
The principal planes are at right angles 61
The focal points, focal planes, focal surface, developables .... 62
The middle point .63
The sheets of the focal surface may be curves or developables ... 64
Congruence of rays meeting a twisted cubic 64
Surfaces connected with a congruence, the limit surface, the focal surface,
the middle surface, the middle envelope, the limit envelope, the
developables, the principal surfaces 65
Parabolic, hyperbolic, and elliptic congruences 65
NORMAL CONGRUENCES 66
A congruence is normal if its rays are normal to a single surface . . 67
Four different ways of expressing the condition that a congruence be
normal 67, 68
Some transformations in which a normal congruence remains normal:
refraction or reflexion (MalusDupin), "deformation," theorems of
Beltrami and Ribaucour 69, 70
Refraction of a normal congruence 70
Rays of normal congruence are tangents to a singly infinite family of
geodesies 70
" Threadconstruction " for normal congruence 71
Congruence of tangent lines to two confocal quadrics .... 71
Directed normal congruences 71
The directing curves of a doublydirected normal congruence are focal
conies ............ 72
The surfaces normal to a doublydirected congruence are cyclides of
Dupin, having circular lines of curvature 73
Cyclides of Dupin ; equations in elliptic and Cartesian coordinates,
threadconstruction, definition as envelopes, effect of inversion . 74
ISOTROPIC CONGRUENCES 74
Parametric condition that a congruence be isotropic .... 75
Defined by means of two applicable surfaces whose corresponding points
are equidistant 75
Generated by means of a sphere 76
CONTENTS, xi
PAOE
Defined as a congruence whose focal surface is an isotropic developable . 78
The generators of either system of a system of confocal onesheeted
hyperboloids form an isotropic congruence 78
The middle envelope of an isotropic congruence is a minimal surface . 78
Equations for principal radii and conditions for minimal surface . . 7'J
SECTION III. RULED SURFACES.
Construction of tangent plane 80
Defined as intersection of two planes containing a variable parameter . 81
Homographic correspondence between points and tangent planes on
generator 81
Normals along a generator generate a paraboloid 82
Lines of striction 88
Coordinates of any point expressed in two parameters . . . .84
Central point, central plane, and parameter of distribution of a generator 84
Application to ruled surfaces of normal, hyperbolic, and elliptic congruences 86
Nature of contact along any generator 86
Existence of double lines on algebraic ruled surface . . . . 87, 88
Class of tangent cone is equal to degree of surface 89
Number of planes through a point that contain two generators equals
number of points in which a plane meets double curve ... 89
Ruled surfaces generated by a line meeting three fixed algebraic curves . 90
By a line meeting one algebraic curve once, and another twice . . 92
By a line meeting one algebraic curve three times 98
Double and multiple generators on these 93
Order of condition that three algebraic surfaces should have a line in
common 94
Nodal curves on these ruled surfaces 96
CHAPTER XIII (b).
TRIPLY ORTHOGONAL SYSTEMS OF SURFACES, NORMAL CONGRUENCES OF CURVES.
TRIPLY ORTHOGONAL SYSTEMS 98
Dupin's theorem 98
Another proof 10*
Differential equation satisfied by / (x, y, z) if the equations r=f(x, y, z)
represent one family of a triply orthogonal system
Form of the differential equation when / (a;, y, z) = X+ Y+Z, where
X, Y, Z are functions of x, y, z respectively Ill
Systems conjugate to the family x'y m z n =r
Systems consisting of bicircular quartics 11:!
A class of systems which are envelopes of a oneparameter family of
surfaces
W. Roberts' system pv = a\
CONTENTS.
Other examples. Systems of Dupin cyclides ...... 115
Lame's curvilinear coordinates ........ . 115
Lame's equations ........... 117
Deduction of the Cartesian differential equation ..... 117
Darboux' extension of Dupin's theorem ....... 118
Derivation of these theorems and Joachimsthal's from more general
principles ............ 119
Complexes of curves, curvature, and torsion ..... 120, 121
Further generalisation of DupinDarboux theorem ..... 122
NORMAL CONGRUENCES OF CURVES ........ 123
Definition of a congruence of curves ........ 123
Condition that a congruence be normal ....... 124
Ribaucour's theorem on " deformation " of plane normal congruences . 124
Cyclic systems, Ribaucour's theorem ....... 125, 12G
Cyclic congruences ........... 12G
The orthogonal surfaces of a cyclic system are a family of Lame . . 127
Examples on cyclic systems ......... 127
CHAPTER XIV.
THE WAVE SURFACE, THE CENTROSURFACE, PARALLEL, PEDAL, AND
INVERSE SURFACES.
WAVE SURFACE 128
Sections by principal planes. Sixteen nodal points, four of which are real 12 ( J
Apsidal surfaces 130
Polar reciprocal of apsidal is apsidal of polar reciprocal .... 132
Wave surface has sixteen planes of circular contact, four of which are real 133
Geometrical proof that four tangent planes touch along circles . . 134
Coordinates of point on wave surface in elliptic functions of two para
meters 135
Equation in elliptic coordinates. The two real sheets .... 137
Expression for angle between tangent plane and radius vector . . . 138
Length and direction of perpendicular on tangent plane .... 138
Construction for perpendicular on tangent plane 140
Exercises.
THE SURFACE OF CENTRES 141
Clebsch's generalised centresurface of quadric 141
Section by principal planes 144
Its cuspidal curves 145
Its nodal curves 146
Centrosurface of surface of m th degree . 148
Order and class of congruence of lines normal to surface of w" 1 degree . 149
Class of centro surface . 149
CONTENTS. xiii
PAGE
Degree of centresurface . . . 150
Grouping of the 28 bitangents of centrosurface of quadric . . . 151
Synnormals, normopolar surface 152
PARALLEL SDKFACES.
Degree, class, nodal, and cuspidal curves of parallel to surface of m th degree 154
Application of parallel to quadric 154
PEDAL SURFACES 155
INVERSE SURFACES 156
Lines of curvature preserved in inversion 158
Degree and class of inverse of surface of m ih degree and of first pedal . 158
Effect of inversion on geodesic torsion 159
Surface of elasticity, lines of curvature 160
First negative pedal of quadric .,,,,,.., 160
CHAPTEB XV.
SURFACES OF THE THIRD DEGREE.
Cubics with a nodal line 163
Cubics with nodes 165
Different kinds of nodes . 166
Reduction in class caused by nodes 167
Higher nodes equivalent to combination of simpler ones .... 168
Segre's method of analysing higher nodes 168
Twentythree species of cubics 170
Cubics with four nodes 172
Cubics with three coalescing nodes 172
CANONICAL FORM THE HESSIAN 173
Sylvester's canonical form 173
Hessian and Steinerian 174
Properties of corresponding points on the Hessian .... 174, 175
Common tangent planes to cubic and Hessian touch cubic along para
bolic curve 175
Relation of the canonical pentahedron to the Hessian .... 176
The canonical form is unique 177
Surface analogous to the Cayleyan 178
Polar cubic of a plane 179
The polar cubic touches the Hessian along a curve 180
The nodes of the Hessian lie on this curve 180, 181
This curve meets the plane in three pairs of corresponding points . . 181
Section by tangent plane, how related to Hessian 182
RIGHT LINES ON A CUBIC 183
Each line meets ten others 184
Number of triple tangent planes , 185
XIV CONTENTS.
PAGE
Symbolism for the lines Schlafli's doublesixes .... 185, 188
Analysis of kinds of cubics based on number of real lines . . . 188
Independent proofs of Schlafli's theorem 189, 190
Lines related to the bitagents of a plane quartic 190
INVARIANTS AND COVARIANTS OP CUBICS 191
Method of obtaining contravariants in five letters ..... 192
Five fundamental invariants 197
Equation of surface determining the twentyseven lines .... 199
CHAPTER XVI.
SURFACES OF THE FOURTH DEGREE.
QUARTICS WITH SINGULAR LINES SCROLLS 202
Scrolls with a triple line 202
Their reciprocals 204
Subforms 205
Quartics with nonplane nodal line are in general scrolls . . . 206
Scrolls with nodal curve of third degree 207
Scrolls with nodal curve of second degree 212
Steiner's quartic 213
Quartics with a nodal line contain sixteen right lines .... 215
Birational transformation of such quartics into a plane .... 216
Different kinds of nodal right lines . . . . . . . . 217
Pliicker's complex surface 218
Its connection with a quadratic complex 220
QUARTICS WITH NODAL CONICS CYCLIDES 221
Segre's enumeration of these quartics 221
Configuration of the sixteen lines on the surface .... 222, 223
Cone of contact from point on nodal conic 224
Quartics containing systems of quadriquadric curves .... 225
Cyclidea generated as envelopes 226, 227, 233
Fivefold generation of cyclides 227, 229
Cases where centrelocus is specially related to Jacobian sphere . 227, 237
The five Jacobian spheres are mutually orthogonal 230
Confocal cyclides 232
Spheroquartics 234
Dupin's cyclide 235
Equations of cyclides with isolated nodes 236
Loria's spherical coordinates 237
Quartics with a cuspidal conic 238
QUARTICS WITH ISOLATED SINGULARITIES 238
Triple points 238
Quartics may have sixteen nodes 239
CONTENTS. XV
PAGE
Quartics with one to seven nodes .... 240
Two kinds of octonodal quartics .241
Dianodal surfaces. Enneadianomes and decadianomes . 242
Weddle's surface , 244
Symmetroids 245
Rummer's quartic 246
Notations for the 16 6 configuration 247, 248
Rummer's quartic is determined by six nodes 247
Tetrahedroids 24'J
Parametric expression of Rummer's quartic 250
Connection with six apolar linear complexes . . . . 250,261
Number of ovals of nonsingular quartics 251
CHAPTER XVII.
GENERAL THEORY OP SURFACES.
SECTION I. SYSTEMS OP SURFACES 253
Jacobian of four surfaces 253
Degree of tactinvariant of three surfaces 254
Degree of condition that two surfaces may touch 255
Degree of developable enveloping a surface along a given curve . . 256
Degree of developable generated by a line meeting two giveu curves . 256
On the properties of systems of surfaces 256
Principle of correspondence . . . 259
SECTION II. TRANSFORMATIONS OF SURFACES 262
Unicursal surfaces . . 263
Correspondence between points of surface and of plane .... 264
Expression for deficiency of a surface 267
CREMONA TRANSFORMATIONS 268
Homaloidal surfaces 270
Quadroquadric transformation 271
Principal system 275
SECTION III. CONTACT OF LINES WITH SURFACES 277
Flecnodal tangents 277
Clebsch's calculation of surface S 278
Inflexional tangents which touch the surface again 286
Triple tangents 287
Tangents which satisfy four conditions 288
SECTION IV. CONTACT OF PLANES WITH SURFACES .... 291
Biflecnodal points 292
Nodecouple curve . 297
XVI CONTENTS.
CHAPTER XVII (a).
THEORY OF RECIPROCAL SURFACES.
PAGE
Ordinary singularities 300
Number of triple tangent planes to a surface 303
Effect of multiple lines on degree of reciprocal 307
Application to developables of theory of reciprocals 309
Singularities of developable generated by a line resting twice on a given
curve 310
Application to ruled surfaces 310
Intersection of ruled surface with its Hessian 313
ADDITION ON THE THEORY OF RECIPROCAL SURFACES .... 313
INDEX OP SUBJECTS 321
INDEX OP AUTHORS CITED 333
CHAPTER XIII.
PARTIAL DIFFERENTIAL EQUATIONS OF FAMILIES OF
SURFACES.
422. LET the equations of a curve
< (x, y, z, c lt c. 2 ...c,,) =0, ^ (x, y, z, c lt c 2 ...c,,) =0,
include n parameters, or undetermined constants ; then it is
evident that if n equations connecting these parameters be
given, the curve is completely determined. If, however, only
n  1 relations between the parameters be given, the equations
above written may denote an infinity of curves ; and the
assemblage of all these curves constitutes a surface whose
equation is obtained by eliminating the n parameters from
the given n + 1 equations ; viz. the n  1 relations, and the
two equations of the curve. Thus, for example, if the two
equations above written denote a variable curve, the motion
of which is regulated by the conditions that it shall intersect
n1 fixed directing curves, the problem is of the kind now
under consideration. For, by eliminating x, y, z between the
two equations of the variable curve, and the two equations
of any one of the directing curves, we express the condition
that these two curves should intersect, and thus have one
relation between the n parameters. And having n  1 such
relations we find the equation of the surface generated in the
manner just stated. We had (Art. 112) a particular case of
this problem.
Those surfaces for which the form of the functions <f> and
i/r is the same are said to be of tlie same family, though the
equations connecting the parameters may be different. Thus,
if the motion of the same variable curve were regulated by
VOL. n. 1
2 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
several different sets of directing curves, all the surfaces
generated would be said to belong to the same family. In
several important cases, the equations of all surfaces belonging
to the same family can be included in one equation involving
one or more arbitrary functions, the equation of any in
dividual surface of the family being then got by particularising
the form of the functions. If we eliminate the arbitrary
functions by differentiation, we get a partial differential equa
tion, common to all surfaces of the family, which ordinarily
is the expression of some geometrical property common to
all surfaces of the family, and which leads more directly
than the functional equation to the solution of some classes
of problems.
423. The simplest case is when the equations of the variable
curve include but two constants* Solving in turn for each
of these constants, we can throw the two given equations
into the form u = c 1 , v = c 2 ; where u and v are known func
tions of x, y, z. In order that this curve may generate a
surface, we must be given one relation connecting c lt c 2 ,
which will be of the form c x = <f> (c 2 ) ; whence putting for c x
and c 2 their values, we see that, whatever be the equation of
connection, the equation of the surface generated must be of
the form u = <f> (v).
We can also, in this case, readily obtain the partial diffe
rential equation which must be satisfied by all surfaces of the
family. For if U= represents any such surf ace, U can only
differ by a constant multiplier from u  <f> (v~). Hence, we
have \U=u  <f> (v), and differentiating
with two similar equations for the differentials with respect
to y and z. Eliminating then X and <f> (v), we get the re
quired partial differential equation in the form of a deter
minant.
* If there were but one constant, the elimination of it would give the
equation of a definite surface, not of a family of surfaces,
DIFFERENTIAL EQUATIONS OF FAMILIES OF SURFACES. H
7,, U t , U 3
HU w 2 , u 3 = 0.
In this case u and v are supposed to be known functions of
the coordinates ; and the equation just written establishes a
relation of the first degree between U lt U 2 , U 3 .
If the equation of the surface were written in the form
z<j) (x, y) = 0, we should have U 3 = l, U l = p, 7 2 = q,
where p and q have the usual signification, and the partial
differential equation of the family is of the form Pp + Qq = R,
where P, Q, R are known functions of the coordinates. And,
conversely, the integral of such a partial differential equation,
which* is of the form u = <}> (v), geometrically represents a
surface which can be generated by the motion of a curve
whose equations are of the form u = c lt v = c 2 .
The partial differential equation affords the readiest test
whether a given surface belongs to any assigned family. We
have only to give to U l , U%, U 3 their values derived from the
equation of the given surface, which values must identically
satisfy the partial differential equation of the family if the
surface belong to that family.
424. If it be required to determine a particular surface of
a given family u = < (v), by the condition that the surface shall
pass through a given curve, the form of the function in this
case can be found by writing down the equations u = c lt v = c 2 ,
and eliminating x, y, z between these equations and those of
the fixed curve; we thus find a relation between q and c.,,
or between u and v, which is the equation of the required
surface. The geometrical interpretation of this process is,
that we direct the motion of a variable curve u = c lt v = c,, by
the condition that it shall move so as always to intersect the
given fixed curve. All the points of the latter are therefore
points on the surface generated.
If it be required to find a surface of the family u = <f> (v)
* Boole's Differential Equations, p. 323, or Forsyth's Differential Equations,
Art. 185.
1*
4 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
which shall envelope a given surface, we know that at every
point of the curve of contact U lf U z , U 3 have the same value
for the fixed surface, and for that which envelopes it. If
then, in the partial differential equation of the given family,
we substitute for U^ U 2 , U 3 their values derived from the
equation of the fixed surface, we get an equation which will
be satisfied for every point of the curve of contact, and which
therefore, combined with the equation of the fixed surface,
determines that curve. The problem is, therefore, reduced to
that considered in the first part of this article ; namely, to
describe a surface of the given family through a given curve.
All this theory will be better understood from the following
examples of important families of surfaces belonging to the
class here considered ; viz. whose equations can be expressed
in the form u = <f> (v ).
425. Cylindrical Surfaces. A cylindrical surface is gene
rated by the motion of a right line, which remains always
parallel to itself. Now the equations of a right line include
four independent constants ; if then the direction of the right
line be given, this determines two of the constants, and there
remain but two undetermined. The family of cylindrical sur
faces belongs to the class considered in the last two articles.
Thus, if the equations of a right line be given in the form
x = lz+p, y = mz + q; I and m which determine the direction
of the right line are supposed to be given ; and if the motion
of the right line be regulated by any condition (such as that
it shall move along a certain fixed curve, or envelope a certain
fixed surface) this establishes a relation between p and q, and
the equation of the surface comes out in the form
x  Iz = <f> (y  mz).
More generally, if the right line is to be parallel to the
intersection of the two planes ax + by + cz, ax + b'y + cz, its
equations must be of the form
ax + by + cz a, ax + b'y + cz = ft,
and the equation of the surface generated must be of the form
+ cz = <f> (ax + b'y
DIFFERENTIAL EQUATIONS } I A Ml I, IKS Ol S. 6
Writing ax + by + cz for u, and a'x + b'y + c'z for r in the
equation of Art. 423, we see that the partial differential equa
tion of cylindrical surfaces is
(be  cb') U l + (ca r  ac") U 2 + (ab f  ba') U y  0,
or (Ex. 3, Art. 44) U t cos a + U% cos ft + U 3 cos 7 = 0, where a, ft,
7 are the directionangles of the generating line. Remember
ing that U lt U 2 , U 3 are proportional to the directioncosines
of the normal to the surface, it is obvious that the geometrical
meaning of this equation is, that the tangent plane to the sur
face is always parallel to the direction of the generating line.
Ex. 1. To find the equation of the cylinder whose edges are parallel to
x = Iz, y = mz, and which passes through the plane curve z = 0, <f> (x, y) = 0.
Ans. <p (x  Iz, y  mz) = 0.
Ex. 2. To find the equation of the cylinder whose sides are parallel to the
intersection of ax + by + cz, a'x + b'y + c'z, and which passes through the
intersection of ox + fly + yz = 8, F (x, y, z) = 0. Solve forx, y, z between the
equations ax + by + cz = u,a'x + b'y + c'z = v, ax + &y + yz = 8, and sub
stitute the resulting values in F (x, y, z) = 0.
Ex. 3. To find the equation of a cylinder, the directioncosinea of whose
edges are /, m, n, and which passes through the curve U = 0, Y = 0. The
elimination may be conveniently performed aa follows: If x', y', z' be the co
ordinates of the point where any edge meets the directing curve, x, y, z those
/p _ y f M /' y jj'
of any point on the edge, we have  = . Calling the
I in n
common value of these functions 0, we have
x'=xl0,y' = y mO, z' z  n0.
Substitute these values in the equations U = 0, V = 0, which x'y'z' must
satisfy, and between the two resulting equations eliminate the unknown 8
and the result will be the equation of the cylinder.
Ex. 4. To find the cylinder, the directioncosines of whose edges are I, m,
n, and which envelopes the quadric Ax* + By* + Cz* 1. From the partial
differential equation, the curve of contact is the intersection of the quadric
with
Alx + Bmy + Cnz = 0.
Proceeding then, as in the last example, the equation of the cylinder is found
to be
(Al n  + Bm" + Cn 2 ) (Ax* + By* + Cz*  1) = (Alx + Bmy + Cfi/)*.
426. Conical Surfaces or Cones. These are generated by
the motion of a right line which constantly passes through a
fixed point. Expressing that the coordinates of this point
satisfy the equations of the right line, we have two relations
6 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
connecting the four constants in the general equations of
a right line. In this case, therefore, the equations of the
generating curve contain but two undetermined constants,
and the problem is of the kind discussed, Art. 423.
Let the equations of the generating line be
x a ^y  ft _z  7
I m n '
where a, ft, 7 are the known coordinates of the vertex of the
cone, and /, m, n are proportional to the directioncosines of
the generating line ; and where the equations, though appar
ently containing three undetermined constants, actually con
tain only two, since we are only concerned with the ratios of
the quantities I, m, n.
Writing the equations then in the form
x  a _ I y  ft _ m
z  7 n' z  7 n'
we see that the conditions of the problem must establish a
relation between I : n and m : n, and that the equation of the
x  a . (11  ft\
cone must be of the form    d> '
z  7 \ z  7/
It is easy to see that this is equivalent to saying that
the equation of the cone must be a homogeneous function of
the three quantities x  a, y  ft, z y, as may also be seen
directly from the consideration that the conditions of the prob
lem must establish a relation between the directioncosines
of the generator ; that these cosines being I : N /J(/ + ?/i' 2 + ;r),
&c., any equation expressing such a relation is a homogeneous
function of I, m, n, and therefore of x  a, y  ft, z  7, which
are proportional to I, m, n.
When the vertex of the cone is the origin, its equation is
x /i/\
of the form  = <); or, in other words, is a homogeneous
z r \z)
function of x, y, z.
The partial differential equation is found by putting
y _ n tfi /"5
u = , v= . in the equation of Art. 423, and when
z  7 z  7
cleared of fractions is
DIFFERENTIAL EQUATIONS OF FAMILIES OF SURFACES. 7
#l. U,, U 3 ,
zy, 0,  (x  a)
0, zy, (yft) =0,
or (x  a) /! + (y  ft) J7 2 + (z  y) U 3 = 0.
This equation evidently expresses that the tangent plane at
any point of the surface must always pass through the fixed
point afty.
We have already given in Ex. 9, Art. 121, the method of
forming the equation of the cone standing on a given curve ;
and (Art. 277) the method of forming the equation of the
cone which envelopes a given surface.
427. Conoidal Surfaces. These are generated by the
motion of a line which always intersects a fixed axis and re
mains parallel to a fixed plane. These two conditions leave
two of the constants in the equations of the line undeter
mined, so that these surfaces are of the class considered (Art.
423). If the axis is the intersection of the planes a, ft, and
the generator is to be parallel to the plane y, the equations of
the generator are a = q/8, y = c 2 , and the general equation of
conoidal surfaces is obviously ^ = <#> (7)
The partial differential equation is (Art. 423)
I U lt U,, U 3
\fta l  aft lt fta. 2  aft. 2 , fta 3  aft 3
I 7i> 7 2 . 7s = 0>
where a = a^x + a 2 y + a^z + o 4 , &c. The lefthand side of the
equation may be expressed as the difference of two deter
minants ft (Uia 2 y s )  a (U l ft. 2 y 3 ) = 0.
This equation may be derived directly by expressing that
the tangent plane at any point on the surface contains the
generator; the tangent plane, therefore, the plane drawn
through the point on the surface, parallel to the directing
plane, and the plane aft  aft' joining the same point to the
axis, have a common line of intersection. The terms of the
determinant just written are the coefficients of x, y, z in the
equations of these three planes.
8 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
In practice we are almost exclusively concerned with
right conoids ; that is, where the fixed axis is perpendicular
to the directing plane. If that axis be taken as the axis of z,
and the plane for plane of xy, the functional equation is
y = x<f> (z), and the partial differential equation is xTJ l + yU< i
= 0.
The lines of greatest slope (Art. 421) are in this case
always orthogonally projected into circles on the directing
plane. For in virtue of the partial differential equation just
written, the equation of Art. 421,
U%dx  Utdy = 0,
transforms itself into xdx + ydy = 0, which represents a series
of concentric circles. The same thing is evident geometric
ally ; for the lines of level are the generators of the system ;
and these being projected into a series of radii all passing
through the origin, are cut orthogonally by a series of con
centric circles.
Ex. 1. To find the equation of the right conoid passing through the
axis of z and through a plane curve, whose equations are x = a, F (y t z) = 0. Eli
minating then a;, y, z between these equations and y = c^x, z = c. 2> we get
F (Cja, c a ) = 0; or the required equation is F ( , z j = 0.
\ x /
Wallis's conocuneus is when the fixed curve is a circle x = a, y + z^ = r 2 .
Its equation is therefore a 2 ;/ 2 + 2 2 2 = r 2 a; 2 .
Ex. 2. Let the directing curve be a helix, the fixed line being the axis of
the cylinder on which the helix is traced. The equation is that given Ex.
1, Art. 371. This surface is often presented to the eye, being that formed by
the under surface of a spiral staircase.
[Ex. 3. A right conoid of special interest in Statics and Dynamics is
Ball's Cylindroid.* If 6 be the angle between the generator and & fixed plane
through the axis, the surface is defined by the relation
z h sin 28
and its equation is therefore
2 (a; 2 + j/ 2 )  Zhxy = 0.
Ex. 4. Any rijht conoid may be expressed by two parameters by equa
tions of the form
x=pcosq, y=ps'\uq, 2
Thus for the conoid of Ex. 2, </> (3) = . (See Ex. 1, Art. 371.)]
* R. S. Ball, The Theory of Screws (Dublin, 1876).
DIFFERENTIAL EQUATIONS OF FAMlLlhs <>i si l;l \. is '.)
Ex. 5. The equation of any surface generated by the motion of a right
line meeting two fixed right lines a/3, yS, must be of the form = f { ).
ft \'/
428. Surfaces of Revolution. The fundamental property
of a surface of revolution is that its section perpendicular to
its axis must always consist of one or more circles whose
centres are on the axis. Such a surface may therefore be
conceived as generated by a circle of variable radius whose
centre moves along a fixed right line or axis, and whose plane
is perpendicular to that axis. If the equations of the axis be
v ^^=y ~ & = Y, then the generating circle in any posi
tion may be represented as the intersection of the plane per
pendicular to the axis lx + my + nz = c lt with the sphere whose
centre is any fixed point on the axis,
These equations contain but two undetermined constants ;
the problem, therefore, is of the class considered (Art. 423),
and the equation of the surface must be of the form
(x  a)' 2 + (y  /9) 2 + (z  y}' 2 = (Ix + my f nz) .
When the axis of z is the axis of revolution, we may take the
origin as the point afiy, and the equation becomes
# 2 + ?/ 2 + z 2 = <f> (z), or z = i/r (x + y).
The partial differential equation is found by the formula of
Art. 423 to be
U lt U 2 , U 3
I, m, n
xa,y 
= 0,
or {m(zi)n(y ft)} Ui
+ {n (x a)l(z 7)} U 2 + {l(y ft)m(z a)} *7 8 =
When the axis of z is the axis of revolution, this reduces to
The partial differential equation expresses that the normal
always meets the axis of revolution. For, if we wish to ex
press the condition that the two lines
xa y @_z
~T m n
10 ANALYTIC GEOMETRY OF THBEE DIMENSIONS.
should intersect, we may write the common value of the equal
fractions in each case, 6 and 6'. Solving then for x, y, z, and
equating the values derived from the equations of each line,
we have
a+W = x'+ Ufl, ft + me = y' + U^, 7 + nS = z + U/ \
whence, eliminating 9, 6', the result is the determinant already
found
ff,, U*, U 3
I, m, n
X  a, y'  0, z'y
=
[Ex. 1. Any surface of revolution ma) be expressed by two parameters as
follows :
x = p cos q, y = p sin q, z = <j> (p).
Ex. 2. By a suitable choice of parameters (u, v) the square of the linear
element (Art. 377) of a surface of revolution may be expressed in the form
ds = f (u) (du* + dv*)
and hence (Art. 390), any surface whose linear element can be thus expressed
is deforniable into a surface of revolution.
For example, the right conoid of Ex. 2, Art. 427, satisfies this condition as
may be seen by using the form given in Ex. 4 of the same Art.]
429. The equation of the surface generated by the revo
lution of a given curve round a given axis is found (Art. 424)
by eliminating x, y, z between
Ix + my + nz = u, (x a)' 2 + (y  /3) 2 + (z  y)' 2 = v,
and the two equations of the curve ; replacing then u and v by
their values. We have already had an example of this (Ex. 3,
Art. 121), and we take, as a further example, to find the sur
face generated by the revolution of a circle
y = Q, (xa)* + z* = r
round an axis in its plane (the axis of z).
Putting z = u, x* + y = v, and eliminating between these
equations and those of the circle, we get
{J(v)a}* + u* = r*, or { J(a? + y*)  aY~ + z 2 = r\
which, cleared of radicals, is
(x 2 + y* + z* + a'* r 2 ) 2 = 4a 2 (z 2 + y}.
It is obvious that when a is greater than r, that is to say, when
the revolving circle does not meet the axis, neither can the
surface, which will be the form of an anchor ring, the space
fcll I I.U..N HAL, EQUATIONS <>! KAMIUl.s t.. M ] 1
about the axis being empty. On the other hand, when the
revolving circle meets the axis, the segments into which the
axis divides the circle generate distinct sheets of the surface,
intersecting in points on the axis z= J(r*a' 2 ), which are
nodal points on the surface.
The sections of the anchor ring by planes parallel to the
axis are found by putting y = constant, in the preceding equa
tion. The equation of the section may immediately be thrown
into the form SS' = constant, where S and S f represent circles.
The sections are Cassinians of various kinds (see fig. Higher
Plane Curves, p. 44). It is geometrically evident, that as the
plane of section moves away from the axis, it continues to cut
in two distinct ovals, until it touches the surface y = a  r
when it cuts in a curve having a double point (Bernoulli's
Lemniscate) ; after which it cuts in a continuous curve.
Ex. Verify that x 3 + y 3 + z 3  3xyz = r 3 is a surface of revolution.
Ans. The axis of revolution is x = y = z.
430. The families of surfaces which have been considered
are the most interesting of those whose equations can be ex
pressed in the form u = <f> (v). We now proceed to the 
when the equations of the generating curve include more than
two parameters. By the help of the equations connecting
these parameters, we can, in terms of any one of them, express
all the rest, and thus put the equations of the generating
curve into the form
F{x,y,z,c,<l>(c'), + (c),&c.} = 0,f{x,y,z,c,<l>(c),1r (c),&c.} = 0.
The equation of the surface generated is obtained by elimi
nating c between these equations; and, as has been already
stated, all surfaces are said to be of the same family for which
the form of the functions F and / is the same, whatever be the
forms of the functions </>, i/r, &c. But since evidently the
elimination cannot be effected until some definite form has
been assigned to the functions </>, ^, &c., it is not generally
possible to form a single functional equation including all
surfaces of the same family ; and we can only represent them,
as above written, by a pair of equations from which there
12 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
remains a constant to be eliminated. We can, however,
eliminate the arbitrary functions by differentiation, and obtain
a partial differential equation, common to all surfaces of the
same family ; the order of that equation being, as we shall
presently prove, equal to the number of arbitrary functions
<f>, ^, &c.
It is to be remarked, however, that in general the order of
the partial differential equation obtained by the elimination
of a number of arbitrary functions from an equation is higher
than the number of functions eliminated. Thus, if an equa
tion include two arbitrary functions </>, ty, and if we differentiate
with respect to x and y, which we take as independent vari
ables, the differential equations combined with the original one
form a system of three equations containing four unknown
functions <f>, i/r, ^>', ty'. The second differentiation (twice
with regard to x, twice with regard to y, and with regard to
x and y) gives us three additional equations ; but, then, from
the system of six equations it is not generally possible to
eliminate the six quantities $, ty, </>', ^', <", ty" '. We must,
therefore, proceed to a third differentiation before the elimi
nation can be effected. It is easy to see, in like manner, that
to eliminate n arbitrary functions we must differentiate
2n  1 times. The reason why, in the present case, the order
of the differential equation is less, is that the functions elimi
nated are all functions of tJie same quantity.
431. In order to show this, it is convenient to consider
first the special case, where a family of surfaces can be ex
pressed by a single functional equation. This will happen
when it is possible by combining the equations of the*
generating curve to separate one of the constants so as to
throw the equations into the form
u = c lf F (x, y, z, c lf c a . . .C H ) = 0.
Then expressing, by means of the equations of condition, the
other constants in terms of c lt the result of elimination is
plainly of the form
F {x, y, z, u, <f> (w), >/r ( u \ &c.} = 0.
DIFFERENTIAL EQUATIONS OF FAMILIES OF SURFACES. 13
Now, if we denote by F lt the differential with respect to x
of the equation of the surface, on the supposition that u is
constant, and similar differentials in y, z by F 2 , F a , we have
But, in these equations, the derived functions <', i/r', &c., only
jp*
enter in the term ; they can, therefore, be all eliminated
all
together, and we can form the equation, homogeneous in
Ui, U 2 , U s ,
=o>
which contains only the original functions <f>, t/r, &c. If we
write this equation V = 0, we can form from it, in like manner,
the equation
u lt u,, u s
V lt F 2 , F 3
Miy 'i/
1 . M/rt. tto V/.
1 7 ^7 ,)
which still contains no arbitrary functions but the original
(f>, i/r, &c., but which contains the second differential co
efficients of U, these entering into F 1? F 2 , F 3 . From the
equation last found we can in like manner form another, and
so on ; and from the series of equations thus obtained (the
last being of the w th order of differentiation) we can eliminate
the n functions $, i/r, &c.
If we omit the last of these equations we can eliminate all
but one of the arbitrary functions, and according to our choice
of the function to be retained, can obtain n different equations
of the order nI, each containing one arbitrary function.
These are the first integrals of the final differential equation
of the n ib order. In like manner we can form $n (n  1) equa
tions of the second order, each containing two arbitrary func
tions, and so on.
432. If we take x and y as the independent variables, and
as usual write dz=pdx + qdy, dp = rdx + sdy, &c., the process
14 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
of forming these equations may be more conveniently stated
as follows : " Take the total differential of the given equation
on the supposition that u is constant,
F^x + F. 2 dy + F 3 (pdx + qdy} = ;
put dy = mdx, and substitute for m its value derived from the
differential of u = 0, viz.
u^dx + u z dy + u 3 (pdx + qdy} = 0."
For, if we differentiate the given equation with respect to
x and y, we get
dF
and the result of eliminating = from these two equations is
the same as the result of eliminating m between the equations
F l +pF 3 + m (F 3 + qFs) = 0, % +pu 3 + m ( 2 + quj = 0.
It is convenient in practice to choose for one of the equations
representing the generating curve its projection on the plane
of xy ; then, since this equation does not contain , the value
of m derived from it will not contain p or q, and the first
differential equation will be of the form
R being also a function not containing^ or q. The only terms
then containing r, s, or t in the second differential equation are
those derived from differentiating p + qm, and that equation
will be of the form
r + 2sm + tm 2 = S,
where S may contain x, y, z, p, q, but not r, s, or t. If now
we had only two functions to eliminate, we should solve for
these functions from the original functional equation of the
surface, and from p + qm = R ; and then substituting these
values in m and in S, the form of the final second differential
equation would still remain
where m' and S' might contain x, y, z, p, q. In like manner
}i we had three functions to eliminate, and if we denote the
DIFFERENTIAL EQUATIONS OF FAMILIES OF SURFACES. 15
partial differentials of z of the third order by a, /9, 7, 8, the
partial differential equation would be of the form
a 4 3m ft + 3w 2 7 + m?8 = T.
And so on for higher orders. This theory will be illustrated
by the examples which follow.
433. Surfaces generated by lines parallel to a fixed plane.
This is a family of surfaces which includes conoids as a par
ticular case. Let us, in the first place, take the fixed plane
for the plane of xy. Then the equations of the generating
line are of the form z = c lt y = c 2 x + c 3 . The functional equa
tion of the surface is got by substituting in the latter equation
for c 2 , <j> (z}, and for c 3 , i/r (z}. Since in forming the partial
differential equation we are to regard z as constant, we may
as well leave the equations in the form z = c lf y = c 3 a? + c r
These give us
According as we eliminate c 3 or c 2 , these equations give us
p + qc 2 = 0, px + qy = qc s . There are, therefore, two equations
of the first order, each containing one arbitrary function, viz.
p + q<l> (z) = 0,px + qy = q^r (z}.
To eliminate arbitrary functions completely, differentiate
p + qm = Q, remembering that since w = c 2 , it is to be regarded
as constant, when we get
r + 2sm + tntf = 0,
and eliminating m by means oip + qm = Q, the required equa
tion is
cfr  %pqs +pH = 0,
Next let the generating line be parallel to ax+ by + cz ; its
equations are
ax + by + cz = c lt y = c< t x + c i ;
and the functional equation of the family of surfaces is got
by writing for c 2 and c 3 , functions of ax + by + cz. Differ
entiating, we have
a + cp + m (b + cq) = 0, m = c. 2 .
The equations got by eliminating one arbitrary function are
therefore
16 ANALYTIC GEOMETEY OF THEEE DIMENSIONS.
a + cp + (b + cq) <f> (ax + by + cz) 0,
(a + cp) x + (b + cq} y=(b + cq) ^ (ax + by + cz).
Differentiating a f bm + c (p + mq) = 0, and remembering that
m is to be regarded as constant, we have
r + 2sm + im? = 0,
and introducing the value of m already found,
(b + cq) 2 r  2 (a + cp) (b + cq) s + (a + cp)t = 0.
434. This equation may also be arrived at by expressing
that the tangent planes at two points on the same generator
intersect, as they evidently must, on that generator. Let
a, /?, 7 be the running coordinates, x, y, z those of the point
of contact ; then any generator is the intersection of the tan
gent plane
with a plane through the point of contact parallel to the fixed
plane
a (a  x) + b (ft  y) + c (7  z) = 0,
whence (a + cp) (a  x) + (b + cq) (/3  y) = 0.
Now if we pass to the line of intersection of this tangent plane
with a consecutive plane, a, ft, 7 remain the same, while
y> Z >P, <? vary. Differentiating the equation of the tangent
plane, we have
(rdx + sdy) (ax) + (sdx + tdy) (fty) = 0.
And eliminating a x, ft y,
(b + cq) (rdx + sdy) = (a + cp) (sdx + tdy).
But since the point of contact moves along the generator
which is parallel to the fixed plane, we have
adx + bdy + cdz = 0, or (a + cp) dx + (b + cq) dy = 0.
Eliminating then dx, dy from the last equation, we have, as
before,
(b + cq) 2 r2 (a + cp) (b + cq) s + (a + cp) 2 t = 0.
435. Surfaces generated by lines which meet a fixed axis.
This class also includes the family of conoids. In the first
place let the fixed axis be the axis of z ; then the equations
of the generating line are of the form y = c^x, z = c. 2 x + c 3 ; and
DIFFERENTIAL EQUATIONS OF FAMILIES OF SURFACES. 17
the equation of the family of surfaces is got by writing in the
latter equation for c 2 and c 3 , arbitrary functions of y : x.
Differentiating, we have m = c l ,p + mq = c%, whence
px + qy = x<f> (\ and z  px  qy = ty (  J.
Differentiating again, we have r+ 2sm + tm* = Q, and putting
?/
for m its value = c 1 = , the required differential equation is
27
rx 2 + Isxy + ty 2 = 0.
This equation may also be obtained by expressing that
two consecutive tangent planes intersect in a generator. As
in the last article, we have for the intersection of two consecu
tive tangent planes
(rdx + sdy} (a  x) + (sdx t tdy) (fi  y} = 0.
But any generator lies in the plane
ay = @x, or (a  x) y = (& ?/) x.
Eliminating therefore,
x (rdx + sdy) + y (sdx + tdy) = 0.
But ~= = J. Therefore, as before, rx 2 + l&xy + ty 2 = 0.
ax a x
More generally, let the line pass through a fixed axis a/9,
where a = ax+ by + cz + d, ft=a'x+ b'y + c'z + d'. Then the
equations of the generating line are a = c a /3, y = c. 2 x + c 3 , and
the equation of the family of surfaces is y = x<f> ^ +
Differentiating, we have
?7i = c 2 , a + cp+ m (b + cq)= c^ (a + c'p + m (b' + c'q}}.
Differentiating again, we have r + 2sm + tm 2 = 0, and putting
in for m from the last equation, the required partial differential
equation is
{(a + cp) 0 (a + c'pW 2 t + {(b + cq) 0 (V + c'q)a}' 2 r
 2 {(a + cp} 0(a' + c'p}a\ \(b + cq)  (b' + cq)a] s = 0.
436. If the equation of a family of surfaces contain n
arbitrary functions of the same quantity, and if it be required
to determine a surf ace of the family which shall pass through
n fixed curves, we write down the equations of the generating
VOL. II. 2
18 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
curve u = c l , F (x, y, z, c lt c 2 , &c.) = 0, and expressing that the
generating curve meets each of the fixed curves, we have a
sufficient number of equations to eliminate c lt c. 2 , &c. Thus,
to find a surface of the family x + y<j> (z} + i/r (z) = which shall
pass through the fixed curves
y = a, F(x,z) = Q; y=a, F l (x,z) = Q.
The equations of the generating line being z = c lt x = yc 3 + c s ,
we have, by substitution,
F (ac. 2 + c 3 , Cj) = 0, F l (c 3  ac.,, Cj) = 0,
or, replacing for c lf f 3 , their values,
F{X + CZ (ay}, z\ = Q, F l {x  c 2 (a + y), z} = 0,
and by eliminating c. 2 between these the required surface is
found.
Ex. Let the directing curves be
y = a, + * = !; t/ = a, x* + * = c*.
We eliminate c. 2 between
Solving for c a from each, we have
a  y a + y
The result is apparently of the eighth degree, but is resolvable into two
conoids distinguished by giving the radicals the same or opposite signs in the
last equation.
437. We have now seen, that when the equation of a
family of surfaces contains a number of arbitrary functions of
the same quantity, it is convenient, in forming the partial
differential equation, to substitute for the equation of the sur
face, the two equations of the generating curve. It is easy
to see, then, that this process is equally applicable when the
family of surfaces cannot be expressed by a single functional
equation. The arbitrary functions which enter into the
equations (Art. 430) are all functions of the same quantity,
though the expression of that quantity in terms of the co
ordinates is unknown. If then differentiating that quantity
gives dy = mdx, we can eliminate the unknown quantity ni,
DIFFERENTIAL EQUATIONS OF FAMILIES OF SURFACES. 19
between the total differentials of the two equations of the
generating curve, and so obtain the partial differential equa
tion required. In practice it is convenient to choose for one
of the equations of the generating curve, its projection on the
plane xy.
For example, let it be required to find the general equation
of ruled surfaces : that is to say, of surfaces generated by the
motion of the right line. The equations of the generating
line are
z = cp + c 3 , y = c<iX + c 4 ,
and the family of surfaces is expressed by substituting for
c a , c 3 , c 4 arbitrary functions of c v Differentiating, we have
p + mq = c lt m = c 2 .
Differentiating the first of these equations, m being proved to
be constant by the second, we have
r + 2sm + tm 2 = 0.
As this equation still includes m or c 2 , the expression for
which, in terms of the coordinates is unknown, we must
differentiate again, when we have
a + 3/3m + 3yni? + &m 3 = 0,
where a, ft, 7, B are the third differential coefficients. Elimi
nating m between the cubic and quadratic just found, we have
the required partial differential equation. It evidently resolves
itself into the two linear equations of the third order got by
substituting in turn for m in the cubic the two roots of the
quadratic.
This equation might be got geometrically by expressing
that the tangent planes at three consecutive points on a
generator pass through that generator. The equation
pdx + qdy = dz
is a relation between^, q,  1, which are proportional to the
directioncosines of a tangent plane, while dx, dy, dz are
proportional to the directioncosines of any line in that plane
passing through the point of contact. If, then, we pass to a
second tangent plane, through a consecutive point on the
same line, we are to make p, q vary while the mutual ratios
of dx, dy, dz remain constant. This gives
2*
20 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
rdx z + Zsdxdy + tdy' 2 = 0.
To pass to a third tangent plane, we differentiate again,
regarding dx : dy constant ; and thus have
adx* + Spdtfdy + Zydxdy* + Bdy 3 = 0.
Eliminating dx : dy between the last two equations, we have
the same equation as before.
The first integrals of this equation are found, as explained
(Art. 431), by omitting the last equation and eliminating all
but one of the constants. Thus we have the equation
p + mq = c 1 , from which it appears that one of the integrals is
p + mq = <f> (in),
where m is one of the roots of r + 2sm + tm 2 = 0. The other two
first integrals are
y  mx = i/r (m), and zpx mqx = x (#0
The three second integrals are got by eliminating m from
any pair of these equations.
438. Envelopes. If the equation of a surface include n
parameters connected by n  1 relations, we can in terms of
any one express all the rest, and throw the equation into
the form
z = F {x, y, c, </> (c), V (c), &c.}.
JT/1
Eliminating c between this equation and 5 = 0, which we shall
write F' = 0, we find the envelope of all the surfaces obtained
by giving different values to c. The envelopes so found are
said to be of the same family aslong as the form of the function
F remains the same, no matter how the forms of the functions
<, $, &c., vary. The curve of intersection of the given surface
with F' is the characteristic (see p. 30) or line of intersection
of two consecutive surfaces of the system. Considering the
characteristic as a moveable curve from the two equations of
which c is to be eliminated, it is evident that the problem of
envelopes is included in that discussed Art. 430, &c. If the
function F contain n arbitrary functions <, i/r, &c., then since
F' contains </>', ^', &c., it would seem, according to the theory
previously explained, that the partial differential equation of
DIFFERENTIAL EQUATIONS OF FAMILIES OF SURFACES. 21
the family ought to be of the 2W" 1 order. But on examining
the manner in which these functions enter, it is easy to see that
the order reduces to the th . In fact, differentiating the equa
tion z = F, we get
JTjl fJW
p = F l + ^c l , q = F 2 + ^c. 2 , that is, p = F t + c l F t q = F 2 + c 2 F',
but since F' = 0, we have p = F lt q = F. 2 , where, since' F, and F 2
are the differentials on the supposition that c is constant, these
quantities only contain the original functions <f>, ty and not the
derived <', i/r'. From this pair of equations we can form
another, as in the last article, and so on, until we come to
the w th order, when, as easily appears from what follows, we
have equations enough to eliminate all the parameters.
439. We need not consider the case when the given equation
contains but one parameter, since the elimination of this be
tween the equation and its differential gives rise to the equation
of a definite surface and not of a family of surfaces. Let the
equation then contain two parameters a, b, connected by an
equation giving b as a function of a, then between the three
equations z = F, p = F lt q = F 2 , we can eliminate a, b, and the
form of the result is evidently / (x, y, z, p, g) = 0.
For example, let us examine the envelope of a sphere of
fixed radius, whose centre moves along any plane curve in the
plane of xy. This is a particular case of the general class of
tubular surfaces which we shall consider presently.
N )w the equation of such a sphere being
(xa)' 2 + (yp)* + z' 2 = r 2 ,
and the conditions of the problem assigning a locus along which
the point aft is to move, and therefore determining /8 in terms
of a, the equation of the envelope is got by eliminating a
between
(xa)*+{y4>(a)}* + z* = v*, (xa) + {y<f> (a)} <fr' () = 0.
Since the elimination cannot be effected until the form of the
function < is assigned, the family of surfaces can only be ex
pressed by the combination of two equations just written,
We might also obtain these equations by expressing that the
22 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
surface is generated by a fixed circle, which moves so that
its plane shall be always perpendicular to the path along which
its centre moves. For the equation of the tangent to the
locus of a8 is
jo
y/3=*^(xa) or y  </> (a) = </>' (a) (x  a).
And the plane perpendicular to this is
(x  a) + [y  $ (a)} $ (a) = 0,
as already obtained. To obtain the partial differential equa
tion, differentiate tjae equation of the sphere, regarding a, /3 as
constant, when we have
x  a +pz = 0, y  ft + qz = 0.
Solving for x  a, y  /3 and substituting in the equation of
the sphere, the required equation is
z z (l+p + q*) = r\
We might have at once obtained this equation as the geo
metrical expression of the fact that the length of the normal
is constant and equal to r, as it obviously is.
440. Before proceeding further we wish to show how the
arbitrary functions which occur in the equation of a family
of envelopes can be determined by the conditions that the
surface in question passes through given curves. The tangent
line to one of the given curves at any point of course lies in
the tangent plane to the required surface ; but since the en
veloping surface has at any point the same tangent plane as
the enveloped surface which passes through that point, it
follows that each of the given curves at every point of it
touches the enveloped surface which passes through that point.
If, then, the equation of the enveloped surface be
z = F(x, y, c lt c a ...O.
the envelope of this surface can be made to pass through n  1
given curves ; for by expressing that the surface, whose equa
tion has just been written, touches each of the given curves,
we obtain n  1 relations between the constants c lt c, &c.,
which, combined with the two equations of the characteristic,
enable us to eliminate these constants.
DIFFERENTIAL EQUATIONS OF FAMILIES OF SURFACES.
For example, the family of surfaces discussed in the last
article contains but two constants and one arbitrary function,
and can therefore be made to pass through one given curve.
Let it then be required to find an envelope of the sphere
(s  a) 2 4 (7/) 3 + *' = /,
which shall pass through the right line x = mz, y = 0. The
points of intersection of this line with the sphere being given
by the quadratic
(mz  a) 2 + * + = r\ or (1 + m^z"  2mza + a 2 + &  r = 0,
the condition that the line should touch the sphere is
(1 + m 2 ) (a 2 + 2  r 2 ) = mV.
We see thus, that the locus of the centres of spheres touching
the given line is an ellipse. The envelope required, then, is
a kind of elliptical anchor ring, whose equation is got by
eliminating a, /3 between
(x  a) 2 + (y  ) 2 + z* = r\ (1 + m 2 ) (a 2 + 2  r 2 ) = W 2 a 2 ,
Cea)Za + (y)d = 0, ada + (1 + w 8 ) d/8 = 0,
from which last two equations we have
The result is a surface of the eighth degree.
441. Again, let it be required to determine the arbitrary
function so that the enveloping surface may also envelope a
given surface. At any point of contact of the required sur
face with the fixed surface z=f(x, y}, the moveable surface
z = F (x, y, c v c 2 , fec.) which passes through that point, has
also the same tangent plane as the fixed surface. The values
then of p and q derived from the equations of the fixed surface
and of the moveable surface must be the same. Thus we have
fi = F lt fi = F a , and if between these equations and the two
equations z = F, z=f, which are satisfied for the point of
contact, we eliminate x,y,z, the result will give a relation
between the parameters. The envelope may thus be made
to envelope as many fixed surfaces as there are arbitrary
functions in the equation.
Thus, for example, let it be required to determine a tubular
surface of the kind discussed in the last article, which shall
24 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
touch the sphere x 2 + y 2 + z' 2 = R 2 . This surface must then
touch (x  a) 2 + (y  yS) 2 + z 2 = r 2 . We have therefore
x : y : z = x  a : y  /3 : z ;
conditions which imply z = 0, fix = ay. Eliminating x and y
by the help of these equations, between the equation of the
fixed and moveable sphere, we get
4 (a 2 + /3 2 ) R 2 = (R 2  r 2 + a 2 + /3 2 ) 2 .
This gives a quadratic for a 2 + /3 2 , whose roots are (7? + r) 2 ;
showing that the centre of the moveable sphere moves on one
or other of two circles, the radius being either R + r or R  r.
The surface required is therefore one or other of two anchor
rings, the opening of the rings corresponding to the values
just assigned.
442. We add one or two more examples of families of en
velopes whose equations include but one arbitrary function.
To find the envelope of a right cone whose axis is parallel to
the axis of z, and whose vertex moves along any assigned
curve in the plane of xy. Let the equation of the cone in its
original position be z 2 = m 2 (x 2 + y 2 ) ; then if the vertex be
moved to the point a, /?, the equation of the cone becomes
z 2 = m 2 {(x  a) 2 + (y  /3) 2 }, and if we are given a curve along
which the vertex moves, /3 is given in terms of a. Differ
entiating, we have pz = m 2 (x  a), qz = m 2 (y  #) ; and elimi
nating, we have
p 2 + q 2 = m 2 .
This equation expresses that the tangent plane to the surface
makes a constant angle with the plane of xy, as is evident
from the mode of generation. It can easily be deduced hence,
that the area of any portion of the surface is in a constant
ratio to its projection on the plane of xy.
443. The families of surfaces, considered (Arts. 439, 442),
are both included in the following : To find the envelope of a
surface of any form which moves without rotation, its motion
being directed by a curve along which any given point of the
surface moves. Let the equation of the surface in its original
DIFFERENTIAL EQUATIONS OF FAMILIES OF SI! 25
position be z = F (x, y}, then if it be moved without turning
so that the point originally at the origin shall pass to the
position a/?7, the equation of the surface will evidently be
z  7 = F (x  a, y  /3). If we are given a curve along which
the point a(3y is to move, we can express a, ft in terms of 7,
and the problem is one of the class to be considered in the
next article, where the equation of the envelope includes two
arbitrary functions. Let it be given, however, that the direct
ing curve is drawn on a certain known surface, then, of the
two equations of the directing curve, one is known and only
one arbitrary, so that the equation of the envelope includes but
one arbitrary function. Thus, if we assume /9 an arbitrary
function of a, the equation of the fixed surface gives 7 as a
known function of a, /?. It is easy to see how to find the
partial differential equation in this case. Between the three
equations
solve for x  a, y  ft, z  7, when we find
xa=f(p, q}, yP=f (p,q), z  7 =/" (p, q).
If, then, the equation of the surface along which afiy is to
move be F (a, @, 7) = 0, the required partial differential equa
tion is
? {x f (p. q}, y f (p, q), * f" (P, ?)} = 0.
The three functions /, /, /", are evidently connected by the
relation df" =pdf+ qdf.
It is easy to see that the partial differential equation just
found is the expression of the fact, that the tangent plane at
any point on the envelope is parallel to that at the corre
sponding point on the original surface.
Ex. To find the partial differential equation of the envelope of a sphere
of constant radius whose centre moves along any curve traced on a fixed equal
sphere
fc2 + ,,2 + ,2 = r 2.
The equation of the moveable sphere is (x  a) a + (y  /3) s + (*  y) 1 = r*>
whence x  a + p (z  y) = 0, y  + q (2  y) = 0,
and we have
x ~ a = ~ ^
26 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
If we write 1 + p* } g 2 = p 2 it is easy to see, by actual differentiation, that
the relation is fulfilled
The partial differential equation is
(x p + prY + (yp + 9>') 2 + (
or (z 8 + if + z*) l + pa + j + 2 x + _ 2 r = 0.
444. We now proceed to investigate the form of the partial
differential equation of the envelope, wJien the equation of the
nwveable surface contains three constants connected by two
relations. If the equation of the surface be z = F(x, y, a, b, c),
then we have p = F lt q = F 2 . Differentiating again, as in Art.
432, we have
r + sm = F u + mF^, s + tm = jF 12 + wF 22 ;
and eliminating m, the required equation * is
(rF ll )(tF,. 2 ) = (sF 1 ^.
The functions F u , F^, F. 2 . 2 contain a, b, c, for which we
are to substitute their values in terms of p, q, x, y, z derived
from solving the preceding three equations, when we obtain
an equation of the form
Rr + 2Ss + Tt+U (rt  s 2 ) = V,
where R, S, T, U, V are connected by the relation
RT+UV=S\
445. The following examples are among the most im
portant of the cases where the equation includes three para
meters.
Developable Surfaces. These are the envelope of the plane
z = ax + by + c, where for b and c we may write </> (a) and
T/T (a) . Differentiating, we have p = a,q = b, whence q = </> (p).
Any surface therefore is a developable surface if p and q are
connected by a relation independent of x, y, z. Thus the
family (Art. 442) for which p % + q* = w 2 , is a family of develop
able surfaces. We have also z  px  qy fy (p), which is the
* I owe to Professor Boole my knowledge of the fact, that when the
equation of the moveable surface contains three parameters, the partial differ
ential equation is ofi the form stated above. See his Memoir, Phil. Trans.,
1862, p. 437.
DIFFERENTIAL EQUATIONS OF FAMILIES OF SURFACES. 27
other first integral of the final differential equation. This
last is got by differentiating again the equations p * a, q 6,
when we have r + sm = 0, s + tm = 0, and eliminating m,
rts 2 = 0,
which is the required equation.
By comparing Arts. 285, 295, 311, it appears that the
condition rt = s~ is satisfied at every parabolic point on a sur
face. The same thing may be shown directly by transforming
the equation rt  s 2 = into a function of the differential
coefficients of U, by the help of the relations
= rU
3
=  tU 3 ;
when the equation rt  s 2 = is found to be identical with the
equation of the Hessian. We see, accordingly, that every
point on a developable is a parabolic point, as is otherwise
evident, for since (Art. 330) the tangent plane at any point
meets the surface in two coincident right lines, the two in
flexional tangents at that point coincide. The Hessian of a
developable must therefore always contain the equation of the
surface itself as a factor. The Hessian of a surface of any
degree n being of the degree 4n  8, that of a developable
consists of the surface itself, and a surface of 3n  8 degree
which we shall call the ProHessian.
In order to find in what points the developable is met by
the ProHessian, I form the Hessian of the developable
surface of the r^ degree (see Arts. 329, 330) xu + y*v = 0,
and find that we get the developable itself multiplied by a
series of terms in which the part independent of x and y i^
f!^  (^\\ This proves that any generator xy
dw i \dwzd/ )
meets the ProHessian in the first place, where xy meets v ;
that is to say, twice in the point on the cuspidal curve (w),
and in r  4 points on the nodal curve (a;), Art. 330 ; and in
the second place, where the generator meets the Hessian
of u considered as a binary quantic ; that is to say, in the
Hessian of the system formed by these r  4 points combined
28 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
with the point on (m) taken three times ; in which Hessian
the latter point will be included four times. The intersection
of any generator with the Pro Hessian consists of the point
on (i) taken six times, of the r  4 points on (#), and of
2 (r  5) other points, in all 3r  8 points.*
446. Tubular Surfaces. Let it be required to find the
differential equation of the envelope of a sphere of constant
radius, whose centre moves on any curve. We have, as in
Art. 443,
(xa?+(yp)*+(z^ = R\
xa+p(zj)=0, y/3 + q (2^ = 0,
whence 1 +p 2 + (z  7) r + m {pq + (z  7) s\ = 0,
pq + (z  7) s + m {1 + q 2 + (z  7) t\ = 0.
And therefore
{l+p* + (zy)r}{l + q* + (z<y)t} = {pq + (z y )s}\
R
Substituting for z  7 its value    from the first three
(l+jr + <n*
equations, this becomes
JR 2 (rt  s 2 )  E {(1 + g 2 ) r  2pqs + (1 +p*) t} x /(l +p*+ g 2 )
which denotes, Art. 311, that at any point on the required
envelope one of the two principal radii of curvature is equal
to B, as is geometrically evident.
447. We shall briefly show what the form of the differ
ential equation is when the equation of the surface whose
envelope is sought contains four constants. We have, as before,
in addition to the equation of the surface, the three equations
p = F,,q = F 2 , (r  F u ) (t  F 22 ) ~ (s  F 12 ) 2 .
* Cayley has calculated the equation of the ProHessian (Quarterly
Journal, vol. vi p. 108) in the case of the dovelopables of the fourth and fifth
orders, and of that of the sixth order considered, Art. 348. The ProHessian
of the developable of the fourth order is identical with the developable itself.
In the other two cases the cuspidal curve is a cuspidal curve also on the Pro
Hessian, and is counted six times in the intersection of the two surfaces. I
suppose it may be assumed that this is generally true. The nodal curve is
but a simple curve on the ProHessian, and therefore is only counted twice
in the intersection.
DIFFERENTIAL EQUATIONS OF FAMILIES OF SURFACES. 29
Let us, for shortness, write the last equation pr = o 2 , and let
us write aF m = A, 0F in = B, yF^ = C, 8F 22!1 = D;
then, differentiating pr = a\ we have
(A + Bm) r + (C + Dm) p2(B + Cm) a = 0.
Substituting for m from the equation <r + rm = 0, and remem
bering that pr = <r 2 , we have
AT*  3Bor 2 + 3<7<r 2 T  Da 3 = 0,
in which equation we are to substitute for the parameters im
plicitly involved in it, their values derived from the preceding
equations. The equation is, therefore, of the form
a + 3w + 37W 2 + Bm 3 = U,
where m and U are functions of x, y, z, p, q, r, s, t. In like
manner we can form the differential equation when the equa
tion of the moveable surface includes a greater number of
parameters.
448. Having in the preceding articles explained how
partial differential equations are formed, we shall next show
how from a given partial differential equation can be derived
another differential equation satisfied by every characteristic
of the family of surfaces to which the given equation belongs
(see Monge, p. 53). In the first place, let the given equation
be of the first order ; that is to say, of the form
/ (x, y, z, p, q) = 0.
Now if this equation belong to the envelope of a moveable
surface, it will be satisfied, not only by the envelope, but also
by the moveable surface in any of its positions. This follows
from the fact that the envelope touches the moveable surface,
and therefore that at the point of contact x, y, z, p, q are the
same for both. Now if x, y, z be the coordinates of any
point on the characteristic, since such a point is the inter
section of the two consecutive positions of the moveable
surface, the equation f(x, y, z,p,q) = Q will be satisfied by
these values of x, y, z, whether p and q have the values
derived from one position of the moveable surface or from the
next consecutive. Consequently, if we differentiate the given
30 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
equation, regarding p and q as alone variable, then the points
of the characteristic must satisfy the equation
Pdp+
Or we might have stated the matter as follows : Let the
equation of the moveable surface be z = F (x, y, a), where
the constants have all been expressed as functions of a single
parameter a. Then (Art. 438) we have p = F^x, y, a),
q = F (2 (x, y, a), which values of p and q may be substituted
in the given equation. Now the characteristic is expressed
by combining with the given equation its differential with
respect to a ; and a only enters into the given equation in con
sequence of its entering into the values for p and q. Hence
we have, as before, P d f. + Q ^ = 0.
da da
Now since the tangent line to the characteristic at any
point of it lies in the tangent plane to either of the surfaces
which intersect in that point, the equation dz=pdx + qdy is
satisfied, whether p and q have the values derived from one
position of the moveable surface or from the next consecutive.
We have therefore ~ dx + ^ dy = 0. And combining this
equation with that previously found, we obtain the differential
equation of the characteristic Pdy  Qdx = 0.
Thus, if the given equation be of the form Pp + Qq = R,
the characteristic satisfies the equation Pdy  Qdx = 0, from
which equation, combined with the given equation and with
dz=pdx + qdy, can be deduced Pdz = Rdx, Qdz = Edy. The
reader is aware * of the use made of these equations in in
tegrating this class of equations. In fact, if the above system
of simultaneous equations integrated give u = c lt v = c 2 , these
are the equations of the characteristic or generating curve in
any of its positions, while in order that v may be constant
whenever u is constant we must have u = <f>(v).
*See Boole's Differential Equations, p. 323, and Forsyth's Differential
Equations, Art. 185.
DIFFERENTIAL EQUATIONS OF FAMILIES OF SURFACES. Ml
Ex. Let the equation be that considered (Art. 439), viz. ** (1 + p* + g)  r 1 ,
then any characteristic satisfies the equation pdy = qdx, which indicates (Art.
421) that the characteristic is always a line of greatest slope on the surface, as
is geometrically evident.
449. The equation just found for the characteristic gener
ally includes p and q, but we can eliminate these quantities
by combining with the equation just found the given partial
differential equation and the equation dz=pdx + qdy. Thus,
in the last example, from the equations z (1 +p* + q) =r 2 ,
qdx = pdy, we derive
z*(dx* + dy* + dz*} = r'W + dy}.
The reader is aware that there are two classes of differential
equations of the first order, one derived from the equation of
a single surface, as, for instance, by the elimination of any
constant from an equation 7=0, and its differential
U^dx + U 2 dy + U 3 dz = 0.
An equation of this class expresses a relation between the
directioncosines of every tangent line drawn at any point on
the surface. The other class is obtained by combining the
equations of two surfaces, as, for instance, by eliminating three
constants between the equations [7 = 0, F = 0, and their differ
entials. An equation of this second class expresses a relation
satisfied by the directioncosines of the tangent to any of the
curves which the system U, V represents for any value of the
constants. The equations now under consideration belong to
the latter class. Thus the geometrical meaning of the equation
chosen for the example is, that the tangent to any of the curves
denoted by it makes with the plane of xy an angle whose
cosine is z : r. This property is true of every circle in a vertical
plane whose radius is r ; and the equation might be obtained
by eliminating by differentiation the constants a, /9, m, between
the equations
(xa^ + (y^ + z^ = r\ x a + m(y  ffi0.
450. The differential equation found, as in the last article,
is not only true for every characteristic of a family of surfaces,
but since each characteristic touches the cuspidal edge of the
32 ANALYTIC GEOMETRY OF THEEE DIMENSIONS.
surface generated, the ratios dx : dy : dz are the same for
any characteristic and the corresponding cuspidal edge ; and
consequently the equation now found is satisfied by the cus
pidal edge of every surface of the family under consideration.
Thus, in the example chosen, the geometrical property ex
pressed by the differential equation not only is true for a
circle in a vertical plane, but remains true if the circle be
wrapped on any vertical cylinder ; and the cuspidal edge of
the given family of surfaces always belongs to the family of
curves thus generated.
Precisely as a partial differential equation in p, q (express
ing as it does a relation between the directioncosines of the
tangent plane) is true as well for the envelope as for the par
ticular surfaces enveloped, so the total differential equations
here considered are true both for the cuspidal edge and the
series of characteristics which that edge touches. The same
thing may be stated otherwise as follows : the system of
equations U=Q, j~ = 0, which represents the characteristic
when a is regarded as constant, represents the cuspidal edge
when a is an unknown function of the variables to be elimin
ated by means of the equation y^ = 0. But the equations
(7=0, ^ =0 evidently have the same differentials as if a
da
were constant, when a is considered to vary, subject to this
condition.
Thus, in the example of the last article, if in the equa
tions (x  a) 2 + (y j8) 2 + = r 2 . ( x ~ a ) + m (y  y9) = 0, we write
/3 = <f> (a), m = <f> (a), and combine with these the equation
1 + <' (a) 2 = (y/3) <f>" (a), the differentials of the first and
second equations are the same when a is variable, in virtue
of the third equation, as if it were constant ; and therefore
the differential equation obtained by eliminating a, ft, m
between the first two equations and their differentials, on the
supposition that these quantities are constant, holds equally
when they vary according to the rules here laid down. And
DIFFERENTIAL EQUATIONS OF FAMILIES OF SURFACES. 33
we shall obtain the equations of a curve satisfying this differ
ential equation by giving any form we please to </> (a), and
then eliminating a between the equations
(x  a)*+\y  <f> (a)}* + * = r f (x  a) + f (a) {y  $ (a)} = 0,
(a).
It is convenient to insert here a remark made by M. Roberts, viz. that if
in the equation of any surface we substitute for x, x + xdx, for y, y + \dy, for
z,z + Kdz, and then form the discriminant with respect to A, the result will be
the differential equation of the cuspidal edge of any developable enveloping the
given surface. In fact it is evident (see Art. 277) that the discriminant ex
presses the condition that the tangent to the curve represented by it touch
the given surface. Thus the general equation of the cuspidal edge of develop
ables circumscribing a sphere is
(x s + y* + z*  a z ) (dx" 5 + di/ 2 + da*) = (xdx + ydy + zdz)*,
or (ydz  zdy)*> + (zdx  xdz)* + (xdy  ydx) = a 2 (dx 1 * + dy* + dz*).
In the latter form it is evident that the same equation is satisfied by a
geodesic traced on any cone whose vertex is the origin. For if the cone be
developed into a plane, the geodesic will become a right line ; and if the dis
tance of that line from the origin be o, then the area of the triangle formed by
joining any element ds to the origin is half ads, but this is evidently the pro
perty expressed by the preceding equation.
451. In like manner can be found the differential equation
of the characteristic, the given partial differential equation
being of the second order (see Monge, p. 74). In this case we
can have two consecutive surfaces, satisfying the given differ
ential equation, and touching each other all along their line
of intersection. For instance, if we had a surface generated
by a curve moving so as to meet two fixed directing curves,
we might conceive a new surface generated by the same curve
meeting two new directing curves, and if these latter directing
curves touch the former at the points where the generating
curve meets them, it is evident that the two surfaces touch
along this line. In the case supposed, then, the two surfaces
have x, y, z, p, q common along their line of intersection and
can differ only with regard to r, s, t. Differentiate then the
given differential equation, considering these quantities alone
variable, and let the result be
Edr + Sds + Tdt = 0,
VOL n. 3
34 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
But, since p and q are constant along this line we have
drdx + dsdy = 0, dsdx + dtdy 0.
Eliminating then dr, ds, dt, the required equation for the
characteristic is
Bdy*  Sdxdy + Tdx z = 0.
In the case of all the equations of the second order, which
we have already considered, this equation turns out a perfect
square. When it does not so turn out, it breaks up into
two factors, which, if rational, belongs to two independent
characteristics represented by separate equations ; and if not,
denote two branches of the same curve intersecting on the
point of the surface which we are considering.
452. In fact, when the motion of a surface is regulated by
a single parameter (see Art. 321), the equation of its envelope,
as we have seen, contains only functions of a single quantity,
and the differential equation belongs to the simpler species
just referred to. But if the motion of the surface be regulated
by two parameters, its contact with its envelope being not
a curve, but a point, then the equation of the envelope will
in general contain functions of two quantities, and the differ
ential equation will be of the more general form. As an
illustration of the occurrence of the latter class of equations in
geometrical investigations, we take the equation of the family
of surfaces which has one set of its lines of curvature parallel
to a fixed plane, y = mx. Putting dy^mdx in the equation
of Art. 310, the differential equation of the family is
m 2 { (1 + g 2 ) s  pqt\ + m { (1 + q 2 ) r  (1 +p*) t]
{(l+p*)spqr} = 0.
As it does not enter into the plan of this work to treat of
the integration of such equations, we refer to Monge, p. 161,
for a very interesting discussion of this equation. Our object
being only to show how such differential equations present
themselves in geometry, we shall show that the preceding
equation arises from the elimination of a, ft between the fol
lowing equation and its differentials with respect to a and yS :
Or a)* + (y )+{* (a + w/8)}*H* (ft ma)} 2 .
DIFFERENTIAL EQUATIONS OF FAMILIES OF SURFACES. 35
Differentiating with respect to a and ft, we have
(x  a) + (z  <) </>' = m
(//  ) + m (z  <j>) </>'=
whence (x  a) + m (y  ft) + (1 + m 2 ) (z  <) </>' = 0.
But we have also
whence (a;  a) = m (y  /3) + (p + wg) (z  <) = 0.
And, by comparison with the preceding equation, we have
p + mq = (1 + w 2 ) <' (a + m/3). If, then, we call a + mft, 7, the
problem is reduced to eliminating 7 between the equations
x + my y+(p + mq) \z  $ (7)} = 0, p + mq = (1 + m' 2 ) <f>' (7).
Differentiating with regard to x and y, we have
(1 +p* + mpq} + (r + ms) {z  <f> (7)} = {1 + (p + mq) <f>'\ 7,,
{m (1 + <7 2 ) +pq} + (s+ mf) \z  $ (7)} = {1 4 (p + mq)
but from the second equation
r + ms : s + mt : : 7 t : y r
Hence, the result is
(1 +p* + mpq) (s + mt) = {m (1 + q z ) +pq] (r + ms),
as was to be proved.
CHAPTER XIII (a).
COMPLEXES, CONGRUENCES, RULED SURFACES.'
453. THE preceding families of cylindrical surfaces, coni
cal surfaces and conoidal surfaces, are all included in the more
general family of ruled surfaces ; but it is natural to consider
these from a somewhat different point of view. We start
with the right line, as a curve containing four parameters.
Considering these as arbitrary, we have the whole system of
lines in space ; but we may imagine the parameters connected
* In W. R. Hamilton's second supplement on Systems of Rays, Transac
tions of the Royal Irish Academy, vol. xvi. 1830, were first investigated the
properties of a congruence other than that formed by the normals to a surface.
As to the theory of complexes and congruences see Pliicker's posthumous work,
Neue Geometrie des Raumes gegrilndet auf die Betrachtung der geraden Linie
als Raumelement, Leipzig, 1868, edited by Klein ; also Kummer's Memoirs,
Crelle, LVII. 1860, p. 189 ; and " Ueber die algebraischen Strahlensysteme, ins
Besondere iiber die der ersten und zweiten Ordnung," Berl. Abh. 1866, pp.
1120; and various Memoirs by Klein and others.
As regards ruled surfaces see Chasles's Memoir, Quetelet's Correspondancc,
t. xi. p. 60, and Cayley's paper, Cambridge and Dublin Mathematical Journal,
vol. vn. p. 171 ; also his Memoir, " On Scrolls otherwise Skew Surfaces,"
Philosophical Transactions, 1863, p. 453, and later Memoirs. [See his Collected
Papers. See also Pliicker, Tlieorie generale des surfaces reglees leur classi
fication et leur construction, Ann. di Mat. n. 1, 1867.]
[The literature of the subject is very extensive, and the reader who wishes
for a more complete bibliography may refer to Loria, H Passato ed il Presente
delle Principals Teorie Geometriche (3rd ed. Turin, 1907), p. 207 sqq., p. 407
sqq. (complexes and congruences), and p. 120 sqq. ; p. 374 (ruled surfaces). We
may refer especially to Jessop, A Treatise on the LineComplex (Cambridge,
1903) for a compendious treatment of algebraic complexes; and to Bianchi,
Lezioni di Geometria Di/erenziale (2nd ed. Pisa, 1902), or to Eisenhart's
Differential Geometry (Boston, 1909), which is based on Bianchi's work, for
summaries of the more interesting differential properties of rectilinear con
gruences. On isotropic congruences see Ribaucour's Memoir cited on p. 75.]
36
COMPLEXES, CONGRUENCES, RULED SURFACES. 37
by a single equation, or by two, three, or four equations
(more accurately, by a onefold, twofold, threefold or four
fold relation). In the last case we have merely a system con
sisting of a finite number of right lines, and this may be ex
cluded from consideration ; the remaining cases are those of
a onefold, twofold, and threefold relation, or may be called
those of a triple, double, or single system of right lines.
A. The parameters have a onefold relation. We have
here what Pliicker has termed a complex of lines. As ex
amples, we have the system of lines which touch any given
surface whatever, or which meet any given curve whatever,
but it is important to notice, as has been already remarked
in Art. 8Qd and in Art. 316 (D), that these are particular
cases only ; the lines belonging to a complex do not in general
touch one and the same surface, or meet one and the same
curve.
We may, in regard to an algebraic complex, ask how
many of the lines thereof meet each of three given lines, and
the number in question may be regarded as the order of the
complex.
B. The parameters have a twofold relation. We have
here a congruence of lines. A wellknown example is that
of the normals of a given surface. Each of these touches at
two points (the centres of curvature) a certain surface, the
centresurface or locus of the centres of curvature of the
given surface, and the normals are thus bitangents of the
centresurface. And so, in general, we have as a congruence
of lines the system of the bitangents of a given surface. But
more than this, every congruence of lines may be regarded as
the system of the bitangents of a certain surface, for each line
of the congruence is in general met by two consecutive lines,
and the locus of the points of intersection is the surface in
question.* The surface may, however, break up into two
separate surfaces, and the original surface, or each or either
of the component surfaces may degenerate into a curve ; we
have thus as congruences the systems of lines,
(1) the bitangents of a surface,
* See Art. 457.
38 ANALYTIC GEOMETRY OF THEEE DIMENSIONS.
(2) lines " through two points " of a curve,
(3) common tangents of two surfaces,
(4) tangents to a surface from the points of a curve,
(5) common transversals of two curves,
the last four cases being, as it were, degenerate cases of the
first, which is the general one.
We may, in regard to [an algebraic] congruence, ask how
many of the lines thereof meet each of two given lines ? the
number in question is the orderclass of the congruence. But
imagine the two given lines to intersect ; the lines of the con
gruence are either the lines which pass through the point of
intersection of the two given lines, or else the lines which lie
in the common plane of the two given lines, and the questions
thus arise : (1) How many of the lines of the congruence pass
through a given point ? the number is the order of the con
gruence. (2) How many of the lines of the congruence lie
in a given plane? the number is the class of the congru
ence.
[A surface generated by the lines of a congruence which
meet a given directing curve may be termed a ruled surface
of the congruence. Let U be a ruled surface whose directing
curve is a right line a. If a right line ft intersects a in P,
the lines of the congruence intersecting both a and ft are
those passing through P, and those lying in the plane con
taining a and ft ; that is, their number is in + n. Hence the
degree of the surface U is in general m + n.
Hence also the orderclass of an algebraic congruence is
equal to the sum of the order and class ; for the number of
rays of the congruence meeting any two right lines a and 7 is
equal to the number (m + ri) of points in which the ruled
surface U meets the line 7.]
C. Eelation between the parameters threefold. We have
here a regulus of lines or ruledsurface, that generated by a
series of lines depending on a single variable parameter.
The order or degree of the system is the number of lines of
the system which meet a given right line.
RECTILINEAR COMPLEXES. 39
SECTION I. RECTILINEAR COMPLEXES.
454. In accordance with Plucker's work on the right line
considered as an element of space, we must therefore first
consider the properties of a rectilinear complex ; that is to
say, of a system of lines which satisfy a single relation be
tween the six coordinates. If this relation be of the n th
degree, the complex is of the w th degree ; all the lines of it
which pass through a given point form a cone of the n* h
order, and those which lie in a given plane envelope a curve of
the n th class (see Art. 8Qd). [The order is twice the degree.]
If, for instance, the complex be of the first degree, all the
lines which pass through a given point lie in a given plane,
the polar plane of the point ; and, reciprocally, those which
lie in a given plane pass through a given point, the pole of
the given plane. To each line in space corresponds a con
jugate or polar line, the points of the one line corresponding
to the planes which pass through the other. Any line which
meets two conjugate lines will be a line of the complex.
When five lines of such a complex are given, it is evident, by
counting the number of constants, that the complex is de
termined ; and what has just been said enables us to construct
geometrically the plane answering to any point. For, taking
any four lines of the complex, the two lines which meet these
four are conjugate lines, and the line passing through the
assumed point and meeting the conjugate lines is a line of
the complex. A second line is determined in like manner,
and the two together determine the plane.
If we consider a series of parallel planes, to each corre
sponds a single point, and the locus of these points is therefore
a line of the first degree, which right line may be called the
diameter of the system of planes. To the plane infinity
corresponds a point at infinity, and through this point all
the diameters pass ; that is to say, they are parallel.
One of the diameters is perpendicular to the corresponding
plane, and this diameter may be called the axis of the com
plex.
40 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
If the axis and a line of the complex be given, the com
plex is determined. If a line of the complex be translated
parallel to the axis or rotated round it, it still belongs to the
complex.
When the line meets the axis we have the limiting case
of a complex, the special complex, consisting of all lines which
meet a given one, the directrix. It will be remembered (Art.
57c) that the condition that a complex shall be of this nature
is that its coefficients shall satisfy the equation
the equation of the complex being
Pp + Qq + Rr + Ss + Tt + Uu = 0.
[Ex. 1. The equation of a linear complex being
Pp + Qq + Rr+ Ss + Tt + Uu = 0,
prove that the coordinates of the polar line of any line (p, q, r, s, t, u,) are
p  \S,q  \T,r  \U, s  \P, t  \Q, u  \R,
_ Pp + Qq + Rr + Ss + Tt + Uu
PS + QT + RU
Ex. 2. A line of the complex coincides with its polar line.
Ex. 3. The polar line of any line with regard to a special complex
coincides with the directrix.]
[454a. Simplest form of equation of linear complex.
Geometrical construction. The polar plane of any point
x'y'z'w' is
P (yz  y'z) + Q (zx  z'x) + R (xy  x'y) + S (xw 1  x'w)
+ T (yw'  y'w} + U (zw r  z'w] = 0.
And therefore, using rectangular Cartesian coordinates, the
poles of the parallel planes whose equations for different values
of ft are
fj. (ax + by + cz) + d = Q
where a, b, c, d are constants, lie on the right line
 Qz + Ry + S ; _ Pz  Ex + T _ Py + Qx+U
a b c
which is the equation of the diameter of this system of planes.
It is easy to see then that the directioncosines of all dia
meters are in tlu ratio P : Q : R.
Since the directioncosines of the perpendicular to the
polar plane of any point xyz are proportional to
THE LINEAR COMPLEX. 41
 Qz + Ey + S, PzEx + T, Py + Qx+ U,
we find by expressing the condition that the diameter through
a point is perpendicular to the polar plane, that the
of the axis of the complex is
_ Py + Qx+U
P Q R
The directioncosines of the perpendicular to the polar
plane of the origin are in the ratio S : T : U, and those of a
diameter through the same point are in the ratio P:Q:R.
Therefore if the axis of z be taken as the axis of the complex,
we must have P = 0, Q = 0, S = Q, T = 0, and the equation of
the complex is reduced to the simple form
R (xy f  yx') + U(z  z) = 0, or
xy'  yx' = h(z  z'}.
From this we can derive a geometrical construction for the
linear complex. Let r be the shortest distance between the
axis and a ray of the complex passing through xyz and
x'y'z, and let 6 be the angle between the ray and the axis.
Then
and tan $ = ** + ??).
z z
Thus xy  yx = (z z) r tan 6
and h = r tan 6.
Thus the linear complex is defined by the property that the
shortest distance between a ray and the axis, multiplied by the
tangent of the angle between the ray and the axis, is constant.
If we put x  x = dx, y  y = dy, z  z' = dz, the
equation of the complex may be written in the differential
form used by Lie,
xdy  ydx + hdz = 0.
This equation is satisfied, p being constant, by
x = r cos <, y = r sn <
that is, the lines of the complex are tangent lines to a series of
helices, whose axes coincide with the axis of the complex.
42 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
Ex. 1. A curve of a linear complex is defined as one whose tangent lines
belong to the complex, therefore the coordinates of its points, when ex
pressed as functions of a parameter, satisfy the equation
ydx  xdy  Jidz = 0.
If a, /3, y be the directioncosines of the tangent line, I, m, n those of the
principal normal, and A, p, v those of the binormal, then
ya,  xfi  hy = 0,
and by differentiating with regard to s, and using the first of the FrenetSerret
formulas (Art. 363 (a)),
yl  xm  hn = 0,
therefore \ : p : v = y :  x :  h.
Hence the polar plane of a point is the osculating plane thereat of all the
curves through the point.
Ex. 2. If we use the equations (Art. 368 (a))
n _ di> _ dv dv dv
r ~ ds = a dx + * dy + y dz
we find since n = \p  pa.,
1 h _ v^
T ~ x* + J/ 2 + & 2 ~ h '
Hence all the curves through a given point have tlie same torsion thereat,
and as Professor MacWeeney has pointed out, the torsion is proportional to
the square of the sine of tlie angle between the polar plane and the axis.]
455. Let us pass now to a complex of the second degree,
the quadratic complex; that is to say, the system of lines
whose six coordinates are connected by a relation of the second
degree. Then, from what has been said, all the lines of the
complex which lie in a given plane envelope a conic, and
those which pass through a given point form a cone of the
second order. We may consider the assemblage of conies
corresponding to a system of parallel planes, and obtain thus
what Pliicker calls an equatorial surface of the complex ; or,
more generally, the assemblage of conies corresponding to
planes which all pass through a given line, obtaining thus
Pliicker's complex surface. It is easy to see that the given
line is a double line on the surface, and that the surface is of
the fourth degree, its section by one of the planes consisting
of the line twice, and of the conic corresponding to the plane.
The surface will be of the fourth class, and Pliicker shows
also that it has eight double points.
[The lines of the complex through a given point lie on a
COMPLEXES OP ANY DEGREE.
quadric cone ; and the singular surface of the complex is the
locus of points at which these cones break up into planes.
The lines of the complex in a given plane envelope a conic
and it may be shown that the envelope of planes for ichich
those conies reduce to pairs of points coincides with the
singular surface. This surface is known as Kummer's quar
tic and will receive attention in subsequent Articles.]
SINGULAR LINES, POINTS, PLANES, AND SURFACES OF
COMPLEXES OF ANY DEGREE.*
[455a. If = be the equation of a complex of the w th
degree, the lines through a fixed point A describe a cone of
the ?i th degree, and the lines in a plane envelope a curve of
the w th class. If A, B, C be any three neighbouring points
their corresponding cones have points in common, and if P
be one of them whose corresponding cone is S, then S has
PA, PB, PC as generators, and hence this cone must have
a double edge which joins P to a point in the immediate
vicinity of A, B, G. Of the three lines PA, PB, PC, two
in general lie on one sheet and one on the other. A point
whose cone has a double edge is called a singular point and
its double edge a singular line.
If x'y'z'w' be such a singular point then replacing p, . . .
by yz  y'z, . . . the cone will have a double edge if
d<f> _ d<f> _ d<}> _ d<j)
dx dy dz dw~
Now denoting ^, ... by fa, fa, fa, fa, fa, fa, these con
ditions reduce to
y'fa  z'fa + wfa = 0, z'fa  x'fa + w'fa = 0,
x'faijfa + w'fa = Q, x'fa + y'fa + z'fa = 0,  (1)
or any three of them.
From these we deduce the necessary conditions for a
singular line of the complex, viz. :
< = 0, ^ = fafa + fafa + fafa = .
It follows therefore from (1) that </> 4 , fa, fa, fa, fa, fa are
Arts. 455ao are due to Mr. R. Russell, F.T.C.D.
44 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
the coordinates of a line (the conjugate line) which intersects
the singular line in the singular point ; and the plane of the
two lines is called the singular plane.
4556. The singular surface is the locus of a point
xy'z'w' which is the vertex of a cone of the complex having
a double edge, and its equation might be obtained as
follows :
If fy = and any two of the equations in (1) be satisfied
the remaining two of (1) will be satisfied and also < = 0. In
addition three equations connect x', y', z, w' withp, q, r, s, t, u,
hence we have six homogeneous equations from which
p, q, r, s, t, u can be eliminated.
By analogous reasoning the singular planes envelope a sur
face, and it will be shown that this is the singular surface.
455c. If xy'z'w be the singular point in 455a and
x"y"z"w" a second point on the double edge PQ of S, then
the cones of all points on PQ have the singular plane of PQ
as a common tangent plane.
The cone of such a point is
4> {y (z'+kz")z(y' + lcy"), . . .} = 0,
and the tangent plane along PQ is
/d(f>\' /d<j>\ f (d$\ /'rf0V
*()+* I?) + *l;7 )+> l:r =0
\dx/ \d/ \dz J \dw/
or S {y (z + ft*") *(y'+ ky")} & = 0.
Making use of the conditions in (1) the tangent plane
reduces to S (yz"  y"z) ^ = 0, and this is the plane of the
singular line and its conjugate.
455<i. Cones of the complex whose vertices lie in a fixed
plane contain homogeneously three parameters, and therefore
in the usual way envelope a surface.
Suppose for simplicity that the plane is w' = 0, then one
of the cones is
<f> (yz  y'z, zx'  z'x, xy  x'y,  x'w,  y'w,  z'w} 0,
and the envelope is obtained by eliminating x'y'z from
COMPLEXES OF ANY DEGREE. 43
*o ^o ** = o.
ax ay dz
But if xyzw be regarded as fixed and x'y'z as variable, these
conditions express that the cone S' whose vertex is xyzw
meets w' Q in a curve having a double point, and this can
not be true unless S' has a double edge. Hence the cones of
the complex whose vertices lie in a fixed plane envelope the
singular surface, and from 455c we see that the singular
plane is the tangent plane to the enveloping cone where it is
met by two consecutive cones. Hence the locus of singular
points, the envelope of singular planes, and the envelope of
cones of the complex are the same singular surface, and the
singular plane touches the surface where the singular line to
which the plane corresponds meets its conjugate. We might
have obtained the same results if we had developed the pro
perty that lines of the complex lying in a plane envelope a
curve of the /t th class, but it is left as an exercise in tangential
coordinates.
455e. Degree and Class of the Singular Surface. The
number of singular points on a given line is obviously the
same as the number of singular planes through it, and these
cases are both included in the conditions that a singular line
and its conjugate meet the given line. If p' ... be the
given line the conditions are
There are therefore 4rc (n  I) 2 cases in all, and the order and
class are each 2n (n  I) 2 .
THE QUADRATIC COMPLEX
a ll p 2 + . . . + 2o 11 p5' + . . . + 2a 36 r< + . . . = 0.
455/. The cone of the complex with a given vertex is
a quadric cone which in the case of a singular point reduces
to two planes. The lines of the complex in a given plane
envelope a conic which in the case of a singular plane reduces
to a pair of points, and the singular surface is of the fourth
degree and class.
46 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
The four points in which the surface is met by any line
are equiinharmonic with the four tangent planes through it.
This property may be established by several methods of
which the following is one : If 2 = 0, 10 = be the given line,
a, 1, 0, any point on it, and 0, 0, v, 1 any plane through it,
then the cone of the point is obtained by substituting  z,
za t u  ay,  aw,  w, for p, q, r, s, t, u in <> = 0, and if
vz + w = is a tangent plane to this cone then
33 [#22 a2 ~ 2a 12 a + On  %v {  <z 24 a 2 + # u  a 25 a + e^J
+ i^(a 44 a 2 + 2a 43 a + a 55 )]
 {a 23 a  a n + v (a.^a + a 35 )} 2 = 0,
a quadriquadric function of a, v.
If the two values of v are equal the four values of a give
singular points on the line, and if the two values of a are
equal the four values of v give singular planes through the
line. But the two biquadratics are known to be equianhar
monic, hence the proposition is proved.
4550. The Singular Quartic has Sixteen Double Points and
Sixteen Planes of Contact. The conic of the complex in any
singular plane reduces in general to a distinct pair of points.
Are there any cases in which they coincide ? If this is so we
might expect that all lines in the plane through the point
would be singular, and that therefore each of the lines would
touch the surface at the singular point on it, and therefore
the plane would touch along a conic section.
Suppose pqrstu a singular line, Piq^iS^u^ its conjugate,
then if 5*  ^ = the conjugate line is also singular. Let
dpi ds l
z be its conjugate, and denote by 12 the quantity
, then we have the following conditions :
<J> 00 = involving 01 =
<f> 01 = involving II = 0, 02 =
02= 5> 1 s 2 = 12 = ^ 11 =
THE QUADRATIC COMPLEX. 47
Now obviously the lines 0, 1, 2 either lie in a plane or meet in
a point, and considering the former case any line in the plane
is \p + np l + vp^. ... If it belongs to the complex, v*fa = 0,
and therefore the two points in the singular plane coincide, or,
all lines in the complex and in the plane pass through a point.
Again the line \p + fip l ... is singular, and its conjugate
is \pi + /jp 2 , . , and the former line touches the quartic
where it is met by the latter, and hence the plane touches
the quartic along a conic.
The five equations of condition give thirtytwo solutions,
of which sixteen refer to planes of contact, and sixteen to
double points.
455/j. The Equation of the Singular Quartic. From the
theorem that two quadratic expressions in n variables can be
reduced simultaneously to the sums of n squares of linear
functions of these variables, it can be immediately inferred
that the most general quadratic complex can be put in the form
2a 36 rw = 0.
The singular quartic is the envelope of cones whose
vertices are in the plane w' = Q. Substituting for p . . .
yz  y'z, zx  z'x, xy'  x'y,  x'w,  y'w,  z'w and forming the
envelope, ^v' 2 divides out, and we find for the equation
a^a^a^x 4 + a^a^a^ h a^a^^a^ + a^a^a^w*
+ (a n yW + a^w 2 } (a^a^ + a^a^  a 2 )
+ Zxyzw (a@y + a n a^a + cu^a^ + a^a^y) =
where a = a 25  a^, j3 = a 36  a 14 , 7 = 0^035.
_
Writing x= */ a n a 55 a 6 x > V s * tAw^ 7, z= ^/a^a^a^ Z,
W,
a ll a 44 ~ a >
and for convenience restoring small letters, it reduces to
x* + y* + z 4 + w* + <2\ (y*z* + A 2 ) + 2/t (^V + y*w*)
Q where (3)
2 2 2  a 2  2 = 2
a = + c  2cX, = c a + a  ca/i, T = a
2/ca6c = afiy + a*a + 6 2 y8 + c?v, a + + 7 = 0.
It is easily seen that a relation exists between X, /*, v, K.
48 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
The conditions just obtained suggest taking four points
ABCD in space such that BC = a, CA = ft, AB = y, AD = a,
BD = b, CD = c and the cosines of the angles at D as X, //., v.
If the points were real ABCD would obviously lie in an
ordinary plane and 1  X 2  /i 2  i> 2 + 2X/w = 0. But this is
not the only interpretation. It is easy to verify that if the
plane ABC touch the circle at infinity then AB + BC + CA =
however ABC are chosen, and cosA = cosB = cosC =  1.
The expression for 6 vol ABCD is abc N /l  A, 2  /* a  v 2 + 2X/ii/,
and putting g = BAD, rf= CAD, it is equal to
afty ,yi  cos 2 !;  cos 2 r)  cos 2 C + 2 cos ^ cos rj cos C
= afiy N /  l(cos %+cos rj)
= ^^ (afiy 4 a 2 a + 6 2 + c 2 ?).
Therefore 2 + 1  X  p*  i/ 2 + "2\/u> = (4)
It may easily be shown that this is the necessary and
sufficient condition that the singular quartic (3) shall have
double points, that is that the equations
x (x 2 + vy 2 + fjiz 2 + Xwr) + Kyzw = 01
(5)
y (vx 2 +y 2 + \z 2 + /iw? 2 ) + Kzxw = '
z (jjus 2 + Xy 2 + z z + i/M? 2 ) + Kxyiv =
w (Xx 2 + /*y 2 + vz 2 + w 2 ) + Kxyz = 1
are simultaneously satisfied.
And if the quartic has one double point it will have
sixteen, obtained by permuting and changing the signs of
JT, i/, z, w so as to leave yz' + x 2 w, z^X' + y^iv, x 2 y 2 + z*w~
xyzw unaltered.
455i. Planes of Contact and Double Points of the Singu
lar Quartic. The equation of the quartic being unaltered by
interchanging x, y and also z, w, etc., if ax + fiy + yz+ 8w =
be one plane of contact, so also are
Sx + yy + ftz + aw = 0, yx + By + az + @w = 0,
/3x + ay + 82 + yw = 0.
Denoting these by P, L, M, N respectively, and
THE QUADRATIC COMPLEX. 49
X 1 + W 2 + Z* + tt> 2
y ^ , yz + xw, zx f yw, xy + zw by S, U, V, W t
a? 4. #' 4. n/'J 4. fr'
and a  E^3  , 7 + o, 7 a + 5, a/9 + 7 g by <r> 0,, 0,, 3 ,
LMNP=(abcdfghlmn) (U, V, W, Sf + kxyzw, . . (6)
where obviously d = 4a@y8, I = 0. Z S .
Now the double transformation
z = /> fy' + iaO, y = p (x' + iy), z = p (w'+iz), w = p(z+iw f )
a' = p(@ + ia), & = p (a + i/3), y = p (8 + isy), &' = p (y + iS)
involves LMNP = L'M'N'P',
U = 2ip*V', V=2ip*U', W=2ip*S',S = 2ip*W",
and therefore the equation (6) becomes
L'M'N'P'=  V (a, 6, c, d,f, g, h, I, m, w) 2 (F', U', S', WJ
Jcp* (V'*+U" i lx'y'zw').
Comparing we have  4/> 4 c = 4a'/9Y8' =  4/3 4 (a 2 + /S 2 ) (y + 5 2 ),
 8p 4 / = 2Z' = 2(9 2 '^ 3 ' =  Sp 4 ^^,
or c = (a 2 4 /S 2 ) ( 7 2 + S 2 ), /= ^cr, Z = OJ r
Substituting these and analogous values if desired in (6) we
see that the six products in pairs of U, V, W, S are the terms
of the square
++
a e.0, ej
and we have the identity
*  LMNP = R[x* + y l + z i + w* + '2X
z*w*) + IKxyzw] (7)
where R, \, p, v, K are to be found, if desired, in terms of
a, , 7, 3.
Now " 2  LMNP = has double points where L = 0,
M = meets * = (), that is at the points (a, ft,  7,  S) and
 & 7,  & It follows therefore that $ = has these as
double points, and
are satisfied by a, /3,  7,  S.
These give equations identical with (5) in 455/i from
which the value of X, /*, v, K can easily be found. In fact
VOL. II. 4
50 ANALYTIC 'GEOMETRY OF THREE DIMENSIONS.
\ 2 (?/V 2 + aW), 2(*V + y V), 2 (arty 2 + aW), 4
aS 2
,/3 7
3 2 a 7 2
fe
MN+LP, NL + MP, LM + NP
A
are linear functions of S, U, V, W, hence W*LMNP =
reduces to the form
+ 2g(NL + MP) + 2A (LAf + #P)} 2  IQkLMNP = 0,
and a relation exists between the coefficients/, g, h, k for double
points. It is k + 1  /*  ^ 2  A 2 + 2/gr A = 0.
455;'. T/K? Singular Quartic is a General Rummer's Quartic
that is one having Sixteen Double Points.
If a quartic have sixteen nodes the tangent cone from one
of them is of the sixth degree, and it has as double edges the
lines joining that node to the fifteen others. It therefore
breaks up into six planes, and each of these planes touches
the surface along a conic.
Each plane through the node is met by the five others in
a nodal line, that is a line joining two double points, so that
six double points lie in every plane of contact, two planes of
contact have two double points on their line of intersection,
and through the line joining two double points two planes of
contact can be drawn.
By drawing four planes of contact that form a tetrahedron
ABCD there are two double points on each of the lines DA,
DB, DC, BC, CA,AB. If we choose those on DA, DB, DC
and one point on each of the conies in which the quartic is
touched by the planes DBC, DCA, DAB, we can describe a
quadric through these nine points, and it will therefore con
tain the conies in the planes DBC, DCA, DAB. It contains
six points on the plane of contact of ABC and therefore also
the conic, and therefore the quartic is reducible to the form
V 2 =
W may be taken of the form
THE QUADRATIC COMPLEX. ', 1
x* + y" + 2 2 + w + Ifyz + 2gzx + %hxy + Zlxw + 2m yw + 2mir,
and the double points on the six edges are
x = Q MJ = (y  az) (y  az) =
y = Q W = Q (ZPX)(ZPX)=Q
z = w = Q (x yy) (x  y'y) =
y = Q Z = Q (x\w) (x\'w)=Q
z = Q x = Q (y  fj.w) (y  pic} =
r = y = (z vw) (z  v'w) = 0.
Now through the point 0/310 in addition to the planes of
contact x = 0, w = four others can be drawn each of which
meets the edges A B, AC, AD in double points on them, there
fore the double points connected with the quantities a y3 7 X
are coplanar : hence a/3y +1 = 0.
Similarly aY +1 = 0, 7 V +1 = 0, yap +1 = 0,
therefore a = a,j3 = ft',y = y', or
a =  a', ft =  {?, y = y
which in volve/=Z, g = m, h = n, or
/=  I, g =  m, h =  n,
and y may be taken
a*' 2 + y' 2 + z 1 + w> 2 + 2/ (yz + xic) + 2g (zx + yw) + 1h (xy +
455 A;. Condition that a line PQRSTU shall be a Conjugate
Line of $ = 0.
Evidently  =*H  ^ = ^=*. . (1)
and pS + qT+rU+sP+tQ + uR = 0.
Eliminating pqrstu we obtain as the condition a sym
metrical determinant equation
. (2)
73
S,' if U,' P,' Q,' $ .
This might be called the conjugate complex of < = 0. It
may be written
+ . . . =0.
where A mn is the minor of #,,, in the discriminant of <f>.
4*
52 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
Cosingular Complexes. From the equations (1), (2) it
appears on solving torp, q, r, s, t, u, or from general considera
tions, that
17 j,
= $* jg**4, etc.
hence every singular line of < = is a conjugate line of $ = 0,
and ever} 7 singular line of $ = is a conjugate line of </>=0,
and the two complexes </> = 0, $ = have the same singular
surfaces.
The original complex might have been written
+ 2/*(ps + qt + ru)  0,
when the conjugate complex $^ = would have been the
same determinant equation as above, only that a u , a 25 , a 36
would have been replaced by <z u + /x, a 2 5 + A t > a se + /* We
thus see that all complexes 3^ = have the same singular
surface as < = 0.
455Z. Double Tangent Lines of the Singular Surface. If
a conjugate line be given there is usually one singular line to
which it corresponds. If p q r s' t' u' be a singular line we
have to consider the equations
< H = 4,; ( n =l, 2, 3, 4, 5, 6), ps + qt + ru = 0,
and in general the only solution is
p, q, r, s, t, u =p', q', r, s f , t', u.
But if fa <j>. 2 <f> 3 < 4 < 3 < 6 are identically connected by a linear
relation, that is if the discriminant of < vanishes, the seven
equations reduce to five of the first and one of the second
degree, of which there are two distinct solutions, and the
conjugate line passes through the point of contact of each
tangent plane.
Now k may be any one of six values if the discriminant of
<j>  %k(ps + qt+ru)=Q vanishes, and hence for any of these six
. forms of the quadratic complex the conjugate lines are double
tangents to the singular surface.
THE QUADRATIC COMPLEX. 53
455m. The Principal Linear Complexes. Suppose pqrstu
are any six quantities, not coordinates of a line, satisfying the
equations
^ = fc = <h = 0i = 02 = <k =
p q r s t u
then A; is a root of the discriminant of <  2k(ps + qt + ru) = 0.
There are therefore six sets of quantities p lt . . .,#, . . .,
p 6 . . . corresponding to the roots k v . . . k f> , and they satisfy
the following relations :
0m,, = k m mn = k n mn, therefore </>, = 0, mn = 0, where m, n have
any values 1, 2, 3, 4, 5, 6, but
Also <f> mm = k m mm = < 2k m (p, n s m + q m t m f r m u m ).
Now consider the complex^ + qt^ + ru^ + sp l + tq l + ru^ = 0.
It pqrstu be any singular line, and < 4 , </> 5 , </>,.,, 0,, <^>. 2) <^> 3
its conjugate, then the line < 4  Op, . . . which lies in the
first of the above linear complexes is such that
< 01  001 = or (k l 0)01 = 0, therefore = k v
But this line is the conjugate of pqrstu with respect to
<f>  2&i (ps+qt + rw) = ; and this is a case in which the line
04 ~ h\P> ... is conjugate to two distinct singular lines.
We see, therefore, that if from a singular point lines be drawn
in the singular plane tangents elsewhere to the singular sur
face they belong to the six principal linear complexes
w=Q(m = l, 2, 3,4, 5, 6).
If we transform the equations <f> = 0, ty = 0, ps + qt + ru =
by. the substitutions
TT
. . . + p u,
/ J
VTT
or their equivalents Pv/TT = P s i + tfi + ru i + s Pi + ^1\ + ru i
we obtain 0s=fc,P 2 + k 2 Q 2 + k 3 R 2 + A; 4 S 2 + k b T 2 + k 6 U' 2 = 0,
54 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
in which P = 0, Q = . . , are the six principal linear com
plexes.
From these equations we may write
and finally obtain parametric expressions for the singular
lines, viz.
p = S
With the introduction of suitable multipliers attached to the
groups (p l q l Ti s 1 ^ uj, (p* & r t % t 2 uj . . . , which do not
require explanation, we may more simply take the represent
ation of singular lines and their conjugates in the forms
(6 _____ \ / 6 \
5 p m V/M  k m Jv  k m ., . . ), ( S p m k m Ju  k m Jv  k m ., . . . ),
i / \ i
in which it follows from what precedes that
P = 0, 1, 2, 3, 4.
The case of u = v is special, for here the singular lines are
p, (v  kj p 2 (vkj . . . p 6 (v  k 6 ), . . ,
and their conjugates are
Pih  &i) PA (  * 2 ) . . . p 6 k (v  k 6 ). . . .
Since there are 32 ways of distributing the signs it follows
that these singular lines lie in one or other of 32 pencils, and
that all the singular lines in each pencil are intersected by
all the conjugate lines, and finally that the singular lines and
their conjugates are either coplanar or concurrent. The
former correspond to the planes of contact of the singular
surface, and the latter to the nodes. To actually divide the
32 special cases into planes of contact and nodes the follow
ing proposition may be used : If 1, 2, 3 be three coplanar,
and 4, 5, 6 three concurrent lines, the point (4 5 6) lies or
does not lie in the plane (1 2 3) according as
CONGRUENCE COMMON TO TWO COMPLEXES. 55
14, 15, 16
24, 25, 26
does or does not vanish.
34, 35, 36
6 6
Now suppose that Sp m , Sq, n . . . and its conjugate determine
6 .;
a coplanar group, and that Spj m , Sq m i m . . . where i m = 1
i i
forms a concurrent group in which the point of concurrence
is outside the plane, then applying the above condition we
find that the determinant becomes
m . mm mm . mm 2k^i m . mm
Sk m i m . mm 2k m i, n . mm Sk? m i m . mm
2k m i m . mm Sk* n i n . mm Sk* m i m . mm
Since Sk" mm=0 (/> = 0, 1, 2, 3, 4) this reduces at once to
a vanishing determinant unless three of the i's are positive
and three negative. There are ten such arrangements giving
rise to ten nodes outside the plane of contact : the remaining
six nodes lie on the conic of contact.
Similarly starting with any node we can at once write
down the ten planes of contact that do not contain it, and
there is no difficulty in arranging the 32 cases into their
groups of nodes and planes of contact.
455n. When a congruence is defined as the system of
rays common to two complexes whose equations are given in
line coordinates the existence of the focal points and planes
(Art. 457) may be deduced as follows. The two complexes
being /= 0, $=0, we have also ps+ qt+ r = 0, and if two
consecutive lines intersect, 8p, Bq, Br, Bs, Bt, Bu are connected
by the following equations :
We may therefore regard Bp, Bq, &r, Ss, 8t, Bu as defining a
line P which intersects L (p, q, r, s, t, ti), and the two lines
A, B whose coordinates are &/ 4 + </> 4 , A/ 5 +$j,, . . . &/ 3 +<k,
where
L intersects, each of the lines A^ B, therefore P i?
56 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
concurrent and coplanar with either L and A or L and B.
These are the points (LA, LB) in which a line is met by a
consecutive line (the focal points), and the planes (LA, LB)
are those of the consecutive intersecting lines (the focal planes).
455o. Congruence of the Quadratic and Linear Complexes,
viz.
$ = 0, 5S& = where SP& 4= 0.
Noting that / 4 , / 5 , / 6 , / lt / 2 , / 3 of the last article are here
P! Q 1 R l S l T! (7 X , we shall define a new line by the equations
fa+JeS^/jiS . . . (f> t + JcP l = fiP . . . and therefore
Solving for pqrstu we have
where $ = is the conjugate complex of </> =
Since SS& = 0, SSp = we have
^$00  $01 = /**oi  **n =
therefore PQRSTU satisfy the complex
2 =
which possesses the property that its discriminant vanishes.
A i i/d$oo, , rf^iA , fdd> {>0 7 ^^ii\
Again e^(^^u  Vajj) = H^  ^ j  ^P.
therefore x e 4 + 8 2 8 5 + e 3 e 6 = 0, or P, $, E, 5, T, C7 is a
singular line of S = 0, and pqrstu is its conjugate, and is
therefore a bitangent to the singular surface of S = 0.
If we compare we see that PQRSTU and its conjugate
with respect to O = form precisely in this the same pencil
of lines as pqrstu and BpSqBrBsStBu in the last article, and
we recognize therefore the identity of the focal surface of the
congruence with the singular surface of
SECTION II. RECTILINEAR CONGRUENCES.
455j9. We have a rectilinear congruence of the second order
when we have two equations each of the first degree between
the six coordinates ; or, in other words, the congruence con
EECTILINEAB CONGRUENCES. 57
sists of the lines common to two given complexes. We may
evidently for either of the two given equations
Ap + Bq + &c. = 0, A'p + &c. = 0,
substitute any equation of the form
U + kA')p + &c. = 0;
and then determine k, so that this equation shall express that
every line of the congruence meets a given line. We have
thus a quadratic equation for k, and it appears that the con
gruence consists of the system of lines which meet two fixed
directing lines. Any four lines then determine a congruence
of this kind; for (see Art. 57d) we have two transversals
which meet all four lines,* and the congruence consists of
all the lines which meet the two transversals. An exception
occurs when these two transversals unite in a single one ; or,
what is the same thing, when the quadratic equation just
mentioned has two equal roots. The lines of the congruence,
then, all meet the single transversal ; but, of course, another
condition is required ; and by considering the transversal as
the limit of two distinct lines we arrive at the condition in
question ; in fact the congruence consists of lines each meet
*The hyperboloid determined by any three of the lines (see Art. 113)
meets the fourth in two points through which the transversals pass. If the
hyperboloid touches the fourth line, the two transversals reduce to a single one
and it is evident that the hyperboloid determined by any three others of the
four lines also touches the remaining one. This remark, I believe, is Cayley's.
If we denote the condition that two lines should intersect by (12), then the
above condition that four lines should be met by only one transversal is ex
pressed by equating to nothing the determinant
 (12), (13), (14)
(21),  (23), (24)
(31), (32),  (34)
(41), (42), (43), 
The vanishing of the determinant formed in the same manner from five lines
is the condition that they may all meet a common transversal. The vanishing
of the similar determinant for six lines expresses that they all belong to a
linear complex, which has been called the " involution of six lines " ; and
occurs when the lines can be the directions of six forces in equilibrium. The
reader will find several interesting communications on this subject by
Sylvester and Cayley, and by Chasles, in the Comptes Rendus for 1861,
Premier Semestre.
58 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
ing a given line, and such that considering the common point
of the given line and a line of the congruence, and the
common plane of the same two lines, the range of points
corresponds homographically with the pencil of planes.
. Instead of treating a rectilinear congruence as a
system of rays whose coordinates satisfy two relations, we
might define it by the property that in general a finite number
of rays, or one only, passes through any point in space ; and
this definition is more convenient in dealing with certain
differential properties.
There are now two methods of procedure. In the first
method, used by Hamilton, we regard the directioncosines (/,
m, n) of the ray through any point (P, x, y, z) as determinate
functions of x, y, z, connected by the relation Z 2 + m 2 + n 2 = 1.
In general I, m, n, if. taken as the directioncosines of the
tangent line to a curve through P will define a congruence of
curves, a finite number of curves passing through each point
of space. But in order that the congruence may be recti
linear the values of I, m, n must not vary as we proceed along
the direction they determine. Now if u be any function of
x> y, ^ we have
du , du , du ,
du = ? ax + T dy + T dz,
dx dy dz
and therefore if u does not vary along the direction I, m, n,
, du du du rt
I 5 + m j + n j = 0.
dx dy dz
If then we substitute I, m, n for u in this equation we get
three conditions to be satisfied in order that the congruence
may be rectilinear. But we have also Z 2 + w' 2 + n 2 = 1, and
if both sides of this equation be differentiated with regard to
x, y, z in turn, it is easy to see that the conditions are equiva
lent to
I _ m n
dn dm ~ dl dn ~ dm dl
dy dz dz dx dx dy
with Z 2 + w 2 + n 2 = ] .
BECTILINEAR CONGRUENCES. 59
Ex. 1. From the preceding condition it follows at once that the necessary
and sufficient conditions that I, m,n should represent a congruence of normals
to a family of surfaces are
dn _dm _dl^_dn _ dm dl _
dy dz ~ ds~ dx~ dx ~ dy = '
or in other words that
Idx + mdy + ndz
should be a perfect differential. See Art. 457rf.
Ex. 2. The condition that three functions /, g, h, of x, y, z may be pro
portional at each point to the directioncosines of a rectilinear congruence is
f df j. * d f JL % df ffyj. A< 3 , J, d 9 * dh , dh i. dh
f dx + 9dy + h dz _ f te + 9Jy + h dz J dx + * dy + h dz
Ex. 3. Prove that for any rectilinear congruence
+m d + n
x dy
.
dy dz J dx
The second method, used by Kummer and succeeding
writers, is an application of Gauss' parametric treatment of sur
faces (Art. 377). In fact the rays of a congruence, like the
points on a plane or surface, form a " doublyinfinite " manifold.
We choose arbitrarily a director surface or surface of reference^
and through each point (x, y, z) thereof draw the correspond
ing ray (I, m, ri). Then x, y, z, I, m, n are functions of two
parameters p and q, since x, y, z lies on a known surface and
/, m, n depend only on the position of the point xyz.]
[456. Limit Points, Principal Planes. There are certain
points and planes associated with each ray of a congruence,
depending on the limiting relations of the ray to those in its
neighbourhood.
Let the equations of a ray through x, y, z be
_
I m n
Then x, y, z, I, m, n, as just explained, may be regarded as
functions of two parameters, the point x, y, z moving on the
surface of reference. If we take a second ray (I'm'n) through
another point (x'y'z') on the surface of reference and con
sider the line joining a point x + Ir, y + mr, z + nr, to a point
60 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
x' + I'r', y' + m'r', z + n'r on the second ray, then the conditions
that the joining line may be perpendicular to both give
I (x'  x) 4 m (y'  y) + n (z  z}  r + r cos =
l'(x x)+m (y' y) + n (z  z) + r r cos 6 = 0,
where is the angle between the rays. If we take the rays
indefinitely near, and so replace x  x, y  y, z  z, r  r, by
dx, dy, dz, dr, we derive from these the equation
_ dxdl + dydm + dzdn .. .
2 * 2
which determines the point where one ray is met by the
shortest distance from a ' ' consecutive ray " through x + dx,
y + dy, z + dz. Keplacing dx, dl, etc., by
dxj dxj dlj dl j
j~dp + rdq, dp + dq, etc.
dp dq dp dq
we find
_ r _ adp* + (b + b') dpdq + cdq
edp 1 + 2/dpdq + gdq*
where, using the suffixes 1 and 2 to denote differentiation
with regard to^? and q,
a = l]X v + m l y l + n l z l , b = l
b' = I 2 x v + m 2 y l + n 2 z l} c = I%
e = If + m^ + n^, /= /^ 2 + m 1 m 2 + n^, g =
Writing t for the ratio dp : dq we have
Since the denominator of th : s fraction is the sum of three
squares it cannot change sign, and r therefore cannot become
infinite, but will lie between two extreme values ; that is to
say, the points on any ray of a congruence wliere it is met by
the stortest distance from a consecutive ray, range on a certain
determinate portion of the line, the extreme points being called
by Hamilton the virtual foci* but now more commonly the
limit points of the ray.
It is easily proved that the values of r for the limitpoints
are roots of the equation
First Supplement Trans. R.I.A., XVI, Part I, p. 52.
RECTILINEAR CONGRUENU s 61
and the values of t corresponding to these are roots of
\et+f at + $(b + b') _fl
\ft+9 $(b + b')t + c~ '
The directioncosines of the right line perpendicular to a
ray Imn, and to its shortest distance from a consecutive
ray are proportional to dl, dm, dn, that is to ^dp + l^dq,
m^dp + m.^dq, n^dp + n^dq, and for the values of dp : dq cor
responding to the limit points they are therefore proportional
respectively to
where t and t' are the roots of the preceding quadratic in t.
The condition that these two lines should be at right angles
is thus
and the quadratic equation shows that this condition is satis
fied. It follows also that the shortest distances corresponding
to the limit points are at right angles.
The planes containing a ray and these extreme shortest
distances are called the principal planes of the ray, and we
infer that the principal planes are at right angles.
Let us now suppose that the consecutive rays correspond
ing to the limit points are those for which dp = and dq = Q,
i.e. they correspond to the parametric lines on the surface of
reference. Then the roots of the quadratic for t are and oo ,
and if we exclude the case (considered in Art. 4570) for
which a, b + b', c are proportional to e, 2/, g, we find
Now the directioncosines of the right line perpendicular
to the plane containing a ray and its shortest distance from a
consecutive ray, are
where t = , and those of the normals to the principal planes
\4/\J
are
62 ANALYTIC GEOMETRY OP THREE DIMENSIONS.
L mi n l i ^2 W 2 n '2
/= ' p' and T' T= I'
ve ^/a ve ve v ve
Hence if be the angle between the shortest distance and a
chosen principal plane (or one of the extreme shortest dis
tance)
sn =
t ' 2 + 9
For the special parameters used the equation (3) deter
mining the distances (r lt rj of the limit points is at once
factorized, and we find
/T>/2 I ft
Now r = ; , hence we reach Hamilton's equation
et*+g
r^=r l cos 2 6 + r. 2 sin 2 0.]
[457. The focal points, focal planes, focal surface, develop
ables. When a congruence consists of normals to a surface
there are two points (centres of curvature) where a given ray
intersects a consecutive ray (Arts. 301, 303, 378). In like
manner we shall now prove that, on each ray of any rectilinear
congruence there are two points where it intersects a consecu
tive ray, and these are called the focal points or foci of the
ray. This is plainly equivalent to the theorem stated in Art.
453, that a congruence is a system of bitangents to a certain
surface.
As before let xyz be the coordinates of a point P on any
chosen surface of reference and Imn the ray through P. If
p be the distance of a focal point from P, its coordinates are
x + lp, y + mp, z + np; and since it lies on the consecutive ray
l + dl, m + dm, n + dn through the consecutive point x + dx,
y + dy t z + dz, we have
dx + Idp + pdl = with two similar equations.
Since dx^x^p + x^dq, dl^l^p + l^dq, etc., these equations
become
RECTILINEAR CONGRUENCES.
(aji f pl^dp + (x t + pljdq + ldp = Q
(y l + pm^dp + (y z + pmjdq + mdp =
(*i + pn^dp + (z z + pn.>}dq + ndp = 0.
Now ll l + ??im l + nn l = 0, U^ + mm 2 +nn., = Q; therefore if
both sides of these three equations be multiplied by l lt m^, r^.
and then by 1 2 , m. 2 , n. 2 , we have by addition
(a + ep} dp + (b +fp) dq = Q
(b' +fp) dp + (c + gp) dq = Q
where a, b, b', c, e, f, g have the meanings assigned to them
in the last article. The elimination of p yields an equation
for determining the values of dp : dq corresponding to the
two focal points, namely
(&'  af) dp 2 + {ec +f (b'  &)  ag] dpdq +(fc bg^dq* = 0.
The focal distances are formed by eliminating dp : dq, and
are roots (/?,, p 2 ) of the equation
a+ep b+fp\ Q
b +fp c + gp\
or (eg  f} p 2 + {ag + ec  f (b + b'} } p + ac  bb' = 0.
Comparing this with the equation (3) in Art. 456 for the
distances of the limit points, we have
and therefore, the point halfway between the limit points
coincides with tliat halfway between the focal points. This
point is called the middle point of the ray.
The focal surface is the locus of the focal points, and is,
like the surface of centres, two sheeted (Art. 306). By the
arguments used in Art. 306, it follows that each ray of the
congruence touches both sheets at the focal points, and the
two focal planes being defined as the planes containing a ray
and a consecutive ray are tangent planes to the focal surface,
which is equally well defined as the envelope of the focal
planes.
Since each ray meets two consecutive rays, there are two
singly infinite families of developables consisting of rays of the
congruence. The corresponding cuspidal edges lie on the two
64 ANALYTIC GEOMETEY OF THREE DIMENSIONS.
sheets of the focal surface, and each sheet is enveloped by the
developables of one family.
Ex. 1. If lt 2 be the angles made by tbe focal planes with the principal
plane, prove from Hamilton's equation (last Art.) that
cos 20j + cos 20 2 =
and hence show that the focal planes and the principal planes have common
bisecting planes.
Ex. 2. The directioncosines of the normal to a focal plane are proportional
to (mi^nnij) s+mn 2 w. 2 ?i, etc., where s is a root of the quadratic determining
the values of dp : dq corresponding to the focal planes.
Ex. 3. A congruence clearly reciprocates into a congruence. The focal
planes and points are interchanged and the focal surface reciprocates into the
new focal surface.
The focal surface may degenerate into a curve or developable, or it may
break up into two surfaces either or both of which may be a curve or develop
able. If all the rays of a congruence intersect a curve that curve is a
" degenerate " sheet of the focal surface, and reciprocally if all rays lie on
tangent planes to a developable, the latter is a sheet of the focal surface.
When the rays intersect two curves or the same curve twice the focal surface
consists of the curve or curves, and the reciprocal theorem may be easily
stated.]
For instance, the degeneration which has been just mentioned of
necessity takes place when the congruence is algebraically of the first order.
In this case, since through each point only one ray of the congruence can in
general be drawn, a point cannot be the intersection of two of the lines unless
it be a point through which an infinity of the lines can be drawn ; and if the
locus of points of intersection were a surface, every point of the surface would
be a singular point, which is absurd. The locus is therefore a curve. If it be
a proper curve, it must by definition be such that the cone standing on it,
whose vertex is an arbitrary point, shall have one and but one apparent
double line. This is the case when the curve is a twisted cubic, and there is
no higher curve which has only one apparent double point. The only con
gruence then, of the first order, consisting of a system of lines meeting a
proper curve twice, is when the curve is a twisted cubic. We might, however,
have a congruence of lines meeting two directing curves, and if these curves
be of the orders m, m', and have a common points, the order of the congruence
will be mm'  a. The only algebraic congruence of the first order of this kind
is when the directing lines are a curve of the n th order, and a right line meet
ing it n  1 times.
[457a. Surfaces connected with a congruence. There
are certain surfaces associated with a congruence to which
attention may be directed: The limit surface is the two
sheeted locus of the limit points; the focal surface (Art. 457)
RECTILINEAR CONGRUENCES. 65
is the twosheeted locus of the foci or envelope of the focal
planes ; the middle surface is the locus of the " middle point "
which lies halfway between the limit points and as we have
seen (Art. 457) coincides with the point halfway between the
foci. The envelope of the plane drawn through the middle
point perpendicular to the ray is called the middle envelope ;
and the twosheeted envelope of the principal planes may be
termed the limit envelope. The coordinates of the points on
the first three surfaces, and of the tangent planes to the last
two can at once be expressed in terms of the parameters p, q
by the equations already given for the limit points and foci
and for the values of t which determine the values of dp:dq
corresponding to the principal planes.
In addition to these unique surfaces, we have the two
families of developables already defined, and the principal sur
faces may also be noticed. These are two singly infinite
families of ruled surfaces generated by rays of the congruence
which are chosen so that the shortest distance between a
generator and the consecutive generator intersects the former
in a limit point, i.e. the limit points describe lines of striction
(Art. 562). Each principal plane touches a principal surface
at the corresponding limit point.
In general the developables meet the focal surfaces in two
conjugate systems of curves. This follows at once from the
definition of conjugate directions (Art. 268), since all the rays
lie in tangent planes to each sheet of the focal surface.
The two focal points (but not the focal planes) will coin
cide on all rays touching the curve of intersection of the focal
sheets. But they may coincide on every ray and in this case
the two families of developables coalesce into one, the focal
planes also coincide, and from the last paragraph or directly
we see that the rays are tangents to one system of asymptotic
lines on the focal surface. Such a congruence is said to be
parabolic. A congruence or a portion thereof is said to be
hyperbolic when the focal points on the rays are real and
distinct ; and elliptic when the rays though real have imagin
ary focal points. This classification is evidently determined
VOL. II. 5
66 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
by the sign of the discriminant of either of the quadratic
equations in Art. 457.*
As an example consider the congruence denned by the equations
I = by, m = ax, n = >Jl  ax  by
the plane 2=0 being the surface of reference. Clearly all real rays emanate
from points within the ellipse a 3 x 3 + b*y 2 = 1. It will be found that the dis
criminant of the quadratic determining the values of dx : dij for which two
consecutive rays intersect is
We leave it to the reader to show that if ab is positive the real portion of the con
gruence is hyperbolic ; and that if ab is negative the equation A = represents
a curve lying within the ellipse, and dividing it into two regions for the outer
of which the corresponding portion of the congruence is hyperbolic and for the
inner elliptic, while the equation A=0 corresponds to the curve of intersection
of the focal sheets.
Ex. 1. The coordinates of a point on the middle surface are expressed in
terms of two parameters p and q by the equations
{ = x + It, i) = y + mt, z + nt
and x, y, z, the point where the ray meets the surface of reference, are known
functions of p and q.
Ex. 2. The condition that the surface of reference may be the middle
surface is
ag + ec  f (b + b') = 0.
Ex. 3. If the surface of reference be the middle surface the coordinates
of a point {, ij, , on the middle envelope are given in terms of p and q by the
equations
dw
f dw
 ~dq
where w = Ix f my + m.]
[457&. Normal Congruences. A normal congruence is one
whose rays are cut orthogonally by a singly infinite system of
surfaces, which are therefore capable of being represented
implicitly or explicitly by equations of the form
/ (x, y, z) = constant,
where the constant varies from surface to surface.
* For the geometrical justification of the terms used see Sannia, Gtometria
differenziale delle congruence rettilinee (Math. Ann., 68, 1910),
BECTILINBAB CONGRUENCES. f>7
The preceding definition is, however, superfluous, for it has
long been known that */ a congruence is normal to a siin/ti
surface it is normal to a singly infinite system of surfaces.
Let the single surface (5 = 0) to which all the rays are normal
be taken as the surface of reference. Any point (Q, , rj, f)
on a ray through (P, x, y, z) is defined by
= x + lt, ij = y + mt, =z + nt,
where t is the distance PQ.
Hence ld + mdrj + nd= dt, since Idx + mdy + ndz = 0. And
if t is constant ld% + mdrj + nd%=0; that is, the surface locus of
Q, as P moves along S = 0, is normal to the rays of the con
gruence. Through any point in space we can describe one
of these normal surfaces, by finding the locus of points equi
distant from S = Q; and so these surfaces form a singly
infinite system, and the proposition stated has been proved.
The directioncosines I, m, n may be supposed to be given
as functions of the current coordinates f, 77, f, of any point
in space. If Idi; + mdrj + nd is a perfect differential dt, it is
clear that the congruence thus defined is normal to the sur
faces t = constant. From what precedes it follows that the
converse is also true, and so, as Hamilton pointed out, the
necessary and sufficient condition that a congruence be normal
is that Idt; + mdrj + nd should be a perfect differential of a
function of three independent coordinates. (Cf. Ex. 1, Art.
Now let the surface of reference be chosen arbitrarily,
and let I, m, n be a normal congruence defined by two para
meters p and q. We then have as before
ld% + mdi) + nd= Idx + mdy + ndz + dt = Pdp + Qdq + dt.
The condition requires that ld% + mdrj + nd, when expressed
in any three independent parameters, should be a perfect
differential, and therefore since
P = lx + my l + nz lt Q = Ix 2 + my 2 + nz^,
we must have, because Pdp+ Qdq is a perfect differential,
v (lx, + my, + nz^\ = (Ix 2 + my^ + nz^
dq dp
and therefore Ix + m. + n l z 2 = l^x l + m, i y l + n 2 z l , i.e. the
68 ANALYTIC GEOMETRY OP THREE DIMENSIONS.
necessary and sufficient condition that a congruence, expressed
by any two parameters with any surface of reference, may be
normal, is
b = b'.
Now if we compare the quadratics for the limit points
(Art. 456) and for the focal distances (Art. 457), we see that
they are of the form
Thus the condition b = b' is equivalent to the condition that
these equations have the same roots. Hence a normal con
gruence is one whose focal points coincide with the limit
points. In this case both coincide with the centres of cur
vature of the orthogonal surfaces, and the focal surface,
limit surface, and limit envelope coincide with the surface
of centres. The focal planes are therefore at right angles.
Again, if we use the expression given in Ex. 2, Art. 457,
for the directioncosines of the normal to a focal plane, we
shall find that the cosine of the angle (0) between the two
focal planes is
edp^dp^ +f(dp l dq% + dp. 2 dq^) + gdq^dq^
J(edpi + 2fdp l dq 1 4 gdq^} (edp/ + ^fdp. 2 dq z + gdq/)
where dp l : dq l and dp. 2 : dq are roots of the quadratic in
dp : dq given in Art. 457. It will be found that when cos 6
is expressed in terms of a, b, b r , c, e,f, g, the numerator of the
fraction after reduction is (eg f) (b  b'), and hence the neces
sary and sufficient condition that a congruence be normal is
that the focal planes cut at right angles.
Ex. 1. By taking the middle surface for the surface of reference and making
dp=0, dq=0 correspond to the focal planes it can be proved that for any con
gruence the sine of the angle between the focal planes is ^, where r is the
distance between the focal points and p that between the limit points.
Ex. 2. The congruence formed by common tangent lines to two confocals
and, in particular, that formed by lines meeting e$ch of two focal conies, a.ro
normal (see Art. 176).
RECTILINEAR CONGRUENCES. (')'.)
The following theorem, due to Mains and Dupin,* is im
portant in geometrical optics : A normal congruence remain*
normal after refraction or reflexion at any surface. The
refracting surface may be taken as the surface of reference,
and if //, be the constant index of refraction f
sin i = /z sin i'
where i and i' are the angles between the normal and the in
cident and refracted rays. Hence if the rays are I, m, n, and
I', m\ n, we have easily
n = \w + fin'
where u, v, w are the directioncosines of the normal to the
surface of reference. Hence, along the surface of reference
lax + mdy + ndz = /z (I'dx + m'dy + n'dz) .
Thus, if ldx + mdy+ ndz is a perfect differential of a function
of p and q the same is true of I'dx + m'dy + n'dz, and there
fore (p. 67) if the first congruence is normal so is the
second.
There is a simple geometrical relation between the normal
surfaces of the two congruences. Let Q (, 77, ) be a point
on the incident ray, P (x, y, z) the point where this ray meets
the refracting surface, and Q' (', v', ') a point on the refracted
ray. If QP = p, PQ' = p', then it is easy to prove
= \(udx + vdy + wdz) +dp
and therefore I'd^' + m'drf + n'd' = 0, provided dp + pdp' = 0.
Hence the refracted rays are normal to the surfaces derived
from any surface normal to the incident rays by taking the
loci of points for which p + ftp' is constant.
Ex. 3. Prove that a surface may be found such that any normal con
gruence will be refracted through it with any index so that (n)the emergent
rays will pass through a given point or more generally (6) coincide with the
rays of any other normal congruence.
* Malus proved that the rays of a star when reflected from a surface are
normal to some surface ; Dupin extended the theorem as here given.
t For reflexion /*=!.
70 ANALYTIC GEOMETRY OP THREE DIMENSIONS.
Thus any normal congruence may be illustrated theoretically at all
events by the refraction through some surface of a system of rays of light
proceeding from a fixed point chosen arbitrarily.
When a surface is deformed a congruence may be said to
be deformed with it if the ray at a point on a surface does not
alter its angular relations to the linear elements of the surface
in the neighbourhood of the point. Beltrami first showed
that a normal congruence remains normal if deformed with
any surface.
Take the surface as that of reference and suppose that the
parametric lines correspond. Then expressing the condition
(Art. 390) that E and G and the angles between the ray and
the parametric lines are unaltered we find that lx l + my l + nz^
and lx. 2 + my. 2 + nz. 2 are unaltered, and therefore
Idx + mdy + ndz, since it = (lx t + my l + nz^) dp + (lx. 2 + my., +
nz. 2 } dq, continues to be a perfect differential.
We may here mention an associated theorem which is a
particular case of a general theorem due to Ribaucour (Art.
486^) . If tangent planes be drawn through the rays of a recti
linear normal congruence to any surface the congruence re
mains normal if the surface be deformed in any manner carry
ing the rays in its tangent planes in the way just indicated.
As the general proof will be given later we do not insert it
here.
The rays of any congruence are tangent lines to a singly
infinite system of curves on either sheet of a focal surface,
and the other sheet is the envelope of the osculating planes
of these curves. If the congruence is normal these curves
are geodesies on the focal surface, which of course is the sur
face of centres (Art. 309). Thus the rays of every normal
congruence are tangent lines to a singly infinite family of
geodesies on some surface. Conversely the tangent lines to a
singly infinite family of geodesies on a surf ace form a normal
congruence ; for one focal plane of a ray is the tangent plane
at the point where the ray touches the surface, while the
second focal plane is the osculating plane to the geodesic at the
RECTILINEAR CONGRUENCES. 71
same point, and since these planes are at right angles (Art.
308) the congruence is normal (p. 68). Since the envelope of
a singly infinite family of curves on a surface is another curve
(real or imaginary) on the same surface, every normal con
gruence consists of the rectilinear prolongations of geodesies
touching a fixed curve on some surface. The envelope may
of course reduce to a point or to a group of points. Thus a
normal congruence may be illustrated by stretching a string
over a fixed surface as the following example shows :
Ex. 4. If a thread be passed tightly round any part of a curve on a sur
face, or attached to a fixed point thereon, and stretched over the surface to a
point in space, the possible positions of its rectilinear portions form a normal
congruence, and any point of the thread describes a normal surface.
Ex. 5. The common tangent lines to two confocal quadrics form a normal
congruence, whose rays continued geodesically on either surface touch their
common line of curvature, which reduces to an umbilic when one of the con
focals reduces to a focal conic. (Cf. Art. 405.)
Ex. 6. Using elliptic coordinates (Art. 421a), the surfaces normal to the
congruence of Ex. 3 are the family
ff(\)d\ + f'f(n) dp + I f (v) dv = constant
where / (t) = _J^IKZ*L
^(a t)(fi t) (y  t)
Use the expression (Art. 421a) for twice the element of length of a ray,
namely
f (\) d\ + f (p) dp + f ( v ) d*.
Ex. 7. If the curves on a surface orthogonal to the family q = constant,
be geodesies, their tangent lines are normal to the family of surfaces
+ p = constant,
where p is the length of a tangent line (Art. 396o, Ex. 1).]
[457c. Directed Normal Congruences. If a sheet of the
focal surface degenerates into a curve (see end of Art. 4.">7>.
all the rays meet this curve and the congruence may be
described as directed. We may apply the term singlydirected
if each ray meets a fixed curve once only, and the term
doubly directed if all the rays meet each of two fixed curves
once, or the same curve twice. A doublydirected congruence
is determined if we are given both the directing curves.
72 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
Suppose thai all the rays of a congruence meet a curve
C r Then one focal plane (L = 0) on any ray is the plane
containing the ray and the tangent line to C v at the point P
where the ray meets the curve. The second focal plane
(M =0) is clearly the tangent plane through the ray to the
cone of rays that pass through P and are .tangent to the
second focal sheet (or which intersect the second directing
curve if there is one). In the case of a normal congruence
the planes L and M are mutually perpendicular (p. 68), and
so the tangent plane along any generator of the cone is
perpendicular to the plane containing the generator and a
fixed line through P. The sections of the cone by planes
perpendicular to that fixed line must therefore be circles,
because the tangent at any point of a section is perpendicular
to the line joining the point to a fixed point in its plane.
Thus the cone is one of revolution, and its axis is the tangent
line to CL. Conversely if every cone is one of revolution round
the tangent line, the congruence is normal. We have proved
then that the necessary and sufficient condition that a directed
congruence be normal is that the tangent cone from any point
of the directing curve to the second sheet of the focal surface
should be a cone of revolution round the tangent line to the
directing curve at the vertex.
It' the rays also meet a second curve (<7 2 ) this forms the
second sheet of the focal surface, and if the congruence be
normal, any cone passing through either curve and having its
vertex on the other is a cone of revolution. Both the curves
must therefore be conies since (Art. 200) it is not otherwise
possible to draw more than four quadric cones through the
same curve. Hence (Art. 184) we have the remarkable
theorem : The necessary and sufficient condition that a
doublydirected congruence should be normal is that the
directing curves should be focal conies of a confocal system of
quadrics.]
[457cZ. Cyclides of Dupin. We shall now consider the
surfaces normal to a doublydirected congruence. Let P be
1, 1 < TILINEAR rONOTUTENCT.S.
a point on a directing curve of a directed normal conf'i u< IMV.
A sphere with P as centre intersects the tangent cone from
P to the second focal sheet in a circle which is clearly ;i line
of curvature on a normal surface. Thus if the normals to a
surface form a directed congruence the lines of curvature of
one system are circles, and it follows that if the normals to
a surface form a doubly directed congruence, the tines of
curvature of both systems are circles. Such surfaces are
known as Cyclides of Dupin.*
Conversely it may be seen by using Lancret's theorem
(Art. 312) that if on a surface the lines of curvature of one
system are circles the centres of curvature for all the points
on a given circle reduce to a single point symmetrically placed
with regard to the circle, and therefore since the circles form
a singly infinite system, the corresponding sheet of the sur
face of centres is a curve. We infer that if the lines of
curvature of both systems are circles the normals to the surface
form a doublydirected congruence.
Corresponding to any quadric we have three systems of
parallel cyclides of Dupin, namely, those whose normals inter
sect a pair of focal conies. Only one system is real, and its
equations may be expressed very simply in elliptic coordinates
by the method suggested in Ex. 6, Art. 4576. In this case
the two quadrics reduce to the focal conies and then (Art.
421a) X = 7, /i = /3. The equation
\f(\)d\+ \f(n} dp + \f(v) dv = constant
then becomes
>Ja  A, + ^/a  //, + Ja  v constant,
or using the notation of Art. 160
a'a"a'" = t
where t is a constant.
If this equation be rationalized it will be found (Art. 1GO)
that the cyclide referred to the axes of the con focal system is
the surface of the fourth order
* See Dupin's Applications de gfomftrie et de mfclianiqne (1'22).
74 ANALYTIC GEOMETRY OP THREE DIMENSIONS.
Or 2 + 2/ 2 + 2 /i 2 & 2 + 2 ) 2
= 4 (t*  A 2 ) if + 4 (P  k 2 ) z + 4 (tx
Eeferring to Art. 4216, we see that the equations
TI = t + x /a  7, r 2 = t + Ja  7, s 1 = t  v /a  , S 2 = *  N /o  /3
represent a cyclide of the system. From this we derive the
following mechanical construction : If a string of constant
length be attached at one end to the focus of a conic and
stretched tight over the conic, the locus of its extremity is a
portion of a Dupin cyclide. This is a special case of the
theorem of Ex. 4, Art. 4576.
The cyclide may be defined in various ways as an envelope. Consider the
two circles of curvature at a point P ; these lie respectively on two spheres
whose centres are on the focal conies, and since the line joining the centres is
the normal to the cyclide at P it passes through a point of intersection of
the spheres ; hence the two spheres touch at P. The spheres also touch the
cyclide which is thus the envelope of a one parameter system of spheres whose
centres are on the focal conies, each sphere of one system touching all those
of the other. But since three spheres are enough to determine an envelope,
the cyclide is definable as the envelope of a sphere touching three fixed spheres
(see also Art. 567).
Conversely every such envelope is a cyclide of Dupin. For if we invert
from a point common to the three spheres, the inverse envelope is the en
velope of a sphere touching three fixed planes and is therefore a right cone.
The lines of curvature of a right cone are the generators and the circular
sections, and therefore since lines of curvature are preserved in inversion
(Art. 384), those on the original envelope are circles.
Ex. 1. Prove that the equation of the envelope of a sphere touching three
fixed spheres S l = 0, S a = 0, S 3 = may be written
tm*J&l + *31 v'^'s + *lN/#3 =
where S l = (x  zj + (y  j/,) 8 + (  *i) 2  ^
and / M * = teag* + (t/ 2  i/ 3 ) 2 + (,  %)*  (r,  7 8 ) 2 .
Ex. 2. Any cyclide of Dupin inverts into another.
Ex. 3. If the directing conies are parabolas in perpendicular planes each
passing through the focus of the other, the corresponding cyclides are of the
third order.
Ex. 4. All circles drawn through a fixed point and cutting orthogonally
a given Dupincyclide, intersect two fixed spherocurves.]
[457e. Ribaucour's Isotropic Congruences. We have seen
(Art. 456) that the points where a ray meets the shortest
RECTILINEAR CONGRUENCES. 75
distance from a consecutive ray lie between the limit points
L l and L. 2 . But all these points may coincide and a ray will
then meet all shortest distances at the middle point M. A
congruence all of whose rays possess this property is said to be
isotropic.* The middle surface coincides with the limit sur
face and is real, but it may be shown that the focal surface
is imaginary.
Since the points where a ray meets its shortest distance
from a consecutive ray is given (Art. 456) by
_ adp 2 + (b + b') dpdq + cdq 1
edp* + Zfdpdq + gdq*
and since in the present case all these values are equal, the
necessary and sufficient condition tJiat a congruence be isotropic
is a = b + b' = c_
~e ~W 9
each of these with sign changed being the distance of the
middle point from the point where the ray meets the surface
of reference. Hence if the middle surface be taken for the
surface of reference we have a = b + b' = c = Qor what amounts
to the same, for all variations on the middle surface didl +
We notice that an isotropic congruence is uniquely de
fined by the property that the lines of striction of all its ruled
surfaces lie on a surface (the middle surface).
Ex. 1. Through each point of the plane z = a ray I, m, n is drawn, where
I = ky, m =  kx, n = / 1  k* (x* + j/ a ).
Prove that the congruence so formed is isotropic, that z = is the middle sur
face and that the middle envelope (Art. 457a) reduces to a point.
Ex. 2. The only normal isotropic congruence is a system of rays passing
through a point.
Ex. 3. If the distance between corresponding points on two applicable
surfaces be constant the joining lines form an isotropic congruence, whose
middle surface is the locus of the middle point of those lines.
Conversely if the same constant length be measured in each direction
* These congruences were fully investigated by Ribaucour in connexion
with minimal surfaces (see Art. 457/) in his titiide des filasso'ides on Surface*
a Courbure moyenne nulle, " Mt'moires Couronnfs par L' Academic Royale de
Belgique," XLIV, 1882.
76 ANALYTIC GEOMETRY OP THBEE DIMENSIONS.
along a ray from the middle surfaces of an isotropic congruence the loci of the
extremities are applicable surfaces.
Ex. 4. The spherical representation of a congruence, or of a portion
thereof, is sometimes defined as the arrangement of points in which radii
through the centre of a fixed sphere and parallel to rays of the congruence
meet the surface of the sphere. Prove that if the congruence be isotropic
corresponding linear elements of any curve C on the middle surface and of
the spherical representation of the rays through points on C are orthogonal.
Eibaucour gives the following method of generating
isotropic congruences. Let p and q be isometric parameters
(Art. 392a) on any sphere, and let da be the element of an
arc of the sphere ; then d<r~ = V 2 (dp* + dq 2 } where X is a
function of p and q. At each point IT of the sphere draw a
tangent line to one of the parametric curves, and measure on
this line a distance IIP = \. Then the ray through P drawn
parallel to the normal (I, m, ri) to the sphere at U generates
an isotropic congruence.
Ex. 5. For example the congruence of Ex. 1 may be defined by taking,
along the tangent line to the parallel of latitude at n, a length nP propor
tional to the cosine of the latitude, and drawing through P a ray parallel
to the radius of the sphere at n. (Cf. Art. 392a, p. 419.)
Conversely every isotropic congruence has the relation just
mentioned to any sphere. There is no difficulty in proving
the original proposition, but we shall indicate a method of
proving the converse. Let IT be the spherical representation
of a ray R of an isotropic congruence, i.e. IT is the point
where a fixed sphere is met by the radius parallel to the ray.
Let the tangent plane to the sphere at IT cut the ray in P.
Then the system of lines PIT envelopes a singly infinite
system of curves on the sphere, and we have to show that
these curves with their orthogonal trajectories on the sphere
form an isothermal system, i.e. they can be represented by
the parametric equations u = constant, v = constant, so that
where da is an element of the spherical arc. Furthermore
PIT is proportional to JE.
KECTILINEAR CONGRUENCES. 77
The ray being I, m, n, and the radius of the sphere unity, the coordinate
of n are Imn. Let nP = r, and let nP be tangent at n to the curve q =
constant. Then if we take the curves p = constant to represent the ortho
gonal trajectories ol q = constant we have
do 2 = edp* + gdq*, and / = 0.
The coordinates of P are x, y, z where
x = I + fil lt y m + fim^ z = n + /^n 1
where n = 7.
ve
The conditions for an isotropic congruence, since / = 0, are
b + b' = 0, a = C 
*
and these will be found to yield the equations
_ 1 dg 1 de
dq ' /t dp ~ g dp e dp
U?&
from which we infer that ^ is a function of p alone, and that is a function
9
of q alone. Thus  is of the form ^4^r. and we find
g f(j)
do 2 = E (dit 2 + di> 2 )
where u is a function of p only, and v of q only, and the parametric lines are
therefore unaltered. Hence the parametric lines are isothermal. Again the
preceding equations show that if e = g then /i is constant, and therefore
JM
is constant.
Taking the middle surface for the surface of reference we
have a = 0, b + b' = 0, c = 0, and the values of dp : dq corres
ponding to the focal points (Art. 457) are given by
edp 2 + 2fdpdq + gdq 2 = 0.
Since the left hand is dl? + dm 2 + dri 2 (Art. 456), the focal
points, planes and surface of the congruence are imaginary.
Moreover (Ex. 2, Art. 457), it is easily seen that if a focal
plane be written in the form
we must have a 2 +/3 2 + 7 >2 = 0. Hence (Art. 211) the focal
planes of an isotropic congruence touch the imaginary circle
at infinity, and therefore the focal surface is a developable
containing the imaginary circle at infinity * The converse
* The nature of a congruence whose focal sheets are developable is easily
understood by comparing it with the reciprocal case of a doublydirected con
gruence (Art. 457c). The rays are the intersections of pairs of tangent
planes.
78 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
may easily be proved. A developable of this type is sometimes
described as isotropic. Thus an isotropic congruence may be
defined as one whose focal surface is an isotropic developable,
or, more generally, two isotropic developables.* The congru
ence is real if the developables are conjugate imaginaries,
otherwise it is imaginary.
Ex. 6. The generators of one system of a family of quadrics passing through
a common curve form a doubly directed congruence determined by that curve.
Reciprocally the generators of quadrics touched by a common developable form
a congruence whose focal surface coincides with that developable.
Hence prove Ribaucour's theorem : The generators of either system of a
system of confocal hyperboloids of one sheet form an isotropic congruence.
It may be noticed that the generators of one system are the reflexions of
the generators of the other system with regard to any confocal quadric of
different species. Ribaucour proves also the converse, that if two congruences
which are mutual reflexions with regard to a surface are isotropic, the surface
is a quadric and the rays of the congruences are generators of quadrics con
focal thereto.]
[457/. One of the most interesting properties of the
congruence considered is this (Kibaucour) : The middle
envelope of an isotropic congruence is a minimal surface, that
is, its mean curvature vanishes (Art. 295, Ex. 1).
The demonstration depends on certain lemmas :
(1) If the directioncosines of the normal at a point Q (, 77, ) on a surface
S = are I, m, n, expressed in terms of two parameters p, q, the condition
for a minimal surface is
eD" + gD  2fD' 
where e = l^ + m^ + n^, f = Ijl^ + m^m^ + n^, g = 1 2 2 + m^ + n 2 *,
D = l^i + w^! + n^j, D' = Zjlj + m^ + n^ = l.^ + m 3 ij l + n^i and
D" = l^ + w 2 Tj 3 + WaCat where the suffixes 1 and 2 denote differentiation with
regard to p and q respectively.
Let Q' ({', TI', f) be a centre of curvature on S = corresponding to Q,
and let r be the principal radius QQ'. We have three equations
f ' =  + rl, i\' = i) + rm, = + rn
and therefore d' = d + rdl + Idr, with two similar equations.
But if ITJ^ be supposed to move along a line of curvature d{' = Idr, etc.,
since dr is an element of the arc along which {' moves, and therefore
d{ + rdl = 0, etc.
Usingrf = jdp + rjjrfg.dZ = Ijdp + lydq, etc., we find that the value of dp : dq
for the principal directions are given by
*Thjs js in fgt the definition from which Ribaucour starts.
RECTILINEAR CONGRUENCES. 79
Ddp + Hdq  r (edp + fdq) =
&dp + D"dq  r(fdp + gdq) =
and the principal radii aro given by the quadratic
(*7 ~ / J ) *'*  (*>" + gD  2fD') r + DD"  D* = 0.
Consequently the condition for a minimal surface is
eD" + gD  2/Z)' =
and if e = 0, g = the condition is D' = or l^ t + m^ f n^ = 0.
(2) Consider an isotropic congruence, and suppose its middle surface
If = is the surface of reference. Let P (x, y, z) be the middle point of a ray
whose directioncosines are I, m, n ; then x, y, z, I, m, n are functions of two
parameters p, q.
Let Q ({, rj, ) be the point of contact of the middle plane through P (i.e.
the plane perpendicular to the ray) with the middle envelope S = 0. Then
, ;, are functions of p and q.
The conditions for an isotropic congruence are
a = ^i + m 1 y l + n i z l = ......... (i)
b + b' = l^ + mfly + n^., + l z x l + m,^ l + n^ v = .... (ii)
c = l,,x. t + TWa?/., + n^ = ......... (iii)
We may choose the parameters p, q so that the parametric lines p = con
stant, q = constant correspond to the " minimal lines " in the spherical repre
sentation of the congruence (Art. 392a). This implies
dP + dm? + dn? = 2/dpdq, and e = g = 0, i.e.
If + mf + V = . ". . . . (iv)
V + m 2 2 + Ha* = ..... (v)
P +m? + n* = 1 ..... (vi)
Also since P and Q both lie on the middle plane
Ix + my + nz = l + mn + n . . . . (vii)
and, because the directioncosines of the normal at Q to S = are I, m, n, we
have
li + m *li + n Ci = ( vi )
and /{, + TOT;., + n, = . . . . (ix)
Using these equations we can prove the following relations, where
/ = IJ^ + mjitif + njn,
l u = f / In with two similar equations . . . (A)
and l a =  If, with two similar equations . . . (B)
Proof of (A). Differentiating both sides of equations (iv) and (vi) with
regard to p, 2^ u = and 21J = 0, and by a second differentiation of the
latter, using (iv), 2W U = 0. Hence since 2lj* = 0, and 2^ = 0, we find
and therefore 2W U = A/.
2^/u
 
But ' = 2Z 2 Z n since 2^/u = by (iv)
therefore
80 ANALYTIC GEOMETKY OF THREE DIMENSIONS.
Proof of (B). The equations (iv) and (v) give 2^i 12 = 0, 2y i2 0, and
since from fvi) 21J = 0, 2W = 0, we have
Z 12 = pi, w 12 = pin, 7i 12 = fi.n,
and by using (vi) p = 2ZZ 12 =  2^ =  /, which proves the formula (B).
(3) We can now prove
and the condition that S = may be a minimal surface is therefore
Zz u + w;/ 12 + nzu = 0.
Differentiating both sides of equation (vii), v\z:"Slx 2Z, with regard to
p and q successively and using equations (ii) and (viii)
But since 2lx = 2Z the equation (B) gives 2/ 12 = 2/ 12 { and consequently
(4) To prove 2lx iy + mij^ + nz n = 0, use the identity
2 (lx a + wj/ 12 + nz^ = JT (Zx 1 + Mty! + nz^ + ^ (Zx
which follows from equation (ii).
Now by (B), /2^ x =  2^! = 2^x ]2 , by (i),
d fb\ , . d Sb'\
Hence lx l + my l + nz^ = = (  1, and similarly, Ix.^ + my^ + nz^ = j~ I 7 )
and therefore, since 6 + b' = 0, it follows that
and the proposition of this article is proved.]
SECTION III. RULED SURFACES.
458. On account of the importance of ruled surfaces, we
add some further details as to this family of surfaces.
The tangent plane at any point on a generator evidently
contains that generator, which is one of the inflexional tan
gents (A.rt. 265) at that point. Each different point on the
generator has a different tangent plane (A.rt. 110), and the
following construction gives the tangent plane and the second
inflexional tangent. We know that through a given point
can be drawn a line intersecting two given lines ; namely, the
intersection of the planes joining the given point to the
given lines. Now consider three consecutive generators,
and through any point A on one draw a line meeting the
other two. This line, passing through three consecutive
RULED SURFACES. 81
points on the surface, will be the second inflexional tangent
at A, and therefore the plane of this line and the generator
at A is the tangent plane at A. In this construction it is
supposed that two consecutive generators do not intersect,
which ordinarily they will not do. There may be on the
surface, however, singular generators which are intersected
by a consecutive generator, and in this case the plane con
taining the two consecutive generators is a tangent plane at
every point on the generator. In special cases also two
consecutive generators may coincide, in which case the
generator is a double line on the surface.
[Any surface may be regarded as generated by the motion of a curve
/L(X, y, z, w, t) = 0, f*(x, y, z, w, t) =
where t is a variable parameter. Eliminating dt from the two equations of the
type
& dx + & dy +^ dz + p dw + ^d< =
dx dy dz dw dt
we find
Xdx + Tdy + Zdz + Wdw =
and it is clear that XYZW &re coordinates of the tangent plane at x, y, z, w.
In the case of a ruled surface f l = and/ 2 = may be taken to be planes
Ol x + j3j!/ + 7j2 + Sjtfl = 0, a.]X + fay + y^z f S t w 
and it will be found that the coordinates of the tangent plane are
a^a  a 2 L 1 , ^3  ft. 2 L lt 7jL 2  y^L lt SjLj  SoLj,
where L lt L 2 are linear in x, y, z, w, and their coefficients are functions of t.
There is thus a homographic correspondence between the points on a generator
and the tangent planes at these points, i.e. if T = 0, T' = be the tangent
planes at xyzw and x' y' Z, w', then T + \ T' = is the tangent plane at
the point x + \x', y + \y', z + \z', 10 + \w'. This is proved synthetically
in the following article.]
459. TJie (inharmonic ratio of four tangent planes pass
ing through a generator is equal to that of their four points
of contact. Let three fixed lines A, B, C be intersected by
four transversals in points aa'a'a", bb'b"b'", cc'c'c". Then
the auharmonic ratio {bb'b"b"'\ = {cc'c'c"}, since either measures
the ratio of the four planes drawn through A and the four
transversals. In like manner {cc'c'c "}= {aa'a'a'"}, either
measuring the ratio of the four planes through B (see Art.
114). Now let the three fixed lines be three consecutive
generators of the ruled surface, then, by the last article, the
VOL. n. 6
82 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
transversals meet any of these generators A in four points,
the tangent planes at which are the planes containing A and
the transversals. And by this article it has been proved that
the anharmonic ratio of the four planes is equal to that of
the points where the transversals meet A.
[Ex. Prove that the generators of a ruled surface are cut equianharinoni
cally by the other system of asymptotic lines.]
460. It is well known that a series of planes through any
line and a series through it at right angles to the former
constitute a system in involution, since the auharmonic ratio
of any four is equal to that of their four conjugates. It
follows then from the last article that the system formed by
the points of contact of any plane, and of a plane at right
angles to it, form a system in involution ; or, in other words,
the system of points where planes through any generator
touch the surface, and where they are normal to the surface
form a system in involution. The centre of the system is the
point where the plane which touches the surface at infinity
is normal to the surface ; and, by the known properties of
involution, the rectangle under the distances from this point
of the points where any other plane touches and is normal,
is constant.
461. The normals to any ruled surface along any generator
generate a hyperbolic paraboloid. It is evident that they are
all parallel to the same plane, namely, the plane perpendicular
to the generator. We may speak of the anharmonic ratio
of four lines parallel to the same plane, meaning thereby that
of four parallels to them through any point. Now in this
sense the anharmonic ratio of four normals is equal to that
of the four corresponding tangent planes, which (Art. 459) is
equal to that of their points of contact, which again (Art. 460)
is equal to that of the points where the normals meet the
generator. But a system of lines parallel to a given plane
and meeting a given line generates a hyperbolic paraboloid,
if the anharmonic ratio of any four is equal to that of the four
RULED SURFACES. R3
points where they meet the line. This proposition follows
immediately from its converse, which we can easily establish.
The points where four generators of a hyperbolic parabo
loid intersect a generator of the opposite kind are the points
of contact of the four tangent planes which contain these
generators, and therefore the anharmonic ratio of the four
points is equal to that of the four planes. But the latter ratio
is measured by the four lines in which these planes are inter
sected by a plane parallel to the four generators, and these
intersections are lines parallel to these generators.
462. The central points of the involution (Art. 460) are, it
is easy to see, the points where each generator is nearest
the next consecutive ; that is to say, the point where each
generator is intersected by the shortest distance between it
and its next consecutive. The locus of the points on the
generators of a ruled surface, where each is closest to the
next consecutive, is called the line of striction of the surface.
It may be remarked, in order to correct a not unnatural
mistake (see Lacroix, vol. in. p. 668), that the shortest distance
between two consecutive generators is not an element of the
line of striction. In fact, if Aa, Bb, Cc be three consecutive
generators, ab the shortest distance between the two former,
then b'c the shortest distance between the second and third
will in general meet Bb in a point b' distinct from b, and
the element of the line of striction will be ab' and not ab.
Ex. 1. To find the line of striction of the hyperbolic paraboloid
Any pair of generators may be expressed by the equations
? + J = A,, ?*;
a 6 a b A
x y x y 1
a + b = ft *' a~5 = M
Both being parallel to the plane *   , their shortest distance is perpendicular
&
to this plane, and therefore lies in the plane
6*
84 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
which intersects the first generator in the point 2 = ^  =j .
When the two generators approach to coincidence, we have for the co
ordinates of the point where either is intersected by their shortest distance
a*  b 3 1 x y a 2  fe 2 1
~ a s + 6* \*' a + I ~ a 2 + 6 2 \'
and hence (o + *) (j + J) = ( 2  *) g  f ), or + = 0.
The line of striction is therefore the parabola in which this plane cuts the
surface. The same surface considered as generated by the lines of the other
system has another line of striction lying in the plane
x y
 = 0.
a j 6*
Ex. 2. To find the line of striction of the hyperboloid
z 2 y* _ z* _
a* + & ~ ^ ~
Ans. It is the intersection of the surface with
oM 2 && _
of t/ 3 * 2 '
11 11^11
where A = r + 5, B =,+  C = r*  .
u C^ Or C~ Or fl
[462a. The parametric method of Gauss may be conven
iently applied to ruled surfaces by the use of a device ana
logous to that employed for congruences (Art. 456). Let x, y, z
be the coordinates of any point (P) on any fixed director
curve, p = constant, lying on the surface. If Q(%, ij, ) be any
point on a generator through P we have
where q = PQ, and I, m, n are the direction cosines of PQ.
If we suppose x, y, z and I, m, n to be functions of p, these
three equations express the coordinates of any point on the
ruled surface in terms of p and q. Using the method and
notation of Art. 456, if r is the distance from P of the point
where a ray is met by the shortest distance from a consecu
tive ray
= l l
The point thus defined is called the central point of the
generator, and the tangent plane thereat the central plane.
RULED SURFACES. 85
Since dg = (x i + ql^dp + Idq, with similar values for drj and
d, the direction cosines of the normal to the tangent plane
at , 77, are proportional to the coefficients of d, drj, d in
the determinant
dt\
and the corresponding ratios for the tangent plane at x, y, z are
formed by replacing q by zero. Now if P be the central point
l l x 1 +m l y l +n l z l = Q, and if in addition the ray is parallel to
the axis of z, l = m = and n = l, and if, thirdly, the director
curve be chosen so that its tangent line at P coincides with
the axis of y, we have x l = z l = Q. If all these conditions
are fulfilled we have also m^ = 0, and the direction cosines of
the perpendicular to the tangent plane at any point , */,
on the ray through P are in the ratio y l :  ql : 0. Thus
the tangent of the angle between the tangent plane at any
point of the generator and the tangent plane at the central
point has a constant ratio to the distance between the points,
The reciprocal of this constant ratio has been called the
parameter of distribution of the ray. It becomes infinite for
a developable surface.
The locus of the central points is the line of striction
(Art. 462) and its parametric equation is, in general,
, ii i !!
2
Ex. 1. In general the parameter of distribution is
+ Q 2 I m n
Ex. 2. Investigate by means of torsion and curvature the central points
and parameters of distribution for the ruled surface generated by the principal
normals of a curve.
If dt represent the shortest distance between two conse
cutive generators, and if da 2 = dl 1 + dm' + dri*, it may be shown
easily that the parameter of distribution is 3. Let us now
consider the ruled surfaces of a rectilinear congruence which
86 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
passes through a given ray. Using the method of Art. 457,
we can show that the parameter of distribution for a ruled
surface corresponding to a given value of dp : dq is
1 (eV  afidp 1 * + {ec  ag +/(b>  b}}dpdq + (cf bgWq*
(eg  / 2 ) k edp* + 2fdpdq + cdq*
and thus, following the method of Art. 456, we can show
7r = "jr l cos 2 6 + 7T 2 sin 2 6
where is the angle which the central plane of the ruled
surface with parameter of distribution TT makes with a fixed
plane. TJ^ and 7r 2 are the extreme values of the parameter of
distribution. It is easy to see that for a normal congruence
7r i + 7r 2 = 0> f r a hyperbolic congruence (Art. 45 la) T^TT., is
negative and for an elliptic positive, while for an isotropic or
circular congruence TT I = 7r 2 .] *
463. Given any generator of a ruled surface, we can de
scribe a hyperboloid of one sheet, which shall have this gener
ator in common with the ruled surface, and which shall also
have the same tangent plane with that surface at every point
of their common generator. For it is evident from the con
struction of Art. 458 that the tangent plane at every point
on a generator is fixed, when the two next consecutive gener
ators are given, and consequently that if two ruled surfaces
have three consecutive generators in common, they will touch
all along the first of these generators. Now any three non
intersecting right lines determine a hyperboloid of one sheet
(Art. 112) ; the hyperboloid then determined by any generator
and the two next consecutive will touch the given surface as
required.
In order to see the full bearing of the theorem here enun
ciated, let us suppose that the axis of z lies altogether in any
surface of the w th degree, then every term in its equation must
contain either x or y ; and that equation arranged according
to the powers of x and y will be of the form
^y* + &c. = 0,
* See an interesting paper by Sannia, " Geonietria differenziale delle con
gruenze rettilinee" (Math. Ann., 68, 1910).
RULED SURFACES. 87
where u n _ lt v,^ denote functions of zot the (  l) th degree, &c.
Then (see Art. 110) the tangent plane at any point on the
axis will be u'^x +v' n _ l y = 0, where u,^ denotes the result of
substituting in u,,^ the coordinates of that point. Conversely,
it follows that any plane y = mx touches the surface in n  1
points, which are determined by the equation u t ._ l + mv H _ l = 0.
If however u n _ lt v,^ have a common factor u p , so that the
terms of the first degree in x and y may be written
u t , (*__! + v n _ / ,_ 1 y) = 0, then the equation of the tangent plane
will be w'_ p _ 1 + v n _ p _^y = 0, and evidently in this case any
plane y = mx will touch the surface only in n  p  1 points.
It is easy to see that the points on the axis for which u p = Q
are double points on the surface. Now what is asserted in the
theorem of this article is, that when the axis of z is not an
isolated right line on a surface, but one of a system of right
lines by which the surface is generated, then the form of the
equation will be
M 2 ( ux + v y) + & c  = 0,
so that the tangent plane at any point on the axis will be the
same as that of the hyperboloid ux + vy, viz. ux + v'y = 0. And
any plane y = mx will touch the surface in but one point. The
factor u n _ 2 indicates that there are on each generator n  2
points which are double points on the surface.
[Ex. The tjangent lines to the second system of asymptotic lines through the
points on the given generator are generators of the hyperboloid.]
464. We can verify the theorem just stated, for an im
portant class of ruled surfaces, viz., those of which any
generator can be expressed, by two equations of the form
at m + bt m ~ l + ct m ~' 2 + &c. = 0, a't" + b't n ~ l + c't n ~* + &c. = 0,
where a, a, b, b', &c. are linear functions of the coordinates,
and t a variable parameter. Then the equation of the surface
obtained by eliminating t between the equations of the genera
tor (see Higher Algebra, Arts. 85, 8G) may be written in the
form of a determinant, of which when m = n the first row and
first column are identical, being (ab f ), (ac'\ (ad f ), &c., or when
m>n, the first row is as before and the first column consists.
88 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
of n such constituents, a and zeros. Now the line aa' is a
generator, namely, that answering to =oo ; and we have
just proved that either a or a will appear in every term,
both of the first row and of the first column. Since, then,
every term in the expanded determinant contains a factor
from the first row and a factor from the first column, the ex
panded determinant will be a function of, at least, the second
degree in a and a, except that part of it which is multiplied
by (ab f ), the term common to the first row and first column.
But that part of the equation which is only of the first degree
in 'a and a determines the tangent at any point of aa ; the
ruled surface is therefore touched along that generator by
the hyperboloid ab'  ba f = 0.
If a and b (or a and b'} represent the same plane, then
the generator aa' intersects the next consecutive, and the
plane a touches along its whole length. If we had b = ka,
b' = ka', the terms of the first degree in a and a' would
vanish, and aa' would be a double line on the surface.
465. Returning to the theory of ruled surfaces in general,
it is evident that any plane through a generator meets the
surface in that generator and in a curve of the (n  l) th order
meeting the generator in n  1 points. Each of these points
being a double point in the curve of section is (Art. 264) in
a certain sense a point of contact of the plane with the surface.
But we have seen (Art. 463) that only one of them is properly
a point of contact of the plane ; the other n  2 are fixed points
on the generator, not varying as the plane through it is
changed. They are the points where this generator meets
other nonconsecutive generators, and are points of a double
curve on the surface. Thus, then, a skew ruled surface in
general has a double curve which is met by every generator in
rc  2 points. It may of course happen, that two or more of
these 7i2 points coincide, and the multiple curve on the
surface may be of higher order than the second. In the case
considered in the last article, it can be proved (see Higher
RULED SURFACES. SO
Algebra, Lesson xviu.) that the multiple curve is of the order
^ (m + n  1) (m + n  2),
and that the number of triple points thereon is
\ (m + n  2) (m + n  3) (m + n  4).
A ruled surface having a double line will in general not
have any cuspidal line unless the surface be a developable,
and the section by any plane will therefore be a curve having
double points but not cusps.
466. Consider now the cone whose vertex is any point,
and which envelopes the surface. Since every plane through
a generator touches the surface in some point, the tangent
planes to the cone are the planes joining the series of gene
rators to the vertex of the cone. The cone will, in general,
not have any stationary tangent planes; for such a plane
would arise when two consecutive generators lie in the same
plane passing through the vertex of the cone. But it is only
in special cases that a generator will be intersected by one
consecutive ; the number of planes through two consecutive
generators is therefore finite ; and hence, one will, in general,
not pass through an assumed point. The class of the cone,
being equal to the number of tangent planes which can be
drawn through any line through the vertex, is equal to the
number of generators which can meet that line, that is to say,
to the degree of the surface (see end of Art. 124). We have
proved now that the class of the cone is equal to the degree of a
section of the surface ; and that the former has no stationary
tangent planes as the latter has no stationary or cuspidal points.
The equations then which connect any three of the singularities
of a curve prove that the number of double tangent planes
to the cone must be equal to the number of double points of
a section of the surface ; or, in other words, that the number
of planes containing two generators which can be drawn
through an assumed point, is equal to the number of points of
intersection of two generators which lie in an assumed plane.*
* These theorems are Cayley's. Cambridge and Dublin Matlwmalical
Journal, vol. vn. p. 171.
90 ANALYTIC GEOMETKY OF THEEE DIMENSIONS.
467. We shall illustrate the preceding theory by an enumer
ation of some of the singularities of the ruled surface generated
by a line meeting three fixed directing curves, the degrees of
which are m^, m. 2 , m s *
The degree of the surface generated is equal to the number
of generators which meet an assumed right line ; it is there
fore equal to the number of intersections of the curve m^ with
the ruled surface having for directing curves the curves m. 2 , m s
and the assumed line ; that is to say, it is m^ times the degree
of the latter surface. The degree of this again is, in like
manner, m. 2 times the degree of the ruled surface whose direct
ing curves are two right lines and the curve w 3 , while b}' a
repetition of the same argument, the degree of this last surface
is 2w 3 . It follows that the degree of the ruled surface when
the generators are curves m lt m. 2 , m 3 , is 2? 1 w 2 w 3 .
The three directing curves are multiple lines on the sur
face, whose orders of multiplicity are respectively m.,m s ,
WgWj, m^m^. For through any point on the first curve pass
w 2 w 3 generators, the intersections, namely, of the cones having
this point for a common vertex, and resting on the curves
w 2 , ra 3 .
468. The degree of the ruled surface, as calculated by the
last article, will admit of reduction if any pair of the directing
curves have points in common. Thus, if the curves m 2 , m^
have a point in common, it is evident that the cone whose
vertex is this point, and base the curve m lt will be included
in the system, and that the degree of the ruled surface proper
will be reduced by rn^, while the curve m^ will be a multiple line
whose order of multiplicity is only m. 2 m 3  1. And generally if
the three pairs made out of the three directing curves have
common respectively a, ft, 7 points, the degree of the ruled
surface will be reduced by n\a + w 2 /8 + wyy.f while the order
* I published a discussion of this surface, Cambridge and Dublin Mathe
matical Journal, vol. vm. p. 45.
fMy attention was called by Prof. Cayley to this reduction, which takes
place when the directing curves have points in eomuiou.
RULED SURFACES. '.)!
of multiplicity of the directing curves will be reduced respec
tively by a, yS, 7. Thus, if the directing lines be two right
lines and a twisted cubic, the surface is in general of the sixth
degree, but if each of the lines intersect the cubic, the degree
is only the fourth. If each intersect it twice, the surface is
a quadric. If one intersect it twice and the other once, the
surface is a skew surface of the third degree on which the
former line is a double line.
Again, let the directing curves be any three plane sections
of a hyperboloid of one sheet. According to the general theory
the surface ought to be of the sixteenth degree, and let us see
how a reduction takes place. Each pair of directing curves
have two points common ; namely, the points in which the
line of intersection of their planes meets the surface. And
the complex surface of the sixteenth degree consists of six
cones of the second degree, together with the original quadric
reckoned twice. That it must be reckoned twice, appears
from the fact that the four generators which can be drawn
through any point on one of the directing curves are two
lines belonging to the cones and two generators of the given
hyperboloid.
In general, if we take as directing curves three plane sec
tions of any ruled surface, the equation of the ruled surface
generated will have, in addition to the cones and to the original
surface, a factor denoting another ruled surface which passes
through the given curves. For it will generally be possible
to draw lines, meeting all three curves which are not gene
rators of the original surface.
469. The degree of the ruled surface being 27W 1 w 2 w 3 , it
follows, from Art. 465, that any generator is intersected by
2ra 1 w 2 w 3  2 other generators. But we have seen that at
the points where it meets the directing curves, it meets
(m 2 m 3  1) + (WgWi  1) + (rn^m.^  1) other generators. Conse
quently it must meet 2w 1 w 2 w 3  (m. 2 m 3 + rn.^ + mjn^ + 1 gene
rators, in points not on the directing curves. We shall establish
this result independently by seeking the number of generators
92 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
which can meet a given generator. By the last article, the
degree of the ruled surface whose directing curves are the
curves m^ m. 2 , and the given generator, which is a line resting
on both, is 2w 1 w 2  m^ m. 2 . Multiplying this number by w 3 ,
we get the number of points where this new ruled surface
is met by the curve m^. But amongst these will be reckoned
(n^m.j  1) times the point where the given generator meets
the curve m%. Subtracting this number, then, there remain
^m^rn.^m^  m. 2 m 3  m^m^  m^m^ + 1
points of the curve w 3 , through which can be drawn a line
to meet the curves m lt m. 2 , and the assumed generator. But
this is in other words the thing to be proved.
470. We can examine in the same way the degree of the
surface generated by a line meeting a curve m^ ttcice, and
another curve w 2 once. It is proved, as in Art. 467, that the
degree is m. 2 times the degree of the surface generated by a line
meeting n^ twice, and meeting any assumed right line. Now
if 7* a be the number of apparent double points of the curve m lt
that is to say, the number of lines which can be drawn
through an assumed point to meet that curve twice, it is
evident that the assumed right line will on this ruled surface
be a multiple line of the order 7^, and the section of the ruled
surface by a plane through that line will be that line 7^ times,
together with the ^n^ (m^  1) lines joining any pair of the
points where the plane cuts the curve 111^. The degree of this
ruled surface will then be 7^ + ^m t (m^  1), and, as has been
said, the degree will be m. 2 times this number, if the second
director be a curve m. 2 instead of a right line.
The result of this article may be verified as follows : Con
sider a complex curve made up of two simple curves m lt m. 2 ;
then a line which meets this system twice must either meet
both the simple curves, or else must meet one of them twice.
The number of apparent double points of the system is
h^ + h. 2 + m^m^ ; * and the degree of the surface generated by
* Where I use h in these formulae Prof. Cayley uses r, the rank of the
system, substituting for h from the formula r=m (!) 2/t. And when the
system is a complex one, we have simply Rr l + r. s .
RULED SURFACES. '.til
a line meeting a right line, and meeting the complex curve
twice, is
1) + A! + A 2
471. The degree of the surface generated by a line whir It
meets a curve three times may be calculated as follows, when
the curve is given as the intersection of two surfaces U, V :
Let x'y'z'w' be any point on the curve, xyzw any point on a
generator through x'y'z'w' ; and let us, as in Art. 343, form
the two equations W + \W + &c. = 0, 8V + \&V + &c. = 0.
Now if the generator meet the curve twice again, these
equations must have two common roots. If then we form
the conditions that the equations shall have two common
roots, and between these and U'=Q, F' = 0, eliminate x'y'z'w',
we shall have the equation of the surface ; or, rather that
equation three times over, since each generator corresponds
to three different points on the curve UV. But since U' and
V do not contain xyzw, the degree of the result of elimination
will be the product of pq the order of U', V by the weight of
the other two equations (see Higher Algebra,Ijesson xvin.).
If, then, we apply the formulae given in that Lesson for find
ing the weight of the system of conditions that two equations
shall have two common roots, putting m=pl, n = ql,
X = 0, V=^7, fi = Q, p=q, the result is
and the degree of the required surface is this number
multiplied by $pq. But the intersection of U, V is a curve
(see Art. 343), for which m=pq, 2h=pq (p  1) (q  1), whence
pq (p + q) = m 2 + m 2A. Substituting these values, the degree
of the surface expressed in terms of m and h is
l(m  2) (6 A + m  w 2 ), or (in  2) A  %m(m  !)(  2),
a number which may be verified, as in the last article.
472. The ruled surfaces considered in the preceding
articles have all a certain number of double generators.
Thus, if a line meets the curve w x twice, and also the curves
m 2 and m a , it belongs doubly to the system of lines which
94 ANALYTIC GEOMETRY OP THREE DIMENSIONS.
meet the curves m lt m. 2 , m. 3 and is a double generator on the
corresponding surface. But the number of such lines is evi
dently equal to the number of intersections of the curve 77*3,
with the surface generated by the lines which meet m^ twice,
and also m. 2 , that is to say, is m. 2 m. A {%m l (rn^  1) + 7<J; the
total number of double generators is therefore
^m^m^m^rn^ + ra 2 + m^  3) + h^m^m^ + h^m^m^ + hjnjn.,.
In like manner the lines which meet m^ three times, and also
w 2 belong triply to the system of lines which meet m^ twice,
and also w 2 ; and the number of such triple generators is seen
by the last article to be m^rn^  2)7^  ^m^in^m^  1) (m^  2).
The surface has also double generators whose number we
shall determine presently, being the lines which meet both
??&! and m 2 twice.
Lastly, the lines which meet a curve four times are
multiple lines of the fourth order of multiplicity on the surface
generated by the lines which meet the curve three times.
We can determine the number of such lines when the curve
is given as the intersection of two surfaces, but will first
establish a principle which admits of many applications.
473. Let the equations of three surfaces U, V, W contain
xyzw in the degrees respectively X, X', X", and x'y'z'w' in
degrees /*, //, p", and let the XX'X" points of intersection of
these surfaces all coincide with x'y'z'w' ; then it is required to
find the degree of the further condition which must be ful
filled in order that they may have a line in common. When
this is the case, any arbitrary plane ax + fiy + yz + Sw must be
certain to have a point in common with the three surfaces
(namely, the point where it is met by the common line), and
therefore the result of elimination between U, V, W and the
arbitrary plane must vanish. This result is of the degree
XX'X" in aftyS, and /zX'X" + /A/'X + /t"XX' in x'y'z'w. The first
of these numbers (see Higlier Algebra, Lesson xvin.) we call
the order, and the second the weight of the resultant. Now,
since the resultant is obtained by multiplying together the
results of substituting in ax + fty + yz + &w, the coordinates
RULED SURF ACER. 9">
of each of the points of intersection of U, V, W, this resultant
must be of the form J7 (ax + $y + yz' + &M/)**'A". The con
dition ax + $y + yz' + 8iv' = 0, merely indicates that the arbit
rary plane passes through x'y'z'w', in which case it passes
through a point common to the three surfaces, whether they
have a common line or not. The condition, therefore, that
they shall have a common line is II = ; and this must be of
the degree
/zX'X" + fi\"\ + fj,"\\'  \\'\" ;
that is to say, the degree of the condition is got by subtracting
tlw order from the weight of the equations U, V, W.
474. Now let x'y'z'w' be any point on the curve of inter
section of two surfaces U, V, xyzw any other point ; and, as
in Art. 471, let us form the equations SU + fa&U + &c. = 0,
&V+ \S 2 F+ &c. = 0. If x'y'x'w be a point through which a
line can be drawn to meet the curve in four points, and xyzw
any point whatever on that line, these two equations in X
will have three roots common. And, therefore, if we form
the three conditions that the equations should have three
roots common, these conditions considered as functions of
xyzw, denote surfaces having common the line which meets
the curve in four points. But if x'y'z'w had not been such a
point, it would not have been possible to find any point xyzw
distinct from x'y'z'w, for which the three conditions would be
fulfilled ; and, therefore, in general the conditions denote
surfaces having no point common but x'y'z'w'. The degree,
then, of the condition which x'y'z'w' must fulfil, if it be a
point through which a line can be drawn to meet the curve
in four points, is, by the last article, the difference between
the weight and the order of the system of conditions, that the
equations should have three common roots. But (see Hig/t< r
Algebra, Lesson xvm.) the weight of this system of con
ditions is found, by making m=pl, n = q  1, \=p, /* = <?,
\' = fi = 0, to be
3  9 + + 2 + 5 + 2
 66 (p + q) + 108} ;
96 ANALYTIC GEOMETRY OF THEEE DIMENSIONS.
while the order of the same system is
ifrV  3pV (p + q) + 2p 8 g 8 + <2pq (p + q*  Spq (p + q)
+ TSpq  36}.
The degree, then, of the condition H = to be fulfilled by
x'y'z'w', being the difference of these numbers, is
I {2p Y  6p V(P + <!) + 3p? (p + q) 2 + I8pq (p + q)
66
The intersection of the surface IT with the given curve de
termines the points through which can be drawn lines to
meet in four points ; and the number of such lines is therefore
of the number just found multiplied by pq. As before,
putting pq = m, pq (p + q) = m? + m 27i, the number of lines
meeting in four points is found to be
_i_ { _ m * + 18m 3  71w 2 + 78m  48mh + 1327i + 12/i 2 }.*
From this number can be derived the number of lines which
meet both of two curves twice. For, substitute in the
formula just written ra x + w 2 for m, and h^ + h 2 + m 1 m 2 for h ,
and we have the number of lines which meet the complex
curve four times. But from this take away the number of
lines which meet each four times, and the number given (Art.
472) of those which meet one three times and the other once ;
and the remainder is the number of lines which meet both
curves twice, viz.
Jirji* + {n^m.2 (Wj  1) (w,  1).
475. Besides the multiple generators, the ruled surfaces
we have been considering have also nodal curves, being the
locus of points of intersection of two different generators. I
do not know any direct method of obtaining the order of
these nodal curves ; but Cayley has succeeded in arriving at
a solution of the problem by the following method. Let m
be one of the curves used in generating one of the surfaces
* It may happen, as Cayley has remarked, that the surface n may al
together contain the given curve, in which case an infinity of lines can be
drawn to meet in four points. Thus the curve of intersection of a ruled sur
face by a surface of the p t h order is evidently such that every generator of the
ruled surface meets the curve in p points.
RULED SURFACES. 07
we have been considering, M the degree of that surface, <f> (m)
the degree of the aggregate of all the double lines on that
surface ; then if we suppose m to be a complex curve made
up of two simple curves m l and m 2 , the surface will consist of
two surfaces M lt M 2 having as a double line the intersection
of M l and M^ in addition to the double lines on each surface.
Thus, then, < (m) must be such as to satisfy the condition
$ (wj + m 2 ) = <f> (mj + <f> (m 2 ) + A^Mj.
Using, then, the value already found for M l in terms of m lt
solving this functional equation, and determining the con
stants involved in it by the help of particular cases in which
the problem can be solved directly, Cayley arrives at the con
clusion, that the order of the nodal curve, distinct from the
multiple generators, is in the case of the surface generated
by a line meeting three curves m lt m 2 , m 3 ,
^m l m. 2 ni. A {'im l m 2 m 3  (m> l m 3 + m 3 m l 4m 1 w 2 )
 2 (m l 4 m 2 + m 3 ) + 5},
in the case of the surface generated by a line meeting m l
twice and m 2 once, is
m^h l (m l  2) (Wj  3) + ^rn^m^  1) (m^  2) (m l  3)}
+ w 2 (w 2  1) \\h* + ih^mS  m^  1) + ^m^  1) (mf  5m l + 10)},
and in the case of the surface generated by a line meeting m l
three times, is
1  5)  k^mf  tonf + 5m^  49w t + 120)
+ ^ ( Wi e _ em^ 4 Sl^ 4  270m, 8 4 868m! 2  408m!).
VOL. II.
CHAPTER XIII (b).
TRIPLY ORTHOGONAL SYSTEMS OP SURFACES.* NORMAL
CONGRUENCES OF CURVES.
476. WE have already given proofs of Dupin's theorem
regarding orthogonal surfaces in Art. 304 ; as this theorem
has led to investigations on systems of orthogonal surfaces,
we proceed to present the proof under a different and some
what more geometrical form as follows. Imagine a given
surface, and on each normal measure off from the surface
an infinitesimal distance I (varying at pleasure from point to
point of the surface, or say an arbitrary function of the posi
tion of the point on the surface) : the extremities of these
distances form a new surface, which may be called the con
secutive surface ; and to each point of the given surface
corresponds a point on the consecutive surface, viz. the point
on the normal at the distance I ; hence, to any curve or series
of curves on the given surface corresponds a curve or series of
curves on the consecutive surface. Suppose that we have on
the given surface two series of curves cutting at right angles,
then we have on the consecutive surface the corresponding two
series of curves, but these will not in general intersect at right
angles.
Take A a point on the given surface ; AB, AC, elements of
the two curves through A ; A A', BB', CC' the infinitesimal
distances on the three normals ; then we have on the con
secutive surface the point A', and the elements A'B', A'C' of
the two corresponding curves ; the angles at A are by hypo
* [By a triply orthogonal system of surfaces is meant three oneparameter
families of surfaces, such that each surface cuts orthogonally all those of
different family. Any one of the families is described as a Lam family.]
98
TRIPLY ORTHOGONAL SYSTEMS OF SURFACES.
99
thesis each of them a right angle ; the angle B'A'C' is not in
general a right angle,
and it may be shown
that the condition of
its being so, is that
the normals BB', A A'
shall intersect, or that
the normals (7(7, A A'
shall intersect, for it
can be shown that if
one pair intersect, the
other pair also intersect. But the normals intersecting, AB,
AC, will be elements of the lines of curvature, and the two
series of curves on the given surface will be the lines of
curvature of this surface.
477. Take x, y, z for the coordinates of the point A ; a, ft, y
for the directioncosines of AA' ; a lt ft^ y l for those of AB,
and a 2 , # 2 , y 2 for those of AC. Write also
Then it will be shown that the condition for the intersection
of the normals A A', BB', is
a^a + fr&fi + 72^7 = 0,
the condition for the intersection of the normals A A', CC' is
a^a + &S 2 + 7^7 = 0,
and that these are equivalent to each other, and to the con
dition for the angle B'A'C' being a right angle.
Taking I, l lt 1 2 for the lengths A A', AB,AC, the coordinates
of A', B, C measured from the point A, are respectively
(la, I ft, fy), (^oj, ^ft, ^7^, (Z 2 a 2 , Z 2 &, ,73).
The equations of the normal at A may be written
X=x + 0a, Y=y + 00, Z = z + 0y,
where X, Y, Z are current coordinates, and 6 is a variable
parameter. Hence for the normal at B passing from the
7 *
100 ANALYTIC GEOMETBY OF THREE DIMENSIONS.
coordinates x, y, z to x + l^, y + 1&, z + ^7^ the equations
are
and if the two normals intersect in the point (X, Y, Z), then
+ #V = 0,
Eliminating and 8,0, the condition is
a,
= 0;
7>
or since a. 2 , /9 2 , 7 2 = ^  ^7, 7^  7^, aft  a^,
this is a^a + /9 2 ^i/^ + 7a^i7 = 0.
Similarly the condition for the intersection of the normals
A A', CC'is
= 0.
We have next to show that
Oa^a + p&fi + 73^7 = a&a + 0^0 + 7^07.
In fact, this equation is
(oA  aj,) a + (J3&  0,8,} + (7^  7l 8 2 ) 7 = 0,
which we proceed to verify.
In the first term the symbol a^  a x 8 2 is
a. 2 (a^ + 0$, + 7^,)  Oj (a^d x + 0^ + 7 2 d),
this is (a^  a^g) d, + (7^2  7^) d, ;
or, what is the same thing, it is
/B&74,
and the equation to be verified is
(#2,  yd,) a + (yd z ad t ) + (ad,  0dJ 7 = 0.
X Y Z
Writing a, 0, 7 = ^, ^, ^,
where if I =f (x, y, z) is the equation of the surface, X, Y, Z
are the derived functions J J( ^, and E= ^(X 2 ^ P + Z 2 ),
the function on the lefthand consists of two parts ; the first
is
TRIPLY ORTHOGONAL SYSTEMS OF SURFA* 101
/,  yd,) X + (yd,  ad,) Y+(ad, $d,} Z\,
that is {a (d,Z  d,Y) + (&X  &) + 7 (d, Y  d,X) } ,
which vanishes ; and the second is
 i {a (fid,  yd,} + fi (yd x  ad,) + y (ad,  ftdj} K,
which also vanishes ; that is, we have identically
a&a 4 fi&fi + 7.5817 = a^a + fr&j/S + 7,8,7,
and the vanishing of the one function implies the vanishing
of the other.
Proceeding now to the condition that the angle B'A'C' shall
be a right angle, the coordinates of B' are what those of A'
become on substituting in them x + l v a v y + l^ lt z + 1^ in
place of x, y, z \ that is, these coordinates are
x + la+ l l a l + IjBtdLa), &c.,
or, what is the same thing, measuring them from A' as origin,
the coordinates of B' are
and similarly those of ' measured from the same origin A
are
Hence the condition for the angle to be right is
+ a8^) (a 2
Here the terms independent of /, 8J, 8. 2 l vanish ; and writing
down only the terms which are of the first order in these
quantities, the condition is
a l (I8. z a+a8 2 l) + a 2 (I8 l a+a8 l l)
+ fit (I8 2 j3+ 8 a Z) + fi t (iW+fiW
+ 7l (I8,y+ y8,l) + y, (I8 l y+y8 l l) = 0,
102 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
where the terms in SJ, B 2 l vanish ; the remaining terms divide
by Z, and throwing out this factor, the condition is
(oAo + frStP + 7 A7) + ( A" + && + 7 . 2 8 l7 )  0.
By what precedes, this may be written under either of the
forms
7 = 0,
7 = 0.
and the theorem is thus proved.
Now in any system of triply orthogonal surfaces taking for
the given surface of the foregoing demonstration any surface
of one family, we have not only on the given surface, but also
on the consecutive surface of the family, two series of curves
cutting at right angles; and the demonstrated property is
that the two series of curves on the given surface (that is on
any surface of the family) are the lines of curvature of the
surface. And the same being of course the case as to the
surfaces of the other two families respectively, we have
Dupin's theorem.
478. In regard to the foregoing proof, it is important to
remark that there is nothing to show, and it is not in fact in
general the case, that A'B', A'C' are elements of the lines
of curvature on the consecutive surface. The consecutive
surface (as constructed with an arbitrarily varying value of Z)
is in fact any surface everywhere indefinitely near to the
given surface ; and since by hypothesis A A' and BB' intersect
and also A A', CC' intersect, then AB and A'B' intersect, and
also AC and A'C' ; the theorem, if it were true, would be, that
taking on the given surface any point A, and drawing the
normal to meet the consecutive surface in A', then the tan
gents AB, A Cot the lines of curvature at A meet respectively
the tangents A'B', A'C' of the lines of curvature through A' ;
and it is obvious that this is not in general the case ; that it
shall be so, implies a restriction on the arbitrary value of the
function I.
Cayley has shown that when the position of the point A
on the given surface is determined by the parameters p, q,
TRIPLY ORTHOGONAL SYSTEMS OF SURFACES. 103
which are such that the equations of the curves of curvature
are p = const., q = const, respectively, then the condition is
that I shall satisfy the same partial differential equation as is
satisfied by the coordinates x, y, z considered as functions of
p, q, viz. the equation (Art. 384)
d'u _ 1 I dE^du _ 1 1 dG du
dpdq 2 E dq dp 2 G dp dq
The above conclusion may be differently stated : taking
r =/ (x, y, zj a perfectly arbitrary function of (x, y, z), the
family of surfaces r=f(x, y, z), does not belong to a system
of orthogonal surfaces ; in order that it may do so the fore
going property must hold good ; viz. it is necessary that
taking a point A on the surface r, and passing along the
normal to the point A' on the consecutive surface r + dr, the
tangents to the lines of curvature at A shall respectively
meet the tangents to the lines of curvature at A'. And
this implies that r, considered as a function of x, y, z, satis
fies a certain partial differential equation of the third order,
Cayley's investigation of which will be given presently.*
* The remark that r is not a perfectly arbitrary function of (x, y, z) was
first made by Bouquet, Liouv. t. xi. p. 446 (1846), and he also showed that in
the particular case where r is of the form r = / (x) + <j> (y) + ^ (z), the necessary
condition was that r should satisfy a certain partial differential equation of
the third order ; this equation was found by him, and in a different manner
by Serret, Liouv. t. xn. p. 241 (1847). That the same is the case generally
was shown by Bonnet (Comptes rendus, LIV 556, 1862), and a mode of obtain
ing this equation is indicated by Darboux, Ann. de Ffcole normale, t. m. p.
110 (1866). His form of the theorem is that in the surface r = f (x, y, z) if a. 0,
y are the directioncosines of a line of curvature at a given point of the surface,
then the function must be such that the differential equation adx + pdy + ydz =
shall be integrable by a factor. The condition as given in the text is in the
form given by Levy, Jour, de I'fcole polyt., XLIII. (1870) ; he does not obtain
the partial differential equation though he finds what it becomes on writing
therein ^=0, r=0; the actual equation (which of course includes as well
dx 'dy
this result, as the particular case obtained by Bouquet and Serret) was ob
tained by Cayley, Camples rendus, t. LXXV. (1872) ; but in a form which (as he
afterwards discovered) was affected with an extraneous factor. [For a fuller
historical account see Levy, op. cit., p. 127.]
104 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
479. Dupin's theorem, and the notion of orthogonal sur
faces are the foundation of Lame's theory of curvilinear
coordinates.* Representing the three families of orthogonal
surfaces by_p = </> (x, y, z\ q = ^r (x, y, z\ r=f (x, y, z), then
conversely x, y, z are functions of p, q, r which are said to
be the curvilinear coordinates of the point. It will be ob
served that regarding one of the coordinates, say r, as an
absolute constant, then p, q are parameters determining the
position of the point on the surface r =/ (x, y, z}, such as are
used in Gauss' theory of the curvature of surfaces ; and by
Dupin's theorem it appears that on this surface the equations
of the lines of curvature are p = const, q = const, respectively ;
whence also (Art. 384) x, y, z each satisfy the differential
equation
cPu 1 1 dE du _ 1 J dG du,
j n JK o n ,1^ /7 ~ '
dpdq 2E dq dp 2 G dp dq
(and the like equations with q, r and r, p in place of p, q
respectively), a result obtained by Laire, but without the
geometrical interpretation.
Conversely we may derive another proof of Dupin's
theorem from these considerations ; taking x, y, z as given
functions of p, q, r, and writing
dx dx dy dy dz dz r
h + = \ V Q\
dp dq dp dq dp dq
dx d 2 x dy d 2 y dz d*z r
I ** " I. ._. I nr\ CiY \ i V (*
dp dq dr dp dq dr dp dq dr **'
ths conditions for the intersections at right angles may be
written
L ./ J * L " _t J * L* * 2 J '
and the first two equations give
dx , dy . dz _ dy dz _ dz dy . dz dx dx dz . dx d y _ dy dx
dr dr dr dp dq dp dq dp dq dp dq dp dq dp dq
Moreover, by differentiating the three equations with respect
tojp, q, r respectively, we find
* Laim, Comptes rendus, t. vi. (1888), and Liouv. t. v. (1840), and various
later Memoirs ; also Lemons sur lea coordonnffs curvilignes, Paris, 1859.
TlilPLY ORTHOGONAL SYSTEMS OF SURFACES.
LOfi
[rp . q] + [pq . r] = 0, [pq . r] + [qr .p]  0, [qr .p] + [rp . q] = 0,
that is [qr.p] = Q, [rp . <?] = r [pq.r] = Q. The last of these
dx
equations, substituting in it for =,
values, becomes
, the foregoing
dr dr
dx
dp
dx
dq
d'x
dy dz
dp dp
dy dz
dq dq
d*y ,!'
dpdq '
dpdq ' dpdq
and the equation [p . q\ = is
dx dx dy dy ,dzdz_^
dp dq dp dq dp dq
These equations are therefore satisfied by the values of x, y, z
in terms of p, q, r', and regarding in them r as a given con
stant but p, q as variable parameters, the values in question
represent a determihate surface of the family r=f (x, y, z};
and it thus appears that this surface is met in its lines of
curvature by the surfaces of the other two families.
480. We proceed now to the investigation of Cayley's
differential equation already referred to. Let P be a point on
a surface belonging to a triply orthogonal system, PN the
normal, P2\, PT 2 the principal tangents or directions of
curvature, then, by Dupin's theorem, the tangent planes to
the two orthotomic surfaces are NPT lt NPT. 2 . Take now a
surface passing through a consecutive point P' on the normal,
and if the surface be a consecutive one of the same orthogonal
family, the planes NPT lt NPT 2 must also meet its tangent
plane at P' in the two principal tangents P'T/, PT 2 '. This
is the condition which we are about to express analytically.
Take rf(x,y,z)=Q for the equation of the family of
the orthogonal system, the given surface being that corre
sponding to a given value of the parameter r ; and let the
differential coefficients of / (or what is the same thing, of r
considered as a function of x, y, z) be L, M, N of the first
106
ANALYTIC GEOMETRY OF THREE DIMENSIONS.
order, and a, b, c, f, g, h of the second order ; and then the
point P being taken as origin, the equation of the tangent
plane at that point is Lx + My + Nz = 0, which we shall call
for shortness T = 0; while the inflexional tangents are de
termined as the intersections of T with the cone
(a, b, c,f, g, h)(x,y,z)* = Q,
which we shall call U = 0. The two principal tangents are
determined as being harmonic conjugates with the inflexional
tangents, and also as being at right angles, that is to say,
harmonic conjugates with the intersection of the plane T with
x* + y 2 + z* = Q, or F=0. Suppose now that we had formed
the equation of the pair of planes through the normal, and
through the inflexional tangents at P', and that this was
(a", b", c"J", g", h")(x, y, *) 2 = 0, or TF=0,
then the planes NPT^ NPT 2 must be harmonic conjugates
with these also, so that the resulting condition is obtained
by expressing that the three cones U, V, W intersect the
plane T in three pairs of lines which form a system in in
volution.
Now we have here evidently to deal with the same
analytical problem as that considered, Conies, Art. 388c, viz.
to find the condition that three conies shall be met by a line
in three pairs of points forming an involution. The general
condition there given is applied to the present case by writing
a'=b' = c' = l,f = g' = h' = 0, and in the determinant form is
a", b", c", 2/', <2g",
2/ , 20 ,
N
a
I
L
b , c ,
1 , 1
,
M ,
N
L
M
L
= 0.
, N, M
We see then that the form of the required condition is
Ha" + 386" + Cc" + Iff + 2<&g" + 2DA" = 0,*
* Cayley has also shown, that if from any surface a new surface be de
rived by taking on each normal an infinitesimal distance = p, where p is a given
function of x, y, z, the condition that the new surface shall belong to the
same orthogonal system is
THINLY OBTHOGONAL SYSTEMS OF SURFACES. 107
where H, B, &c., are the minors of the above written deter
minant, and it still remains to determine a", b", &c.
481. It may be observed, in the first instance, that the
equation of the pair of planes passing through the normal,
and the first pair of inflexional tangents is got by elimi
nating 6 between T + 0T' = Q, ?7+ 2JI0 + W = 0, where T
. His
and U' is L 2 + b
The equation of the pair of planes is therefore
Now the consecutive point P' is a point on the normal
whose coordinates may be taken as XL, \M, \N, \ being
an infinitesimal whose square may be neglected, and the cor
responding differential coefficients for the new point are
L + \SL, M+\SM, N+\SN, a + \&a, &c., where 8 denotes
the operator
T d ,, d , T d
L j + M T + NT.
dx dy dz
Hence the equation of the tangent plane at P' referred to that
point as origin, is L'x + M'y + N'z = 0, or T + XST = 0, where
BT means xBL + yBM + zBN, and it is to be observed that BT
is the same as what we have just called II. And the equation
of the cone which determines the inflexional tangents is
U+\BU=Q. The equations of this plane and cone referred
to the original axes are T+ \ (BT  T') = 0, U+ X (BU  2/7) = 0,
but it will be seen presently that the terms added on account
of a change of origin do not affect the result. In order to
form the equation of the pair of planes through the normal
and through these inflexional tangents, we have to eliminate
6 between
T + \ (U  T) + 6 (T' + &c.) = 0,
U + X (8 U  2JI) + 16 (U + &c.) + 0*(V' + &c.) = 0.
** 1 9. *
and that this condition is equivalent to that given in the text.
108 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
Now since we are about to express the condition that the
resulting equation shall denote a surface intersecting T in a
pair of lines belonging to an involution, to which the intersec
tion of U by T also belongs, we need not attend to any terras
in the result which contain either T or U ; nor need we attend
to any terms which contain more than the first power of X.
The terms then, of which alone we need take account, are
 2IIT' (IT  T) + T" 2 (SUII) = 0,
or dividing by T', T'B U  2IT 2 = 0.
We have thus a"= (L 2 + Af2+ N*)Sa 2 (8L), &c., and the
required condition is
= 2(H, 3B, C, f, <B, D)(8L, BM, S.Y) 2 .
Cayley has shown that the condition originally obtained by
him in a form equivalent to that just written, contains an
irrelevant factor, the righthand side of the equation being
divisible by L + M' 2 + N' 2 . This we proceed to show.
482. We may in the first place remark, that since the
united points or foci of an involution given by the two equa
tions M = (a, h, b)(x, y), v = (a, h', b')(x, y)' 2 , are determined
by the equation
therefore w, = 5 vr 3"
dx N dz
= 0, Conies, Art. 342 ; if 11 and r be
given as functions of x, y, z, where Lx + My + Nz = 0, and
we find immediately that the
foci of the involution are given by the equation
u lt u z , u 3
L,M,N = 0.
Thus then, or as in Art. 297, the two principal tangents are
determined as the intersections of the tangent plane with
the cone
ax + hy + gz, Jix + by +fz, gx +fy + cz
x , y , z
L , M , N = 0.
We shall write this equation
TRIPLY ORTHOGONAL SYSTEMS OF SURFACES.
$ (a, b, c, f, g, h)(z, y, *) 2 = 0,
that is to say,
a = 2 (Mg  Nti), b = 2 (Nh  Lf), c = 2 (L/  My),
a) \LhNf,
109
It is useful to remark that the conic derived from two
others, according to the rule just stated, viz. which is the
Jacobian of two conies and of an arbitrary line, is connected
with each of the two conies by the invariant relation 6 = 0;
that is to say, the two relations are
A&+ .Bb + Cc + 2FI + 2Gg f 2jEfh = 0,
where A, B, &c., are the reciprocal coefficients bcf 2 , &c. ;
and ^4'a+&c. = 0, which, in the particular case under con
sideration, reduces to a+ b+ c = 0, which is manifestly true.
Again, referring to the condition, Art. 480, that three
conies U, V, W should be met by a line in three pairs of
points forming an involution, it is geometrically evident that
if W be a perfect square (\x + f*>y + vz}' 2 , this condition can
only be satisfied if \x + py + vz passes through one of the foci
of the involution, and hence we are led to write down the
following identical equation which can easily be verified :
L, M, N
(H, JS, C, 3f, (3, 1b)(X, /A, i/) 2 = 2 u lt u 2 , u a
where in u lt &c., we are to write for x,y,z, pN  vM, vL  \N,
\M  pL ; that is to say, in the case we are at present con
sidering, the determinant is
L, M, N,
vL  \N, \M  fiL ,
aL' + hM' + gN', hU + bM' +fN', gL' +fM' + cN'
where we have written L', &c., for fiNvM, &c. This
determinant may be otherwise written
L, M, N
L', M', N'
X, L, a, h, g
^ M, h, b, f
v, N, g, f, c
110 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
But in the particular case where X = SL = aL + hM + gN, &c.,
this determinant may be reduced by subtracting the last
three columns multiplied respectively by L, AT, N from the
first ; then observing that LL' + MM'+ NN' = Q, we see that,
as we undertook to show, the determinant is divisible by
L 2 + M' 2 + N*, the quotient being
L' M' N'
L, a, h, g
M,h, b, f
N, g, f, c
483. The quotient is obtained in a different and more
convenient form by the following process given by Cayley.
The following identities may be verified, B, &c., a, &c.,
having the meanings already explained :
C = c (L 2 +M*+ N*) + 2N (MSL  LSM),
f = t(L*+M*+N*) + M (MSL  LSM) + N (LSN  NSL\
= g(L 2 +M*+N 2 ) + N (NSM MSN) + L (MSL  LSM),
Hence we have
(BSL + DSM + (3S2V) = (aSL + hSM + gSN) (L 2 + M 2 +
+ (LSL + M8M + NSN) (NSM  MSN},
with corresponding values for
DSL + JBSM + fSN, 6SL + fSM + GSN,
and hence immediately
(H, B, C, f, <B, D)(SL, SM, SNf
= (L 3 + M* + N*) (a, b, c, f, g, h)(8L, SM, SNy 2 .
Hence the equation, Art. 481, omitting the factor L 2 + M 2 + N 2 ,
becomes
= 2 (a, b,c, f,g,h)(SL, SM, SN)*.
484. There is still another form in which the result may be
expressed. Writing, as usual, in the theory of conies bcf* = A,
&c., the determinant at which we arrived at the end of Art.
482 is, when expanded,
TRIPLY ORTHOGONAL SYSTEMS OF SURFACES. Ill
 {ALL' + BMM' + CNN' + F (MN'+ M'N)
+ G (NL'+ N'L) + H (LM'+ L'M)}.
Now, from last article
2LZ/ = a  (L 2 + M* + AT 2 ) a, &c.,
MN' + M'N = jf  (L 2 + M* + N*) f , &c.,
and remembering that A&+ &c. = 0, the expanded determinant
last written is seen to be
2U + BB + CC + 2JTF+ 2<BG + 2t>H,
and thus eventually the differential equation is given in the
form
485. As a particular case of this equation of Cayley's
may be deduced that which Bouquet had given (Liouville, xi.
p. 446) for the special case where the equation of the system
of surfaces is r = X+ Y+ Z, where X, Y, Z are each functions
of x, y, z respectively only. In this case then we have
L = X', M=Y, N = Z', a = X", 6=7", c = Z",f=g = h = Q;
A = Y"Z",B = Z"X", C = X"Y", F=G = H=0',
H= (Y'  Z") X'YZ', JB = (Z"  X") X'Y'Z',
<L = (X"Y") X'Y'Z';
&a = X'X", Sb= Y'Y", 8c = Z'Z'",
and the differential equation being divisible by X'Y'Z' is re
duced to
X'X'" (Y"  Z") + Y'Y" (Z"  X") + Z'Z'" (X"  Y")
42(7" Z"} (Z"  X"} (X"  7") = 0.
486. Even when the equation of condition is satisfied by
an assumed equation it does not seem easy to determine the
two conjugate systems. Thus Bouquet observed that the
condition just found is satisfied when the given system is of
the form tfy m z n = r, but he gave no clue to the discovery of
the conjugate systems. This lacuna was completely supplied
by Serret, who has shown much ingenuity and analytical
power in deducing the equations of the conjugate systems, when
112 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
the equation of condition is satisfied.* The actual results are,
however, of a rather complicated character. We must con
tent ourselves with referring the reader to his memoir, only
mentioning the two simplest cases obtained by him, and which
there is no difficulty in verifying a posteriori. He has shown
that the three equations,
represent a triple system of mutually orthogonal surfaces
The surfaces (r) are hyperbolic paraboloids. The system (jp)
is composed of the closed portions, and the system (q) of the
infinite sheets, of the surfaces of the fourth order,
(* 2  ?/ 2 ) 2  2p 8 (z 2 + 7/ 2 + 2z 2 ) +p* = 0.
Serret has observed that it follows at once from what has
been stated above, that in a hyperbolic paraboloid, of which
the principal parabolas are equal, the sum or difference of the
distances of every point of the same line of curvature from
two fixed generatrices is constant.
He finds also (in a somewhat less simple form) the follow
ing equations for another system of orthogonal surfaces,
P =
r = (z 2 + W7/ 2 + wV)!  O 2 + to 2 */ 2
where &> is a cube root of unity.
[In continuation of the work of Serret, Darboux shows
that the two families associated with the family x'y m z n =p are
obtained by eliminating X between the equation
( T 2\l / 7/2\'
\ + * } (\+ y ~)
I / \ ml
 = constant
n
and its derivative with regard to \ (Darboux, Systemes Ortho
gonaux, Livre I., chap, vi.).]
An interesting system of orthogonal surfaces, and very
* [The subject is dealt with by Darboux (see below), and also by Forsyth,
Differential Geotnetry, p. 451.]
TRIPLY ORTHOGONAL SYSTEMS OP SURFACES. 113
analogous to the system of confocal quadric surfaces, is given
by Darboux in his Memoir referred to (Art. 478, note), namely,
the system of cyclides
where a, b, c, d are given constants, and in place of \ we
are to write successively the three parameters p, q, r. The
formulae for x, y, z in terms of p, q, r, are
(a 2 
(a  6) (a  c)
(b  c) (b a)
hp) (c + q) (
(c  a) (c b)
where, writing for shortness,
(<2d+p) C2d+q") (2d+r} (2dp) (2d  q) (2d  r)
~ Id (2d  a) (2d  b) (M  c) ' n ~ 4d (2d+ a) (2rf+ b) (2d+ c)'
A/72
we put M =
If d = oo , the system of surfaces is
z l
a+"X + 6+x"c+X^ ? "'
which is in effect the system of confocal quadrics : a slight
change of notation would make the constant term become  1.
[A very general system of surfaces discussed by Darboux
(Systemes Orthogonaux) consists of the three families of
surfaces which are the envelopes, for all values of /, m, n, t,
of the surfaces represented by the equations
I / \ m/ \ n>
Through any point three surfaces of the family may be
drawn, corresponding to the three values of X determined by
the equation = 0, namely,
(tn.
y* ' z \
m n
VOL. II.
114 ANALYTIC GEOMETEY OF THREE DIMENSIONS.
If \, X 2 are two roots of this equation the condition of
orthogonality is
which is identically satisfied, as may be verified by elimi
nating t between the two equations which express that \ and
\. 2 are roots of the cubic.]
W. Roberts, expressing in elliptic coordinates the condi
tion that two surfaces should cut orthogonally, has sought
for systems orthogonal to L + M+ N = r, where L, M, N are
functions of the three elliptic coordinates respectively. He
has thus added some systems of orthogonal surfaces to those
previously known.* Of these perhaps the most interesting,
geometrically, is that whose equation in elliptic coordinates
is fj,v = a\, and for it he has given the following construc
tion : Let a fixed point in the line of one of the axes of a
system of confocal ellipsoids be made the vertex of a series
of cones circumscribed to them. The locus of the curves of
contact will be a determinate surface, and if we suppose the
vertex of the cones to move along the axis, we obtain a
family of surfaces involving a parameter. Two other systems
are obtained by taking points situated on the other axes as
vertices of circumscribing cones. The surfaces belonging to
these three systems will intersect, two by two, at right angles.
It may be readily shown that the lines of curvature of the
abovementioned surfaces (which are of the third order) are
circles,! whose planes are perpendicular to the principal planes
of the ellipsoids. Let A, B be two fixed points, taken re
spectively upon two of the axes of the confocal system. To
these points two surfaces intersecting at right angles will corre
spond, and the curve of their intersection will be the locus
of points M on the confocal ellipsoids, the tangent planes at
which pass through the line AB. Let P be the point where
* Complex rendus, 53, 1861, and Journal fUr Math., 62, 1868.
t Thus they are special forms of Dupin cyclides (Art. 457d).
TRIPLY ORTHOGONAL SYSTEMS OF SURFACES. 1 1 :")
the normal to one of the ellipsoids at M meets the principal
plane containing the line AB, and because P is the pole of
AB in reference to the focal conic in this plane, P is a given
point. Hence the locus of M, or a line of curvature, is a
circle in a plane perpendicular to the principal plane contain
ing AB.
[The following are simple examples of triply orthogonal systems.
Ex. 1. A family of parallel surfaces forms one of a triply orthogonal
system, the other two families consisting of the developables generated by
normals along a line of curvature.
Ex. 2. By inversion of the preceding system we see that every surface
belongs to a triply orthogonal system, two of the families consisting of surfaces
whose lines of curvature of one system are circles passing through any chosen
point.
In the case of a Dupin cyclide (Art. 457d) the parallel
surfaces are Dupin cyclides and the developables of Ex. 1
become right cones having their vertices on the focal conies,
and by inverting this system we can construct any number
of triply orthogonal systems consisting of Dupin cyclides.]
[486a. Lamt's Ctirrili/icar Coordinates. lip, q, r are three
independent functions of xyz, we have in general
x (f> (p, q, r), y = ^ (p, q, r), z = x(P> <?' r )
and p, q, r are said to be curvilinear coordinates of the point
x, y, z. It is usual to assume with Lame (cf. Art. 479) that
the surfaces represented by p = a, q = b, r = c are a triply
orthogonal system. Elliptic coordinates (Arts. 409, 421a)
illustrate this mode of representation.
If ds is the element of any arc in space, we easily find
ds* = H 2 dp* + K'dq 1 + L*dr
where H'' = ( 7 ) + (  ) + ( 5 ) = x? + ?// + z?
\dp/ \dp/ \dps_
with similar values for K and L, the suffixes 1, 2, 3 denoting
differentiation with regard to^?, q, r respectively ; the cosine
of the angle between two elements ds and ds' is
Hdpdp' + Kdqdq'+ Ldrdr'
8
116 ANALYTIC GEOMETEY OF THEEE DIMENSIONS.
The directioncosines of the tangent lines to the curve
q = constant, r= constant and the two orthogonal curves are
*1 yi Zj_. X fa Z^. % 1/3 *3
H' H' H' K' K' K' L' L' L'
and thus we have six equations of the type
<T* Z /*, 2 n 2
^. + ?L + 5L=1
H* K* L*
XiVi , rJ Wh , M_3 = n
H* K* L*
If we use the conditions of orthogonality x^., + y^., + z^z* =
with two similar equations, it is easy to prove
x i x n + y\y\\ + z i z n = HH i
ff ff
Multiplying in turn by ^, vJ, 3 and adding we find
ti^ K. Li
i
ff _ rf _ f  _ ~ _ f _ _ T ill
*\\ *! TT *8 j^2 ** T2 ' ' V X y
The same equation is satisfied by the differentials of y and z,
and we obtain altogether nine equations by interchanging
the numerals 1, 2, 3 and the corresponding magnitudes
H, K, L.
Differentiating with regard to x, y, z the three expressions
of the type x%x 3 + 7/ 2 y 3 + z 2 z s , and equating the results to zero
we find
Also x 2 x Z3 + y 2 y 23 + z^ 3 = KK 3
and as before we find three equations of the form
which are satisfied by x, y, and z, with six other equations
obtained by cyclical interchange, which become, when p, q,
or r are constants, the differential equations of the lines of
curvature on each surface (cf. Art. 479).
If we now substitute for the second differentials their
TRIPLY ORTHOGONAL SYSTEMS OF SURFACES. 117
values given by (1) and (2) in terms of the first differentials,
in the identities
dxu = dxu dyn^dyu dz u = dz lt
dq dp ' dq dp ' dq dp
we reach three equations
Rx 2 + P'x 3 = 0, Ry z + Py 3 = 0, Rz 2 + P> 3 =
where P, Q, R, F, Q', R' contain only H, K, L and their
differentials with regard to p, q, and r. We thus arrive at
Lame's equations of which the first three are represented by
d_(l dL\ d(l dK\ ldKdL_
dq\Kdq r dr\L drJ^IPdp dp ~
and the second three by
p , = ^H^\_dK dH_ldL cM =Q (4
dqdr K dr dq L dq dr
These six equations must be satisfied by H , K, L in order
that ds~ may be expressible in the form given, and it can be
shown further* that functions H, K, L satisfying the equa
tions determine a triply orthogonal system except as to its
position and orientation in space. f
It is worthy of remark that the last written equation
of Lame may be used to deduce one form of the differential
equation (Arts. 480 sqq.) which expresses the condition that
the surfaces represented by p = constant may belong to a
triply orthogonal system. The equation (3) may be written

K* dq VL* dr L*~dr \K* dq
For convenience let us represent H , regarded as a function of
x, y, z, by T. It is easy to prove the relation
\dy
Lame's equation is then equivalent to
(Ss+ss r=o
* See Forsyth, Differential Geumetry, Art. 251.
tFor applications of Lame's equation to determine particular triply or
thogonal systems, see Darboux, Systemes Orthogonaux, Bk. IT, which deals
with curvilinear coordinates.
118 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
Id d d d
where o, = = = =p l +p t +j>,
the suffixes of p, q, r, now representing differentiation with
regard to x, y, x. It may be shown without difficulty that
not only T but also p, x*+ y*+ jr*, px, py, and ps, satisfy the
equation
which may be written, after using the conditions of ortho
gonality, in the form
d d*
=a
We have thus six linear equations in the coefficients of this
operator, and by eliminating them, after a slight linear trans
formation we obtain Gayley's equation
Pn PP PaPn Pit
*Pi 00 p, p t
% 3 p, p l
%, l, ft
111 000
0.
Darboux uses this equation freely, and observes that it ia
satisfied by any function p satisfying the equation
where X, a, ft, 7, are arbitrary functions of j>.*J
[4866. There is an interesting theorem on triply orthogo
nal surfaces, due to Darboux, who describes it as the '* recip
roque " of Dupin's. The two theorems, Darboux shows, may
be collectively expressed as follows : If all ths fvrfacts of one
singly infinite family of surfaces be cut orthogonally by all
those of a sffond, the necessary and sufficient condition that
the ttro families should belong to a triply orthogonal system is
that the curve of intersection of any surface of the one f amity
* For further deductions M Darboax, Sy&mn Ortkajtmumr.
TRIPLY ORTHOGONAL SY>JKM Kfc.
with any surface of the other should always be a line of curva
ture on either.*
Darboux' contribution consists in showing that these con
ditions are sufficient. This amounts to proving that if /, m, n
be the directioncosines of the common tangent line at x, y t z
to any two surfaces, one from each of the two families,
then, from the fact that the surfaces are always orthogonal
and intersect in a line of curvature, it will follow that the
Idx + mdy + ndz =
satisfies the condition of integrability f and thus represents
a singly infinite family of surfaces W= constant, which cut
the other surfaces orthogonally. Instead of proving this
theorem separately we shall indicate a method of showing
that Dupin's, Darboux', and Joachimsthal's theorems* are
special cases of much wider relations connecting the torsions
of systems of curves, each system g defined by Ui'
property that the directioncosines (/, m, n) of the principal
normal at any point (P, x, y, z) in space, are given functions
of z, y, z. It is clear that the system of geodesies on a singly
infinite family of surfaces, U= constant, is a special case of
those considered, and arises when the equation
ldx+mdy+ ndz = ()
is integrable in the form U= constant, that is when
dl dn\ dm dl
In general if PN be the principal normal (I, m, n) belong
ing to the system S, it may be proved by the FrenetSerret
formulae (Art. 368a), that one and only one curve of S can
be drawn through P in any chosen direction perpendicular
to PN, and if  and , are the torsions of two mutually
T T
orthogonal Scurves through P, then
* See Darbooz, L#p* tvr let SytUmet OrOtoyonauz (Paris, 1910).
t Foryth'i Differential EytuU**u, Act. 152.
tThe second theorem proved in Art. 304, p. 310.
120 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
(D
Now suppose we have two complexes * of this kind, S l and
S 2 , defined by the directioncosines l lt m^, n t and Z 2 , m. 2 , w 2 . Let
(?! and C 2 be the curves, one from each complex, having a
common tangent line at P, whose directioncosines are there
fore m^n^  m 2 n v njl 2  n 2 l lt Ipi^  l^n^. Let be the torsion
T 12
of <7j and ' that of C 2 ; let 12 represent the angle between
T 21
the principal normals l v m lt % and Z 2 , w 2 , w 2 ; and let s 12
represent the arc of the curve through P whose tangent line
is the common tangent line just mentioned. Then by using
the FrenetSerret formulae we can prove
 = + _il ^ ^ ^ ^2)
T 12 T 21 " S 12
Dupin's theorem is a special case of the relations (1) and
(2) when applied to three systems S lt S 2 , S 3 . For suppose
the condition of integration is satisfied for each of these
families, namely, I I = 0, 7 2 = 0, 7 3 = where the suffixes denote
the values of I for each system ; and suppose further that
7T
^23 = #31 = ^12 = 9 Thus the three complexes represent the
systems of geodesies on a triply orthogonal system of sur
faces. Then the relation (2) gives the three equations
T 23 T 32 T 31 T 13 T 12 T 21
and the relation (1) gives the three equations
J_ + A=o, J+J=o, I+ia
T 12 T 13 T 23 T 21 T 31 T 32
Hence these six torsions vanish and therefore the corre
sponding curves, which are clearly lines of intersection of
* In general when only one curve of a system passes through each point in
space the curves are said to form a congruence (extending the sense of the word
used in Art. 453). In like manner the curves of one of our systems S may be
said to form a complex, since all those passing through a given point lie on a
surface associated with the point.
TRIPLY ORTHOGONAL COMPLEXES OF CURVES. 1 J 1
surfaces of different families, are lines of curvature (Art. 3966,
Cor. 1).
Darboux' theorem also follows very simply from the two
fundamental relations. For if /S l? S 2 consist of geodesies on
two mutually orthogonal families of surfaces, we have Jj = 0,
/ 2 = and 6 n is a right angle everywhere ; and if S a represents
the complex of curves whose principal normals are the tan
gent lines to the mutual intersections of the surfaces of these
families (one from each family), then B 23 and zl are right
angles. Hence the first five of the preceding six equations
hold good and the sixth is replaced by
T 31 T 32
The theorem now amounts to proving, what is algebraically
obvious, that if =0, i.e. if the mutual intersections of
T 12
curves of the two families of surfaces are lines of curvature
on the first, then we must also have
and therefore S 3 is a complex of geodesies on a singly infinite
family of surfaces which are clearly orthogonal to the surfaces
of the other two families.] *
[486c. The curvature and torsion of the curves at a point
P of a complex S may be investigated in the same manner
as in the case of geodesies. The FrenetSerret formulce give
a ft 7
I m n
dl dm dn
ds ds ds
Since I, m, n are functions of x, y, z we have
1 _ _ / rf/ ..dm dn\ 1 _
p \ ds ds ^ ds) ' r
dl dl Q dL dl
j = aj + /8. j + 7= ; etc.
ds dx dy 'dz
*For detailed proofs of the above relations and for other connected
theorems see R. A. P. Rogers, Some Differential Properties of the Orthogonal
Trajectories of a Congruence of Curves, ivith an application to Curl and Diver
gence of Vectors, Proceedings of the Royal Irish Acedemy, 29, A, 6 (1912).
122 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
If the point P be taken for origin and the axis of z for the
normal I, ?n, n, we have, neglecting higher powers of x, y, z,
and remembering that Idl + mdm + ndn = 0,
I = a^ + a^y + a 3 z, m=b l x+b. i y+b 3 z, n=\ t
and by choosing suitable axes of x and // we can make
2 + >i = 0, and therefore 1= ^ = = 26, =  2a,. If we now
dx dy
put a = cos#, ft=sinO, y=Q, the expressions for p and T
become
1 cos0 sin 2 1 1 . / 1 1 \
 +  , =/+   sin cos 6 (1)
P Pi P> r 2 \fr p 2 /
p l and p. 2 are the two extreme radii of curvature and their
directions are at right angles. When these directions always
coincide with those of zero torsion (which are not otherwise at
right angles) we have 1= 0, and the curves are geodesies on
a oneparameter system of surfaces.*
The curves whose tangent lines are the directions of
extreme curvature form two "congruences" mutually ortho
gonal. From the equations (1) combined with the relations
given in Art. 486a we can now deduce without difficulty
the following generalisation of the theorems of Dupin and
Darboux : If S v S. 2t S 3 are three complexes of curves whose
normals at each point are mutually ortJwgonal, the necessary
and sufficient condition that the direction of a curve common
to S l and S. 2 sJiould be a direction of extreme curvature on S 1
is / 2 = 7 3 . We have thus another proof of the theorems
mentioned which assert that when I l = Q and I. 2 = Q we must
have / 3 = 0.]
* It may be mentioned that when the orthogonal trajectories of the curves
of the complex S form a rectilinear congruence (the condition for which is
given in Ex. 2, Art. 455;)), the value of  for any direction is the parameter of
distribution of the corresponding ruled surface and the second of the equations
(1), which may evidently be reduced to the form =  + isecuiva
T TI T a
lent to the relation (Art, <M>2a) * = *, cos''<j> + sin>.
NORMAL CONGRUENCES OF CUR\ 123
[486d. Normal Curve Congruences. A family of curves is
described as a congruence * when one passes through each
point of space ; I, m, n being functions of x, y, z the con
gruence is defined by
dx _ dy _ dz
I m n
or by two equations of the form
fi (x, y, z, a, b) = 0, / 2 (x, y, z, a, 6) =
where a and b are constant for each curve.
When the condition
; (dn dm\ /dl dn\ /dm dl\ n
'(dy'dz) + m (dz~dx) +n (dx~dy) =
is always satisfied the curves are orthogonal to a singly infinite
system of surfaces, and the congruence is said to be normal.
Another mode of representation is by means of three
parameters p, q, r. The coordinates of any point in space
satisfy
* = (P, ?. r), y = ^r(p, q, r), z = x(p, q, . (1)
p and q being constant the point describes a curve, and the
entire assemblage of such curves for different constant
values of p and q forms a congruence. We may evidently
regard p, q as the coordinates of a point on the director
surface, or surface of reference (cf. Art. 455p). The direction
dx dy dz
cosines of any curve are clearly proportional to v f* r,'
and thus the curves of the congruence are orthogonal tra
jectories of the curves satisfying
dx , dy , dz ,
j dx+ ,'' dii+ . dz=Q.
dr dr dr
(I '/* ft or ct y
Replacing dx by r dp + r dq + = dr, and dy and dz by the
corresponding values, the preceding equation takes the form
Pdp+ Qdq+ RdrQ ... (2)
* Darboux, Surfaces, vol. n. Ch. I. Lilienthal, Uber die Krummung der
C itrvenschaaren (Math. Ann., 32, 1888). Bibaucour, Mf moire sur la th
gi'nerale des surfaces courbes (Journal des Math., iv. 7, 1891). Eisenhart,
Congruences of Curves (Trans. Amor. Math. Soc., 4, 1903). Soo also notes Art.
4836.
124 ANALYTIC GEOMETRY OF THEEE DIMENSIONS.
where P, Q, and R are each functions of p, q, r. We can
thus express the condition that the congruence may be the
orthogonal trajectories of a singly infinite family of surfaces,
namely
dQ dE\ dE dP\ dP dQ\_ ,

We can use the condition just given to prove a beautiful
theorem due to Ribaucour* relating to normal congruences
consisting of plane curves. In the first place it is clear that
the planes of the curves being a doubly infinite system en
lope a surface. The congruence may be said to be deformed
when its envelope is deformed without stretching and the
tangent planes are carried along with it so that the curves
preserve their positions relative to the deformed lines on the
surface. The theorem then is that a normal congruence of
plane curves continues to be normal after deformation.
Take the envelope of the planes as the surface of refer
ence. Let A (x, y, z] be the point of contact of the plane of
a curve and let B (, 77, ) be any point on the curve. If
AB = r and its direction cosines are I, m, n, we have since the
curve lies in the tangent plane
lz + Xa^f/^, V) = y + \y l + fjuy 2 , =z+\z l + /^ 2
where x, y, z are functions of p and q and X, fi are functions
of^>, q, and r. E, F, G being the coefficients at A in the
square of the linear element (Art. 377) of the surface of re
ference, and the suffixes 1, 2, 3 denote differentiation with
regard to p, q, and r, we find without difficulty
P = X, (1 + Xi) E+ (/* 3 + Vig + X,/^) F4 /j^f^G
dE dE dF dE\ dG
with a similar value for Q, and
R = \3 2 E+ 2\3/X3 F + fJi^ G.
Now if B, <f> be the angles between AB and the parametric
tangent lines at A we have
* Mtm. BUT la theorie generate des surfaces courbes (Journ. de Math.,
IV. 7, 1891).
NORMAL CONGRUENCES OP CURVES. 125
rjE
and cos<f> =
In the deformation in question 0, <j>, and r are the same for
corresponding points on the surface and curve, and if (as in
Art. 390) we make the parametric lines correspond E, F, G are
likewise unaltered by deformation. Thus \, /*, and therefore
P, Q, B, are unaltered, and hence the condition of integrability
is unaffected, which proves the theorem.
If the point B lies on the surface .2 = 0, defined by the
equation Pdp + Qdq + Rdr = 0, the corresponding point B' on
the "deformed" congruence will lie on the surface 5" = 0,
defined by the equation Pdp + Qdq + Rdr = ; for we have
shown that P, Q, and R are unaltered. Now 2" and 2" are
orthogonal to their respective congruences ; thus all the points
on any surface orthogonal to the original congruence are trans
formed into points on a surface orthogonal to the new con
gruence.
When the envelope is a curve the theorem becomes : If all
the planes are tangent to a curve C, the congruence remains
normal when the curve is twisted without change of curva
ture, the planes and curves preserving their positions relative
to the corresponding points of contact, tangent lines, and
osculating planes of C.]
[486e. Cyclic Systems. A congruence of circles normal to
a family of surfaces is described by Bibaucour as a cyclic
system* In general the coordinates of any point on a circle
whose centre is a, ft, 7 and radius p, may be written
x=a + p (X cos 0+\' sin 0)
y = ft+ p (p cos 0+ /*' sin 0)
z = y+p ( v cos 6+v sin 0}
where \, p, v, V, //, v are the directioncosines of two per
 .  T~ 
* Sur les systfrnes cycliques (Comptes rendus de VAcad. des Sciences, 76,
1873), also Bianchi, Giorno. di Mat., 21, 1888 (and op. tit.), and Guischard, Ann.
EC. Norm., HI. 14, 15, 20 (1897).
126 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
pendicular lines in the plane of the circle ; 6 is then the angle
a radius makes with a fixed radius in the plane. If a, ft, 7,
p, \, //,, v, V, //, v are functions of two parameters p and q,
the preceding equations will represent as just explained a
congruence of circles in the most general form, the parameter
6 now taking the place of r in the equations (1). It will be
found that the equation (2) takes the form
Pdp+Qdq + pd0 = ... (4)
and the condition (3) becomes
I = L cos 0+Msin 0+N=0 . . (5)
where L, M, N are functions of p and q alone. If a value of
satisfying this equation be substituted in (4) it becomes say
Rdp+Sdq = 0, and if it happens that R = S=0 for all values
of^> and q, the congruence is normal to the surface J=0.
For given values of p and q not more than two values of
satisfy the equation 7=0, unless L = M^= N = 0, when every
value satisfies. Thus, if for all values of p and q more than
two values of # satisfy 1=0, this equation must be an identity.
Hence we have Ribaucour's theorem : If the circles of a con
gruence are normal to more than two surfaces they constitute
a cyclic system, tliat is, they are normal to a singly infinite
family of surfaces.
This method may also be used to prove that the congru
ence of circles normal to a plane and to any arbitrary surface
form a cyclic system, and by inversion, the circles normal to
a sphere and to any arbitrary surface form a cyclic system.']
[486/. A cyclic congruence is defined as the rectilinear con
gruence consisting of the axes of the circles of a cyclic system,
that is, of the lines through the centres of the circles perpendi
cular to their planes. These axes must form a congruence
since one corresponds to each circle, and they therefore form
a doubly infinite system of lines in space. Each ray of the
cyclic congruence meets two consecutive rays (Art. 457), and
thus through each ray we have two deyelopables of the con
gruence. Now it may be shown by using the conditions
deducible from (5) of the foregoing article, viz. L = 0, M = 0,
CYCLIC SYSTEM^ 127
$ = 0, that the developables of the cyclic congruence corre
spond to the lines of curvature of any surface orthogonal to
the circle of the cyclic system.* Thus to each developable (J)
corresponds a surface (2) generated by the associated circles,
and this surface contains lines of curvature of all the sur
faces orthogonal to the circles. Hence by Darboux' theorem
(Art. 4866) or directly, the two singly infinite families of
the type 2, form with the original orthogonal surfaces a
triply orthogonal system; in other words, a singly infinite
family of surfaces loJiose orthogonal trajectories are circles
belongs to a triply orthogonal system (Ribaucour).
Amongst other interesting properties of cyclic systems
the following may be mentioned and can be proved by the
foregoing methods :
(1) The tangents to the lines of curvature on an orthogonal surface meet
the corresponding axis in its focal points.
(2) The osculating circles of the curves of intersection of two families of a
triply orthogonal system belong to a cyclic system.
(3) The circles normal to a sphere and tr> any surface are orthogonal to a
singly infinite system of surfaces belonging to a triply orthogonal system.
(Cf. Art. 48&J.)
If the surface chosen is a cyclide of Dupiu all the surfaces of the system
are cyclides of Dupin.
(4) If all the planes of the circles of a cyclic system pass through a fixed
point, each of the circles is orthogonal to a fixed sphere having that point as
centre.
(5) Any four surfaces orthogonal to the circles of a cyclic system meet the
circles in four points whose anharmonic ratio is constant.
For further information with regard to triply orthogonal systems the
reader should refer to Lucien Levy, Stir les Systimes de Surfaces Tripltinent
OrtJiogonaux in the Memoires Courontu'S, publ. par 1'Acad. 11. de Belgique, uv.
(1896). This memoir contains an historical resum& of the whole subject. See
in particular Darboux, Lemons sur Les Systemes Orthogonaux (Paris, 1910), in
which this distinguished mathematician deduces a number of systems from
the general differential equations.]
* For detailed proof see Ribaucour (op. tit.) or Blanch i, Lesioni, vol. 11.
CHAPTER XIV.
THE WAVE SURFACE, THE CENTBOSURFACE, PARALLEL,
PEDAL, AND INVERSE SURFACES.
487. BEFORE proceeding to surfaces of the third order
we think it more simple to treat of some special surfaces,
the theory of which is more closely connected with that
explained in preceding chapters. We begin by denning and
forming the equation of Fresnel's Wave Surface*
[The Wave Surface has received a great deal of attention from geometers,
partly owing to its optical interest, and partly to the fact that it is a special
type of Rummer's quartic surface containing sixteen nodes and sixteen double
tangent planes (see Ch. XVI). Among the distinguished mathematicians
who have investigated its properties we find, besides Fresnel, Cauchy, Herschel,
Hamilton, MacCullagh, H. Lloyd, Pliicker, Lame, Bertrand, Cayley, Brioschi,
W. Roberts, Mannheim (who has made it a special study), Darboux (who with
others investigates its asymptotic lines and lines of curvature ; Comptcs
Rendits, 1881, 1885), and Weber, who shows how to represent it by means of
elliptic functions. Wolffing (Bibliotheca Matliematica, in. 3, 1902) gives a
resum6 of the previous work done on the subject. Models of the surface
have been constructed by Brill, and (independently) by Cotter, whose model
may be seen in the museum of the Engineering School, Trinity College,
Dublin. For a more detailed bibliography see Loria, op. cit.]
THE WAVE SURFACE.
If a perpendicular through the centre be erected to the
plane of any central section of a quadric, and on it lengths be
taken equal to the axes of the section, the locus of their ex
tremities will be a surface of two sheets, which is called the
Wave Surface. Its equation is at once derived from Arts.
101, 102, where the lengths of the axes of any section are
expressed in terms of the angles which a perpendicular to
See Fresnel, Mfmoires de I'lnstitut, vol. vn. p. 186, published 1827.
128
THE WAVE SURFACE. 129
its plane makes with the axes of the surface. The same
equation then expresses the relation which the length of a
radius vector to the wave surface bears to the angles which
it makes with the axes. The equation of the wave surface
is therefore
V 6V
""
where r 2 = x 1 + y~ + z' 2 . Or, multiplying out,
 {a 2 (6 2 + c 2 ) x 2 + 6 2 (c 2 + a 2 ) y 2 + c 2 (a? + 6 2 ) ^} + aW = 0.
From the first form we see that the intersection of the wave
surface by a concentric sphere is a spheroconic.
488. The section by one of the principal planes (e.g. the
plane z) breaks up into a* circle and ellipse
(x + y z  c 2 ) (a?x~ + b*y 2  a s & 2 ).
This is also geometrically evident, since if we consider any
section of the generating quadric through the axis of z, one
of the axes of that section is equal to c, while the other axis
lies in the plane xy. If, then, we erect a perpendicular to
the plane of section, and on it take portions equal to each of
these axes, the extremities of one portion will trace out a
circle whose radius is c, while the locus of the extremities of
the other portion will plainly be the principal section of the
generating quadric, only turned round through 90. In each
of the principal planes the surface has four double points ;
namely, the intersection of the circle and ellipse just men
tioned. If x, y be the coordinates of one of these intersec
tions, the tangent cone (Art. 270) at this double point has
for its equation
4 (xx'+ yy  c 2 ) (a 2 xx' + byy'  a*b*) + z 2 (a s  c 2 ) (6*  c 2 ) = 0.
The generating quadric being supposed to be an ellipsoid, it
is evident that in the case of the section by the plane z, the
circle whose radius is c, lies altogether within the ellipse
whose axes are a, b ; and in the case of the section by the
plane x, the circle whose radius is a, lies altogether without
the ellipse whose axes are b, c. Real double points occur
VOL. IL 9
130 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
only in the section by the plane y ; they are evidently the
points corresponding to the circular sections of the generating
ellipsoid.
The section by the plane at infinity also breaks up into
factors # 2 + i/ 2 + 2 , a^+b' 2 y' 2 +c 2 z z , and may therefore also be
considered as an imaginary circle and ellipse, which in like
manner give rise to four imaginary double points of the sur
face situated at infinity. Thus the surface has in all sixteen
nodal points, only four of which are real.
489. The wave surface is one of a class of surfaces which
may be called apsidal surfaces. Any surface being given, if
we assume any point as pole, draw any section through that
pole, and on the perpendicular through the pole to the plane
of section take lengths equal to the apsidal (that is to say,
to the maximum or minimum) radii of that section ; then the
locus of the extremities of these perpendiculars is the apsidal
surface derived from the given one. The equation of the
apsidal surface may always be calculated, as in Art. 101. First
form the equation of the cone whose vertex is the pole, and
which passes through the intersection with the given surface
of a sphere of radius r. Each edge of this cone is proved
(as at Art. 102) to be an apsidal radius of the section of the
surface by the tangent plane to the cone. If, then, we form
the equation of the reciprocal cone, whose edges are perpen
dicular to the tangent planes to the first cone, we shall obtain
all the points of intersection of the sphere with the apsidal
surface. And by eliminating r between the equation of this
latter cone and that of the sphere, we have the equation of
the apsidal surface.
490. If OQ be any radius vector to the generating surface,
and OP the perpendicular to the tangent plane at the point
Q, then OQ will be an apsidal radius of the section passing
through OQ and through OR which is supposed to be per
pendicular to the plane of the paper POQ. For the tangent
plane at Q passes through PQ and is perpendicular to the
THE WAVE SURFACE.
131
plane of the paper ; the tangent line to the section QOR lies
in the tangent plane, and is
therefore also perpendicular to
the plane of the paper. Since
then OQ is perpendicular to the
tangent line in the section QOR,
it is an apsidal radius of that
section.
It follows that OT, the radius
of the apsidal surface corresponding to the point Q, lies in the
plane POQ, and is perpendicular and equal to OQ.
491. The perpendicular to the tangent plane to the apsidal
surface at T lies also in the plane POQ, and is perpendicular
and equal to OP*
Consider first a radius OT' of the apsidal surface, inde
finitely near to OT, and lying in the plane TOR, perpendicular
to the plane of the paper. Now OT' is by definition equal
to an apsidal radius of the section of the original surface by
a plane perpendicular to OT', and this plane must pass through
OQ. Again, an apsidal radius of a section is equal to the
next consecutive radius. The apsidal radius therefore of a
section passing through OQ, and indefinitely near the plane
QOR, will be equal to OQ. It follows, then, that OT= OT,
and therefore that the tangent at T to the section TOR is
perpendicular to OT, and therefore perpendicular to the plane
of the paper. The perpendicular to the tangent plane at T
must therefore lie in the plane of the paper, but this is the
first part of the theorem which was to be proved.
Secondly, consider an indefinitely near radius OT" in the
plane of the paper; this will be equal to an apsidal radius
of the section ROQ', where OQ' is indefinitely near to OQ.
But, as before, this apsidal radius being indefinitely near
to OQ' will be equal to it, and therefore OT" will be equal
* These theorems are due to MacCullagh, Transactions of the Royal
Irish Academy, vol. xvi., in his collected works, p. 4, etc.
9*
132 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
as well as perpendicular to OQ'. The angle then T"TO is
equal to Q'QO, and therefore the perpendicular OS is equal
and perpendicular to OP.
It follows from the symmetry of the construction, that if
a surface A is the apsidal of B, then conversely B is the apsidal
of A.
[The preceding results may be proved analytically. Let
xyz be the coordinates of an apse Q on a section of U= 0,
drawn through the origin, and let f, 77, be the coordinates
of the corresponding point T on the apsidal surface. Using
the condition that OQ is a limiting radius in the section it
will be found that, as x, y, z satisfy the equation
U^dx + U 2 dy + U 3 dz =
> *7> (T will satisfy the equation
x y z d drj d% __
Ui U 2 U 3 U { U a U 3 ''
and the directioncosines of the normal at T to the apsidal
surface are therefore proportional to the coefficients of
dg, drj, d in this equation.]
492. The polar reciprocal of an apsidal surface, with re
spect to the origin 0, is the same as the apsidal of the recip
rocal, with respect to 0, of the given surface.
For if we take on OP, OQ portions inversely proportional
to them, we shall have Op, Oq, a radius vector and corre
sponding perpendicular on tangent plane of the reciprocal of
the given surface. And if we take portions equal to these
on the lines OS, OT which lie in their plane, and are respec
tively perpendicular to them, then, by the last article, we
shall have a radius vector and corresponding perpendicular on
tangent plane of the apsidal of the reciprocal. But these
lengths being inversely as OS, OT are also a radius vector,
and perpendicular on tangent plane of the reciprocal 'of the
apsidal. The apsidal of the reciprocal is therefore the same
as the reciprocal of the apsidal.
In particular, the reciprocal of the wave surface generated
from any ellipsoid is the wave surface generated from the
reciprocal ellipsoid.
THE WAVE SUKFAt 133
We might have otherwise seen that the reciprocal of a
wave surface is a surface also of the fourth degree, for the
reciprocal of a surface of the fourth degree is in general of
the thirtysixth degree (Art. 281); but it is proved, as for
plane curves, that each double point on a surface reduces the
degree of its reciprocal by two ; and we have proved (Art. 488)
that the wave surface has sixteen double points.
To a nodal point on any surface (which is a point through
which can be drawn an infinity of tangent planes, touching
a cone of the second degree) answers on the reciprocal surface
a tangent plane, having an infinity of points of contact, lying
in a conic. From knowing then, that a wave surface has four
real double points, and that the reciprocal of a wave surface
is a wave surface, we infer that the wave surface has four
tangent planes which touch all along a conic.* [Further, each
of these conies is a circle. For since the wave surface, as its
equation shows, passes through the imaginary circle at infinity,
every plane section passes through the I and / points in its
plane, and is therefore in general a " bicircular quartic ".
Hence jf the plane section reduces to two coincident conies
it must be a circle.]
[Ex. 1. The four real tangent planes with circular contact are represented
by
* Nto*  6* *V&  c 8 bj a *  <? = 0.
Ex. 2. If the equations of the four tangent planes as just expressed are
written
Tj = 0, T, = 0, T 3 = 0, TI =
the equation of the wave surface may be written
r 1 r,r s r 4 = w*
where W= a s z* + b*y* + c 9 * 2  o6 + 6*(x + y* ! **  a s ) + 6(a*  c).]
* Sir W. R. Hamilton first showed that the wave surface has four nodes,
the tangent planes at which envelope cones, and that it has four tangent
planes which touch along circles. Transactions of the Royal Irish Academy,
vol. xvi. (1837), p. 132. Dr. Lloyd experimentally verified the optical theorems
thence derived, Ibid. p. 145. The geometrical investigations which follow are
due to Professor MacCullagh, Ibid. p. 248. See also Pliicker, " Discussion de
la forme gn6rale des ondea lumineuses," Crelle, t. six. (1839), pp. 144 and
91, 92.
134 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
493. We shall now prove 'geometrically that the four
tangent planes touch along a circle. It is convenient to
premise the following lemmas :
LEMMA I. " If two lines intersecting in a fixed point, and
at right angles to each other, move each in a fixed plane, the
plane containing the two lines envelopes a cone whose sections
parallel to the fixed planes are parabolas." The plane of the
paper is supposed to be parallel to one of the fixed planes
and the other fixed plane is supposed to pass through the line
MN. The fixed point in which the two lines intersect is
supposed to be above the paper, P being the foot of the per
pendicular from it on the plane of the paper. Now let OB
be one position of the line which moves in the plane OMN,
then the other line OA, which is parallel to the plane of the
paper being perpendicular to OB and to OP, is perpendicular
to the plane OB P. But the plane OAB intersects the plane
of the paper in a line BT parallel to OA, and therefore per
pendicular to BP. And the envelope of BT is evidently a
parabola of which P is the focus and MN the tangent at the
vertex.
LEMMA II. "If a line OC be drawn perpendicular to OAB,
it will generate a cone whose circular sections are parallel
to the fixed planes " (Ex. 4, Art. 121). It is proved, as in Art.
125, that the locus of C is the polar reciprocal, with respect
to P, of the envelope of BT. The locus is therefore a circle
passing through P.
LEMMA III. " If a central radius of a quadric moves in
a fixed plane, the corresponding perpendicular on a tangent
plane also moves in a fixed plane." Namely, the plane per
pendicular to the diameter conjugate to the first plane, to
which the tangent plane must be parallel.
494. Suppose now (see figure, Art. 490) that the plane
OQR (where OB is perpendicular to the plane of the paper)
is a circular section of a quadric, then OT is the nodal radius
of the wave surface, which remains the same while OQ moves
THH WAVE SURFACE. 135
in the plane of the circular section ; and we wish to find the
cone generated by OS. But OS is perpendicular to OR which
moves in the plane of the circular section and to OP which
moves in a fixed plane by Lemma III, therefore OS generates
a cone whose circular sections are parallel to the planes FOR,
QOR. Now T is a fixed point, and TS is parallel to the
plane POR, therefore the locus of the point S is a circle.
The tangent cone at the node is evidently the reciprocal
of the cone generated by OS, and is therefore a cone whose
sections parallel to the same planes are parabolas.
Secondly, suppose the line OP to be of constant length,
which will happen when the plane POR is a section perpen
dicular to the axis of one of the two right cylinders which
circumscribe the ellipsoid, then the point S is fixed, and it is
proved precisely, as in the first part of this article, that the
locus of T is a circle.
495. The equations of Art. 251 give immediately another
form of the equation of the wave surface. It is evident
thence, that if 0, 0' be the angles which any radius vector
makes with the lines to the nodes, then the lengths of the
radius vector are, for one sheet,
1 _cos 2 j (0ff) sin'j (0  6'}
p 2 c 2 a 2
and for the other
1 = cos'j (0+ 0'} , sin 2 j (0+ ff)
p* m c* a*
1 1 /I 1\ .
while ;>= (5 ) s" 1 sin 0.
p 2 p V a/
It follows hence also that the intersections of a wave surface
with a series of concentric spheres are a series of confocal
sp heroconies. For, in the preceding equations, if p or p' be
constant, we have 00' constant.
[495a. The coordinates of a point on the wave surface may be
expressed as elliptic functions of two parameters as follows :
* Appell and Lacour, Fonctions Elliptiques (1897), Art. 119.
136 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
Let x 1 + ?/ 2 + & = a
aV+Z>y + cV = /3.
Then on the wave surface
2 (Z> 2 + c 2 )z 2 + Z> 2 (c 2 + aV+ c 2 (a 2 + 6V = a/3+ aW.
a  6 2 3  c 2 a 2
T ,
if we put
we find
= a
72 
where & 2 = 3  and P
3  nri  
a 2  c 2 6 2 (a 2  c 2 )
Hence in the Jacobian notation
x = b sn (u, &) dn (v, I)
y = a en (u, Jc) en (v, I)
z = a dn (u, k) sn (v, I)
If we calculate the value of F in the linear element (Art.
377) we shall find that it vanishes, and hence the parametric
curves cut at right angles. They consist of a system of
spheroquartics, a = constant, and of a system of quartic
curves ft = constant.]
496. The equation of the wave surface has also heen
expressed as follows by W. Roberts in elliptic coordinates.
The form of the equation
99 799 49
afx 1 b*y z c z z z _
o k> * I ' **% a I q a ~~ V/
a 2 T o r c r
shows that the equation may be got by eliminating r 2 between
the equations
y? if~ 2
=  5+ . y ,.,+ , .,= 1, and x i + if+z* = r 1 .
r a r o r  cr
Giving r 2 any series of constant values, the first equation
denotes a series of confocal quadrics, the axis of z being the
primary axis, and the axis of x the least ; and for this system
(Art. 160) h?= b 2  c 2 , k*=* a?  c 2 . Since r 2 is always less than
a 2 and greater than c 2 , the equation always denotes a hyper
THE WAVE SURFACI . 137
boloid, which will be of one or of two sheets according as r*
is greater or less than b. The intersections of the hyper
boloids of one sheet with corresponding spheres generate one
sheet of the wave surface, and those of two sheets the other.
Now if the surface denote a hyperboloid of one sheet, and
if X, ft, v denote the primary axes of three con focal surfaces
of the system now under consideration which pass through
any point, then the equation gives us r 2  cr = /j?, but ( A.rt. 161)
whence the equation of one sheet in elliptic coordinates is
\+v* = c*+h+k* = a*+b' 2 c*.
In like manner the equation of the other sheet is
The general equation of the wave surface also implies
p? + v* = a 2 + 6 2  c 2 , but this denotes an imaginary locus.
Since, if X is constant, /A is constant for one sheet and v
for the other, it follows that if through any point on the sur
face be drawn an ellipsoid of the same system, it will meet
one sheet in a line of curvature of one system of the ellipsoid,
and the other sheet in a line of curvature of the other
system.
If the equations of two surfaces expressed in terms of
X, fi, v, when differentiated give
Pd\+Qdn+Rdv = Q, P'd\ + Q'dp+R'dv = Q,
the condition that they should cut at right angles is (cf.
Art. 410)
(X 2  A 2 ) (X 2  & 2 ) QQ' <
(A* **)(&**)
which is satisfied if P = 0, Q = Q, R' = Q. Hence any surface
v= constant cuts at right angles any surface whose equation
is of the form < (X, /*) = (). The hyperboloid therefore,
v = constant, cuts at right angles one sheet of the wave sur
face, while it meets the other in a line of curvature on the
hyperboloid.
138 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
497. The plane of any radius vector of the ivave surface
and the corresponding perpendicular on the tangent plane,
makes equal angles with tlie planes through the radius vector
and the nodal lines. For the first plane is perpendicular to
OR (Art. 490) which is an axis of the section QOE of the
generating ellipsoid and the other two planes are perpendicular
to the radii of that section whose lengths are 6, the mean axis
of the ellipsoid, and these two equal lines make equal angles
with the axis. The planes are evidently at right angles to
each other, which are drawn through any radius vector, and
the perpendiculars on the tangent planes at the points where
it meets the two sheets of the surface.
Reciprocating the theorem of this article, we see that the
plane determined by any line through the centre and by one
of the points where planes perpendicular to that line touch
the surface, makes equal angles with the planes through the
same line and through perpendiculars from the centre on the
planes of circular contact (Art. 494).
498. If the coordinates of any point on the generating
ellipsoid be x'y'z, and the primary axes of confocals through
that point a, a" ; then the squares of the axes of the section
parallel to the tangent plane are a 2  a' 2 , a 2  a"' 2 , which we
shall call p 2 , p'~. These, then, give the two values of the
radius vector of the wave surface, whose directioncosines are
shall now calculate the length and the
a c
directioncosines of the perpendicular on the tangent plane at
either of the points where this radius vector meets the surface.
It was proved (Art. 491) that the required perpendicular is
equal and perpendicular to the perpendicular on the tangent
plane at the point where the ellipsoid is met by one of the
axes of the section ; and the directioncosines of this axis are
*D % *D 1] *D Z
ft* Tfi, ^jr The coordinates of its extremity are then
these several cosines multiplied by p, and the directioncosines
of the corresponding perpendicular of the ellipsoid are
THE WAVE SURFACE. 139
Ppj^, Ppjjy, Ppj[f t ,
I , ( x* y'* z'*
where ^r, = p~p  ^ ^ + frr/i + ~rf
1" (a a* tro* c*c
Now if the quantity within the brackets be multiplied by
(or  a') 2 , we see at once that it will become 5+ t . Hence
151J = ~ 2 2 ' ^H" * ~ j , /2
This then gives the length of the perpendicular on the
tangent plane at the point on the wave surface which we are
considering. Its directioncosines are obtained from the con
sideration that it is perpendicular to the two lines whose
directioncosines are respectively
P X P"l/' P Z y, P X T> P I/ T> P %
Forming, by Art. 15, the directioncosines of a line perpendic
ular to these two, we find, after a few reductions,
pp a/' pp \ , pp
In fact, it is verified without difficulty, that the line whose
directioncosines have been just written is perpendicular to
the two preceding.
It follows hence also, that the equation of the tangent
plane at the same point is
In like manner the tangent plane at the other point where
the same radius vector meets the surface is
499. If be the angle which the perpendicular on the tan
gent plane makes with the radius vector, we have P = p cos 6 ;
but we have, in the last article, proved P 2 = p ,. 2 . Hence,
cos 2 0= ~ 75, tan 2 0=^ This expression may be trans
fr+p* p*
140 ANALYTIC GEOMETKY OF THREE DIMENSIONS.
formed by means of the values given for p and p (Art. 165).
We have therefore
. _ aW ,, _ (<*&(&&(<?&
^ 2 ' 2  2
Whence tan 2 =
In this form the equation states a property of the ellipsoid,
and the expression is analogous to that for the angle between
the normal and central radius vector of a plane ellipse, viz.
In the case of the wave surface it is manifest that tan 6
vanishes only when p = a, b, or c, and becomes indeterminate
when p = p'=b.
500. The expression tan #= leads to a construction for
the perpendiculars on the tangent planes at the points where
a given radius vector meets the two sheets of the surface.
The perpendiculars must lie in one or other of two fixed
planes (Arts. 497, 498), and if a plane be drawn perpendicular
to the radius vector of the wave surface at a distance^), it is
evident from the expression for tan 0, that p is the distance
to the radius vector from the point where the perpendicular on
the tangent plane meets this plane. Thus we have the con
struction, " Draw a tangent plane to the generating ellipsoid
perpendicular to the given radius vector, from its point of
contact let fall perpendiculars on the two planes of Art. 497,
then the lines joining to the centre the feet of these perpen
diculars are the perpendiculars required."
We obtain by reciprocation a similar construction, to de
termine the points where planes parallel to a given one touch
the two sheets of the surface.
THE WAVE SURFACE. 141
Ex. 1. To transform the equation of the surface, as at Art. 174, so as to
make the radius vector to any point on the surface the axis of z, and the axes
of the corresponding section of the generating ellipsoid the axes of x and y we
find
(x + y* + s 2 ) {p 2 * 2 + (p' a + p 2 ) x 2 + (p"* + p' 2 ) y z + Zpp'xz + 2pp"yz + C 2p'p"xy}
 p*z* (p 2 + p' 2 )  x 2 (pV + p' 2 p' 2 +p" 2 p 2 + p 2 p' 2 )
 y'' (pV 2 + P'*P* + P"V + pV 2 )  2j>/p 2 z2  2pp"p*y* + 2>W 2 = o.
It is easy to see that if we make x and y = in the equation thus trans
formed, we get for z 2 the values p 2 and p' 2 as we ought. If we transform the
equation to parallel axes through the point z = p, the linear part of the equa
tion becomes
2pp (p 2  p' 2 ) (pz + p'x),
from which the results already obtained as to the position of the tangent plane
may be independently established.
Ex. 2. To transform similarly the equation of the reciprocal of the wave
ace
we find
x 2
surface obtained by writing for o, &c., in the equation of the wave surface
2  2pp"p z yz + 2 2 (p' 2 p' 2 +p'V + PV 2 )}
" 2 +p' 2 ) X 2  \ (p + p t + p 2) y* _ A 4 (p '2 + p 2 + p 2 + p 'J) ^
t/ + 2\*pp'xz + 2\*pp"yz + X 8 = 0.
Wa know that the surface is touched by the plane pz = A 2 , and if we put
in this value for z, we find, as we ought, a curve having for a double point the
point y =0, ppx = p'\ 2 If in the equation of the curve we make y = 0, we
get
from which we learn that that chord of the outer sheet of the wave surface
which joins any point on the inner sheet to the foot of the perpendicular from
the centre on the tangent plane is bisected at the foot of the perpendicular.
The inflexional tangents are parallel to
{p'V 2 + & (p' 2  p 2 )} x 2  'Wp"fxy + {pV + P 2 (P' 2  P 2 )} y 2 '
a result of which I do not see any geometrical interpretation.
THE SURFACE OF CENTRES.
501. We have already shown (Art. 206) how to obtain the
equation of the surface of centres of a quadric. We consider
the problem under a somewhat more general form, as it has
been discussed by Clebsch (Crelle, vol. LXII. p. 64), some of
whose results we give, working with the canonical form ;
and we refer to his paper for fuller details and for his method
of dealing with the general equation. By the method of Art.
227, we may consider the normal to a surface as a particular
142 ANALYTIC GEOMETKY OF THREE DIMENSIONS.
case of the line joining the point of contact of any tangent
plane to the pole of that plane with respect to a certain fixed
quadric. The problem then of drawing a normal to a quadric
from a given point may be generalized as follows : Let it be re
quired to find a point xyzw on a quadric U(ax 2 + by + cz 2 + die},
such that the pole, with respect to another quadric V,
of the tangent plane to U at xyzw, shall lie on the line joining
xyzw to a given point x'y'z'w'. The coordinates of any point
on this latter line may be written in the form
x  \x, y  \y, z \z, w'  \w,
and expressing that the polar plane of this point, with regard
to V, shall be identical with the polar plane of xyzw, with
respect to U, we get the equations
x = (ap + X) x, y' = (t>n+\}y, z'=(cp,+ \}z, w'=(dp + \}w.
And since xyzw is a point on U, X : /* is determined by the
equation
ax* by" 2 cz'' 2 dw"* _~
~
X) 2
When X : p is known, x, y, z, w are determined from the pre
ceding system of equations, and since the equation in X : /* is
of the sixth degree, the problem admits of six solutions. If
we form the discriminant, with regard to X : /A, of this equa
tion, we get the locus of points x'y'z'w for which two values
of X : (* coincide, and rejecting a factor x'y" 1 z' z w' t (which in
dicates that two values coincide for all points on the principal
planes), we shall have a surface of the twelfth degree answer
ing to the surface of centres.
502. The problem of finding the surface of centres itself is
easily made to depend on an equation of like form ; for (Art.
197) the coordinates of a centre of curvature answering to
any point x'y'z on an ellipsoid are
_ b" 2 y' _ c*z
~
X
f~ , T5~ > ~
a* b* c 2
Solve for x', y', z' from these equations, and substitute in the
equations satisfied by x'y'z, viz.
GENERALIZED CENTROSURFACE OF QUADRIC. 143
(I ~ r) ('" if
now for a' 2 write a 2  h?, &c., and we get
( ri'l /J^\* //)* J)^\** //2 /i\2 >
\Cv iv ) \U fv j ((/ ~~ fv )
(a?~h?)* t (6~A*) 3 + (c 3  I 2 ) 3 = '
These two equations represent a curve of the fourth degree,
which is the locus of the centres of curvature answering to
points on the intersection of the given quadric with a given
confocal. The surface of centres is got by eliminating /t 2 be
tween the equations ; or (since the second equation is the
differential of the first with respect to 7i 2 ) by forming the dis
criminant of the first equation.
503. I first showed, in 1857 (Quarterly Journal, vol. n.
p. 218), that the problem of finding the surface of centres is
reducible to elimination between a cubic and a quadratic, and
Clebsch has proved that the same reduction is applicable to
the problem considered in its most general form. In fact, let
A denote the discriminant of pU + \V ; which for the canon
ical form (Art. 141), is (ap+\) (bp + \) (c/x + X) (dp + \), and
let H denote the reciprocal of pU+ XF, viz.
(bp + X) (cp + X) (dp + X) x 2 + (cp + X) (dp + X) (ap + X) y* 4 <fcc.
then we have r =   + , . + &c.
A
Now, if we differentiate the righthand side of this equation
with respect to p, we obtain the equation (Art. 501) which
determines X : p which therefore may be written
II = A dU
dp dp
This last equation, which is the Jacobian of fl and A, being
the result of eliminating m between A + wX/2 and its differ
ential,* will be verified when A + m\fl has two equal factors.
* The factor \ is introduced to make A + m\tl a homogeneous quartic func
tion of ft.: \.
144 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
Its differential again ft. ,^ = A ^^ being the result of elimina
tion of m between A + m\ft, and its second differential, will be
verified when A + m\ft, has three equal factors. But both
Jacobian and its differential vanish when both A and ft, vanish.
Thus then, as was stated (Note, p. 143), the discriminant of
the Jacobian of two algebraic functions A, O, contains as a
factor the result of eliminating X : /j, between A and ft, ; and
as another factor, the condition that it shall be possible to
determine m, so that A + m\ft, may have three equal factors.
In the present case the eliminant (with respect to X : /*) of
A, ft,, gives the factor x^ifz^w^, and it is the other condition
which gives the surface answering to the surface of centres.
And this condition is formed, as in Art. 206, by eliminating m
between the S and T of the biquadratic A + m\ft.
504. The discriminant of any algebraic function
ty (X) + (X  a) 2 </> (X),
must evidently be divisible by k ; and if after the division we
make A; = 0, it can be proved that the remaining factor is
ty (a) </> (a) 3 multiplied by the discriminant of <f> (X). Thus,
then, the section of Clebsch's surface by the principal plane 10 is
CLX^ O1J CZ^
the conic  _ , 2 + , _ ^ + . _ , 2 taken three times, together
with the curve of the sixth degree, which is the reduced
discriminant of
ax 2 by 2 r:
(a+X) 2+ (6T\7 2+ (c + X) 2 '
Clebsch has remarked that this conic and curve touch each
other, and the method we have adopted leads to a simple
proof of this. For evidently the discriminant of
ax 2 by' 2 cz* ^
~ '
may be regarded as the envelope of all conies which can be
represented by this equation, and therefore touches every par
ticular conic of the system in the four points where it meets
the conic represented by the differential of the equation with
regard to X, viz.
GENERALIZED CENTROSUBFACE OF QUADRIC. 145
ax* by* cz 2 _*
(oTx) 3 + (6T\)" 3+ (c + X) 3 ~
The coordinates of these points are ax? = (a + X) 3 (6  c),
bif = (6 + X) 3 (c  a), C2 2 = (c + X) 3 (a  6) ; and the equations of the
common tangents at them to the conic and its envelope are
In the case under consideration X=  d. If, then, we use the
abbreviations
(a 6) (ac) (ad) = 4 s , (6 a) (6  c) (6d)=  B s ,
(ca) (c6) (cd) = <7 2 , (da) (<!&) (dc) = D 2 ,
the equations of the common tangents to the conic and the
envelope curve, are
^+^+^ =
A ~ B ~ C
The reasoning used in this article can evidently be applied
to other similar cases. Thus, the surface parallel to a quadric
(Art. 202, Ex. 2) is met by a principal plane in a curve of the
eighth order and a conic, taken twice, which touches that
curve in four points ; and again, the four right lines (Art. 216)
touch the conic in their plane.
505. Besides the cuspidal conies in the principal planes,
there are other cuspidal conies on the surface, which are
found by investigating the locus of points for which the
equation of the sixth degree (Art. 501) has three equal roots.
Differentiating that equation twice with regard to X, we arrive
at a system of equations reducible to the form
ax 2 by* cz 1 duP _
6V
a 3 x 2 W* _
~
The result of eliminating X between these three equations
will be a pair of equations denoting a curve locus. Now
solving these equations, we get
VOL. II. 10
146 ANALYTIC GEOMETRY OP THREE DIMENSIONS.
whence a+X, b+\, &c., are proportional to ahAl, &c.
Substituting from these in the equation (Art. 501)
. by . cz " . dw _n
'.fl. < v \5! "" / i \\2~' / J , v\2 ^>
alx
weget
whence we learn that the locus which we are investigating
consists of curves situated in one or other of eight planes ;
and that these planes meet the principal planes in the
common tangents to the conic and envelope curve considered
in the last article.*
But if we eliminate X between the three equations
so as to form a homogeneous equation in x, y, z, we get
aM* (b  c) a* + fc*B*(c  a) y* + c*C* (a  6) ** = 0,
which denotes a cone of the second degree touched by the
planes x, y, z. Hence, the cuspidal curves in the eight
planes are conies which touch the cuspidal conies in the
principal planes.
506. There will be a nodal curve on the surface answering
to the points for which the equation of Art. 501 has two
pairs of equal roots. Now we saw (Art. 503) that the con
dition for a single pair of equal roots is got by eliminating in
*The existence of these eight planes may be also inferred from the con
sideration that the reciprocal of the surface of centres has an equation of the
form (Art. 199) U 2 = VW, and has therefore as double points the eight points
of intersection of U, V, W. The surface of centres then has eight imagin
ary double tangent planes, which touch the surface in conies (see Art. 271).
The origin of these planes is accounted for geometrically, as Darboux has
shown, by considering the eight generators of the quadric which meet the
circle at infinity (Art. 139). The normals along any of these all lie in the
plane containing the generator and the tangent to the circle at infinity at the
point where it meets it, and they envelope a conic in that plane. In like
manner a cuspidal plane curve on the centresurface will arise every time
that a surface contains a right line which meets the circle at infinity.
GENERALIZED CENTROSURPACE OP QUADRIC. 147
between a quadratic and a cubic equation, namely, the S
and T of the biquadratic A + m\fl. If we write these equa
tions
a + bm + cm* = 0, A + Bm + Cm* + Dm 3 = 0,
it will be found that the degrees in x, y, z, w of these co
efficients are respectively 0, 2, 4 ; 0, 2, 4, 6 ; and the result of
elimination is, as we know, of the twelfth degree. Now the
condition that the equation of Art. 501 may have two pairs
of equal roots, is simply that this cubic and quadratic may
have two common values of m. Generally, if the result of
eliminating an indeterminate m between two equations
denotes a surface, the system of conditions that the equations
shall have two common roots will represent a double curve
on that surface. Thus the result of eliminating m between
two quadratics a+ bm+ cm?, a + b'm+c'm 2 is
(ac  caj + (ba  ab') (be  cb') = 0.
But if we remember that a (be  cb') = 6 (ac  ca') + c (ba  ab'),
this result may be written
a (ac  ca')' + b (ac  ca') (ba  ab') + c (ba  ab') = 0,
showing that the intersection of ac  ca', ba'  ab' (which must
separately vanish if the equations have both roots common),
is a double curve on the surface.
And to come to the case immediately under consideration,
if we have to eliminate between
a+bm+cm = 0, A +Bm+ Cm z + Dm s = 0,
we may substitute for the second equation that derived by
multiplying the first by A, the second by a, and subtracting,
viz.
(aB  bA) + (aCcA)m+ aDm* = 0,
and thus, as has been just shown, the result of elimination
may be written aP 2  bPQ + cQ* = 0, where
P=bcA acB + aD, Q=(ac 6 ! ) A + abB  a'O.
We thus see that the curve PQ is a double curve on the sur
face of centres ; but since P is of the sixth degree and Q of
the fourth, the nodal curve PQ is of the twentyfourth.
10*
148 ANALYTIC GEOMETRY OF THEEE DIMENSIONS.
Further details will be found in Clebsch's paper already re
ferred to.*
507. It is convenient to give here an investigation of some
of the characteristics of the centrosurface of a surface of the
w th degreaf We denote by n the class of the surface, or the
degree of its reciprocal, which, when the surface has no
multiple points, is m (in  I) 2 (see Art. 281) ; and we denote
by a the number of tangent lines to the surface which both
pass through a given point and lie in a given plane, which is
in the same case m (m 1), Art. 282, this characteristic being
the same for a surface and for its reciprocal.
Let us first examine the number of normals to a given
surface (bitangents to the centrosurface, see Art. 306) which
can be drawn through a given point. This is solved as the
corresponding problem for plane curves. (See Higher Plane
Curves, p. 94, and Cambridge and Dublin Mathematical
Journal, vol. n.) Taking the point at infinity, the number
of finite normals which can be drawn through it is the same
as the number of tangent planes which can be drawn parallel
to a given one ; that is to say, is n. To this number must be
added the number of normals which lie altogether at infinity.
Now it is easy to see that the normal corresponding to any
point of the surface at infinity lies altogether at infinity, and
is the normal to the section by the plane infinity, in the ex
tended sense of the word normal (Higher Plane Curves, Art
109) . The number of such normals that can be drawn through
a point in the plane is m+ a (Higher Plane Curves, Art. Ill),
since a is the order of the reciprocal of a plane section. The
total number of normals therefore that can be drawn through
* See also a Memoir by Cayley (Cambridge Philosophical Transactions,
vol. xii.) in which this surface is elaborately discussed. He uses the nota
tion explained, note, Art. 409, when the equations of Art. 197 become
 /3yax = (a 8 + j>) (a s + q),  ya&V = (&* + P) 3 (& + 3),
***.(*+ jp 4* + 4),
a, /3, y having the same meaning as in Art. 206.
f This investigation is derived from a communication by Darboux to the
French Academy, Comptes Rendus, t. LXX. (1870), p. 1328.
THE CENTROSURFACE. 149
any point is m+ n+ a ; or, when the surface has no multiple
points, is m 3  m? + m.
Next let us examine the number of normals which lie in
a given plane. The corresponding tangent planes evidently
pass through the same point at infinity, viz. the point at
infinity on a perpendicular to the given plane. And the cor
responding points of contact are evidently the intersections
by the given plane of the curve of contact of tangents from
that point, and are therefore in number a or m (m  1).
The normals to a surface constitute a congruence of lines
(see Art. 453), and the two numbers just determined are the
order and class of that congruence.
508. To find the locus of points on a surface, the normals
at which meet a given line,
Ax + By + Cz + D = 0, A 'x + B'y+ G'z +D'= 0.
Substituting in these equations the values for the coordinates
of a point on the normal (Art. 273), x = x'+0U l , y = y'+0U t ,
z = z'+0U 3 , and eliminating the indeterminate 0, we see that
the point of contact lies on the curve of intersection of the
given surface with
(Ax + By + Cz+D) (A'U l + B'U,+ C'U Z )
= (A'x + B'y + C'z+ D') (AU 1 + BU< 1 + (717,),
a surface also of the m th order, and containing the given line.
The section of this curve by any plane through that line con
sists of the a points whose normals lie in the plane, and the
m points where the line meets the surface.
509. We can hence determine the class of the centrosur
face. A tangent plane to that surface contains two infinitely
near normals to the given surface (Art. 306) ; and therefore
the tangent planes to the centro surf ace which pass through a
given line will touch the locus determined in the last article.
Now the number of planes which can be drawn to touch
the curve of intersection of two surfaces of the w th order,
being equal to the rank of the corresponding developable, is
(Arts. 325, 342) w 2 (2w  2) ; but, since in this case the line
150 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
through which the tangent planes are drawn meets the curve
in m points, this number must be diminished by 2w. The
class of the centresurface therefore is 2w (w 2 m 1).
510. Darboux * investigates as follows the degree of the
centresurface. Let JJL and v be the two numbers determined
in Art. 507, viz. the order and class of the congruence formed
by the normals ; let M and N be the degree and class of the
centrosurface.
Now take any line and consider the correspondence between
two planes drawn through it such that a normal in one plane
intersects a normal in the other. Drawing the first plane
arbitrarily, any of the v normals in that plane may be taken
for the first normal, and at the point where it meets the
arbitrary line, /*  1 other normals may be drawn ; we see then
that to any position of one plane correspond v(pl) positions
of the other. It follows then, from the general theory of
correspondence, that there will be '2v (/j,  1) cases of coincid
ence of the two planes. Now let us denote by x the number
of points on the line such that the line is coplanar with two
of the normals at the point ; then the cases of coincidence
obviously answer either to points x or to points on the
centrosurface, since for each of the latter points two of the
normals drawn from it coincide. We have then
2 1/ (/JL  1) = x + M.
But in like manner consider the correspondence between
points on the line such that a normal from one is coplanar
with a normal from the other, and we have
2/t (v  1) = x + N,
whence MN=2(pv)
and putting in the values already obtained for /*, v, N, we have
M=2m (w1) (2m  1).
* Similar investigations were also made independently by Lothar Marcks.
(See Math. Annalen, vol. v.) The investigation may be regarded as establish
ing a general relation (which seems to bo due to Klein) between the order and
class of an algebraic congruence, and the degree and class of its " focal surface "
(see Art. 457).
THE CENTROSURFACE. 151
511. The number thus found for the degree of the centre
surface may be verified by considering the section of that
surface by the plane infinity. Consider first the section of the
surface itself by the plane infinity ; the corresponding normals
lie at infinity, and their envelope will (Higiier Plane Curves,
Art. 112) be a curve of the degree 3a + *. And besides (as in
Art. 198) the centresurface will include the polar reciprocal
of the section with regard to the circle at infinity. The degree
of this will be a, and it will be counted three times. Consider
now the finite points of the surface. In order that one of these
should have an infinitely distant centre of curvature, two con
secutive normals must be parallel, and therefore the point
must be on the parabolic curve. It is easy to see that the
normals along the intersection of the surface by another whose
order is m, generate a surface of the degree m 2 m' ; therefore
the normals along the parabolic curve generate a surface
whose degree is 4w 2 (m  2). But the section of this surface
by the plane infinity includes the 4wt (m  2) normals at
the points where the parabolic curve itself meets the plane
at infinity. The curve locus therefore at infinity answer
ing to finite points on the parabolic curve is of the degree
4w (m  1) (m  2) . The total degree then of the section of the
centrosurface by the plane infinity, is
3m (m  1)+ 3m (m  1) + 4w (m  1) (m  2),
or 2w (m  1) (2?n.  1) as before.
511a. In general 28 bitangents can be drawn to the centro
surface of a quadric from any point. In fact the reciprocals
are bitangents to a plane section of the reciprocal surface
which is of the fourth degree. F. Purser * has shown that
these 28 lines resolve into three groups, the six normals which
can be drawn from the point to the surface, the six pairs of
generators of the six quadrics of the system
ftV 6V cV
('  A 2 )* (6*  AT (c*  A*)*
* Quarterly Journal of Mathematics, vol. xiu. p. 338.
152 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
which pass through the point, and the ten synnormals through
the point. To explain what these last are ; the six feet of
normals from any point to a quadric may be distributed in
ten ways into pairs of threes, each three determining a plane.
The two planes of a pair are simply related and besides each
plane touches a surface of the fourth class, or, in other words,
the pole of such a plane with regard to the quadric moves on
a surface of the fourth degree, to which the name normopolar
surface has been given. The analysis which establishes this,
easily shows that three intersecting normals to the quadric at
points of such a plane section meet in a point which describes
a definite right line when the plane section remains unaltered,
which locus line corresponding to any two correlated planes
satisfying the condition of the fourth order, is called a syn
normal. There are therefore ten synnormals through a
point.*
PARALLEL SURFACES.
512. We have discussed, Art. 202, the problem of find
ing the equation of a surface parallel to a quadric, and
we investigate now the characteristics of the parallel to
a surface of the n ib degree. We confine ourselves to the
case when the surface has no special relation to the plane
or circle at infinity. The same principles are used as in the
corresponding investigation for plane curves (HigJwr Plane
Curves, p. 101). The degree of the parallel is found by
making k the modulus = in its equation, which will not
affect the terms of highest degree in the equation. The re
sult will represent the original surface counted twice, together
* In 1862 Desboves published his " Theorie nouvelle des normales aux
surfaces du second ordre," in which the locus line and the related surface are
discussed under the names synnormal and normopolar surface. Purser
independently arrived at the same results (Quarterly Journal, vol. vin. p. 66)
and showed the equivalence of the relation of the fourth order with the in
variant relation in piano that three feet of normals from a point to a quadric
form a triangle inscribed in one and circumscribed to another given conic ;
and gave a construction for any synnormal through a point,
PARALLEL SURFACES. 153
with the developable enveloped by the tangent planes * to the
surface drawn through the tangent lines of the circle at
infinity, this developable answering to the tangents from the
foci of a plane curve (Art. 146). Now it will be seen (Chap.
xvn. post} that the rank of a developable enveloping a sur
face and a curve is nm'+ar', where a, n are characteristics
of the surface and m', r of the curve. In the present case
m'r'=2>, and the rank of the developable is 2 (n+d). The
degree of the parallel surface is therefore 2(w+w+a) or
2 (w 3 w 2 +w) ; in other words, it is double the number of
normals that can be drawn from a point to the surface
(Art. 507).
513. If the equation of the tangent plane to a surface be
ax + fry + yz + 8 = 0, and if the surface be given by a tangential
equation between a, ft, 7, 8, then the corresponding equation of
a parallel surface is got by writing in this equation for B, B + kp,
where p* = a 2 + ft 2 + T 2 . This equation cleared of radicals will
ordinarily be of double the degree of the primitive equation ;
hence the class of a parallel is in general double the class
of the primitive. More generally, to a cylinder enveloping
the primitive corresponds a cylinder enveloping the parallel
surface, and being the parallel of the former cylinder. Hence
the characteristics of the general tangent cone to the parallel
are derived from those of the general tangent cone to the
primitive by the rules for plane curves (Higher Plane Curves,
Art. 117a). Thus then, since (Art. 279 et seq.) we have for
the tangent cone to the primitive,
/jt=a=m(ml), v = n=m (m  I) 2 ,
K = 3m(m Y)(m 2), i=4m(m 1) (m 2),
we have for the tangent cone to the parallel (HigJier Plane
Curves, 1. c.)
2w 2 (w 1), v=2n,
* It is to be noted that every parallel to any of these planes coincides
with the plane itself. The paper of Mr. S. Roberts which I use in this
article is in Proceedings of the London Mathematical Society, 1873.
154 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
Again, the reciprocal of a parallel surface is of the degree 2w,
having a cuspidal curve of the order Sm (m  1) (m  2), and
a nodal of the order
m (in  1) (2w 4  6w 3 + 6w 2  16w+ 25).
The parallel surface will ordinarily have nodal and cuspidal
curves. In fact, since the equation of the parallel surface may
also be regarded as an equation determining the lengths of
the normals from any point to the surface, if we form the
discriminant of this with regard to k (see Conies, p. 337), it
will include a factor which will represent a surface locus, from
each point of which two distinct normals of equal length can
be drawn to the surface. Such a point will be a double point
on the parallel surface whose modulus is equal to this length.
In like manner, each parallel surface will have a determinate
number of triple points. The discriminant just mentioned
will also include a factor representing the surface of centres ;
and plainly to those points on the primitive at which a principal
radius of curvature is equal to the modulus, will correspond
points on the surface of centres which will form a cuspidal
curve on the parallel surface. Roberts determines the order
of the cuspidal curve as double that of the surface of
centres, and confirms his result by observing, that in the
limiting case k = oo , the locus of points on the surface of
centres for which a principal radius of curvature = k, is the
section of the surface of centres by the plane infinity, counted
twice, since k may be oo . The singularities of the parallel
surface here assigned are sufficient to determine the remainder
by the help of the general theory of reciprocal surfaces here
after to be explained.
In the case of the parallel to a quadric, it appears from
what has been stated, that the reciprocal is of the fourth
degree, and having no cuspidal curve, but having a nodal conic.
The parallel itself is of the twelfth degree ; its cuspidal curve
is of the twentyfourth order, being the complete intersection
of a quartic with a sextic surface. The nodal curve is of the
twentysixth order, and includes five conies, one in each of
the principal planes, and two in the plane infinity, namely,
PEDALS. 155
the section of the quadric itself and the circle at infinity.
The remainder of the nodal curve consists of 16 right lines,
each meeting the circle at infinity.*
PEDALS.
514. The locus of the feet of perpendiculars let fall from
any fixed point on the tangent planes of a surface, is a derived
surface to which French mathematicians have given a dis
tinctive name, "podaire," which we shall translate as the
pedal of the given surface. From the pedal may, in like
manner, be derived a new surface, and from this another, &c.,
forming a series of second, third, &c., pedals. Again, the
envelope of planes drawn perpendicular to the radii vectores
of a surface, at their extremities, is a surface of which the
given surface is a pedal, and which we may call the first
negative pedal. The surface derived in like manner from this
is the second negative pedal, and so on. Pedal curves and
surfaces have been studied in particular by W. Roberts,
Liouville, vols. x. and xn., by Tortolini, and by Hirst,
Tortolini's Annali, vol. n. p. 95; see also the corresponding
theory for plane curves, Higher Plane Curves, Art. 121. We
shall here give some of their results, but must omit the
greater part of them which relate to problems concerning
rectification, quadrature, &c., and do not enter into the plan
of this treatise. If Q be the foot of the perpendicular from
on the tangent plane at any point P, it is easy to see that
the sphere described on the diameter OP touches the locus
of Q ; and consequently the normal at any point Q of the
pedal passes through the middle point of the corresponding
radius vector OP. It immediately follows hence, that the
perpendicular OR on the tangent plane at Q lies in the plane
POQ, and makes the angle QOR = POQ, so that the right
angled triangle QOR is similar to POQ; and if we call the
angle QOR, a, so that the first perpendicular OQ is connected
* The parallel to a curve in space might also have been discussed. This
is a tubular surface (Art. 446).
156 ANALYTIC GEOMETBT OF THEEE DIMENSIONS.
with the radius vector by the equation p = p cos a, then the
second perpendicular OR will be p cos 2 a, and so on.*
It is obvious that if we form the polar reciprocals of a
curve or surface A and of its pedal B, we shall have a curve
or surface a which will be the pedal of b ; hence, if we take
a surface S and its successive pedals S v S 2 , ...S n , the recipro
cals will be a series S', S'_ l9 S'_ 2 , ...'., those derived in the
latter case being negative pedals.
It is also obvious that the first pedal is the inverse of the
polar reciprocal of the given surface (that is to say, the sur
face derived from it by substituting in its equation, for the
radius vector, its reciprocal) ; and that the inverse of the
series S lt S 2 , ...S n will be the series S', S'_ lt ...S'_ n _i.
INVEESE SURFACES.
515. As we may not have the opportunity to return to
the general theory of inversion, we give in this place the
following statement (taken from Hirst, Tortolini, vol. n.
p. 165) of the principal properties of inverse surfaces (see
Higher Plane Curves, Arts. 122, 281).
(1) Three pairs of corresponding points on two inverse
surfaces lie on the same sphere (and two pairs of correspond
ing points on the same circle) which cuts orthogonally the
unit sphere whose centre is the origin.
(2) By the property of a quadrilateral inscribed in a circle
the line db joining any two points on one curve makes the
same angle with the radius vector Oa, that the line joining
the corresponding points a'b' makes with the radius vector
Ob'. In the limit then, if ab be the tangent at any point a,
the corresponding tangent on the inverse curve makes the
same angle with the radius vector.
* Thus the radius vector to the n th pedal is of length p cos" a, and makes
with the radius vector to the curve the angle no.. Using this definition of the
method of derivation, Roberts has considered fractional derived curves and
surfaces. Thus for n = J, the curve derived from the ellipse is Cassini's oval.
An analogous surface may be derived from the ellipsoid.
INVERSE SURFACES. 157
(3) In like manner for surfaces, two corresponding tangent
planes are equally inclined to the radius vector, the two cor
responding normals lying in the same plane with the radius
vector, and forming with it an isosceles triangle whose base
is the intercepted portion of the radius vector.
(4) It follows immediately from (2), that the angle which
two curves make with each other at any point is equal to
that which the inverse curves make at the corresponding
point.
(5) In like manner it follows from (3), that the angle
which two surfaces make with each other at any point is
equal to that which the inverse surfaces make at the corre
sponding point.
(6) The inverse of a line or plane is a circle or sphere
passing through the origin.
(7) Any circle may be considered as the intersection of a
plane, and a sphere A through the origin. Its inverse, there
fore, is another circle, which is a section of the cone whose
vertex is the origin, and which stands on the given circle.
(8) The centre of the second circle lies on the line joining
the origin to a, the vertex of the cone circumscribing the
sphere A along the given circle. For a is evidently the
centre of the sphere B which cuts A orthogonally. The plane,
therefore, which is the inverse of A cuts B' the inverse of B
orthogonally, that is to say, in a great circle, whose centre
is the same as the centre of B'. But the centres of B and of
B' lie in a right line through the origin.
(9) To a circle osculating any curve, evidently corresponds
a circle osculating the inverse curve.
(10) For inverse surfaces, the centres of curvature of two
corresponding normal sections lie in a right line with the
origin. To the normal section a at any point m corresponds
a curve a situated on a sphere A passing through the origin ;
and the osculating circle c of a is the inverse of c the oscu
latin circle of a. If now a be the normal section which
158 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
touches a at the point m', then, by Meunier's theorem, the
centre of c is the projection on its plane of the centre of ^
the osculating circle of a v But the normal m'c^ evidently
touches the sphere A at m' so that q is the vertex of the cone
circumscribed to A along c, and theorem (10) therefore
follows from theorem (8).
(11) To the two normal sections at m whose centres of
curvature occupy extreme positions on the normal at in, will
evidently correspond two sections enjoying the same pro
perty ; therefore to tJie two principal sections on one surface
correspond two principal sections on tlie other, and to a line of
curvature on one, a line of curvature on the other*
In the case where the surface has no special relation to the
plane or circle at infinity it is easy to see, as at HigJier Plane
Curves, p. 106, that the inverse of a surface is of the degree 2w,
and class 3m + 2a+n = m 3 +'2,m, that it passes m times through
the origin and m times through the circle at infinity ; and
hence that the degree and class of the first pedal are
and of the first negative pedal 3m + la + n and 2w.
[515a. If a system of curves satisfy the equation
Idx + mdy + ndz =
the inverse system (the origin being the centre of inversion
and the radius of inversion unity) will satisfy the equation
l l dx l + m l dy l + n l dz l =
* Hart's method of obtaining focal properties by inversion (explained
HigJier Plane Curves, Art. 281) is equally applicable to curves in space and to
surfaces. The inverse of any plane curve is a curve on the surface of a sphere,
and in particular the inverse of a plane conic is the intersection of a sphere
with a quadric cone. And as shown (Hig)ter Plane Curves, Art. 281) from
the focal property of the conic p + p' = const, is inferred a focal property of
the curve in space lp + mp' + np" = 0. So, in like manner, the inverse of a
bicircular quartic is a curve in space with similar focal properties. (See
Casey on Cyclides and SpheroQuartics, Phil. Trans., vol. 161 ; Darboux, Sur
une classe remarquabU de courbes et de surfaces algibriques, Paris, 1873.) A
surface which is its own inverse with regard to any point has been called an
anallagmatic surface.
INVERSE SUEFACKS. 159
where
l\ = l> ~ t t L
r 1
with similar values of m 1 and i^ where
r 2 = x* + y* + z 2 , P=lx+ my + n z ; also /f + m^ + n j a = 1 .
The torsion of a curve whose normal is I, m, n at all
points is (Art. 368a) given by
dx dy dz
_^_= I m n
T dl dm dn .
And the corresponding torsion ( J (of the curve whose
normals are l v m lt n^) for the inverse direction dx lt dy lt dz^ is
given by a like formula. By using these expressions we can
prove
(t *?
Now  l = = and may be described as the linear mag
ds r r
uification. Thus the new torsion with changed sign is equal
to the original torsion divided by the linear magnification.
In particular if U= constant represents a surface (or a
family of surfaces) the equation dU=Q gives an equation of
the form
Idx + mdy + ndz =
and the curves whose normals are I, m, n are geodesies. Thus
if a surface be inverted from any point the new geodesic tor
sion for any given direction is equal with changed sign to the
old geodesic torsion of the corresponding direction divided
by the linear magnification. Since geodesic torsion vanishes
for a line of curvature we see that (11) of Art. 515 is a special
case of this theorem.]
x 2 J 1 &
516. The first pedal of the ellipsoid 8 +fj+=l, being
CL \) C~
the inverse of the reciprocal ellipsoid, has for its equation
aV + b*y* + cz* = (* + if + * 2 ).
160 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
This surface is Fresnel's "Surface of Elasticity". The in
verse of a system of confocals cutting at right angles is evi
dently a system of surfaces of elasticity cutting at right angles ;
the lines of curvature therefore of the surface of elasticity are
determined as the intersection with it of two surfaces of the
same nature derived from concyclic quadrics.
The origin is evidently a nodal point on this surface, and
the imaginary circle in which any sphere cuts the plane at
infinity is a nodal line on the surface.
NEGATIVE PEDALS.
517. Cayley first obtained the equation of the first negative
pedal of a quadric, that is to say, of the envelope of planes
drawn perpendicular to the central radii at their extremities.
It is evident that if we describe a sphere passing through the
centre of the given quadric, and touching it at any point x'y'z,
then the point xyz on the derived surface which corresponds
to x'y'z' is the extremity of the diameter of this sphere, which
passes through the centre of the quadric. We thus easily
find the expressions
= *,' (
2
where t = x' 2 + y'* + z" 1 .
Solving these equations for x, y' t z and substituting their
values in the two equations
xx' + yy' + zz = x' 3 + y' 1 + z
Now the second of these equations is the differential, with
respect to t, of the first equation ; and the required surface
is therefore represented by the discriminant of that equation,
SURFACES DERIVED FROM QUADRICS. 161
which we can easily form, the equation being only of the fourth
degree. If we write this biquadratic
it will be found that A and B do not contain x, y, z, while
C, D, E contain them, each in the second degree. Now the
discriminant is of the sixth degree in the coefficients, and is
of the form A(f>+ B*ty; consequently it can contain x, y, z
only in the tenth degree. This therefore is the degree of the
surface required.
It appears, as in other similar cases, that the section by
one of the principal planes z consists of the discriminant of
a: 2 y z
7* 7 '
9 9
a  r, & T~a
a b 2
which is a curve of the sixth degree, and is the first negative
pedal of the corresponding principal section of the ellipsoid,
together with the conic, counted twice, obtained by writing
t = 2c 2 , in the last equation. This conic, which is a double
curve on the surface, touches the curve of the sixth degree in
four points. The double points on the principal planes evi
dently answer to points on the ellipsoid, for which
t = x" 1 + y' 2 + z' 2 = 2a 2 or 26 2 or 2c 2 .
There is a cuspidal conic at infinity, and, besides, a finite cus
pidal curve of the sixteenth degree.
VOL. II. 11
CHAPTER XV.
SURFACES OP THE THIRD DEGREE.
519. THE general theory of surfaces, explained in Chap. XL,
gives the following results, when applied to cubic surfaces.
The tangent cone whose vertex is any point, and which en
velopes such a surface, is, in general, of the sixth degree, having
six cuspidal edges and no ordinary double edge. It is con
sequently of the twelfth class, having twentyfour stationary,
and twentyseven double tangent planes. Since then through
any line twelve tangent planes can be drawn to the surface,
any line meets the reciprocal in twelve points ; and the recip
rocal is, in general, of the twelfth degree. Its equation can
be found as at Higher Plane Curves, Art. 91. The problem
is the same as that of finding the condition that the plane
ax + fty + <yz + Sic =
should touch the surface. Multiply the equation of the surface
by 8 s , and then eliminate 810 by the help of the equation of
the plane. The result is a homogeneous cubic in x, y, z, con
taining also a, j3, 7, 8 in the third degree. The discriminant
of this equation is of the twelfth degree in its coefficients,
and therefore of the thirtysixth in a, /8, 7, 8 ; but this consists
of the equation of the reciprocal surface multiplied by the
irrelevant factor S 24 . The form of the discriminant of a homo
geneous cubic function in x, y, z is 64S 3 + T 2 (Higher Plane
Curves, Art. 224). The same, then, will be the form of the
reciprocal of a surface of the third degree, S being of the fourth,
and T of the sixth degree in a, ft, 7, 8 ; (that is to say, S and
T are contravariants of the given equation of the above
degrees). It is easy to see that they are also of the same
degrees in the coefficients of the given equation.
162
SURFAC1 s OF THE THIRD ]>!:< ;; I
520. Surfaces tuay have either multiple points or multiple
lines. When a surface has a double line of the degree p,
then any plane meets the surface in a section having ^ double
points. There is, therefore, the same limit to the degree of
the double curve on a surface of the n ih degree that there is
to the number of double points on a curve of the n' h degree.
Since a curve of the third degree can have only one double
point, if^a surface of the third degree has a double line, that
line must be a right line* A cubic having a double line is
necessarily a ruled surface, for every plane passing through
this line meets the surface in the double line, reckoned twice,
and in another line ; but these other lines form a system of
generators resting on the double line as director. If we make
the double line the axis of z, the equation of the surface will
be of the form
(ax 3 + 3bx 2 y + Sexy* + dy s ) + z (ax 1 + 2b'xy + c'# 2 )
+ (a"x* + %b"xy + c"y' 2 ) = 0,
which we may write u s + zu z + v 2 = 0. At any point on the
double line there will be a pair of tangent planes z'u 2 + r 2 = 0.
But as z varies this denotes a system of planes in involution
(Conies, Art. 342). Hence the tangent planes at any point on
the double line are tioo conjugate planes of a system in involu
tion.
There are two values of z , real or imaginary, which will
make z'u 2 + v 2 & perfect square ; there are, therefore, two points
on the double line at which the tangent planes coincide ; and
any plane through either of them meets the surface in a section
having this point for a cusp. If the values of these squares
be X' 2 and F 2 , it is evident that ?/ 2 and v. 2 can each be expressed
in the form IX' 2 + mY 2 . If, then, we turn round the axes so
* If a surface have a double or other multiple line, the reciprocal formed
by the method of the last article would vanish identically ; because then every
plane meets the surface in a curve having a double point, and, therefore, the
plane ax + 0y + yz + Sw is to be considered as touching the surface, inde
pendently of any relation between o, j8, y, S. The reciprocal can be found in
this case by eliminating x, y, z, w between u = 0, a = tt p = ,, y = n s ,
8 = u,.
11*
164 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
as to have for coordinate planes the planes X, Y, that is to
say, the tangent planes at the cuspidal points, then every term
in the equation will be divisible by either x 2 or y*, and the
equation may be reduced to the form zx* = wy 2 .*
In this form it is evident that the surface is generated by
lines y = \x, z = \*io, intersecting the two directing lines xy,
zw ; and the generators join the points of a system on zw to
the pairs of corresponding points of a system in involution
on xy, homographic with the first system. Any plane through
zw meets the surface in a pair of right lines, and is to be re
garded as touching the surface in the two points where these
lines meet zw. Thus, then, as the line xy is a line, every
point of which is a double point, so the line zw is a line, every
plane through which is a double tangent plane. The reciprocal
of this surface, which is that considered in Art. 468, is of like
nature with itself.
The tangent cone whose vertex is any point, and which
envelopes the surface, consists of the plane joining the point
to the double line, reckoned twice, and a proper tangent cone
of the fourth degree. When the point is on the surface the
cone reduces to the second degree.
521. There is one case, to which my attention was called
by Prof. Cayley, in which the reduction to the form zx* = icy
is not possible. If u% and v. 2 , in the last article, have a common
factor, then choosing the plane represented by this for one
* It is here supposed that the planes X, T, the double planes of the
system in involution, are real. We can always, however, reduce to the form
w(x* j/ 2 ) + 2zxy, the upper sign corresponding to real, and the lower to
imaginary, double planes, for
(z  w){(x + y)* + (x  j/)*}+ 2(z + w)(x + y)(x  y) = 4(x*z  wy*).
In the latter case the double line is altogether " really " in the surface, every
plane meeting the surface is a section having the point where it meets the
line for a real node. In the former case this is only true for a limited portion
of the double line, sections which meet it elsewhere having the point of meet
ing for a conjugate point, the two cuspidal points marking these limits on
the double line. A right line, every point of which is a cusp, cannot exist on
a cubic unless when the surface is a cone.
SURFACES OF THE THIRD Dl.iiKI I . 165
of the coordinate planes, and ex + dy for another, we can
easily throw the equation of the surface into the form
y 3 + x (zx + wy) = 0.
The plane x touches the surface along the whole length of
the double line, and meets the surface in three coincident right
lines. The other tangent plane at any point coincides with
the tangent plane to the hyperboloid zx + wy. This case may
be considered as a limiting case of that considered in the last
article ; viz., when the double director xy coincides with the
single one wz. The following generation of the surface may
be given : Take a series of points on xy, and a homographic
series of planes through it, then the generator of the cubic
through any point on the line lies in the corresponding plane,
and may be completely determined by taking as director a
plane cubic having a double point where its plane meets the
double line, and such that one of the tangents at the double
point lies in the plane which corresponds to the double point
considered as a point in the double line.* [The reciprocal of
this surface is also of like nature with itself.]
522. The argument which proves that a proper cubic curve
cannot have more than one double point does not apply to
surfaces. In fact, the line joining two double points, since it
is to be regarded as meeting the surface in four points, must
lie altogether in the surface ; but this does not imply that the
surface breaks up into others of lower dimensions. The con
sideration of the tangent cone, however, supplies a limit to
the number of double points on the surface. We have seen
(Art. 279) that the tangent cone is of the sixth degree, and
has six cuspidal edges, and it is known that a curve of the sixth
degree having six cusps can have only four other double points.
Since, then, every double point on the surface adds a double
edge to the tangent cone, a cubical surface can at most have
four double points.
* The reader is referred to an interesting geometrical memoir on cubical
ruled surfaces by Cremona, Atte del Reale Institute Lombardo, vol. 11. p. 291.
166 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
It is necessary to distinguish the various kinds of node
which the surface may possess. (^4) At an ordinary node *
(Art. 283) the tangent plane is replaced by a quadric cone.
The line joining the node to any assumed point, is, as has
been said, a double edge of the tangent cone from the latter
point ; and since to the tangent cone from any point corre
sponds a plane section of the reciprocal surface, this double
edge evidently reduces by two the degree of the reciprocal, or
the class of the given surface. (B) The quadric cone may
degenerate into a pair of planes. Such a node may be called
a binode ; the planes the biplanes, and their intersection the
edge. In the case first considered, it is easy to see that the
tangent planes to any tangent cone along its double edge are
the planes drawn through this line to touch the nodal cone.
When, therefore, the nodal cone reduces to two planes, these
tangent planes coincide, and the line to the binode is a
cuspidal edge of the tangent cone. A binode, therefore,
ordinarily reduces the class of the surface by three. A cubic
cannot have more than three binodes, since a proper sextic
cone cannot have more than nine cuspidal edges. But there
may be special cases of binodes. (1) At an ordinary binode
B. 3 the edge does not lie on the surface ; but if it does,f the
binode is special B^ and reduces the class of the surface by
four. Thus, let xyz be the binode, x, y the biplanes, the
general equation of the surface will be of the form u 3 + xy = Q,
where U 3 = c z 3 + 3c l z 2 x + 3c. 2 z' 2 y + &c. The case where c =
is the special one under consideration. This kind of binode
may be considered as resulting from the union of two
conical nodes. (2) In the special case last considered, the
surface is touched along the edge by a plane cx + c. 2 y, which
commonly is distinct from one of the biplanes ; but it may
* Cayley calls the kind of node here considered a cnicnode, and it is
referred to accordingly as C 2 . [It is now usually called a conic node.]
f [For the general surface the distinction between B 3 and B 4 is that in the
former case the six lines of closest contact = u s = are distinct from the
edge, but in the latter two coincide with this edge. Thus JB 4 and all higher
binodes are of a specialised nature on the cubic.]
SURFACES OF THE THIRD DEGREE. 167
coincide with one of them, that is to say, we may have either
Cj or Co = 0. In this case, the binode I? 5 reduces the class of
the surface by five. Such a point may be considered as re
sulting from the union of a conical node and binode. (3)
Lastly, we may have x a factor in all the terms of w 3 except if,
and we have then a binode ,., which may be regarded as
resulting from the union of three conical nodes, and which
reduces the class of the surface by six. In this case the edge
is said to be oscular* (C] The two biplanes may coincide,
when we have what may be called a unode C7 6 , which reduces
the class of the surface by six ; the equation then being re
ducible to the form w 3 + # 2 = 0. The uniplane x meets the
surface in three right lines, which are commonly distinct ;
but either, two of these may coincide, or all three may
coincide, when we have special cases of unodes, U 7 , U s which
reduce the class of the surface by seven and eight respectively.
U 6 may be regarded as equivalent to three conical nodes, U 7
to two conical and a binode, U s to two binodes and a conical.
[522a. The reduction in class effected by these various
kinds of nodes can be seen by inquiring how many tangent
planes can be drawn through the line zw, distinct from the
plane z. Thus for the case B b the points of contact are the
intersections of the three surfaces
xyw + kz' 2 x + zu + v =
yw + kz' 2 + zu' + v' =
xw + zu" + v" =
where u and v are respectively a binary quadratic and cubic
in x and y, and the accents denote differentiation . Thus they
are the intersections of the cubic with the cubic cones
* In general, if a surface is touched along a right line by a plane, the right
line counts twice as part of the complete intersection of the surface by the
plane, the remaining intersection being of the degree n  2. The line may,
however, count three times, the remaining intersection being only of the
degree n  3. Cayley calls the line torsal in the first case, oscular in the
second. He calls it scrolar if the surface merely contain the right line, in
which case there is ordinarily a different tangent plane at each point of the
line.
168 ANALYTIC GEOMETRY OF THEEE DIMENSIONS.
z(u  xu') + v  xv
Jcz 2 x + z(u  yu") + v yv" = 0.
These cones have seven common edges other than xy which
is a double edge on the first, and hence the class of the cubic
surface is seven. For a general discussion applicable to any
surface see Schlafli (Phil. Trans., 1863).
5226. The equivalence of higher nodes to certain combina
tions of C.} and .B 3 can been seen by Segre's method (522c) or
in an elementary manner, thus
xy + z(z  &)(cj# + c 2 7/) + z(ax 2 + Zhxy + by 2 ) + u a = Q
represents a cubic having conic nodes at (0, 0, 0) and (0, 0, A).
Let k diminish to zero and we get a cubic with a B point.
Similarly if Cj be put zero we see that B b = C2 + B 3 . If c x
and a be both zero, there are binodes at both points and
coalescence gives rise to B e .
This result is apparently in conflict with the statement
B e = 3C 2 , but the difficulty is explained by considering the
tangent cone from any external point. On it we have two
cuspidal edges uniting into an oscnodal edge, but an oscnode
in the theory of plane curves has to be considered as equivalent
to three nodes, not two cusps (see Basset's Surfaces* p. 134),
if Pliicker's formulae are to apply. And in this sense B 6 = 3C 2
and not 2 3 . It will also be shown in Ex. 3 below that three
conic nodes may coalesce to form either B 6 or U 6 .
522c. Segre (Ann. di Mat. (2) xxv.) has developed a
powerful method of decomposing complex multiple points of
any order into simpler constituents. He shows that an sple
point at may be replaced, as regards the determination of
the intersections of the surface with any other surface or
curve through 0, by the following collection of fictitious
points :
(a) An sple point at ;
* A Treatise on the Geometry of Surfaces, by A. B. Basset (Cambridge,
1910).
SURFACES OF THE THIRD DEGREE. 109
(6) Points Oj, O., ... of order .s lt .<?.,, ... at points con
secutive to along certain singular generators of the nodal
cone at O (s r <^ s) ;
(c) Points O pl , O p2 , ... of order S M , * l>2 ... at points
consecutive to O in certain specified directions (s j>r < s,,) ;
and so on.
To determine in how many points a curve meets the sur
face at we have then only to count with their proper
multiplicity the number of these fictitious points that lie
on it.
The following very brief sketch of the proof of these state
ments is sufficient for the purpose of application to the cubic
surface ; the full proof depends on the principles of Birational
Transformation mentioned in the last chapter of this book
ERRATUM.
Page 169, line 14, for "mentioned in the last chapter of this book," read
described in Chapter xvn."
X y (i)
Points on the plane W correspond to the various elements
of direction round 0, and any curve through will meet <f>
in s + k points at if the point on W corresponding to the
direction of its tangent is a Arple point on $.
For instance, if u. has a nodal generator (7) which is also
a generator of u, + l , $ will have a conic node at the point on
W corresponding to 7, and any curve touching 7 at O will
meet <f> in s + 2 points. Should we in this way obtain on $ a
special point O p of any kind, its analysis is continued by
another transformation, but it is to be observed that singular
points arising from special relations of fl to the U functions
are not considered, for /2 is completely at our disposal and
170
ANALYTIC GEOMETRY OF THEEE DIMENSIONS.
such points may be prevented. Of course the conic Q itself,
a singular curve of order n  s, is irrelevant.
Applying now the foregoing theory to the cubic surface
it transforms into
XYfl + W{c Z 3 + ( Cl X+c,Y)Z* + U,Z + U 3 ] = 0.
If c be zero XYW is a conic node, and if in addition c x
vanishes this point is a binode.
This gives us B 4 =2C, and B b =C. 2
The further
condition that the edge of the binode may lie on the cone ob
tained by equating to zero the coefficient of Z, that is, that
the binode on <f may be B 4 , will be found to be that u. 2 con
tains y as a factor or that </> has a B 6 point. Hence
ft  a +, 4 =,.
Unodes may be analysed in a similar way.]
523. Distinguishing cubic surfaces according to the
Singularities described in the preceding articles, we can
enumerate twentythree possible forms of cubics, which are
exhibited in the following table. [The last column gives the
number of distinct lines on the surface.]
No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
Nodes.
2C 2
4
'<
3C,
2C
u,
? 4 + 2C 2
B + Co
Class.
[Lines.]
12
27
10
21
9
15
8
16
8
10
7
11
7
6
6
12
6
7
6
7
6
a
6
8
5
8
5
4
5
3
4
9
4
5
4
5
4
2
4
1
3
3
SURFACES OF THE THIRD DEGREE. 171
The number twentythree is completed by the two kinds
of ruled surfaces or scrolls described in Arts. 520, 521, each
of which is of the third class.*
Ex. 1. What is the degree of the reciprocal of xyz = w 3 ?
Ans. There are three biplanar points in the plane w, and the reciprocal is
a cubic.
Ex. 2. What is the reciprocal of + + " +  = 0?
x y z w
Ans. This represents a cubic having the vertices of the pyramid xyzw for
double points ; and the reciprocal must be of the fourth degree.
The equation of the tangent plane at any point x'y'z'w' can be thrown into
the form p + ^ + ^ + ^ = 0, whence it follows that the condition that
x y z vu
ax + 0y + yz + Siv =
should be a tangent plane is
(Za)i + (mfli + (ny)i + (p5)J = 0,
an equation which, cleared of radicals, is of the fourth degree.f Generally the
reciprocal of ax" + by" + cz" + dw" is of the form
Aci^t + Bfr^ 1 + Cy~~i + DS^~ l = 0.
(Higher Plane Curves, p. 73.)
The tangent cone to this surface, whose vertex is any point on the surface,
* The effect of the nodes C 3 , B 3 , U A on the class of the surface was pointed
out by me, Cambridge and Dublin Mathematical Journal, 1847, vol. n. p. 65 ;
and the twentyseven right lines on the surface were accounted for in each
case where we have any combination of these nodes, Cambridge ana Dublin
Mathematical Journal, 1849, vol. iv. p. 252. The special cases B v B^, B\, f7 7 , U t
were remarked by Schlafli, Phil. Trans., 1863, p. 201. See also Qayley's
Memoir on Cubic Surfaces, Phil. Trans., 1869, pp. 231326. [The standard
forms to which the equations of these various kinds of cubics can be reduced
and the equations of the lines on them are given in The Twentyseven Dines
upon the Cubic Surface, by A. Henderson (Cambridge, 1911). This work,
Blythe's Models of Cubic Surfaces (Cambridge, 1905) and Klein's Memoir
(Math. Ann. vi.) should be consulted to obtain an idea of the shapes of thes
surfaces.]
f W'riting x, y, z, w in place of la, mf), ny, pt respectively, the equation of
the reciprocal surface is
x/(*) + x'fo) + >/(*) + >/() = 0
which rationalised is
(x 8 + y 1 + z* + it) 3  2yz  2zx  2xy  2xw  2yw  20) 8  64xy*w; = 0,
the surface commonly known as Steiner's quartic. It has three double lines
meeting in a point; every tangent plane cuts it in two conies. [See Art.
554a.J
172 ANALYTIC GEOMETRY OP THREE DIMENSIONS.
being of the fourth degree, and having four double edges, must break up into
two cones of the second degree.
A cubic having four double points is also the envelope of
oa s + 6/3 2 + cy 2 + 2J07 + 2mya + 2na0,
where a, b, c, I, m, n represent planes ; and a : y, ft : y are two variable para
meters. It is obvious that the envelope is of the third degree ; and it is of
the fourth class ; since if we substitute the coordinates of two points we can
determine four planes of the system passing through the line joining these
points.
Generally the envelope of aa" + b/3" + &c. is of the degree 3 (nl) J and of
the class n 8 . The tangent cone from any point is of the degree Sn (n  1). It
has a cuspidal curve whose order is the same as the order of the condition that
U + \V may represent a plane curve having a cusp, U and V denoting plane
curves of the 71 th degree ; or, in other words, is equal to the number of curves
of the form U + \V + pW which can have a cusp. The surface has a nodal
curve whose order is the same as the number of curves of the form
U + \V + (t.W which can have two double points. For these numbers, see
Higher Algebra, Lesson xvm.
[Ex. 3. Show that three conic nodes coalesce into a B 6 or a U 6 point ac
cording as the ultimate directions of the lines joining them are coincident or
distinct.
The most general cubic with three conic nodes is
pw 3 + (ax + by + cz)w n  + (fyz + gzx + hxy)w + kxyz = 0.
Write in this for x, y, and z, ty + z, 2ty + z, and 2/ 2 x + 3ty + z respectively, so
that the three nodes are at the points o, t, '2t of the plane conic
a; : y : z : w : : 1 :  / : t : 0.
If the nodes be made to coincide by making t approach zero we simply get
three planes through zw unless some of the constants are made to become in
definitely great at the same time. We will get a surface with a B 6 point if
a = ^b = c = At*
and
/= _fc = fc = **.
For the equation becomes in the limit
pw s + 2Axw* + 1Fy z w  2Fxzw + kz* =
and yzw is a B 6 point.
If, however, we write y + z + tx for x in the general 3nodal form and
make t zero we get the case where the lines joining the nodes remain distinct.
Putting
a=  b =  c = At 1
we get
pup + AxwP + w(f + g + hyz + gz* + hy~) + li(ipz + yz*) =
on which yzw is a U 6 point.
Ex. 4. Cubic surfaces of the third class cannot have conic nodes. For
a cone of the third class cannot have a nodal edge. This is why combinations
like 2C 2 + J5 5 are impossible.
Ex. 5. 2 4 is inadmissible because a quartic cone cannot have two
tacnodal edges.
SURFACES OF THE THIRD DEGREE. 173
Ex. 6. A cubic scroll can be generated by a right line meeting two
conies with one common point and a line meeting both.
Ex. 7. Three tangent planes to a cubic scroll meet it in a circle, namely,
the real tangent planes, one through each of the real lines joining intersections
of the scroll with the imaginary circle at infinity.
Ex. 8. The polar quadric with respect to a cubic scroll of a point on a
cuspidal tangent plane is a cone, and conversely.]
CANONICAL FORM. THE HESSIAN.
524. The equation of a cubic having no multiple point
may be thrown into the form ax 3 + by 3 + cz 3 + dv 3 + ew 3 = 0,
where x, y, z, v, w represent planes, and where for simplicity
we suppose that the constants implicitly involved in x, y, &c.,
have been so chosen, that the identical relation connecting
the equations of any five planes (Art. 38) may be written in
the form x+y + z + v + w = Q. In fact, the general equation of
the third degree contains twenty terms, and therefore nineteen
independent constants, but the form just written contains
five terms and, therefore, four expressed independent con
stants, while, besides, the equation of each of the five planes
implicitly involves three constants. The form just written,
therefore, contains the same number of constants as the
general equation. This form given by Sylvester in 1851
(Cambridge and Dublin Mathematical Journal, vol. vi., p.
199) is very convenient for the investigation of the properties
of cubic surfaces in general.*
* It was observed (Higher Plane Curves, Art. 25) that two forms may ap
parently contain the same number of independent constants, and yet that one
may be less general than the other. Thus, when a form is found to contain
the same number of constants as the general equation, it is not absolutely de
monstrated that the general equation is reducible to this form ; and Clebsch
has noticed a remarkable exception in the case of curves of the fourth order
(see note, Art. 235). In the present case, though Mr. Sylvester gave his
theorem without further demonstration, he states that he was in possession of
a proof that the general equation could be reduced to the sum of five cubes,
and in but a single way. Such a proof has been published by Clebsch (Crelle,
vol. LIX. p. 193). See also Gordan, Math. Annalen, v. 341 ; and on the general
theory of cubic surfaces Cremona, Crelle, vol. 'LXVIII. ; Sturm, Synthetische
Untersiichungen iibzr Flticlien dritter Ordnung. Clebsch erroneously ascribes
the theorem in the text to Steiner, who gave it in the year 1856 (Crelle, vol.
LIII. p. 133) ; but this, as well as Steiner's other principal results, had been
known in this country a few years before.
174 ANALYTIC GEOMETRY OF THEEE DIMENSIONS.
525. If we write the equation of the first polar of any
point with regard to a surface of the n ih degree,
x'L + y'M + z'N + w'P = 0,
then, if it have a double point, that point will satisfy the
equations
ax + liy + gz + lw' = 0, hx' + by' +fz' + mw = 0,
gx +fy' + cz + nw = 0, lx + my' + nz + dw = 0,
where a, b, &c., denote second differential coefficients corre
sponding to these letters, as we have used them in the general
equation of the second degree. Now, if between the above
equations we eliminate x'y'z'w' ', we obtain the locus of all
points which are double points on first polars. This is of the
degree 4 (n  2), and is, in fact, the Hessian (Art. 285). If
we eliminate the xyzw which occur in a, b, &c., since the
four equations are each of the degree (n  2), the resulting
equation in x'y'z'w' will be of the degree 4 (n  2) 3 , and will
represent the locus of points whose first polars have double
points. Or, again, H is the locus of points whose polar
quadrics are cones, while the second surface, which (see
Higher Plane Curves, Art. 70) may be called the Steinerian,
is the locus of the vertices of such cones. In the case of
surfaces of the third degree, it is easy to see that the four
equations above written are symmetrical between xyzw and
x'y'z'w ; and, therefore, that the Hessian and Steinerian are
identical. Thus, then, if the polar quadric of any point A
with respect to a cubic be a cone whose vertex is B, the polar
quadric of B is a cone whose vertex is A. The points A and
B are said to be corresponding points on the Hessian (see
Higher Plane Curves, Art. 175, &c.).
526. The tangent plane to the Hessian of a cubic at A is
the polar plane of B with respect to the cubic. For if we
take any point A' consecutive to A and on the Hessian, then
since the first polars of A and A' are consecutive and both
cones, it appears (as at Higher Plane Curves, Art 178) that
their intersection passes indefinitely near B, the vertex of
either cone ; therefore the polar plane of B passes through
SURFACES OF THE THIRD DEGREE. 17.~>
AA' ; and, in like manner, it passes through every other point
consecutive to A. It is, therefore, the tangent plane at A.
And the polar plane of any point A on the Hessian of a sur
face of any degree is the tangent plane of the corresponding
point B on the Steinerian. In particular, the tangent planes
to U along the parabolic curve are tangent planes to the
Steinerian ; that is to say, in the case of a cubic the develop
able circumscribing a cubic along the parabolic curve also
circumscribes the Hessian. If any line meet the Hessian in
two corresponding points A, B, and in two other points C, D,
the tangent planes at A, B intersect along the line joining the
two points corresponding to C, D. [For if these be C', D', the
polar quadrics of all points along AB have ABC'D' as self
conjugate tetrahedron, and so BC'D' is polar plane of A with
respect to its polar quadric in particular.]
527. We shall also investigate the preceding theorems by
means of the canonical form. The polar quadric of any point
with regard to ax 3 + by 3 + cz 3 + dv 3 + ew' A is got by substi
tuting for 10 its value  (x + y + z + v), when we can pro
ceed according to the ordinary rules, the equation being then
expressed in terms of four variables. We thus 6nd for the
polar quadric ax'x* + by'y'* + cz'z* + dv'v* + ew'w 2 = Q. If we
differentiate this equation with respect to x, remembering that
dw= dx, we get ax'x = ew'w; and since the vertex of
the cone must satisfy the four differentials with respect to
x, y, z, v, we find that the coordinates x', y', z, v', w' of any
point A on the Hessian are connected with the coordinates
y> z > v, w of B, the vertex of the corresponding cone, by
the relations
ax'x = by'y = cz'z = dv'v = eic'w.
And since we are only concerned with mutual ratios of co
ordinates, we may take 1 for the common value of these quan
tities and write the coordinates of B, 1 7, =/, ;.
ax by cz dv
Since the coordinates of B must satisfy the identical relation
176 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
x + y + z + v + w = 0, we thus get the equation of the
Hessian
+ + + _
ax by cz dv ew
or
bcdeyzvw + cdeazvwx + deabvwxy + eabcwxyz + abcdxyzv = 0.
This form of the equation shows that the line vw lies al
together in the Hessian, and that the point xyz is a double
point on the Hessian ; and since the five planes x, y, z, v, w
give rise to ten combinations, whether taken by twos or by
threes, we have Sylvester's theorem that the five planes form
a pentahedron wJiose ten vertices are double points on the
Hessian and whose ten edges lie on the Hessian. The polar
quadric of the point xyz is dvv* + ew'w 2 , which resolves
itself into two planes intersecting along vw, any point on
which line may be regarded as the point B corresponding to
xyz ; thus, then, there are ten points whose polar quadric s
break up into pairs of planes ; these points are double points
on the Hessian, and the intersections of the corresponding
pairs of planes are lines on the Hessian. It is by proving
these theorems independently * that the resolution of the
given equation into the sum of five cubes can be completely
established.
The equation of the tangent plane at any point of the
Hessian may be written
X II Z V W n
i y i. __ i i _ _ o
f* * ~L ^5 '> * j ^ '> "j
ax  by* cz dv* ew 
which, if we substitute for x',  &c., becomes
CLJC
ax' 2 x + by'y + cz'*z + dv'*v 4 ew'*w = 0,
but this is the polar plane of the corresponding point with
regard to U.
* It appears from Higher Algebra, Lesson xvin., that a symmetric deter
minant oip rows and columns, each constituent of which is a function of the
n" 1 order in the variables, represents a surface of the np th degree having
J P (P 3  1) n * double points ; and thus that the Hessian of a surface of the
n" 1 degree always has 10 (  2) 3 double points.
SURFACES OF THE THIRD DEOKI ! . 177
[527a. Two recent papers prove the possibility and the
uniqueness of the analytical reduction to this form in a
comparatively simple manner. Baker* reduces the form
(see Art. 533) ace = bdfto 2x 3 r = 0, where Sx, = and Zh^ r = 0,
i i i
and Bennett f reduces this last form to the sum of five
cubes. The following geometrical proof given by way of
illustration of the theorems of the last few articles, assumes
the result from the Higher Algebra mentioned in the footnote
to Art. 527, namely, that the ten quadrics represented by the
first minors of the Hessian determinant have ten common
points. The polar quadrics of these common points are
therefore such that all the first minors of their discriminants
vanish, or in other words, they are a pair of planes. These
points are double points on the Hessian since their coordin
ates make 5, &c., vanish. Let A be one of these ten points
and let L be the line of intersection of the pair of planes form
ing its polar quadric. The polar quadrics of points on L are a
family of cones U kV with vertices at A, and three members
of this family break up into planes. Accordingly three of the
ten beforementioned points must lie on L and there are three
corresponding lines passing through A. Thus the ten points
lie by threes on lines which pass by threes through them, so
there are ten lines like L. Now let the three lines through
A be denoted by L lt L. 2 , L 3 , and let B lt C l ; B 2 , C 2 ; J3 3 , C, be
respectively the other pair of points on each. Let X, Y, Z
be the remaining three points. Consider the other two lines
through B^ ; one at least of the four points B. 2 , C. 2 , B 3 , C 3 must
lie on one of them, for there are not enough other points avail
able ; let it be B 2 and let Z be the third point on the line B 1 B. 2 .
Consider the section of the Hessian by the plane of L l and L a .
It consists of the three lines L lf L 2 , B^B^Z and of another.
But this other line must pass through C lt Cj, and Z since these
three points are double points on the curve of section, hence
* Proc. Land. Math. Soc., vol. is. flbui., vol. x.
VOL. II. 12
178 ANALYTIC GEOMETRY OP THREE DIMENSIONS.
<?!, C 2 , Z, are collinear. In this way we see that the plane
of any two lines contains a sixth point in addition to the
five on the two lines. The points thus lie by sixes in planes
of which three pass through each point, so there are five
planes. If these be taken as planes of reference the equa
tions of the Hessian and the cubic must be of the form given
in Art. 527, and we see that the pentahedron of reference
is unique,]
[5276. K. Russell* has discussed the focal surface of the con
gruence of lines joining corresponding points on the Hessian.
This surface is the analogue of the Cayleyan in plane cubics.
The following is an outline of his main results :
Let P and P' be two corresponding points (x and J on
the Hessian and let U and U' be the other two points
(6 \
x 1 and x  2 ) where PP' meets the Hessian.
ax ax/
The line PP' will touch the focal surface at points Tand T'
respectively harmonically conjugate to U and U' with respect
9
to P and P', viz. the points x +  and x + , l and 2 being
(i,f QjX
the roots of
5 x _
* ax*e~
If V and V are the correspondents to U and U', the planes
PVV and P'VV touch the Hessian at P and P and the two
tangent lines at P and at P' to the intersection of these planes
with the Hessian give the directions in which we must pro
ceed from P or P' to obtain the two consecutive lines of the
congruence which meet PP'.
V and V are each on the tangent plane to the Hessian
at the other, and conversely two points U and U', which are
the correspondents to two points having this relation, connect
through two corresponding points P and P'.
* Proceedings of tJie Royal Irish Academy, Ser. 3, vol. v. p. 462.
SURFACES OF THK TIIIUM IM.illlEE. 179
(These results can be easily verified by forming the co
ordinates of the various points and remembering that by virtue
of the equations satisfied by x the equation for may be
written
v
* ax(ax i  &) = A ' B ' a C bemg arbltrai 7)
Now V being given six points V can be determined, since
six tangents can be drawn to a uninodal plane quartic from the
node ; so that U being fixed there are six points U' such that
the other two points of the Hessian lying on UU' are corre
spondents. These six lines UU' and the line UV make seven
bitangents through U so that the order of the congruence of
bitangents is seven. The class is three, for the lines joining the
points ayS and 78, &c., in Ex. 3, Art. 529, are three bitangents
lying in an arbitrary plane. The reader should consult the
paper cited in the footnote for further details.]
528. If we consider all the points of a fixed plane, their
polar planes envelope a surface, which (as at Hi(/ln > riun>
Curves, Art. 184) is also the locus of points whose polar
quadrics touch the given plane. The parameters in the
equation of the variable plane enter in the second degree ;
the problem is therefore that considered (Ex. 2, Art. 523) and
the envelope is a cubic surface having four double points. The
polar planes of the points of the section of the original cubic
by the fixed plane are the tangent planes at those points,
consequently this polar cubic of the given plane is inscribed
in the developable formed by the tangent planes to the cubic
along the section by the given plane (Higher Plane Curves,
Art. 185). The polar plane of any point A of the section of
the Hessian by the given plane touches the Hessian at the
corresponding point B (Art. 526), and is, therefore, a common
tangent plane of the Hessian and of the polar cubic now
under consideration. But the polar quadric of J5, being a
cone whose vertex is A, is to be regarded as touching the
given plane at A ; hence B is also the point of contact of the
polar plane of A with the polar cubic. We thus obtain a
12*
180 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
theorem of Steiner's that the polar cubic of any plane touches
the Hessian along a certain curve. This curve is the locus
of the points B corresponding to the points of the section of the
Hessian by the given plane. Now if points lie in any plane
Ix +my + nz+pv + qw, the corresponding points lie on the
surface of the fourth degree h r + H ^ + . Also the
ax by cz dv ew
intersection of this surface with the Hessian is of the six
teenth degree, and includes the ten right lines xy, zw, &c. The
remaining curve of the sixth degree is the curve along which
the polar cubic of the given plane touches the Hessian. The
four double points lie on this curve ; they are tJie points whose
polar quadrics are cones touching the given plane.
[The tangent cone to the polar cubic from any point A on
it, being a cone of the fourth degree with four double edges,
breaks up into the two quadric cones which are the envelopes
respectively of the polar planes of points on the two lines in
which the polar quadric of A meets the given plane. Now if
A is a double point these two cones coincide and become the
nodal cone ; accordingly the two lines just mentioned must
also coincide, which shows that the polar quadric of A becomes
a cone touching the given plane along a line.]
529. If on the line joining any two points x'y'z', x"y"z",
we take any point x + \x", &c., it is easy to see that its polar
plane is of the form P n + 2A,Pi 3 + A^P^, where P u , P 22 are the
polar planes of the two given points, and P 12 is the polar
plane of either point with regard to the polar quadric of the
other. The envelope of this plane, considering A variable, is
evidently a quadric cone whose vertex is the intersection of
the three planes. This cone is clearly a tangent cone to the
polar cubic of any plane through the given line, the vertex of
the cone being a point on that cubic. If the two assumed
points be corresponding points on the Hessian, P 12 vanishes
identically ; for the equation of the polar plane, with respect
to a cone, of its vertex vanishes identically. Hence the polar
plane of any point of the line joining two corresponding points
SURFACES OF THE THIRD DEGREE. I > I
on the Hessian passes through the intersection of the tangent
planes to the Hessian at these points.* [More simply, the
polar plane is the same as the polar plane with respect to its
polar quadric, and therefore passes through G'D' as at the end
of Art. 526.] In any assumed plane we can draw three lines
joining corresponding points on the Hessian ; for the curve
of the sixth degree considered in the last article meets the
assumed plane in three pairs of corresponding points. The
polar cubic then of the assumed plane will contain three right
lines ; as will otherwise appear from the theory of right lines
on cubics, which we shall now explain.
[Ex. 1. The polar quadric of a double point on the Hessian with respect to
the cubic is a pair of planes.
This is the converse of Art. 527o, and may be seen by expressing that the
points (0, 0, 1, 0) and (0, 0, 0, 1) are corresponding points. By properly
choosing the planes x and y the equation of cubic is
w 3 + w 3 (Ix + my) + hxyw + (xyz) 3 = 0.
The Hessian of this will have a double point at (0, 0, 0, 1) if a function,
which is the discriminant of the polar quadric of this point, vanishes, disregard
ing an irrelevant factor which could only be zero if (0, 0, 1, 0) were a binode.
Ex. 2. The polar cubic ol\x + ny + t>z + pw + <rv = with respect to
2ox 3 = is 2(A.  /i) 2 cdezwv = which may be also written
ox,
and from this form the results of Art. 528 are obvious. In particular the four
nodes are the correspondents of the points where the Hessian is met by
2\ 2 x = 2AX = 0, and if (x r . . . r r ) be these latter points the line of contact of
the cone whose vertex is at the point x l is the intersection of the given plane
with
r having the values 2, 3, or 4.
Ex. 9. The six points in which the sextic of Art. 528 meets the plane are
the intersections of the four lines of contact of the polar quadrics of the
nodes ; and opposite intersections are corresponding points on the Hessian.
Let the double points be A, B, C, D, and let the corresponding points be
A'B'C'D', and the lines of contact of the cones be a, /3, y, 8. A', B', C' t D'
lie on the line 2A.x = 2A. 2 x = 0, and ABCD is the self con jugate tetrahedron of
* Steiner says that there are one hundred lines such that the polar plane
of any point of one of them passes through a fixed line, but I believe that his
theorem ought to be amended as above.
182 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
the system of polar quadrics of points on this line. The polar plane of B with
respect to the polar quadric of A' is the plane ACD, which must be identical
with the polar plane of A' with respect to the polar quadric of B. But this
latter plane contains the line /8. Thus ;8 lies in the plane ACD and similarly
a in the plane BCD, so that CD meets the intersection of o and ft, say the
point aj8. But CD lies altogether in the polar cubic, so that a/8 is in the polar
cubic. By the argument just used, we can see that both the polar plane of A
and that of B with respect to the polar quadric of a/3 must both be the plane
2A&, and therefore this quadric is a cone ; and since it must pass through all
the intersections of the polar quadrics of C and D and also have its vertex on
2,\x (because a/3 is on the polar cubic), its vertex must be 76. So that o/3 and
78 are corresponding points on the Hessian.]
529&. It is known that in a plane cubic the polar line, with
respect to the Hessian, of any point on the curve, meets on
the curve the tangent at that point. Clebsch has given as
the corresponding theorem for surfaces, The polar plane, with
respect to the Hessian, of any point on the cubic, meets the tan
gent plane at that point, in the line which joins the three points
of inflexion of the section by the tangent plane. It will be
remembered that the section by a tangent plane is a cubic
having a double point, and therefore having only three points
of inflexion lying on a line. If w be this line, xy the double
point, the equation of such a curve may be written
x s + y z + 6xyw = 0.
Writing the equation of the surface (the tangent plane being #),
3? + y 3 + 6xyw + zu = 0, where u is a complete function of the
second degree u = dz 2 + Qlxw + 6myw + Snzw + &c., of which
we have only written the terms we shall actually require ; and
working out the equation of the Hessian, we find the terms
below the second degree in x, y, z to be d*io* + d(n  %lm)zw 3 .
The polar plane then of the Hessian with respect" to the point
xyz is 4:dw + (n  %lm}z, which passes through the intersection
of zw, as was to be proved.
If the tangent plane z = pass through one of the right lines
on the cubic, the section b}' it consists of the right line x and
a conic, and may be written x 3 + Qxyw = ; and, as before, the
polar plane of the point xyz with respect to the Hessian passes
through the line zw, a theorem which may be geometrically
SURFACES OF THE THIRD DEGREE. 183
stated as follows : When tJie section by the tangent plane is a
line and a conic, the polar plane, with respect to the Hessian,
of either point in which the line meets the conic, passes through
the tangent to the conic at the other point. If the tangent
plane passes through two right lines on the cubic, the section
reduces to xyw, and the polar plane still passes through zw,
that is to say, through the third line in which the plane meets
the cubic. If the point of contact is a cusp, it is proved in
like manner that the line through which the polar plane
passes is the line joining the cusp to the single point of in
flexion of the section.
The conclusions of this article may be applied with a slight
modification to surfaces of higher degree than the third : for
if we add to the equation of the surface with which we have
worked, terms of higher degree in xyz than the third, these
will not affect the terms in the equation of the Hessian which
are below the second degree in x, y, z. And the theorem is
that the polar plane, with respect to the Hessian, of any point
on a surface intersects the tangent plane at that point, in the
line joining the points of inflexion of the section, by the tan
gent plane, of the polar cubic of the same point.
THE RIGHT LINES ON A CUBIC.
530. We said (Art. 49) that a cubical surface neces
sarily contains right lines, and we now enquire how many in
general lie on the surface.* In the first place it is to be observed
that if a right line lie on the surface, every plane through it is a
double tangent plane because it meets the surface in a right
line and conic ; that is to say, in a section having two double
points. The planes then joining any point to the right lines
on the surface are double tangent planes to the surface, and
* The theory of right lines on a cubical surface was first studied in the
year 1849, in a correspondence between Prof. Cayley and me, the results of
which were published, Cambridge and Dublin Mathematical Journal, vol. rv.
pp. 118, 252. Prof. Cayloy first observed that a definite number of right lines
must lie on the surface ; the determination of that number as above, and the
discussions in Art. 533 were supplied by me.
184 ANALYTIC GEOMETKY OF THREE DIMENSIONS.
therefore also double tangent planes to the tangent cone whose
vertex is that point. But we have seen (Art. 519) that the
number of such double tangent planes is twenty 'seven.
This result may be otherwise established as follows : let
us suppose that a cubic contains one right line, and let us
examine in how many ways a plane can be drawn through
the right line, such that the conic in which it meets the
surface may break up into two right lines. Let the right
line be wz ; let the equation of the surface be wU = zV; let
us substitute 10 = 112, divide out by z, and then form the dis
criminant of the resulting quadric in x, y, z. Now in this
quadric it is seen without difficulty that the coefficients of
a; 2 , xy, and y 2 only contain //, in the first degree ; that those of
xz and yz contain /* in the second degree, and that of z z in
the third degree. It follows hence that the equation obtained
by equating the discriminant to nothing is of the fifth degree
in p ; and therefore that through any right line on a cubical
surface can be drawn five planes, each of which meets the sur
face in another pair of right lines ; and, consequently, every
right line on a cubic is intersected by ten others. Consider
now the section of the surface by one of the planes just re
ferred to. Every line on the surface must meet in some point
the section by this plane, and therefore must intersect some
one of the three lines in this plane. But each of these lines
is intersected by eight in addition to the lines in the plane ;
there are therefore twentyfour lines on the cubic besides the
three in the plane ; that is to say, twentyseven in all.
We shall hereafter show how to form the equation of a
surface of the ninth degree meeting the given cubic in those
lines.
531. Since the equation of a plane contains three inde
pendent constants, a plane may be made to fulfil any three
conditions, and therefore a finite number of planes can be
determined which shall touch a surface in three points. We
can now determine this number in the case of a cubical sur
face, We have seen that through each of the twentyseven
SURFACES OF THE THIRD DEGREE. 1 *.'
lines can be drawn five triple tangent planes : for every plane
intersecting in three right lines touches at the vertices of the
triangle formed by them, these being double points in the
section. The number 5 x 27 is to be divided by three, since
each of the planes contains three right lines ; there are there
fore in all fortyfive triple tangent planes.
532. Every plane through a right line on a cubic is
obviously a double tangent plane ; and the pairs of points of
contact form a system in involution. Let the axis of z lie on
the surface, and let the part of the equation which is of the
first degree in x and y be (az 1 + bz + c)x + (a'z 2 + b'z + c'}y ; then
the two points of contact of the plane y = fix are determined
by the equation
(az 1  + bz + c) + fjj(a'z 2 + b'z + c) = 0,
but this denotes a system in involution (Conies, Art. 342). It
follows hence, from the known properties of involution, that
two planes can be drawn through the line to touch the surface
in two coincident points ; that is to say, which cut it in a line
and a conic touching that line. The points of contact are
evidently the points where the right line meets the parabolic
curve on the surface. It was proved (Art. 287) that the right
line touches that curve. The two points then, where the line
touches the parabolic curve, together with the points of con
tact of any plane through it, form a harmonic system. Of
course the two points where the line touches the parabolic
curve may be imaginary.
533. The number of right lines may also be determined
thus. The form ace = bdf (where a, b, &c., represent planes)
is one which implicitly involves nineteen independent con
stants, and therefore is one into which the general equation of
a cubic may be thrown.* This surface obviously contains nine
lines (ab, cd, &c.). Any plane then a = fjib which meets the
surface in right lines meets it in the same lines in which it
* It will be found in one hundred and twenty ways. [See Henderson's
work cited in note to Art. 523.J
186 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
meets the hyperboloid pee = df. The two lines are therefore
generators of different species of that hyperboloid. One meets
the lines cd, ef, and the other the lines cf, de. And, since
/* has three values, [other than and oo ,] there are three lines
which meet ab, cd, ef. The same thing follows from the
consideration that the hyperboloid determined by these lines
must meet the surface in three more lines (Art. 345).
Now there are clearly six hyperboloids, ab, cd, ef; ab, cf,
de, &c., which determine eighteen lines in addition to the nine
with which we started, that is to say, as before, twentyseven
in all.
If we denote each of the eighteen lines by the three which
it meets, the twentyseven lines may be enumerated as fol
lows : there are the original nine ab, ad, af, cb, cd, cf, eb,
ed, ef; together with (ab.cd.ef)^ (ab.cd.ef)^ (ab.cd.ef) s , and
in. like manner three lines of each of the forms ab.cf.de,
ad.bc.ef, ad.be.cf, af.bc.de, af.be.cd. The five planes which
can be drawn through any of the lines ab are the planes
a and b, meeting the surface respectively in the pairs of lines
ad, af; be, be ; and the three planes which meet the surface
in (ab.cd.ef) 1 , (ab.cf.de\; (ab.cd.ef) 2 ,(ab.cf.de) 2 ; (ab.cd.ef) s ,
(ab.cf.de} z . The five planes which can be drawn through
any of the lines (ab.cd.ef)^ meet the surface in the pairs
of lines, ab, (ab.cf.de\ ; cd, (af.cd.be) 1 ; ef, (ad.bc.ef) l ; and in
(ad.be.cf)z, (af.bc.de\; (ad.be.cf) 3 , (af.bc.de) 2 .
534. Schlafli * has made a new arrangement of the lines
which leads to a simpler notation, and gives a clearer con
ception how they lie. Writing down the two systems of six
nonintersecting lines
ab, cd, ef, (ad.be.cf)^ (ad.be.cf}.^ (ad.be.cf) 3 ,
cf, be, ad, (ab.cd.ef)^ (ab.cd.ef) 2 , (ab.cd.ef) 3 ,
it is easy to see f that each line of one system does not inter
* Quarterly Journal of Mathematics, vol. n. p. 116.
f [(ad.be.cf) 1 and (ab.cd.cf) 1 are each met by (ad.bc.ef)^ and the planes
through the last and the former pair meet the cubic again in ad and ef,
showing that the first two lines do not form one of the pairs lying in planes
through the third.]
SURFACES OF THE THIRD DEGREE. 187
sect the line of the other system, which is written in the
same vertical line, but that it intersects the five other lines
of the second system. We may write then these two
systems
1 2> a 3> #4 tt 5> <*(,'
b lt b. 2 , b a , 6 4 , 6 5 , 6,5,
which is what Schlafli calls a "doublesix". It is easy to
see from the previous notation that the line which lies in the
plane of a lt 6 2 , is the same as that which lies in the plane of
2. &r Hence the fifteen other lines may be represented by
the notation c 12 , c 34 , &c., where c 12 lies in the plane of a lt 6 2 ,
and there are evidently fifteen combinations in pairs of the
six numbers 1, 2, &c. The five planes which can be drawn
through c 12 are the two which meet in the pairs of lines
a^b 2 , ajbi, and those which meet in c^c^ C 35 c 46 , c 36 c 45 . There
are evidently thirty planes which contain a line of each of
the systems a, b, c ; and fifteen planes which contain three
c lines. It will be found that out of the twentyseven lines
can be constructed thirtysix " doublesixes ". [They are the
original, 15 of the type
**! "1> 23* C 24 C 25 C M>
#2 ^2> C 13> C 14 C 15 C 16
and 20 of the type
!, a 2 , a 3 , c 66 , Cg4, c 45 ,
C 23> C 31> C 12 "4 ^5' ^G'J
535. We can now geometrically construct a system of
twentyseven lines which can belong to a cubical surface.
We may start by taking arbitrarily any line a l and five
others which intersect it, 6 2 , b 3 , 6 4 , 6 5 , b 6 . These determine
a cubical surface, for if we describe such a surface through
four of the points where a^ is met by the other lines and
through three more points on each of these lines, then the
cubic determined by these nineteen points contains all the
lines, since each line has four points common with the
surface. Now if we are given four nonintersecting lines, we
can in general draw two transversals which shall intersect
188 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
them all ; for the hyperboloid determined by any three meets
the fourth in two points through which the transversals
pass (see Art. 53d and note to Art. 455). Through any four
then of the lines b s , 6 4 , 6 5 , 6 6 we can draw in addition to the
line a l another transversal 2 , which must also lie on the
surface since it meets it in four points. In this manner we
construct the five new lines 0%, 3 , 4 , a 5 , a 6 . If we then take
another transversal meeting the four first of these lines, the
theory already explained shows that it will be a line 6 X which
will also meet the fifth. We have thus constructed a " double
six ". We can then immediately construct the remaining
lines by taking the plane of any pair aj) 2 , which will be met
by the lines b lt <% in points which lie on the line c 12 .
536. Schlafli has made an analysis of the different
species of cubics according to the reality of the twentyseven
lines. He finds thus five species : A. all the lines and planes
real ; B. fifteen lines and fifteen planes real ; C. seven lines
and five planes real ; that is to say, there is one right line
through which five real planes can be drawn, only three of
which contain real triangles ; D. three lines and thirteen planes
real : namely, there is one real triangle through every side of
which pass four other real planes ; and, E, three lines and
seven planes real.
I have also given (Cambridge and Dublin Mathematical
Journal, vol. iv. p. 256) an enumeration of the modifications
of the theory when the surface has one or more double points.
It may be stated generally, that the cubic has always twenty
seven right lines and fortyfive triple tangent planes, if we
count a line or plane through a double point as two, through
two double points as four, and a plane through three such
points as eight. Thus if the surface has one double point,
there are six lines passing through that point, and fifteen
other lines, one in the plane of each pair. There are fifteen
treble tangent planes not passing through the double point.
Thus 2 x 6 + 15 = 27 ; 2 x 15 + 15 = 45.
Again, if the surface have four double points, the lines are
SURFACES OF THE THIRD DEGREE. 189
the six edges of the pyramid formed by the four points (6 x 4),
together with three others lying in the same plane, each of
which meets two opposite edges of the pyramid. The planes
are the plane of these three lines 1, six planes each through
one of these lines and through an edge (6 x 2), together with
the four faces of the pyramid (4x8).
The reader will find the other cases discussed in the paper
just referred to, and in a later memoir by Schlafli in the
Philosophical Transactions for 1863.
[536#. Since the fourth edition of this book a considerable
amount of literature has been published dealing with the lines
on cubic surfaces. The principal results are presented in a
connected form in the work by A. Henderson, mentioned
in the note to Art. 523, a book which is also valuable for the
bibliography of the subject which it contains. The theorem,
generally known as Schlafli's theorem, that five lines a 2 , 3 ,
4 , a 5 , a 6 constructed as in Art. 535 have a common transversal
fc lf can of course be proved independently of the theory of cubic
surfaces, and a variety of such proofs have been published.
An elementary geometrical proof recently published by H. F.
Baker * is contained in Exs. 13 below. R. Russell has com
municated the proof contained in Ex. 4 which connects the
theorem with Poncelet's theorem on coaxal circles ; to him
also is due the arrangement of the proof by line coordinates
contained in Ex. 5.
Ex. 1. If ten lines be related as regards intersections like the lines a,, a a , a 3 ,
a 4 , a 5 , 6,, b.,, b 3 , b 4 , b 6 , the tetrad of points a r b^, fl.,6,, a.Jb 4 , a 4 b.j are coplanar. This
follows from the fact that if two quadrics have two intersecting lines common
their remaining common points lie in a plane conic. Now the two hyper
boloids determined by o 1? a a , a 5 (or b 3 , b t , 6 6 ) and o 3 , o 4 , o s (or 6,, 6, b t ) have
a 5 and b 6 common, and the points mentioned are common to both. Similarly
o 1 6 3 , O 3 6 lt a%b 4 , a 4 fe 2 are coplanar.
Ex. 2. If all the lines of Ex. 1 but b l are given, b L is uniquely determined.
For 6 1 is constructed by joining the point where o 4 meets the plane of 0,63,
aj6 4 , a 4 & 3 , to the point where o s meets the plane of o,6j, a.J> 4 , a t b y
*Proc.R. S. (A), 84 (1911).
190 ANALYTIC GEOMETBY OP THREE DIMENSIONS.
Ex. 3. Show from Ex. 2 that b 1 of Art. 535 meets a 2 , o :! , o 4 , o 5 , a 6 . The
line 6j as derived in Ex. 2 from the nine lines d lt a a , a 3 , a 4 , & 2 , 6 3 , 6 4 , a 5 , a g is
clearly the same as that derived from the first seven and a 6 , 6 5 .
Ex. 4 If a skew hexagon can be drawn whose sides in order are alternately
generators of two quadrics U and V, any number of such hexagons can be
drawn.
The six planes of pairs of consecutive sides of the hexagon are common
tangent planes to U and V and therefore their traces in a principal plane are
tangent to the conic in which the common tangent developable intersects that
plane (Art. 216) and form a hexagon whose vertices are alternately on two
conies and whose sides touch a third conic touched by the common tangents
of the first two. Reciprocate Poncelet's theorem which states that if all the
sides but one of a polygon inscribed in a circle touch coaxal circles, the en
velope of the remaining side is another circle of the system ; we then see that
any number of such plane hexagons can be described and hence any number
of the skew hexagons. Schlafli's theorem at once follows.
Ex. 5. Two quadrics U and V which each have with a third quadric W
the relation
00'  4 AA' =
are related as in the preceding example (see Schur, Math. Ann., xvin , or
Bennett, London Math. Soc. (2) ix.).
Ex. 6. Prove Schlafli's theorem by the use of line coordinates.
Let (12345) denote the determinant of the fifth order whose constituents
arep a s/j'+ <M/3 + r^Uft + s a pp + /a2/3 + 1'a^ft and which is zero when the five
lines have a common transversal. (Note to Art. 455.)
Let^a, &c., be the coordinates of a a , and^'a, &c., those of b a .
It is required to prove that (23456) = is a consequence of the equation
(2'3'4'5'6') = and the fact that the lines a a , bp (a 4=0 an< * J8=t= 1) intersect.
Now generally (12345) = regarded as an equation connecting the co
ordinates of 5 expresses the fact that 5 meets one of the two common trans
versals of 1234 and therefore regarded as a quadratic function of p$, &c.,
(12345) is resolvable into two linear factors. Applying this to (23452') and
remembering that 32' = 42' = 52' = we get
84 45 53 22' 2 = A. In.' 2 7 6 7 and similarly
24 45 52 33' 2 = \ S 7 !' W&
"34 46"63^2' 2 = /* 2T 2*5'
24 46 62 33' 2 = M Wl' 3^5'
and so on. Substitute from these for the various constituents of (2'3'4'5'G') =
and we find (23456) k = where A; is a nonvanishing factor.]
[537. There is an intimate connection between the theory
of lines on a cubic surface and the bitangents of a plane
quartic which may be exhibited by taking a point on the
cubic, drawing the tangent cone from to the cubic, and con
sidering the plane quartic in which this cone is intersected
SURFACES OF THI. THIRD Dl:<;iii 191
by any plane, L, parallel to the tangent plane, P, to the cubic
at 0. Since the planes joining to the lines on the surface
are double tangent planes to the cone, they meet L in 27 of
the bitangents of the quartic. The remaining bitangent is
the line at infinity in L, its points of contact being given by
the directions of the two asymptotic lines at 0.
Any three coplanar lines on the cubic project into three
bitangents which, with that at infinity, form a set of four
bitangents whose points of contact lie on a conic. For if .///
is a line on the surface and is the point xyz the equation
of the surface may be written
the tangent cone is then V' 2  xUP = 0, which is touched twice
by the six plane pairs of the system k 2 xP + 2& U + F = 0, and
clearly these six pairs project into a group as explained in
Higher Plane Curves, Art. 354.
But one of these pairs is xP and the others are easily seen
to be the planes joining O to the lines on the cubic coplanar
with ju\ Denoting the bitangent at infinity by d and the
others by the symbols of the corresponding lines on the cubic,
the group of a l and d are the five pairs b r c lr ; that of b l and d
the five pairs a r c lr ; whence aj) r and b^ r belong to the same
group (that of dc lr ), and hence ajb r belongs to the group of
aj) v or in other words, every doublesix of lines on the cm'n'r
projects into twelve bitangents of a plane quartic belonging to
the same group.]
INVARIANTS AND COVARIANTS OP A CUBIC.
538. We shall in this section give an account of the
principal invariants, covariants, &c., that a cubic can have.
We only suppose the reader to have learned from the Lessons
on Higher Algebra, or elsewhere, some of the most elementary
properties of these functions. An invariant of the equation
of a surface is a function of the coefficients, whose vanishing
expresses some permanent property of the surface, as for
example that it has a nodal point. A covariant, as for
192 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
example the Hessian, denotes a surface having to the original
surface some relation which is independent of the choice of
axes. A contravariant is a relation between a, ft, y, 8, ex
pressing the condition that the plane ax + fty + yz + Bw shall
have some permanent relation to the given surface, as for
example that it shall touch the surface. The property of
which we shall make the most use in this section is that
proved (Lessons on Higher Algebra, Art. 139), viz. that if we
substitute in a contravariant for a, ft, &c., 5, r 1 , &c., and
then operate on either the original function or one of its
covariants, we shall get a new covariant, which will reduce
to an invariant if the variables have disappeared from the
result. In like manner, if we substitute in any covariant for
x, y, &c., j, 95, &c., and operate on a contravariant, we get
a new contravariant or invariant.
Now, in discussing these properties of a cubic we mean to
use Sylvester's canonical form, in which it is expressed by the
sum of five cubes. We have calculated for this form the
Hessian (Art. 527), and there would be no difficulty in calcu
lating other covariants for the same form. It remains to
show how to calculate contravariants in the same case. Let
us suppose that when a function U is expressed in terms of
four independent variables, we have got any contravariant in
, ft, y, 8 ; and let us examine what this becomes when the
function is expressed by five variables connected by a linear
relation. But obviously we can reduce the function of five
variables to one of four, by substituting for the fifth its value
in terms of the others, viz. w =  (x + y + z + v). To find then
the condition that the plane ax + fty + yz + Bv + eio may have
any assigned relation to the given surface, is the same problem
as to find that the plane (a  e)x +(/3 e)*/ + (7  e)z + (B e)v
may have the same relation to the surface, its equation being
expressed in terms of four variables ; so that the contravariant
in five letters is derived from that in four by substituting
a  e, ft  e, 76. B respectively for a, ft, 7, B. Every
SURFACES OP THE THIRD DEGREE. 193
contravariant in five letters is therefore a function of the
differences between a, ft, 7, 8, e. This method will be better
understood from the following example :
Ex. The equation of a quadric is given in the form
ax* + by" 1 + cz 2 + dv* + ew 2 = 0,
where x + y + z + v + w = Q; to find the condition that ax + fiy + yz + Sv + tto
may touch the surface. If we reduce the equation of the quadric tea function
of four variables by substituting for w its value in terms of the others, the
coefficients of x 1 , y*, z*, v* are respectively o + e, b + e, c + e, d + e, while every
other coefficient becomes e. If now we substitute these values in the equation
of Art. 79, the condition that the plane ax + &y + yz + 6v may touch, becomes
a?(bcd + bee + cde + dbe) + fp(cda + cde + doe + ace) + y*(dab + doe + abe + bde)
+ 8 2 (a6c + abe + bce + cae)  2e(adfty + bdya + cdaft + bcaS + caps + abyt) = 0.
Lastly, if we write in the above for a, 0, &c., a  ,  f, &c., it becomes
bcd(af)* + cda(0  e)* + dab(y e) 2 + a&c(5  e) 2 + 6ce(a 8) s + ca3(/3  5) 2
+ abe(y 5) 2 + ade(/3  y)* + bde(a  y)*+cde(a  ft 2 =0,
a contravariant which may be briefly written 2cde(a  /3) 2 = 0.
539. We have referred to the theorem that when a con
travariant in four letters is given, we may substitute for
a, ft, 7, 8 differential symbols with respect to x, y, z, w ; and
that then by operating with the function so obtained on any
covariant we get a new covariant. Suppose now that we
operate on a function expressed in terms of five letters
x, y, z, v, w. Since x appears in this function both explicitly
and also where it is introduced in w, the differential with
d dw d . , , , , , ,
respect to x is 3 + 3 3, or, in virtue of the relation con
tic dx dw
necting w with the other variables, , r . Hence, a contra
variant in four letters is turned into an operating symbol in
five by substituting for
But we have seen in the last article that the contravariant
in five letters has been obtained from one in four, by writing
for a, a  e, &c. It follows then immediately that if in any
contravariant in five letters ice substitute for a, ft, 7, 8, e,
, =.  3 , =, 5, we obtain an operating symbol, with
(Lx dy dz dv dw
VOL. n. 13
194 ANALYTIC GEOMETBY OF THREE DIMENSIONS.
which operating on the original function, or on any covariant,
we obtain a new covariant or invariant. The importance of
this is that when we have once found a contravariant of the
form in five letters we can obtain a new covariant without
the laborious process of recurring to the form in four letters
Ex. We have seen that ~Sde(a.  /3) 2 is a contravariant of the form
ox 2 + by 2 + C2 2 + dv* + ew\
If then we operate on the quadric with ~Sfde t^ = Y, the result, which only
differs by a numerical factor form
bcde + cdea + deab + eabc + abed,
is an invariant of the quadric. It is in fact its discriminant, and could have
been obtained from the expression, Art. 67, by writing, as in the last article,
a + e, b + e, c + e, d + e for a, b, c,.d, and putting all the other coefficients
equal to c.
540. In like manner it is proved that we may substitute
in any covariant function for x, y, z, v, w, differential symbols
with regard to a, ft, 7, B, e, and that operating with the func
tion so obtained on any contravariant we get a new contra
variant. In fact if we first reduce the function to one of four
variables, and then make the differential substitution, which
we have a right to do, we have substituted for
d d d d , / d d d d
' * Z '*' W ' da' dp$f dS' and  ( + 35 + ^ +
But since the contravariant in five letters was. obtained from
that in four by writing a  e for a, &c., it is evident that the
differentials of both with regard to a, ft, 7, S are the same,
while the differential of that in five letters with respect to e
is the negative sum of the differentials of that in four letters
with respect to a, ft, 7, B. But this establishes the theorem.
By this theorem and that in the last article we can, being
given any covariant and contravariant, generate another,
which again, combined with the former, gives rise to new ones
without limit.
541. The polar quadric of any point with regard to the
cubic ax 3 + by 3 + cz 3 + dv 3 + ew 3 is
ax'x 2 + by'y* + cz'z* + dv'v 1 + ew'w 2 = 0,
SURFACES OF THE THIRD DEGREE. 195
Now the Hessian is the discriminant of the polar quadric.
Its equation therefore, by Ex., Art. 539, is 2bcdeyzvw = Q, as
was already proved, Art. 527. Again, what we have called
(Art. 528) the polar cubic of a plane
ax + fty + yz + Bv + ew,
being the condition that this plane should touch the polar
quadric is (by Ex., Art. 538) Scdezvw (a) 2 = 0. This is
what is called a mixed concomitant, since it contains both
sets of variables x, y, &c., and a, ft, &c.
If now we substitute in this for a, ft, &c., =, &c.,
dx dy
and operate on the original cubic, we get the Hessian ; but
if we operate on the Hessian we get a covariant of the fifth
order in the variables, and the seventh in the coefficients, to
which we shall afterwards refer as $,
< = abcde2abx z y 2 z.
In order to apply the method indicated (Arts. 539, 540) it
is necessary to have a contravariant ; and for this purpose I
have calculated the contravariant a, which occurs in the
equation of the reciprocal surface, which, as we have already
seen, is of the form Q^a 3 = r 2 . The contravariant or expresses
the condition that any plane ax + fty + &c. should meet the
surface in a cubic for which Aronhold's invariant S vanishes.
It is of the fourth degree both in a, ft, &c., and in the coeffi
cients of the cubic. In the case of four variables the leading
term is a 4 multiplied by the S of the ternary cubic got by
making x = in the equation of the surface. The remaining
terms are calculated from this by means of the differential
equation (Lessons on Higher Algebra, Art. 150). The
form being found for four variables, that for five is calculated
from it as in Art. 538. I suppress the details of the calcula
tion, which, though tedious, present no difficulty. The re
sult is
a = Sabcd(a  e) (  e) (y  e) (S  e) . [1].
For facility of reference I mark the contravariants with
numbers between brackets, and the covariants by numbers
13*
196 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
between parentheses, the cubic itself and the Hessian being
numbered (1) and (2). We can now, as already explained,
from any given covariant and contravariant, generate a new
one, by substituting in that in which the variables are of
lowest dimensions, differential symbols for the variables, and
then operating on the other. The result is of the difference
of their degrees in the variables, and of the sum of their
degrees in the coefficients. If both are of equal dimensions,
it is indifferent with which we operate. The result in this
case is an invariant of the sum of their degrees in the co
efficients. The results of process are given in the next
article.
542. (a) Combining (1) and [1], we expect to find a con
travariant of the first degree in the variables, and the fifth in
the coefficients ; but this vanishes identically.
(6) (2) on [1] gives an invariant to which we shall refer as
invariant A,
A = 26W 2 e 2  ZabcdeZabc.
If A be expressed by the symbolical method explained
(Lessons on Higher Algebra, xiv., xix.), its expression is
(1235) (1246) (1347) (2348) (5678) 2 .
(c) Combining [1] with the square of (1) we get a covariant
quadric of the sixth order in the coefficients
abcde(ax 2 + by* + cz 2 + dv* + eiv*} . (3),
which expressed symbolically is (1234) (1235) (1456) (2456).
(d) (3) on [1] gives a contravariant quadric
a 2 6VW2(a  /9) 2 . . . {2].
(i) [2] on (1) gives a covariant plane of the eleventh order
in the coefficients
a 2 6 2 c 2 ^ 2 e 2 (ax + by + cz + dv + ew] . (4).
(/) (3) on [2] gives an invariant B,
a 3 b 3 c s d?e*(a + b + c + d + e).
(</) Combining with (3) the mixed concomitant (Art. 541)
we get a covariant cubic of the ninth order in the coefficients
abcde^cde (a + 6) zvw , , (5),
SURFACES OP THE THIRD DEGREE. 197
(k) Combining (5) and [1] we have a linear contravariant
of the thirteenth order in the coefficients
abcdeS(a  b) (a  )( + b)c' 2 d' 2 c  abcde(cd + de + ec}\ [3].
It seems unnecessary to give further details as to the steps
by which particular concomitants are found, and we may
therefore sum up the principal results.
543. It is easy to see that every invariant is a symmetric
function of the quantities a, b, c, d, e. If then we denote the
sum of these quantities, of their products in pairs, &c. f by
p, q, r, s, t, every invariant can be expressed in terms of these
five quantities, and therefore in terms of the five following
fundamental invariants, which are all obtained by continuing
the process exemplified in the last article,
A = s 2  rt, B = t?p,C = t*s, D = 1q,E = t s ;
whence also C' 2  A E = 4V.
We can, however, form skew invariants which cannot be
rationally expressed in terms of the five fundamental invari
ants, although their squares can be rationally expressed in
terms of these quantities. The simplest invariant of this
kind is got by expressing in terms of its coefficients the dis
criminant of the equation whose roots are a, b, c, d, e. This,
it will be found, gives in terms of the fundamental invariants
A, B, C, D, E an expression for t' M multiplied by the product
of the squares of the differences of all the quantities a, b, &c.
This invariant being a perfect square, its square root is an
invariant F of the onehundredth degree. Its expression in
terms of the fundamental invariants is given, Philosophical
Transactions, 1860, p. 233.
The discriminant of the cubic can easily be expressed in
terms of the fundamental invariants. It is obtained by elimi
nating the variables between the four differentials with respect
to x, y, z, v, that is to say,
ax 2 = by* = cz z = dv z = ew*.
Hence x", y 1 , &c., are proportional to bcde, cdea, &c. Sub
stituting then in the equation x + y + z + v + w = 0, we get the
discriminant
198 ANALYTIC GEOMETRY OP THREE DIMENSIONS.
J(bcde) + J(cded) + J(deab} + J(eabc) + J(abcd) = 0.
Clearing of radicals, the result, expressed in terms of the
principal invariants, is
(A' 2  64) 2 = 16384(D + 2AC).
544. The cubic has four fundamental covariant planes of
the orders 11, 19, 27, 43 in the coefficients, viz.
L = PSax, L' = PSbcdex, L" = t*2ax, L'" = t s Sa 3 x.
Every other covariant, including the cubic itself, can, in
general, be expressed in terms of these four, the coefficients
being invariants. The condition that these four planes should
meet in a point, is the invariant F of the onehundredth
degree.
There are linear contravariants, the simplest of which, of
the thirteenth degree, has been already given ; the next being
of the twentyfirst, t*S (a  6) (a  $) ; the next of the twenty
ninth, t 5 2cde(a b) (a yS), &c.
There are covariant quadrics of the sixth, fourteenth,
twentysecond, &c., orders; and contravariants of the tenth,
eighteenth, &c., the order increasing by eight.
There are covariant cubics of the ninth order fZcde(a + b}
zvw, and of the seventeenth, t?2a?x 3 , &c.
If we call the original cubic U, and this last covariant F,
since if we form a covariant or invariant of U+\V, the
coefficients of the several powers of X are evidently covariants
or invariants of the cubic : it follows that, given any covariant
or invariant of the cubic we are discussing, we can form from
it a new one of the degree sixteen higher in the coefficients,
by performing on it the operation
.IS/ 2 d 79^ 9 d JO ^ o d \
M a ^~ + b Ji + ^ :r + Pjj + e " j )
\ da do dc da dej
Of higher covariants we only think it necessary here to
mention one of the fifth order, and fifteenth in the coefficients
Pxyzvw, which gives the five fundamental planes ; and one
of the ninth order, & the locus of points whose polar planes
with respect to the Hessian touch their polar quadrics with
respect to U. Its equation is expressed by the determinant,
SURFACES OP THE THIRD DEGREE. 199
Art. 79, using a, ft, &c., to denote the first differential coeffici
ents of the Hessian with respect to the variables, and a, b,
&c., the second differential coefficients of the cubic.
The equation of a covariant, whose intersection with the
given cubic determines the twentyseven lines, is & = 4H$,
where $ has the meaning explained, Art. 541. I verified
this form, which was suggested to me by geometrical con
siderations, by examining the following form, to which the
equation of the cubic can be reduced, by taking for the planes
x and y the tangent planes at the two points where any
of the lines meet the parabolic curve, and two determinate
planes through these points for the planes, w, z,
z'ij + it)x + %xyz + %xyw + ax^y + by^x + cx z z + dy 2 w = 0.
The part of the Hessian then which does not contain either
x or y is z 2 w 2 ; the corresponding part of $ is  2(cz 5 + dw 5 ) ,
and of 9 is  Sw W(cz 5 + dw b }. The surface 6  4H4> has
therefore no part which does not contain either x or y, and
the line xy lies altogether on the surface, as in like manner
do the rest of the twentyseven lines.* Clebsch obtained the
same formula directly, by the symbolical method of calcula
tion, for which we refer to the Lessons on Higher Algebra.
* This section is abridged from a paper which I contributed to the Philoso
phical Transactions, 1860, p. 229. Shortly after the reading of my memjir,
and before its publication, there appeared two papers in Crelle's Journal, vol.
LVIII., by Clebsch, in which some of my results were anticipated ; in particular
the expression of all the invariants of a cubic in terms of five fundamental
invariants, and the expression given above for the surface passing through
the twentyseven lines. The method, however, which I pursued was different
from that of Clebsch, and the discussion of the covariants, as well as the
notice of the invariant F, I believe were new. Clebsch has expressed his
last four invariants as functions of the coefficients of the Hessian. Thus
the second is the invariant (1234) 4 of the Hessian, &c.
CHAPTER XVI.
SURFACES OF THE FOURTH DEGREE.
545. THE theory of the general quartic surface has hitherto
been little studied. [Eohn has investigated the shapes of
such surfaces as regards the number of closed ovals they
may possess ; Sisam the possibility of expressing their
equations as the sum of the squares of five quadrics, and
Schmidt the properties of their polar quadrics and the
surfaces corresponding to the polar cubic of a plane with
respect to a cubic (Art. 528).*] The quartic developable,
or torse, has been considered, Art. 367. Other forms of
quartics, to which much attention has been paid, are the
ruled surfaces or scrolls which have been discussed by
Chasles, Cayley, Schwarz, Cremona, [Segen and Williams] ; f
and quartics with a nodal conic which have been studied, in
their general form, by Kummer, Clebsch, Korndorfer, [Segre,
Zeuthen,] and others ; and in the case where the nodal curve
is the circle at infinity (under the names of cyclides and
anallagmatic surfaces) by Casey, Darboux, Moutard, [Lori a,
Bertrand], and others. + In 'fact, in the classification of sur
* Rohn, Math, Ann., xxix. ; Leipziger Berichte, LXIII. 1911 ; Sisam, Amer.
Math. Soc. Bull, (2), 14; Schmidt, Prize Dissertation, Breslau, 1908, " Uber
Zweite Polarflachen einer allgemein Flache 4 Ordnung ".
t Chasles, Comptes Rendus, 1861 ; Cayley, Phil Trans., 1864, or Collected
Papers, v. 201, and Phil. Trans., 1869, or Collected Papers, vi. 312 ; Cremona,
Mem. di Bologna, vm. 1868 ; Segen, Crelle, exit. ; Williams, Proc. Amer.
Acad., 36.
J Kummer, Berlin Monatsberichte, July, 1863 ; Crelle, LXIV. ; Clebsch,
Crelle, LXIX. ; Korndorfer, Math. Ann., i., n., HI. ; Zeuthen, Ann. diMat., xiv. ;
Segre, Math. Ann., xxiv. ; Casey and Darboux as cited in note to Art. 515.
See also list of memoirs given in Darboux's work. Loria, Mem. Ace. Turin,
xxxvi. 1884 ; Bertrand, Nouv. Ann. de M. (3), 9 (Dupin's Cyclide).
200
SURFACES OF THE FOURTH DEGREE. 201
faces according to their degree, the extent of the subject
increases so rapidly with the degree, that the theory for ex
ample of the particular kind of quartics last mentioned may
be regarded as coextensive with the entire theory of cubics.
[Quartics with a nodal right line have been studied by
Zimmerman, those with a triple point by Rohn, and those
with isolated conical points by Cayley and Rohn, while the
plane representation of the last class of surfaces and the
treatment of curves lying on them by aid of hyperelliptic
functions have been the subjects of papers by Remy and
others.*
The sixnodal quartic (Weddle's surface) has been recently
discussed by Baker, Bateman, and by Morley and Conner,
and the Steiaerian surfaces of several special quartics by Van
der Vries.f
Steiner's quartic (Art. 523) is treated by Kummer and
many others. J
The sixteennodal quartic, known as Kummer's, is the
subject of a recent book by Hudson, who has summarised
into a remarkably concise and readable volume much of the
immense literature that has grown up in connection with this
surface. The reader is referred to this work for references to
the original memoirs and for fuller information about the sur
face than is contained in the sections at the end of this
chapter.]
* Zimmerman, Prize Dissertation, Breslau, 1904 ; Rohn, Math. Ann., xxiv.
(on triplepoint quartics), and Math. Ann., xxix. (on nodal quartics) ; Cayley,
Three Memoirs on Quartics, in vol. vir. of collected works ; Remy, Comptts
Rendus, 143, 148, 149, and on same subject, Traynard, CM., 140 ; Gamier,
C.R., 149; Chillemi, Bend. Circ. Mat. Palermo, 29; Maroni, Lomb. Inst.
(2), 38, and Fano, ibid., 39.
t Baker, Proc. Lond. Math. Soc. (2), 1 ; Bateman, ibid. (2), 3 ; Morley and
Conner, Amer. J. of M., 31 ; Van der Vries, ibid., 32.
J Kummer and Schroter, Berl. Monatsb., 1863 ; Crelle, LXIV. ; Cremona,
Crelle, LXIII. ; Cayley, ibid., LXIV. ; Clebsch, ibid., LXVII. ; Gerbaldi, La Super
fide di Steiner, Turin, 1881 ; Vahlen, Act. Math., xix. ; Lacour, Nou. Ann. (3),
xvn. ; Cotty, ibid. (4), vm. ; Montesano, Nap. Rend. (3), v.
Rummer's Quartic Surface, by R. W. T. Hudson. Cambridge, 1905.
202 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
QUARTICS WITH SINGULAR LINES SCROLLS.
546. The highest singularity which a quartic can possess
is a triple line, which is necessarily a right line. Every such
surface is a scroll, for it evidently contains an infinity of
right lines, since every plane section through the triple line
consists of that line counted thrice and another line. The
equation may be written in the form u^ = zu 3 + wv 3 , where
u n u 3 , v 3 are functions of the fourth and third orders respec
tively in x and y, and xy denotes the triple line. The three
tangent planes at any point on the triple line are given by the
equation z'u 3 + w'v 3 = 0. Forming the discriminant of this
equation, we see that there are in general four points [known
as "pinch" points] on the triple line, at which two of its
tangent planes coincide. We may take z and w as planes
passing each through one of these points, and x and y as the
corresponding double tangent planes, when the equation
becomes
Further, by substituting for z, z + ax + fty, and for w,
w + <yx + 8y, we can evidently determine a, ft, y, 8, so as to
destroy the terms # 4 , x 3 y, y^x, ?/ 4 in u 4 ; and so, finally, reduce
the equation to the form
mx 2 y 2 = z(ax 3 + bxy) + w(cxy* + dy 3 ).
The planes z, w evidently touch the surface along the whole
lengths of the lines zy, wx, respectively ; and we see that the
surface has four torsal generators, see note, Art. 522.
The surface may be generated according to the method of
Art. 467, the directing curves being the triple line, and any
two plane sections of the surface ; that is to say, the directing
curves are two plane quartics, each with a triple point, and the
line joining the triple points, the quartics having common the
four points in which each is met by the intersection of their
planes. [If we put 1, 4, 4, 4, 3, 3 for m 1} m+, m 3 , a, ft, 7 in
the formulae of Art. 468, we get m^m^  a = 12. This number
includes the line considered nine times among the lines joining
any point on itself to the apparent intersections of the quartics,
SURFACES OP THE FOURTH DEO1M I 203
so that the relevant multiplicity is three as it should he.] But
the generation is more simple if we take each plane section
as one made by the plane of two generators which meet in the
triple line. This will be a conic in addition to these lines ; and
the scroll is generated by a line whose directing curves are
two nonintersecting conies, and a right line meeting both
conies.
The equation of a quartic with a triple line may also be
obtained by eliminating, between the equations of two planes,
a parameter entering into one in the first, into the other in
the third degree ; for instance,
\x + y = 0, \ 3 w + \ z v + \iv + z = ;
that is to say, the generating line is the intersection of one of
a series of planes through a fixed line with the corresponding
one of a series of osculating planes to a twisted cubic, or tan
gent planes to a quartic torse. The four points where the
torse meets the fixed line are the four pinch points already
considered.
547. Returning to the equation
mx 2 y* = z(ax 3 + bx' 2 y) + w(cxy 2 + dy' A )
there is an important distinction according as m does or
does not vanish ; or, in the form first given, according as
M 4 is or is not capable of being expressed in the form
(ax + fiy)u s + (yx + 8y)v s . When m vanishes (II) the surface
contains a right line zw which does not meet the triple line ;
otherwise (I) there is no such line. The existence of such a
line implies a triple line on the reciprocal surface and vice
versa. In fact, we have seen that every plane through the
triple line contains one generator ; to it will correspond in
the reciprocal surface a line through every point of which
passes one generator ; that is to say, which is a simple line
on the surface. Conversely, if a quartic scroll contain a
director right line, every plane through it meets the surface
in a right line and a cubic, and touches the surface in the
three points where these intersect. Every plane through the
right line therefore being a triple tangent plane, there will
204 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
correspond on the reciprocal surface a line every point of
which is a triple point. In the case, therefore, where m
vanishes the equation of the reciprocal is reducible to the
same general form as that of the original.
In the general case (I) we can infer as follows the nature of
the nodal curve in the reciprocal. At each point on the triple
line can be drawn three generators. Consider the section
made by the plane of any two ; this will consist of two right
lines and a conic through their intersection ; and the plane
will touch the surface at the two points where the lines are
met again by the conic. Hence, at each point of the triple
line three bitangent planes can be drawn to the scroll ; and
reciprocally every plane through the corresponding line meets
the nodal curve of the reciprocal surface in three points. We
infer then that this curve is a skew cubic, and we shall con
firm this result by actually forming the equation of the recipro
cal surface. It will be observed how the argument we have
used is modified when the scroll has a simple director line,
the three generators at any point of the triple line then lying
all in one plane. If we substitute y = \x in the equation
of the scroll, we see that any generator is given by the
equations
y = \x, m\*x = (a + b\*)z + (cX 2 + d\ 3 )u\
and joins the points
x = a + b\, y = \(a + &X), z = wX 2 , w = 0,
x = c + d\ y = X(c + <ZX)> 2 = 0, w = m.
The reciprocal line is therefore the intersection of
(x + \y) (a + b\) + m\~z = 0, (x + \y} (c + dX) + mw = 0,
and the equation of the reciprocal is got by eliminating X
between these equations. But if we consider the scroll gene
rated by the intersection of corresponding tangent planes to
two cones
X 2 z + \y + z = 0, \ z u + \v + w = 0,
this will be a quartic (x w  uzf = (yw  zv) (xv  yu) which has
a twisted cubic for a nodal line, since the three quadrics re
presented by the members of this equation have common a
twisted cubic, as is evident by writing their equations in the
SURFACES OF TIIE FOURTH DEGREE. 205
form  =  = . In the case actually under consideration,
x y z
the equation of the reciprocal is
{rrfzw + mczx + mbyw + (be  ad)xy} 2
= \mdzx + mczy + (be  ad)y*\ {mbxw + amyw +(bc ad)x\.
This equation would become illusory if m vanished ; and
we must in that case (II) revert to the original form of the
equations of a generator, which gives
y = \r, (a + b\)z + \ 2 (c + d\)w = 0.
The generator of the reciprocal scroll will be
\y + x = 0, X 2 (c + d\)z = (a + b\)ic,
and the reciprocal is obviously of like nature with the original.
[Quartics of class II may be generated by a line meeting a
skew cubic, one of its chords, and another line, the points
where the chord meets the cubic being two of the pinch
points.]
The two classes of scrolls we have examined each include
two subforms according as either b or c, or both, vanish. In
these cases the triple line has either one or two points at which
all three tangent planes coincide. According to the mode of
generation, noticed at the end of last article, the fixed line
touches the torse, and either one pair or two pairs of the pinch
points coincide.
548. Besides the two classes of quartic scrolls with a triple
line, already mentioned, we count the following :
III. u 3 and v 3 may have a common factor, which answers
to the case ad = bc in the equation already given : which is
then reducible to the form
mx*y~ = (ax + by) (zx 2 + wy 1 }.
In this case also, in the method of Art. 546, the fixed line
touches the torse. The generator of the scroll in one position
coincides with the triple line, ax + by being the corresponding
tangent plane which osculates along its whole length. Also
the equation of the reciprocal scroll being
(mzw f axz + bywf  ziv(ay + bx)~,
206 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
we see that it has as nodal lines the plane conic ay + bx,
mzw + axz + byw, and the right line zw which intersects that
conic. This class contains as subform the case where % + \v 3
includes a perfect cube. The equation may then be reduced
to the form
my* = x(zy? + wy 2 \
the reciprocal of which is
(xz  nwo' 2 ) 2 = y*zw.
IV. Again, u z and v 3 may have a pair of common factors
and the equation is reducible to the form
x*y" = (ax* + bxy + cy 2 ) (xz + yw),
an equation which is easily seen by the same method, as
before, to have a reciprocal of like form with itself.
V. Lastly, u 3 and v s may have common a square factor, the
equation then taking the form
x 2 y 2 = (ax + by) 2 (xz + yiri) ,
which is also its own reciprocal.* In this case two of the
three sheets, which meet in the triple line, unite into a single
cuspidal sheet. The case where u 3 and v 3 have three common
factors need not be considered, as the surface would then be
a cone. [Surfaces of classes III, IV, and V have respectively
three, two, and one distinct pinch points.]
549. We come now to quartic scrolls with only double
lines. If a quartic have a nonplane nodal line, it will ordin
arily be a scroll. For take any fixed point on the nodal line,
and there is only one condition to be fulfilled in order that
the line joining this to any variable point on the nodal line
may lie altogether in the surface, a condition which we can
ordinarily fulfil by means of the disposable parameter which
regulates the position of the variable point. There being thus
an infinite series of right lines, the surface is a scroll. But a
* The first four classes enumerated answer to Cayley's ninth, third, twelfth,
sixth, respectively ; the last might be regarded as a subform of that preceding,
but I have preferred to count it as a distinct class.
SURFACES OF THE FOURTH DEGREE. 207
case of exception occurs, when the surface has three nodal
right lines meeting in a point. Here the section by the plane
of any two consists of these lines, each counted twice, and
there is no intersecting line lying in the surface. This is
Steiner's quartic mentioned note Art. 523 [see Art. 554a].
We consider now the other cases of quartics with nodal
lines, commencing with those in which the line is of the third
order. The case where the nodal lines are three right lines,
no two of which are in the same plane, need not be considered,
since it is easy to see that then the quartic is nothing else than
the quadric, counted twice, generated by a line meeting these
three director lines.
Let us commence with the ease where the nodal line is a
twisted cubic (VI and VII). Such a cubic may be represented
by the three equations xz  y 2 = 0, xw yz = Q, yw  z = ;
the planes x and w being any two osculating planes of the
cubic. The coordinates of any point on it may be taken as
x : y : z : w = X 3 : X 2 : X : 1. If the three quantities xz  y 2 ,
xw  yz, yw  z 2 are called a, ft, 7 respectively, any quartic
which has the cubic for a nodal line will be represented by a
quadratic function of a, ft, 7, say
acC + bft' 2 + C7 2 + 2ffty + 2</ 7 a + Shaft = 0.
Now consider the line joining two points on the cubic X, /z ;
the coordinates of any point on it will be of the form X 3 + e?/t 3 ,
X 2 + 0fji 2 , \ + 0p, 1 + 0. If we substitute these values in a, ft, 7,
they become, after dividing by the common factor 0(\  /*) 2 ,
X/<t, X + /A, 1. Consequently the condition that the line should
lie on the surface is
aXV + 6(X + fji)' 2 + c + 2/(X + fjC) + 1g\p. + 2/iX^(X + /*) = 0.
Thus if either point be given, we have a quadratic to determine
the position of the other ; and we see that the surface is a
scroll, and that through each point of the nodal line can be
drawn two generators, each meeting the cubic twice. The
six coordinates (Art. 57a) of the line joining the points X, p.
are easily seen to be (omitting a common factor X  /*)
X + X/i + /jL 2 , X + /tt, 1, X/i,  X/i(X + fjC), X/A',
208 ANALYTIC GEOMETRY OF THBEE DIMENSIONS.
and as the condition just found is linear in these coordinates,
we may say that a quartic scroll is generated by a line meeting
a twisted cubic twice and whose six coordinates are connected
by a linear relation, or, in other words, by the lines of a linear
complex which join two points on a twisted cubic.
In fact, if p, q, r, s, t, u be the six coordinates, we have
the relation
bp + Ifq + cr +(b + 2</)s  %ht + an, = 0.
We saw (Art. 57c) that a particular case of the linear relation
between the six coordinates of a line is the condition that it
shall intersect a fixed line ; and from what was there said, and
from what has now been stated, it follows immediately that
all the generators of the scroll will meet a fixed line, provided
the quantities multiplying^), q, &c., in the preceding equation
be themselves capable of being the six coordinates of a line ;
that is to say (VII), provided the condition be fulfilled,
When this condition is fulfilled, it appears, from Art. 547, that
the reciprocal of the scroll will have a triple line, the reciprocal
in fact belonging to the first class of scrolls with a triple line
there considered.
550. In order to find the equation of the reciprocal in the
general case VI, we observe that to the generator joining the
points, whose coordinates are X 3 , X 2 , X, 1 ; /i 3 , p 2 , //,, 1, will cor
respond on the reciprocal scroll the generator whose equations
are
x\ 3 + yX 2 + z\ + w = 0, Xfj? + yp? + zp + w = 0,
and the equation of the reciprocal is got by eliminating X, //,
between these equations and the relation already given
connecting X, /*. This elimination has been performed by
Cay ley ; the work is too long to be here given, but the result
is that the equation of the reciprocal scroll is of the same form
and with the same coefficients as the original ; so that the
scroll which has been defined as generated by a line in invo
lution twice meeting a skew cubic may also be defined as
generated by a line in involution lying in two osculating planes
SURFACES OF THE FOURTH DEGREE. 209
of a skew cubic. Thus then the fundamental division of scrolls
with a nodal cubic is into scrolls whose reciprocals are of like
form (VI), and scrolls whose reciprocals have a triple line (VII).
It is to be noted that the general form of the equation of the
reciprocal contains as a factor the quantity b* + %bg  4fh + ac,
the vanishing of which implies that the scroll belongs to the
latter class. The two classes of scrolls may be generated by
a line twice meeting a skew cubic, and also meeting, in the
one case, a conic twice meeting the cubic ; in the other, a
right line.*
551. If we put X = /i in the equation of Art. 549, we obtain
the points at which a generator will coincide with a tangent to
the cubic ; and this equation being of the fourth degree we see
that the intersection of the scroll with the torse 4ay  @ 2 = 0,
of which the cubic is the cuspidal edge, is made up of the
cubic [counting four times] together with four common
generators. There will be four pinchpoints on the cubic,
these points being obtained by arranging the condition already
obtained
tf (aX 2 + 2AX + b) + 2/*{ /iX 2 + (b + 0)X +/} + Z>X 2 + 2/X + c = 0,
and forming the discriminant
(aX 2 + 2/tX + b) (bX 2 + 2/X + c) = {hX 2 + (b + #)X +/} 2 .
[The plane of a generator and the tangent line to the cubic
is a tangent plane to the scroll, and this equation therefore
expresses that the two tangent planes at X coincide.] We
might have so chosen our planes of reference that one of
these four points should correspond to X = 0, the other ex
tremity of the generator through that point being p. = oo , and
in this case /= 0, 6 = 0; or the equation of the scroll may
always be transformed to the form
aa 2 + C7 2 + Zgya + 2haf3 = 0.
Or, again, by choosing the planes of reference so that two of
the four points may be X = 0, X = oc , the equation may be
changed to the form (aa + bP + cy)' 2 = 4w 2 7a.
* These classes, my sixth aud seventh, answer to Cayley's tenth and eighth.
VOL. II. 14
210 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
We have a subform of the scroll, if either a or c = in this
equation ; for in this case two of the four pinchpoints on
the nodal curve coincide, the generator at this point being
also a generator of the torse, and there is a common tangent
plane to scroll and torse along this line.
A third of the pinchpoints would unite if we had b = m ;
and if along with this condition we have both a and c = 0, the
surface is the torse /3 2  4ya = 0.
552. The next species of scrolls to be considered is when
the nodal curve consists of a conic and right line (VIII and IX).
The line necessarily meets the conic, which includes every
point of the section of the scroll by its plane. This scroll
may be generated by a line meeting two conies which have
common the points in which each is met by the intersection
of their planes, and also a line meeting one of the conies.
[A plane through the nodal line will meet the quartic in a
pair of right lines in addition, and for two positions of the
plane these lines will coincide, or the plane will be a trope.
Let y = be one of these planes and let y = 0, z = be the
coincident lines. Take w = as the plane of the conic and let
the planes of x and z cut this plane in the tangents to the
conic where y meets it.] Then it is easy to see that the most
general equation of the scroll can be reduced to the form
(xz  y 2 ) 2 + myw(xz  ?/ 2 ) + w\axy + by' 2 ) = 0,
where xz  y 2 , w is the nodal conic, xy the double line, and
yz is one position of the generator. Take then any point on
the conic, whose coordinates are X 2 , \, 1, ; and any point
z = fj.w on the line xy, and the line joining these points will lie
altogether on the surface if
\ 2 /x 2 + m\/j, + a\ + b = 0.
Thus two generators pass through any point of either nodal
line or nodal conic. The reciprocal is got by eliminating be
tween \ 2 x + \y + z = Q,/j + w = Q, and the preceding equation,
and is
(bxz  w? 2 ) 2  y(bxz  uP}(by + mw  az) + xz(by + mw  az) 2 = 0,
SURFACES OF THE FOURTH DEGlil I 211
which for b not equal is a scroll of the same kind having the
nodal conic, bxz  w*, by + mw  az, and the nodal line zw ; this
is VIII. [There are two pinchpoints on the line, namely,
where it meets (w2^)z + aw and two on the conic, the
points for which \= oo or a\ = m  46.] There is a subform
when ?ri 1 = 4b, that is to say, when the equation is of the
form
(xz  y + myw} = aw z xy.
[In this case there is but one pinchpoint on the line and
but one on the conic distinct from the point xyw.]
If b = we have case IX ; the reciprocal quartic has here
a triple line and is of the third class already considered.*
There is one pinchpoint on the line and two on the conic.
553. The next case (X) is where the conic degenerates
into a pair of lines, in other words, where there are two non
intersecting double lines, and a third cutting the other two.
This class is a particular case of that next to be considered,
viz. where the scroll is generated by a line meeting two non
intersecting right lines. If in any case two positions of the
generator can coincide we have a double generator, and the
scroll is that now under consideration. Thus, for example,
the scroll generated by a line meeting two lines not in the
same plane and also a conic is (Art. 467) of the fourth degree
and has the two right lines as double lines ; but two positions
of the generator coincide with the line joining the points
where the directing lines meet the plane of the conic, which
is accordingly a third double line on the scroll. The general
equation may be written as in the last article,
x 2 z 2 + mxzyw + w\axy + by 2 ) = ;
the line x = \y, z = fiw will be a generator if
X/U 2 + wiX/i + a\ + b = 0,
and the reciprocal is
y*w z + mxzyw + xz 2 (bx  ay) = 0,
* These two species, my eighth and ninth, are Cayley's seventh and
eleventh respectively.
14*
212 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
that is to say, is of the same nature as the original. This
is Cayley's second species. As before, the form
(xz  yw) z = axyic
may be regarded as special.
[In the general case there are two pinchpoints on each of
the lines xy and zw, and in the special case there is one pinch
point on each of these lines, while every point of xw is a pinch
point, or in other words this line has become a cuspidal line.
These species of scrolls are the subjects of the memoir by
Segen quoted in the footnote to Art. 545.]
554. Next let us take the general case (XI) (Cayley's first
species) where there are two nonintersecting double lines.
This scroll may be generated by a line meeting a plane bi
nodal quartic, and two lines, one through each node. When
the quartic has a third node we have the species of last
article. The most general equation is
x(az* + 2hz w + bur} + < 2xy(az + 2h'zw + Vw*)
+ y'^a'z* + lli'zw + b"w) = 0,
the reciprocal of which is easily shown to be of like form.
There are obviously four pinchpoints on each line, and sub
forms may be enumerated according to the coincidence of
two or more of these points.
But again, (XII) in the generation by the binodal quartic
just mentioned two of the nodes may coalesce in a tacnode ;
and we have then a scroll with two coincident double lines
(Cayley's fourth species), the general equation of which may
be written
u 4 + (yw  xz} u z + (yw  xz) 2 = 0,
where u^ u 2 are a binary quartic and quadratic in x and y ;
and the reciprocal is of like form. [This scroll is generated
by a line meeting the quartic and a line through the tacnode
and further determined by the condition of lying in a plane
which has a (1, 1) correspondence with the point on the director
line through which the generators lying in it pass. The
director line is a tacnodal line. If the quartic have an ad
ditional node we get the next case (XIII) in which there is a
SUEFACES OF THE FOURTH DEGREE. 213
double generator.] This will be the case if any factor y  ax
of 2 enters twice into u 4 . In that case it is obvious that the
line y  ax, aw  z is a double line on the surface. This is
Cayley's fifth species.
Every quartic scroll may be classed under one of the
species which we have enumerated.
[554a. Steiners Quartic. It was shown (Art. 523, Ex. 2)
that the reciprocal of a fournodal cubic surface is a quartic
with three nodal lines meeting in a point. Conversely, every
such quartic is the reciprocal of such a cubic, for by proper
choice of the tetrahedron of reference and the implied con
stants the equation of the quartic may be taken as
S = ifz* + 2 2 z 2 + x^if  2xyzw = 0.
The coordinates of any point on the surface are then
x:y:z:w:i 2/87 : 2ya : 2a : a 2 + 2 + 7 2
a, , 7 being parameters which may be considered as the
trilinear coordinates of a point on a plane whose points thus
have a (1, 1) correspondence with the points on S.
To the section of S by the plane \x + py + vz + pw will
correspond the conies of the family
C = p(a i + ff 2 + 7 2 ) + 2X#y + 2/*7a + 2i/a = 0,
to the tangent planes of S will correspond the linepairs of C,
and to coincident linepairs of C will correspond tropes, i.e.
planes touching S all along a conic.
Now the members of G which are perfect squares are
(a /9 7) 2 , so that S has four tropes, namely,
x+y+z+w=0 xyz+w=Q
yzx+w=Q zxy + w = 0,
and hence the reciprocal of S (which is clearly of the third
degree as is seen by forming the discriminant of C) has four
conic nodes.
If the tropes be taken for planes of reference the equation
of the surface takes the form given in the note to Ex. 2,
Art. 523.
There are two special cases of Steiner's quartic arising from
the coincidence of a pair or of all three nodal lines.
214 ANALYTIC GEOMETRY OP THREE DIMENSIONS.
The equation of the surface when the nodal lines are xy
counted twice and xz can be obtained from the general form
ayz 2 + bz' 2 x* + cx 2 y~ + dxyzw =
by putting x + \z for z in this and then conceiving \ to tend
to zero and a and c to infinity in such a way that a + c tends
to zero and a\ remains finite.
We thus obtain an equation of the form
x 4 + y*z 2 + x' 2 yw = 0,
which shows that the line xy is a tacnodal line, and two of
the tropes coincide with the plane y which touches all along
the tacnodal line.
If the three nodal lines coincide they become an oscnodal
line, that is, one such that any plane section of the quartic
has an oscnode when its plane meets the line. Taking y 2  zx
as the cone which osculates the surface all along this line (xy\
and also passes through the conic of contact of the solitary
trope w which has not coincided with the tangent plane along
xy, we see that the equation of the surface must be
(?/ 2  zxf = x s w,
a form which can be verified by the method of the examples
below.
Steiner's quartic is the only surface other than quadrics
and cubic scrolls which contains an infinity of conies passing
through any point on it. This was shown by Darboux and
generalised by Castelnuovo (Rend. Lined, 1894), who states
that outside ruled surfaces Steiner's quartic is the only surface
irhich is cut in nonproper curves by all planes of a doubly
infinite series.
Ex. 1.* The locus of a point whose coordinates are proportional to quadric
functions of three homogeneous parameters is a Steiner quartic.
From what proceeds it will be sufficient to show that in the family of conies
Xt7j + fiU, + rU 3 + pU 4 = 0,
where U r = (a^)^ T f^) r h r ) (a. /3 7)*, there are four members which reduce to a
pair of coincident lines.
This is so because a linear tangential system of conies 2 + *2' can be
determined such that the invariant e vanishes between any member of this
* These examples are chiefly from the papers cited Art. 545.
SURFACES OF THE FOURTH DEGREE. 215
system and the former. Now it is easy to see that this relation is satisfied for
a conic and the square of the equation of any of its tangents ; consequently
the four common tangents of the second system each counted twice must be
members of the first.
Ex. 2. If the system 2 + *2' of Ex. 1 has two coincident common tangents,
we get the second kind of Steinert quartic. Let 2 = TJ 2' (ij + f).
The conies of the first system may be taken to be then
A.a 2 + n& + yy 3 + pa(0  7) = 0,
whence w = *Jx (Jy  Jz), an equation which, when rationalised, is equiva
lent to that already given.
Ex. 3. If the system 2 + *2' has three common tangents, the same method
gives rise to the third kind of Steiner quartic. Here proceeding similarly
we find x : y : z : w : : & : y 3 : ay : a 2  fly whence (yw  z)* = xy 3 .
Ex. 4. The second and third kind of Steiner's surface are reciprocal to
cubic surfaces with the singularities 2C 2 + B t and C 2 + JB 6 respectively. This
is seen by forming the discriminant of the system \U l + p.U 3 + vU 3 + pU t .
Ex. 5. The locus of poles of a given plane with respect to conies on
Steiner's surface is another Steiner's surface. (Montcsano.)]
555. The only quartics with nodal lines which have not
been considered are those which have a nodal right line or
a nodal conic. In either case the surface contains a finite
number of right lines. For take an arbitrary point on the
nodal line, and an arbitrary point on any plane section of the
surface, and the line joining them will only meet the surface
in one other point. We can, by Joachimsthal's method,
obtain a simple equation determining the coordinates of that
point in terms of the coordinates of the extreme points. In
order that the line should lie altogether on the surface, both
members of this equation must vanish ; that is to say, two
conditions must be fulfilled. And since we have two para
meters at our disposal we can satisfy the two conditions in a
finite number of ways.* In the case where the quartic has
a nodal right line xy, substituting y = \x in the equation, and
* The same argument proves that if a surface of the n" 1 order have a
multiple line of the (n  2) th order of multiplicity, the surface will contain
right lines. If the multiple line be a right line it is easily proved, as in Art.
530, that the number of other right lines is 2 (3  4). If the multiple line be
not plane, or if the surface possess in addition any other multiple line, the
surface is generally a scroll. See a paper by R. Sturm, Math. Anrialen, t. iv,
(1871).
216 ANALYTIC GEOMETEY OF THKEE DIMENSIONS.
proceeding, as in Art. 530, we find that eight planes can be
drawn through the nodal line which meet the surface, each
in two other right lines, and thus that there are sixteen right
lines on the surface besides the nodal line.
[Ex. 1. The section of the surface in this case by any triple tangent plane
must be a pair of conies intersecting on the nodal line and at the points of
contact.
Ex. 2. These conies are intersected, the one by one, and the other by the
second of the two right lines lying in any of the eight planes of this Article.
(If both lines met the same conic the intersection of their plane and the
triple tangent plane would meet that conic in three points.)]
[555a. Birational Transformation into a Plane. The
general theory of this process is treated in the next chapter,
but it is easy to see from the facts just established that a
oneone correspondence can be set up between the points in
an arbitrary plane and the points on a quartic surface with a
nodal line.
When this is a right line (D) we have seen that there are
conies lying on the surface which intersect D.
Let P be an arbitrary point in a fixed plane. Through
P one line can be drawn meeting D and a certain specified
one of these conies, and this meets the surface again in one
point Q which is uniquely determined by P, and conversely,
Q determines P. When D is a conic we have seen that
there are right lines meeting it lying on the surface, and an
exactly equivalent process is possible, the only difference
being that here the conic is double and the line single.
If the nodal line is of higher order than unity, and is not
plane, the surface is a scroll. It will be found that similar
reasoning is valid in the case of all these except those
numbered XI and XII.
These two and quartics having isolated nodes do not
permit of the transformation in question.]
556. We do not attempt to give a complete account of the
different kinds of nodal lines on a quartic, the varieties being
very numerous, but merely indicate some of the cases which
SURFACES OF THE FOURTH DEGREE. 217
would need to be considered in a complete enumeration.*
The general equation of a quartic with a nodal right line may
be written
w 4 + zu z + tvv 3 + z^t 2 + zwu% + w 2 v% = 0,
where w 4 , M 3 , &c., are functions in x and y of the order indicated
by the suffixes. Now, attending merely to the varieties in the
last three terms, and numbering the general case (1), we have
the following additional cases ; (2) the three quantities t. 2 ,u 2 ,v z
may have a common factor. In this case one of the tangent
planes is the same along the double line, and one of the sixteen
lines on the surface coincides with that line ; (3) the last terms
may be divisible by a factor not containing x or y, and so be
reducible to the form (az + bw} (zu^ + wv z } ; (4) there may be
both a factor in x and y and also in z and w, the terms being
reducible to the form (ax + by)(az + b'w) (xz + yw) ; (5) we may
have 2 , u 2 , v 2 only differing by numerical factors, in which case
there are two fixed tangent planes along the double line, and
the case may be distinguished when the factor in z and w? is a
perfect square, that is to say, we have the two cases : (5a) the
terms of the second degree reducible to the form xyzw, and
(56) reducible to the form xyz 2 ; (6) the three terms may break
up into the factors (xz  yw)(zu^ + wv^ ; (7) the terms may
form a perfect square (xz 4 ywj*, in which case the line is
cuspidal, the two tangent planes at each point coinciding but
varying from point to point ; (8) the cuspidal tangent plane
may be the same for every point ; the three terms being
reducible to the form (8a), x 2 zw, or (86), xz 1 . This enumera
tion does not completely exhaust the varieties ; and we have
not taken into consideration the varieties resulting from taking
into account the preceding terms, as for instance, if a factor
xz + yw divide not only the last three terms but also the
terms zu 3 + wv 3 . From the theory of reciprocal surfaces after
wards to be given, it appears that a quartic with an ordinary
double line is of the twentieth class, and that when the line
is cuspidal the class reduces to the twelfth. It would need to
* On the subject of multiple right lines on a surface, the reader may con
sult a memoir by Zeuthen, Moth, Annalen, iv. (1871).
218 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
be examined whether the class might not have intermediate
values for special forms of the double line, and, again, what
forms of the double line intervene between the cuspidal and
the tacnodal for which we have seen that the surface is a
scroll, the class being the fourth. [In the general case there
are four pinchpoints ; in (2) these have coincided in pairs ;
in (3) there are two pinchpoints and a triple point, while in
(4) the pinchpoints have coincided ; finally in (5a) and (56)
we have two distinct or coincident triple points.
On cuspidal lines there are singularities called tacnodal
points, that is, points at which any section has a tacnode
instead of a cusp. For these and other details of special
nodal lines, Basset's Surfaces (Arts. 255264, and Chapter V
generally) should be consulted.]
557. A quartic with a nodal line may have also double
points. Two of the eight planes which meet the surface in
right lines will coincide with the plane joining the nodal line
to one of the nodal points. It is easy to write down the
equation of a quartic with a nodal line and four nodal points.
For let U, V, W represent three quadrics having a right line
common and consequently four common points, then any
quadratic function of U, V, W represents a quartic on which
the line and points are nodal.
There are in the case just mentioned four planes, each
passing through the nodal line and a nodal point, each such
plane meeting the surface in the nodal line twice, and in two
lines intersecting in the nodal point. There are at most four
planes containing a nodal point, but any such plane may meet
the surface in the nodal line twice, and in a twofold line having
upon it two nodal points ; the surface may thus have as many
as eight nodal points. The quartic with eight nodes and a
nodal line is Pliicker's Complex Surface (Art. 455), and its
equation is
x, a, h, g
y, h, b,f
1. <7/> c
SURF.U'KS OK THfi FOURTH DEGREE. 219
where a, b, h are of form (z, w}' 1 ; /, g of form (z, w?) 1 , and
c is constant. There are through the nodal line four planes,
the section by each of them being a twofold line, and on each
such twofold line there are two nodes. [For the plane
10 = Qz meets the surface in zw counted twice and in a conic
whose tangential equation is (ab'c'f'g'h')(\fjLvy 2 = 0, where for
instance a means the result of putting 0z = for w in a and
dividing by z 1 . The discriminant of this is of the fourth
order in 0, hence there are four planes for which this equa
tion represents a pointpair. These are double points, for
every line through them in the corresponding planes must
be considered a tangent line to the surface. The section by
these planes is given by the pointequation corresponding to
the pointpair and is accordingly the line joining them counted
twice.]
Suppose that the pairs of nodes are 1,2; 3, 4 ; 5, 6 ; 7,8;
so that 12, 34, 56, 78 each meet the nodal line. For a node 1,
the circumscribed sextic cone is P 2 t/ 4 = 0, where P is the plane
through the double line this should contain the lines 12, 13,
14, 15, 16, 17, 18 each twice ; but P contains the line 12, and
therefore P contains it twice ; hence, C7 4 should contain the
remaining six lines each twice, that is, it breaks up into four
planes A BCD which intersect in pairs in the six lines. Taking
in like manner P' 2 A'B'C'D' = Q for the sextic cone belonging
to the node 2, the eight nodes lie by fours in the eight
planes A, B, C, D, A', B', C', D', and through each of the
nodes there pass four of these planes ; it is easy to construct
geometrically such a system of eight points lying by fours in
eight planes ; the figure may be conceived of as a cube divested
of part of its symmetry.
A special case would arise if one or more of the nodal
points were to coincide with the nodal line. Thus the
equation
ax* + bx 3 y + cx 2 y 2 + dxy i (y  mw) + ey' 2 (y  mw)' 2 + (Ax 3 + Bx 2 y
+ Cxy^z
+ Difz(y  mw) + (A'x 3 + B'x 2 y)w + C'xyw(y  mio}
+ (ax 1 + ftxy + jif)z + (ax* + ft'xy)zw + a"x*w* = 0,
'220 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
represents a quartic having the line xy as nodal and the point
x, z, y  mw as a nodal point ; and if in the above we make
m = Q, the point will lie on xy. The kind of nodal line here
indicated appears to be different from any of those previously
considered.
[Ex. 1. The equation of Pliicker's Complex Surface can be written
>J(m  n)xX + N /(n  l)yY + V /(J  m)zZ = 0,
where X=u'y  fz + p'w =
is the plane joining (1, 0, 0, 0) to the nodal line (p', q', r', s', t', ') with similar
meanings for Y and Z.
It is easy to verify that this is the surface just discussed. The following
proof starts from the definition given in Art. 455 and thus establishes the
identity of the surface of the present article with the general Pliicker's Complex
Surface.
Let * = be the given quadratic complex and p', q', etc., the line co
ordinates of the given line. It will make no difference if instead of * we take
the complex
* + (Ap + Bq + Cr + Ds + Et + Fu) (s'p+t'q + r'u+p's + q't + r'u) = Q.
By a proper choice of the planes of reference, we can reduce the lefthand
member of this equation to the form Ips + mqr + nst, for we have at our dis
posal the six quantities A, B, etc., and the twelve implied in the planes of
reference to satisfy the eighteen equations obtained by equating to zero the
coefficients of the remaining terms. Writing then
* = Ips + mqr + nst = 0,
ps+ qr + st=0,
we have ps : qr: st : : mn: nl: I m.
The lines of * which lie in ax + by + cz + dw 0, satisfy the equation
as + W + ctt = and therefore satisfy
a(m  n) + b(n l) + c(lm) =Q
p q r
and hence lie in tangent planes to the cone
N /a(m  n)x + ^/b(n  l)y + *Jc(l  m)z = 0.
Flicker's surface is accordingly the locus of the conic in which this cone meets
the plane ax + by + cz + dw = 0. Solve for the ratios a : b : c between the
equation of the plane and any two of the relations expressing that it passes
through the given line (Art. 576) and substitute in the equation of the cone,
when the required equation of the surface is obtained. It may be noted that
this is one of four similar forms, another being
*/(l  n)yY + V(  l)*% + *J(m  n)wW = 0.
Ex. 2. The planes X, Y, Z, W are those for which the complex conic
reduces to a pair of points, aqd they touch the surface all along the joining
line. Reciprocally the four points when the nodal line meets the coordinate
planes are those for which the complex cone is a pair of planes. Every plane
SURFACES OP THE FOURTH DEGREE. 221
through one of these points lias it as a cusp on the curve in which the plane
meets the surface.
Ex. 3. The locus of poles of the nodal line with respect to the complex
conies lying in planes through it is a right line.]
QUARTICS WITH NODAL CONICS CYCLIDES.
[558. We come now to quartics with a nodal conic, in
cluding the case where the conic breaks up into a pair of
right lines. Segre (Math. Ann., xxiv.) first made a complete
enumeration of the different species of this class of quartic.
His method of classification depends on considering such
surfaces as the " projections " on to a threedimensional
space of the " surface " common to two quadratic ' varieties "
in fourdimensional space.
He obtains seventysix different species, but this number
includes several surfaces which have already been considered,
namely, the scrolls VIII, IX, X, XIII, and the subforin of X ;
the three kinds of Steiner's quartic ; and thirteen included in
the classification of Art. 556. Rejecting these there remain
fiftyfive thus divided : sixteen with a proper nodal conic ;
seven with a proper cuspidal conic ; eighteen with two nodal
lines not intersecting in a triple point, and fourteen when
they intersect in a triple point. The subforms arise from the
isolated nodes that may exist in addition and from one or both
of the nodal right lines becoming cuspidal or torsal. The
theory of the ordinary case of two intersecting nodal right
lines is included in that when the nodal line is a proper conic.]
559. In this case any arbitrary plane meets the surface in
a binodal quartic ; if the plane be a tangent plane the quartic
will be trinodal ; if the plane be doubly a tangent plane the
quartic will break up into two conies.* If the plane touch
three times, the section must have an additional double point ;
that is to say, one of the conies must break up into two right
lines ; and since a surface has in general a definite number
* It was from this point of view these surfaces were studied by Kummer,
viz. as quartics on which lie an infinity of conies.
222 ANALYTIC GEOMETRY OP THREE DIMENSIONS.
of triple tangent planes we see, as we have already inferred
from other considerations, that the surface contains a definite
number of right lines. This number is sixteen, as may be
shown by the method indicated, Art. 555, but we do not
delay on the details of the proof, as we shall have occasion
afterwards to show how the theorem was originally inferred
by Clebsch. Each of the sixteen lines is met by five others,
the relation between the lines being connected by Geiser and
Darboux with the twentyseven lines of a cubic surface as
follows : If on a cubic surface we disregard any one line and
the ten lines which meet it, then the sixteen remaining lines
are, in regard to their mutual intersections, related to each
other as the sixteen lines on the quartic.
In fact this is easily shown by the method of inversion in
the case where the nodal conic is the circle at infinity, a case
to which the general form can always be reduced by homo
graphic transformation. [The equation of such a quartic
can be written (see Art. 562)
ax* + If + cz 2 + llx + Zmij + 2nz + (j 2 + if + z 1 ) = 0.
The inverse of this is the circular cubic
1 + ax 1 + bif + cz" + 2(lx + my + nz) (x 2 + y~ + z*} = 0.
Now a right line passing through the circle at infinity inverts
into a right line also passing through the circle at infinity.
Any lines that exist on the quartic meet this circle and there
fore invert into lines on the cubic which meet the circle.]
Of the twentyseven right lines on this cubic, one lies in the
plane at infinity, ten meet that line, and the remaining
sixteen meet the circle at infinity ; and these last, and these
only, are inverted into right lines on the quartic.
The lines may be grouped in "double fours," such that in
a double four each line of the one four meets three lines of
the other four ; but no two lines of the same four meet each
other. There are in all twenty double fours, each line there
fore entering into ten of them. [This theorem follows from
the notation for the lines explained in the last chapter. If
the eleven lines we omit are C M and the ten which meet it, the
twenty " double fours " are
Sl'K FACES OP THE FOURTH DEG1M I 223
,

and six each of the following types
There are altogether 120 tetrads of lines such that no two of
a tetrad intersect. Forty of these are those just mentioned ; the
remaining eighty have the property that there is one and only
one line of the remaining twelve lines which fails to meet any
of the tetrad. For instance, a l a 2 a s c t& is not met by c 46 . It is
easy to see that each line meets five others. The plane of
two lines is a triple tangent plane and therefore there are
^(16 x 5) or forty such planes. In the examples which follow
some of these theorems are proved independently of the theory
of the cubic surface.
Ex. 1. Every line on a quartic having a nodal conic meets five other lines
on the surface. Let w = 0, Fyz + Gzx + Hxy = be the conic and ya the
line, so that the quartic is
fei/ 2 + C2 2 + 2fyz + Igzx + 2hxy + 2myw + 2nzw +
{Fyz + Gzx + Hxy I {Fyz + Gzx + Hxy + w(px + q y + rz)\ = 0.
Put y =/jiZ in this and write down the conditions that the result should permit
of the factor
Oiv + pFz + (G + /*H)x.
It will be found that 6 must satisfy two equations of the form
6 s + K^ + K.J0 + AI/UJ =
2 + Vjfl + Aj/>i =
where the suffixes denote the degree of the coefficients in p. The eliminant of
these two equations is of the 6th degree in /x, but contains the factor A, which
the work shows to be irrelevant.
Ex. 2. Infer the existence of sixteen lines from the fact that each line
meets five others.
Let 2, 3, 4, 5, G all meet 1. A unique quadric can be described through
1, 2, 3 and the nodal conic, and its intersection with the quartic consists of
the conic twice and 1, 2, 3, and therefore of one other line say (23) which
meets 2 and 3 since it cannot meet 1. We get ten Hues of the type (23),
making sixteen in all. Further (23) meets (45) because the quartic and the
two quadrics determining these lines must have one point of intersection not
absorbed by the common curve, the line 1 and the conic (see Art. 355). It will
now be found that of the set of sixteen lines each is met by five of the set.
Ex. 3.* The transformation x : y : z : w : : X* : XY : XZ : YW  Z* trans
forms a cubic surface though y = 0, xw + z* = into a quartic having X = 0,
See Geiser, Crelle, LXX., and Cremona, Rend. Inst. Lomb., 1871.
224 ANALYTIC GEOMETRY OP THREE DIMENSIONS.
YW  Z* = for a nodal conic. The sixteen lines of the cubic meeting the
conic become lines on the quartic ; the line of the cubic in the plane y becomes
the point XYZ ; the other ten lines become conies touching Fat XYZ. (The
cubic surface is assumed not to touch x at xyz.)]
[559a. It is shown as in Art. 562 that there are five
quadric cones (known as Kummer's cones after their dis
coverer) all of whose edges are bitangent to the surface.
Consider a point where one of the sixteen lines meets one of
these cones. The tangent plane to the quartic must contain
the line, but it is also the tangent plane to the cone. Hence
all the lines touch all the cones. We can now prove a very
interesting theorem due to Zeuthen : * The cone of contact to
the quartic from a point on the nodal conic is a nonsingular
quartic cone whose bitangent planes are (a) the sixteen planes
joining the vertex to the lines, (b) the ten tangent planes to
the cones from the vertex, (c) the nodal tangent planes to the
surface at the vertex.
In fact the sixteen planes are of course tangent to the
surface at two points on the corresponding line ; every tangent
plane of a Kummer cone is a bitangent plane ; and finally the
tangent plane of a surface is always a bitangent plane of the
tangent cone from its point of contact, namely, along the two
inflexional tangents, and so when there is a nodal line the
tangent planes to both sheets are bitangent planes.
Since the existence of twentyeight bitangent planes has
been demonstrated, it further follows that the cone is non
singular.
If the vertex is taken at a point where one of the lines
L meets the nodal conic, L is a nodal edge of the cone, for
every plane through it touches the quartic twice. The cones
have now only sixteen proper bitangents, namely, those five
tangent planes to Kummer's cones which do not contain L,
the ten planes to the ten lines which do not meet L, and the
tangent plane to that sheet of the quartic which does not con
tain L. The tangent plane to the other sheet, and the planes
*Ann. di Mat., xiv. p. 34.
SURFACES OF THE FOURTH DEGREE. 225
joining L to the five lines which meet it are the six tangent
planes to the cone through the nodal line.]
560. In what follows, we suppose the surface to be a
cyclide, as the term is used by Casey and Darboux.that is to say,
having the circle at infinity as the nodal conic : and in order
to generalise the results, it is only necessary in the equations
of the nodal line w = 0, x* + y 1 + z* = 0, to suppose x, y, z, w
to be any four planes ; while in the special case w is at infinity,
and x, y, z are ordinary rectangular coordinates. The pro
perties of the cyclide may be studied in exactly the same
manner as the properties of bicircular quartics were treated
(Higher Plane Curves, Arts. 251, 272, &c.). Consider any
qnartic whose equation may be written (X, Y, Z, TF) 2 = 0,
where X, Y, Z, W represent quadrics, and we equate to zero a
complete quadratic function of these quantities. By a linear
transformation of these quantities we may reduce this equation
as the general equation of the second degree was reduced, and
so bring it to either of the forms aX* + bY* + cZ' 2 + dW* = Q,
or XY= ZW* only in the latter case the separate factors are
not necessarily real. From the latter form it is apparent
that there are on such a quartic at least two singly infinite
series of quadriquadric curves, and that through two curves
belonging one to each system can be drawn a quadric
\fjiX \Z  fji,W+ Y=0, touching the surface in the eight
points where these curves intersect. And, generally, the
quadric aX+ j3Y+<yZ + BIV will touch the quartic, provided
a, /?, 7, 8 satisfy the familiar relation of Art. 79. All quadrics
included in this form have a common Jacobian on which will
lie all possible vertics of cones involved in the system. Thus,
* It has been shown by Valentiner, Zeutlien Tidsskrift (4), in., that
the form of the equation of a quartic here considered is not of the greatest
generality, and in fact that any surface of the ?i th degree which contains the
complete curve of intersection of two surfaces must be a special surface
when n exceeds 3. The equation of a quartic which contains a quadriquadric
curve depends on only 33 independent constants. [See as well Sisam (l.c.
Art. 645) who shows also that a quadric function of five quadrics is sufficiently
general to represent any quartic.]
VOL. II. 15
226 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
through each of the quadriquadric curves just spoken of, can
be drawn four cones whose vertices lie on the Jacobian.
A special case is when the equation of the quartic can
be expressed in terms of three quadrics only (X, Y, ZJ = 0.
This cannot happen unless the quartic have double points, since
all points common to the three quadrics X, Y, Z are double
points on the quartic. In this case the equation can be brought
by linear transformation to either of the forms
aX* + b Y 2 + cZ* = 0, or XZ = Y 1 .
Such a quartic is evidently the locus of the system of curves
7= XX, Z = \Y, and the quadric X 2 X2\7+Z touches the
quartic along the whole length of this curve. The generators
of any quadric of this system are bitangents to the quartic.
561. To apply this to the cyclide, it is easy to see that
if X, Y, Z, W be four spheres, the equation (X, Y, Z, JF) 2 =
is general enough to represent any cyclide. Since the Jacobian
of four spheres is the sphere which cuts them at right angles,
all spheres of the system aX + @Y + yZ + &W cut a fixed
sphere orthogonally. Further, the coordinates of the centre
of any such sphere are easily seen to be proportional to linear
functions of a, ft, y, 8 ; and, reciprocally, these quantities are
proportional to linear functions of these coordinates. Thus
the condition of contact (Art. 79) being of the second degree in
a, ft, 7, B, establishes a relation of the second degree in these
coordinates. Hence we have a mode of generation for cyclides
corresponding to that given for bicircular quartics (Higher
Plane Curves, Art. 273), viz. a cyclide is the envelope of a
sphere whose centre moves on a fixed quadric F, and which
cuts a fixed sphere J" orthogonally. From this mode of
generation several consequences immediately follow. First,
the cyclide is its own inverse with regard to the sphere J ; for
any sphere which cuts J orthogonally is its own inverse in
respect to it, so that the generating sphere not being changed
by inversion, neither is the envelope. Thus, the cyclide is an
anallagmatic surface, see note, Art. 515. Secondly, the inter
section of F and J is a focal curve of the cyclide ; for the
SURFACES OF THE FOURTH DEGREE. 227
Jacobian J is the locus of all pointspheres belonging to the
system aX+ ftY+yZ + SW; and therefore, from the mode of
generation, every point of the curve FJ is a pointsphere
having double contact with the quartic ; that is to say, is a
focus. Thirdly, in the case where the centre of the enveloped
sphere is at infinity on F, the sphere reduces to a plane
through the centre of J (or more strictly to that plane, to
gether with the plane infinity). It follows then, that if a cone
be drawn through the centre of J whose tangent planes are
perpendicular to the edges of the asymptotic cone of F, these
tangent planes are double tangent planes to the quartic, which
they meet therefore each in two circles, while the edges of
this cone are bitangent lines to the quartic.
[561a. The following is a direct elementary proof of the
main properties of cyclides, starting from their definition as
the envelope of a sphere whose centre moves on a quadric F
and which cuts a fixed sphere ./ orthogonally.
By the ordinary rules for finding envelopes the equation of the cyclide is
found to be
S=[z 2 + i/ 2 + z 2 + r 2  a 2  2  r>J 2  4[a 2 (z  a) 2 + 6 2 (j/  0) 2 + c*(z  7 ) 2 ] =
ilF=~ + + 2  1 and J=(x  ) 2 + (y  )8) 2 + (z  7 ) 2  r 2 .
Let us enquire if other values a', b', c', a.', $', y', r' could be given to the con
stants in S and yet leave its equation unaltered.
This requires
a' 2  a 2 = 6' 2  6 2 = c' 2  c 2 = \ say . . . . (j)
r' 2  a' 2  0' 2  7 ' 2 = r 2  a 2  2  7 2 . . . (ii)
a'(i 2 + A)  aa 2 = 0'(6 2 + A)  0& 2 = y'(c 2 + A)  yc 2 = . (Hi)
The last equation shows that in general four other ways of generating S
exist ; (i) shows that the four new quadrics will be confocal with F, and from the
last three equations it is easy to show that each of the four new spheres are
orthogonal to J and each other.
The following special cases may be noted :
1. J is a pointsphere and hence two roots of (iv) are zero, and the point lies
on the other three quadrics.
2. J touches F, and two roots of (iv) are equal by Art. 202.
3. J is a pointsphere on F, and three roots of (iv) are zero, and the point
lies on the other two quadrics.
15*
228 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
4. J and F have stationary contact, and three roots of (iv) are equal by
Art. 206.
The first two cases represent of course the same kind of cyclide, and so do
the last two, the only difference being the system from which we start. Tho
point / in case 1 is easily seen to be a conic node and in case 3 a binode.
Hence we see that cyclides with a node have only four systems of enveloping
spheres, and one of the J sphere* is a pointsphere at the node lying on the other
three F quadrics, and that a corresponding theorem is true for cyclides having
a binode obtained by changing "four" to " three " and " three " to " two ".]
562. We have thus far considered the equation of the
cyclide expressed in terms of four quadrics ; but it is even
more obvious that the equation can be expressed in terms of
three quadrics. In fact, the equation of a quadric having for
nodal line the intersection of the quadric U by the plane P,
may obviously be written U 2 = P 2 V. Or, again, if we write
down the following most general equation of a quartic, having
as a nodal line the intersection of # 2 + i/ 2 + .z 2 , and w,
(x 2 + y 2 + * 2 ) 2 + 2wM 1 (z 2 + y 2 + * 2 ) + tcX = ;
this can obviously at once be written in the above form as,
We can simplify this equation by transformation to parallel
axes through a new origin, so as to make the w x disappear,
and we may suppose the axes of coordinates to be parallel to
the axes of the quadric v 2 , so that u 2 does not contain the
terms yz, zx, xy. It appears then, from what has been said,
that the cyclide, the general equation being reduced to the
form
(x 2 + y* + * 2 ) 2 = ax 2 + by*
is the envelope of the quadric V + 2X(x 2 + y* + 2*) + X 2 = 0, every
quadric of this system touching the quartic at every point
where it meets it. The discriminant of this quadric equated
to zero gives
T^^+ =d + X 2
6 + 2X c + 2X
and this equation being a quintic in X, we see that there are
five values of X for which this quadric reduces to a cone, and
therefore five cones whose edges are bitangents to the quartic.
Taking this in connection with what was stated at the end
SURFACES OF THE FOURTH DEGREE. 229
of the last article, it may be inferred that there are five spheres
J, each of which combined with a corresponding quadric F
gives a mode of generating the cyclide. And this may be
shown directly by investigating the condition that the sphere
x 2 + y' z + '  Wj should have double contact with the cyclide,
or meet it in two circles. For, substituting in the equation
of the cyclide we get u^  V=Q, and if we subtract from this
2\(x 2 + y 2 + 2 2  Wj) and determine \ by the condition that the
difference shall represent two planes, we get the same quintic
as before for \; and we find also that the centre of the
sphere must satisfy the equation
from which we see that there are five series of double tangent
spheres ; that the locus of the centre of the spheres of each
series is a quadric, and that the five quadrics are confocal.
It appears from what has been said that through any
point can be drawn ten planes cutting the cyclide in circles,
namely, the pairs of tangent planes which can be drawn
through the point to the five cones.*
563. The fivefold generation may be shown in another
way. If we suppose the quadric locus of centres F to be
identical with the sphere J which is cut orthogonally, we
evidently get for the cyclide J itself counted twice. Again, if we
have two cyclides both expressed in the form (X, Y, Z, W)'* = 0,
it appears from the theory of quadrics that by substituting
for X, Y, Z, W linear functions of these quantities both can
be expressed in the form aX i + b Y 2 + cZ* + dW 2 . Thus then
it is possible to express the equation of any cyclide in the
form a'X 1 + b'Y* + c'Z 2 + d'W*, while at the same time we have
an identical equation J* = aX* + bY 2 + cZ* + dlV*. For the
actual transformation we refer to Casey, p. 599, Darboux,
p. 135, but we can show in another way what this identical
* [The details of the work will be found in a paper by J. Fraser, Proc. R.I.A.,
xxiv., A, No. 10, in which also the actual reduction to the form of Art. 563 is
effected.]
230 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
p2
x
_ y f
 z,
1
d,
ll
m,
n,
1
d'
 1',
m',
n\
1
d",
I",
m",
n",
1
d'",
r,
",
ri",
1
equation is. Multiply by the ordinary rule the two deter
minants
1, 2#, 2y, 1z,
1, 22, 2w, 2n,
1O7 *.))/) O'M
. ' , _//( . _/( ,
1, 2Z", 2w", 2n",
1, 2r, 2w'", 2w"',
(where we have written for brevity /a 2 instead of # 2 + y 2 + .z 2 ,
and where either determinant equated to zero gives the equa
tion of the sphere cutting orthogonally four spheres), and the
product is
Y, Z,
(12), (13),
2/ 2 , (23),
(23), 2r" 2 ,
o,
X,
7,
Z,
W,
to*,
(12),
(13),
(14),
(24), (34), 
where (12) is d + d'  211'  2mw'  2ww', and vanishes if the
two spheres cut each other orthogonally. On the supposition
then that each pair of the four given spheres cut orthogonally,
the square of the equation of the sphere cutting them at right
angles is proportional to
0, X, Y,
X,  2r 2 , 0,
Y, 0, 2r
z, o, o,
W, 0, 0,
o,
o,
o,
W
 2r'" 2
whence it immediately follows that if five spheres cut each
other orthogonally, the identical relation subsists
X 2
[To complete the argument it is necessary to show conversely that if a
linear relation connects the squares of five spheres they are mutually ortho
gonal. This is shown at the end of Art. 566. J, X, Y, Z, W being now proved
orthogonal, inversion of the cyclide with respect to any of them obviously leaves
its equation unaltered.]
It may be noted in passing, that in virtue of this identity, the
equation W=Q may be written in the form
SURFACES OP THE FOURTH DEGREE. 231
showing that the sphere W meets the four others in four
planes, which form a selfconjugate tetrahedron with respect
to W. To return to the cyclide, it having been proved that
its equation may be written in the form
l
and that it may be generated as the envelope of a sphere
cutting W orthogonally, we may, by the help of the identity
just given, eliminate any other of the quantities X, Y, &c.,
and write for example the equation in the form
a'Y 2 + b'Z* + c'V 2 + d'W* = 0,
and generate the cyclide as the envelope of a sphere cutting
X orthogonally.
564. The condition that two surfaces whose equations are
expressed in terms of the five spheres X, Y, Z, V, W should
cut each other orthogonally, admits of being simply expressed.
It is in the first instance
dd>dX yd^dX . \
~ + &c  j j~ + &c  )
dx /
j v j~  jv j~
\dX dx /\dX dx
(dd> dX db dX
+ 1 , ,
,y , + &C, )( J^ J~ + &C. ) + &C. = 0.
\dX dy /\dX dy
This equation is reduced by the two following identities,
which are easily verified,
'
. . ,
j j (J ) =4X+4r 2 ,
dx ) \dy / \dz /
++=
dx dx dy dy dz dz
The condition may then be written
+ 2
The first two groups of terms vanish, because < and <fy, which
are satisfied by the coordinates of the point in question, are
homogeneous functions of X, Y, &c. The condition there
fore is
dXdX
232 ANALYTIC GEOMETKY OF THREE DIMENSIONS.
We may simplify the equations by writing X instead of X: r,
&c., so that the identity connecting the five spheres becomes
X i +Y* + Z 1 +V*~+W i = Q,
and the condition for orthogonal section
d$d d$_d =Q
+ +
a condition exactly similar in form to that for ordinary co
ordinates.
565. We can now immediately, after the analogy of quad
rics, form the equation of an orthogonal system of cyclides.
For write down the equation
X 2 Y 2 Z* V z W 2
I L . I I __ Q
Xa X6 Xc \d Xe
in which X is a variable parameter ; and, in the first place, it
is easy to see that three cyclides of the system can be drawn
through any assumed point : for the equation in X, though in
f orm of the fourth degree, is in reality only of the third , the
coefficient of X 4 vanishing in virtue of the identical equation.
And from the condition just obtained, it follows at once, in the
same manner as for confocal quadrics, that any two surfaces
of the system cut each other at right angles.*
[The equation aX+ $Y + yZ + B V=Q will represent a point
sphere on W, which will also be a focus of the cyclide X if
a, /9, 7, B satisfy the two conditions necessary for it to be an
enveloping sphere of the cyclide and of W. But these are, as
in Art. 560,
X a , ^ ~ & / . X c X d.
~ 7 ~
a  e be c e' d e
=
Now it will be found that the values of a : /8 : 7 : 8 which satisfy
these are independent of X. Hence the above system of^
Casey and Darboux seem to have independently made this beautiful
extension to three dimensions of Hart's theorem for the corresponding plane
curvei, Higher Plane Curves, Art. 278.
SURFACES OF THE FOURTH DEGlil I . 233
cyclides are confocal, there being a common focal curve on
each of the five spheres. It is evident from what has been
proved, that confocal cyclides cut each other in their lines of
curvature, i
[Ex. 1. If A.J, \. t , A.J, be the values of \ corresponding to the members of
the family which pass through a point, show that the values of the functions
Z*, Y*, &c., for this point are proportional to f(a), f(b) &c., where
f(a)={(a  ^(a  \J(a  A 3 )}=f(a)
+(0) = (e  a)(0  b)(6  c)(0  d)(0  e).
(Use the method indicated at the end of Art. 206.)
a (
Ex. 2. The coordinates of the point are proportional to 2 N //(a), 0,/3,y,
being the centre of X, 0*$^ that of Y, etc. (Darboux, op. cit. note to Art. 515,
p. 140).]
566. The mode of generating cyclides as the envelope of
a sphere admits of being stated in another useful form. All
spheres whose centres lie in a fixed plane, and which meet a
given sphere orthogonally, pass through two fixed points, there
being two linear relations connecting the coefficients. And it
is easy to see what the fixed points are, for since the spheres
cut at right angles every sphere through the intersection of
the fixed sphere and the plane, they contain the two point
spheres of that system, or the limit points (Conies, Art. Ill)
of the plane and the fixed sphere, these points being real only
when the sphere and plane do not intersect in a real curve.
In the case, then, where the centre of the movable sphere
lies in a fixed surface, it follows, obviously, that the envelope
may be described as the locus of the limit points of each
tangent plane to the fixed surface and of the fixed sphere.
We are thus led to a mode of transformation in which to a
tangent plane of one surface answer two points on another ;
or, if we take the reciprocal of the first surface, it is a (1, "2)
transformation, in which to one point on one surface answer
two on the other. Dr. Casey has easily proved, p. 598, that
the results of substituting the coordinates of one of these
limit points in the equations of the spheres of reference are
proportional to the perpendiculars let fall from the centres of
234 ANALYTIC GEOMETBY OP THREE DIMENSIONS.
these spheres on the tangent plane. Thus, if the surface
locus of centres be given by a tangential equation between the
perpendiculars from the four centres < (X, /*, v, p) = 0, the de
rived surface is < (X, Y, Z, W) = ; and if the first be the
equation of a quadric, the second will be the corresponding
cyclide. [If the tangential equation of J, the common or
thogonal sphere to X, Y, Z, and W, be >/r (X, p, v, p} = 0, it follows
by taking >/r as the locus of centres that J 2  ^(X, Y, Z, W) .
Now if the term \fi is absent from ty the centres of X and Y
are conjugate points with respect to J, and hence X and Y are
orthogonal. This proves the converse of the theorem of Art.
563.]
567. From the construction which has been given an
analysis has been made by Casey and Darboux of the different
forms of cyclides according to the different species of the
quadric locus of centres, and the nature of its intersection
with the fixed sphere. We only mention the principal cases,
remarking in the first place that the spheres whose centres
lie along any generator of the quadric all pass through the
same circle, namely, that wh : ch has for its antipoints the in
tersections of the line and the sphere. The circle in question
is part of the envelope, which may, therefore, be regarded as
the locus of the circles answering to the several right lines
of the qnadric, there being, of course, two series of circles
answering to the two series of right lines.
Now if the quadric be a cone, these circles all lie on the
same sphere, that which has its centre at the vertex of the
cone and which cuts the given sphere orthogonally, and the
cyclide may be regarded as degenerating into the spherical
curve which is the envelope of those circles, that curve being
the intersection of the sphere by a quadric, which curve has
been called a spheroquartic. Strictly speaking, the cyclide
locus of these circles is an annular surface flattened so as to
coincide with the spherical area, which is bounded by the
apheroquartic curve. The properties of these spheroquartics
have been investigated in detail by Casey and Darboux.
SURFACES OP THE FOURTH DEGREE. 235
These curves may be inverted into plane bicircular quartics,
and therefore (see note, Art. 515) have four foci, the distances
from which to any point of the curve are connected by linear
relations.
If the quadric be a paraboloid the cyclide degenerates into
a cubic surface passing through the circle at infinity. If the
quadric be a sphere the cyclide is the surface of revolution
generated by a Cartesian oval round its axis : but Darboux
has given the name Cartesian to the more general cyclide
generated when the quadric is a surface of revolution.
The cyclide may have one, two, three, or four double
points. The nodal cyclides present themselves as the inverse
of quadrics, the inverse of the general quadric being a cyclide
with one node, that of the general cone one with two, of the
general surface of revolution one with three, of the cone of
revolution one with four. The lastmentioned, or tetranodal
cyclide, is the surface to which the name cyclide was origin
ally given by Dupin, and may therefore be called Dupin's
Cyclide. According to its original conception this was the
envelope of the spheres, each touching three given spheres ;
or, more accurately, we have thus four cyclides, for the
tangentspheres in question form four distinct series, those of
each series enveloping a cyclide. The spheres of each series
are distinguished as having their centres on a given conic ;
[namely, for a sphere touching S lt S%, S 3 , all internally or all ex
ternaliy, the conic in which the plane 5*(r 2  r 3 )Si = meets the
quadrics of type JS.^  JS 3 = (r 2  r 3 )], and we thus arrive
at a better definition ; viz. the cyclide is the envelope of a
series of spheres each having its centre on a given conic and
touching a given sphere.
In the last definition the given sphere is not unique but it
forms one of a singly infinite series ; in fact, we may, without
altering the cyclide, replace the original sphere by any sphere
of the series ; the new series of spheres have their centres on
a conic. It is to be added that instead of the series of spheres
having their centres on the first conic, we may obtain the
same cyclide as the envelope of a series of spheres having
'236 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
their centres on the second conic,* and touching a sphere
having its centre at any point of the first conic.
The two conies have their planes at right angles, and are
such that two opposite vertices of each conic are foci of the
other conic ; these conies are focal conies of a system of con
focal quadric surfaces, one of them is an ellipse and the other
a hyperbola.
The relation of the ellipse and hyperbola is such that
taking
(1) Two fixed points on the ellipse, the difference of the
distances of these from a variable point on the hyperbola is
constant, = +c if the variable point is on one branch,  c if
it is on the other branch of the hyperbola (the value of c of
course depending on the position of the two fixed points).
(2) Two fixed points on the hyperbola, if on different
branches, the sum, but if on the same branch, the difference
of their distances from a variable point on the ellipse is con
stant, the value of this constant, of course, depending on the
position of two fixed points.
And using these properties, we see at once how the same
surface can be obtained as the envelope of a series of spheres
having their centre on either conic, and touching a sphere
having its centre at any point of the other conic.
Dupin's Cyclide is also the envelope of a series of spheres
having their centres on a conic, and cutting at right angles
a given sphere ; for instead of the quadric surface in the con
struction for the general cyclide, we have here a conic.
[567a. Nodal cyclides cannot be reduced to the canonical
form of Art. 563, for in their case the quadric F and sphere J
touch in one or more points, and no common selfconjugate
tetrahedron exists. The simplest form of the equation of
such cyclides is obtained by choosing such a tetrahedron of
reference as will give < and ^ of Art. 566 simple forms.
* [If the contacts are all similar this conic is a hyperbola passing through
the centres of S,, S,, S,, and having as its foci the centres of the circles in its
plane which touch the spheres.]
SURFACES OF THE FOURTH DEGREE. 237
By observing that when the term \p is absent from i/r,Xand
Y are orthogonal, and that if the vertex X = is on J, X is a
pointsphere on /, we can infer the number of nodes and the
nature of the inverses of the cyclide from them and derive
other properties mentioned above. It will suffice to take one
case to illustrate the method, say that in which F and J touch
in three distinct points so that their intersection is a circle and
two imaginary generators. We may take as simple forms
v = and A, 2 + /* 2 = being the points of contact. From these
ri) ZW.
The form of S shows that the pointsphere Z, is a node, for
being orthogonal to X and to Y it lies on these spheres.
Similarly the pointspheres X+tY, XiY are nodes on the
surface. We also see that from Z the cyclide inverts into
a quadric of revolution, and that the spheres of selfinversion
X, Y, J respectively invert into two rectangular planes
through the axis of revolution and into the central plane
perpendicular thereto. The reader will find it instructive to
work out the cases of other geometrical relations between F
and J" in a similar manner.
The first complete enumeration of the different kinds of
cyclides was made by Loria (Turin Acad., 1884), and his
eighteen species agree with the number obtained by Segre ;
of these only ten are real. His researches depend on a system
of coordinates in which a sphere plays the part of a line in
Pliicker's system. 6
It is clear that if S^ . . . S b are five fixed spheres, 2x r S r =
is an equation which is general enough to denote any sphere ;
x l . . . x 5 are called the coordinates of the sphere, and a
single relation between them represents a complex of spheres
and two a congruence. Loria shows that a cyclide is the
locus of the sphere of a certain congruence consisting entirely
of pointspheres.
238 ANALYTIC GEOMETEY OF THEEE DIMENSIONS.
A slightly different system of spherical coordinates will
be found explained in Darboux, op. cit., p. 256.
5676. Quartics with a Cuspidal Conic. Let the conic be
the intersection of a quadric V with the plane w, then the
equation must be w 3 P  V' 2 = 0, when P is a plane. By
change of coordinates we may write this without loss of
generality,
w*x = (fyz + gzx + hxy + Ixic + myw + nzwY,
and from this form we see that the tangent plane at yzw, an
arbitrary point on the conic, touches V and hence envelopes
a quadric cone whose vertex is the pole of w with respect to
V. The plane x is clearly a trope and any plane through
either point of intersection of the trope and conic has a
tacnode, as is seen by putting w = \x + p,y t when we get
2 2 Uj 2 + ZUjUz 4 U^ = 0,
as the form of the equation of the section by any plane
through xy w ; u lt 2 , tt, 4 , being binary quantics in x and y. For
the tangent plane at the point xyw, ^ disappears and the
equation represents four lines. Hence a cuspidal conic con
tains eight lines passing in fours through the two tacnodal
points.]
QUARTICS WITH ISOLATED SINGULARITIES.
568. [Passing now to quartic surfaces without singular
lines, the highest singularity they can possess is a triple point.
Such surfaces have been studied in detail by Eohn* who
has exhaustively classified them according to the different
singularities of the tangent cone at the triple point. The
memoir contains a number of diagrams illustrating the shapes
of the surfaces in the neighbourhood of this point.
These surfaces have for equation the monoid "f
u 3 + &)W 4 = 0,
the twelve lines t^ = 0, u 4 = lie on the surface, and the con
dition that there should be a node C 2 elsewhere is easily found
* Math. Ann., 24. f Art. 316 (B).
SURFACES OF THE FOURTH DEGREE. 239
to be that two of these lines should coincide. More generally
Rohn shows that the existence of a binode of type B k involves
the coincidence of k of these lines, and that unodes can only
occur on a singular edge of the cone u 3 . Hence all the types
of surfaces for which u a is nonsingular are obtained by par
titioning the number 12 ; for instance
12=5+3+2+1+1
means that there is a surface having, in addition to the triple
point, the singularities B 5 = l,B 3 = l t (7 2 = 1.] Quartic surfaces
with isolated double points only may have any number of
ordinary conical points up to 16 ; each such node diminishes
the class by 2, so that for the surface with 16 nodes the class
is 36  2 x 16, = 4. Some of the nodes may be replaced by, or
may coalesce into, binodes or unodes, but the theory does not
appear to have been investigated.
The general cone of contact to a quartic is, by Art. 279,
of the twelfth degree, having twentyfour cuspidal and twelve
nodal lines, and sixteen is the greatest number of additional
nodal lines it can possess without breaking up into cones of
lower dimensions. When the surface has sixteen nodes, the
cone of contact from each node is of the sixth degree, and
has the lines to the other fifteen as nodal lines ; from which
it follows that this cone breaks up into six planes.
569. It is to be observed that the equation of a quartic
surface contains thirtyfour constants, that is, the surface may
be made to satisfy thirtyfour conditions ; and that if a given
point is to be a node of the surface, this is equivalent to four
conditions. It would, therefore, at first sight appear that we
could with eight given points as nodes determine a quartic
surface containing two constants ; but this is not so. We
have through the eight points two quadric surfaces U= 0, V =
(every other quadric surface through the eight points being
in general of the form U+\V=Q) and the form with two
constants is in fact U' 2 + aUV + ftV* = 0, which breaks up into
two quadric surfaces, each passing through the eight points.
It thus appears that we can find a quartic surface with at
most seven given points as nodes.
240 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
570. The cases of a surface with 1, 2, or 3 nodes may be
at once disposed of ; taking, for instance, the first node to be
the point (1, 0, 0, 0), the second the point (0, 1, 0, 0), and
the third the point (0, 0, 1, 0), we can at once write down
an equation U=0, with 30, 26, or 22 constants, having the
given node or nodes. We might in the same manner take
the fourth node to be (0, 0, 0, 1) and write down the equation
with 18 constants ; but, in the case of four nodes and in re
ference to those which follow, it becomes interesting to con
sider how the equation can be built up with quadric functions
representing surfaces which pass through the given nodes.
In the case of 4 given nodes we have six such surfaces
p = 0, Q = 0, R = 0, S = 0, T = 0, 7=0, every other quadric
surface through the four points being obtained by a linear
combination of these ; and we have thence the quartic equa
tion (P, Q, E, S, T, E7) 2 = containing apparently 20
constants. The explanation is that the six functions, although
linearly independent, are connected by two quadric equations,
and the number of constants is thereby reduced to 20  2,
= 18, which is right. [Taking yz, zx, xy, xw, yw, zw as the six
quadrics these are connected by the equations PS = QT = RU.]
In the case of 5 given nodes we have through these the
five quadric surfaces P = 0, Q = 0, .8 = 0, S = 0, T=0, and we
have the quartic surface (P, Q, R, S, T) 2 = 0, containing, as it
should do, 14 constants. [One relation connects the quadrics
but it is a cubic relation. See Cayley's Collected Mathematical
Papers, vn. 142.]
571. In the case of 6 given nodes, we have through these
the four quadric surfaces P = 0, Q = 0, R = Q, S = 0, and the
quartic surface (P, Q, R, S) 2 = contains only 9 constants ;
there is in fact through the six points a quartic surface,
the Jacobian of the four functions, J(P, Q, R, S) = 0, not
included in the foregoing form, and the general quartic surface
with the six given nodes is
(P, Q, R,S? + ej(P, Q, R,S) = Q,
containing, as it should do, 10 constants.
SURFACES OF THE FOURTH DEGREE. 241
The foregoing surface J(P, Q, R, S) = 0, where P = 0, Q = 0,
R = 0, S = are any quadric surfaces having six common
points, is a very remarkable one ; it is in fact the locus of the
vertices of the quadric cones which pass through the six points.
It hereby at once appears that the surface has upon it
15 + 10, = 25 right lines, namely, the 15 lines joining each
pair of the six points, and the 10 lines each the intersection
of the plane through three of the points with the plane
through the remaining three points. [This is Weddle's sur
face (Art. 233, note), further discussed in Art. 572a.]
In the case of 7 given nodes we have through these three
quadric surfaces P = 0, Q = 0, R = Q; but forming herewith
the equation (P, Q, .R) 2 = 0, this contains only five constants ;
that it is not the general surface with the 7 given nodes
appears also by the consideration that it has, in fact, an eighth
node, for each of the intersections of the three quadric sur
faces is a node on the surface. We can without difficulty
find a quartic surface not included in the form, but having
the seven given nodes ; for instance, this may be taken to be
V = 0, where v i g made up of a cubic surface having four of
the points as nodes and passing through the remaining three
points, and of the plane through these three points. And the
general equation then is
(P, Q, B) + 0v = 0,
containing, as it should do, 6 constants.
572. Passing to the surfaces with 8 nodes, only seven of
these can be given points ; the eighth may be the remaining
common intersection of the quadric surfaces through the
seven points, and we thus have a form of surface
(P.QltfO,
with eight nodes, the common intersection of three quadric
surfaces ; this is the octadic 8nodal quartic surface. [Octadic
surfaces can acquire two additional nodes, see Cayley,
vn. 153.]
Among the surfaces of the form in question are included
the reciprocals of several interesting surfaces ; for example,
VOL. II. 16
242 ANALYTIC GEOMETRY OF THBEE DIMENSIONS.
order six, parabolic ring; order eight, elliptic ring; order
ten, parallel surface of paraboloid, and first central negative
pedal of ellipsoid ; order twelve, centrosurface of ellipsoid and
parallel surface of ellipsoid the surfaces include also the
general torus or surface generated by the revolution of a conic
round a fixed axis anywhere situated. [See Cayley, vn.
155.]
There is, however, another kind of 8nodal surface, called
the octodianome, for which the eighth node is any point what
ever on a certain surface determined by means of the seven
given points, and called their dianodal surface.
The lastmentioned surface may be made to have another
node, which is any point whatever on a certain curve de
termined by means of the eight nodes; we have thus the
enneadianome ; and finally this may be made to have a new
node, one of a certain system of [thirteen] points determined
by means of the nine nodes ; this is the decadianome. But
starting with seven given points as nodes, the number of
nodes of the quartic surface is at most = 10.
[The dianodal surface of seven points is obtained by the
condition that the 7nodal quartic should have another node
and is J{P, Q, R, y} = 0, a surface of the sixth order. The
dianodal surfaces of the seven points 1 to 7 and of the seven
2 to 8 (8 being on the first surface) intersect in the fifteen lines
joining 2, 3, 4, 5, 6, 7 and the skew cubic containing these
points, and therefore in a residual intersection of the eighteenth
order which is the dianodal curve of the eight points. The
general forms of octodianomes and enneadianomes are
respectively (P, Q) 2 + #v and P 2 + 8^.]
A. kind of 10nodal surface is the Symmetroid, which is
represented by means of a symmetrical determinant
a, h, g, I
h, b, f, m
9, f, c, n
I, m, n, d
where the several letters represent linear functions of the co
ordinates ; such a surface has ten nodes [see Art. 572 (6)], for
=
SURFACES OF THE FOURTH DEGREE. 243
each of which the circumscribed sextic cone breaks up into
two cubic cones ; and thus the ten nodes form a system of
points in space, such that joining any one of them with the
remaining nine, the nine lines are the intersections of two
cubic cones ; these are called an ennead, and the ten points
are said to form an enneadic system.
Some of the kinds of surfaces with 11, 12, and 13 nodes,
and the surfaces with 14, 15, and 16 nodes were considered by
Kummer.* Reverting to the consideration of the circum
scribed cone having its vertex at a node, observe that for a
surface with 16 nodes, this is a sextic cone with fifteen nodal
lines, or it must break up into six planes, say the sextic cone
is (1, 1, 1, 1, 1, 1); and the form being unique, this must be
the case for the cone belonging to each node of the surface,
say the surface is the 16nodal 16 (1, 1, 1, 1, 1, 1).
Similarly, in the case of 15 nodes, the sextic cone has
fourteen nodal lines, or it breaks up into a quadricone and
four planes, say it is (2, 1, 1, 1, 1) ; which form being also
unique, the surface is the 15nodal 15 (2, 1, 1, 1, 1).
In the case of 14 nodes, the cone has thirteen nodal lines,
it must be either a nodal cubic cone and three planes, or else
two quadricones and two planes; that is (3, 1, 1, 1) or
(2, 2, 1, 1). It is found that there is only one kind of surface,
having eight nodes of the first sort and six nodes of the
second sort ; say this is the 14nodal
8(3,1, 1, 1) + 6(2, 2, 1,1).
In the case of 13 nodes, the cones are (4 3 , 1, 1), (3 : , 2, 1),
(3, 1, 1, 1), or (2, 2, 2), viz. (4 3 , 1, 1) is a 3nodal quartic
cone and two planes, and so (3 1} 2, 1) is a nodal cubicone, a
quadricone, and a plane. It is found that there are two forms
of surface, the 13(a)nodal
3(4 3 , 1, 1) + 1 (3,1, 1,1) + 9 (3,, 2,1),
and the 13(yS)nodal 13 (2, 2, 2).
The like principles apply to the cases of twelve, eleven, &c.,
*[And also by Cayley, vn. 279; Rohn, Math. Ann., xxix., and Jesaop,
Quart. J., 31.]
16*
244 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
nodes, [but for their discussion the reader is referred to Rohn's
memoir mentioned in the footnote to this article].
[572a. Some Properties of Weddle's Surface. Let C be
the skew cubic through the nodes (^ . . . N^) and let (P . (7)
denote the cone joining a point P to C.
1. Any plane through two nodes cuts the surface in a
cubic on which the nodes are corresponding points. For the
nodal cone at N l contains N^,, and therefore is (A^ . C).
The plane meets C in one point P other than the nodes, and
the lines joining P to the nodes are generators of each nodal
cone and therefore touch the plane cubic of section at the
nodes.
2. Let PN lt PN 2 meet the quartic again in P x , P 2 . Then
from the properties of a plane cubic A^Pi and ^P 2 meet on
the quartic, say in P 12 . We derive thus six points P r aud
fifteen points P n . Baker shows that N 9 P r , and N r P gt meet on
the quartic in a point P yr ,, that P 123 is the same point as P 466 ,
and thus we have a closed system of thirtytwo points on the
quartic lying in pairs on ninetysix lines through the nodes.
3. "Weddle's Surface may be defined as the locus of points
whose polar planes with respect to all quadrics through the
six nodes are concurrent, and in this sense is made up of pairs
of corresponding points. Now if P and Q are such a pair
the line PQ must be a common tangent line at P to all
quadrics through ^ . . . N 6 and P, or in other words, the eighth
common point of these quadrics coincides with P. This leads
to the definition: Weddle's Surf ace is the locus of a point
which is itself tlie eighth intersection of all quadrics through
it and six fixed points.
4. The line PQ meets C twice and is divided harmonically
by it, for three independent quadrics can be described through
the nodes and the two points where the chord of C through
P meets C.
5. If the coordinates of a point of C are taken to be
1 : X : X 2 : X s , and if \ r is the parameter of N r , and we put
/Or) = {(x  XJC*  X 2 )(x  \)(x  X 4 )(*  X 5 )(*  X 6 )}  *
SURFACES OF THE FOURTH DEGREE. 245
then it is easy to see from the preceding theorem that the
coordinates of a point on Weddle's Surface may be expressed
in terms of two parameters as follows
x:y:z:w::f(8) /(</>) : 0/(0)
For other properties see the papers cited in Art. 545 from
which the foregoing are taken.
5726. Symmetroids. The existence of ten nodes is con
nected with the existence of ten planepairs of the syzygy of
quadrics
a^ 1 + /3t7 2 + 7?7 3 + 8C7 4 = 0. (U r = a^ + &c.).
For from the four equations obtained by differentiating we
can either eliminate a, /3, 7, 8 and obtain the Jacobian (J) ; or
eliminate x, y, z, w and obtain, considering a, @, 7, B as co
ordinates, a symmetroid (S) as written in Art. 572, where
a = aa l + /3a 2 + ya^ + Ba 4 ,
and so on. By means of the four equations the coordinates
of a point on J are given as rational functions of a point on
S, and vice versa, so there is a oneone correspondence of points
on S and J.
But the values of a, /3, 7, 8, which make aU l + &c. a plane
pair make every first minor of S zero, so that to the whole
edge, which lies in J, corresponds a node on S ; thus the ten
nodes correspond to the ten edges.
If a = the point h = g = I is an eleventh node on S. This
will be the case if the four quadrics all pass through the point
yzw, and therefore / has a node there.
Making a = b = c = d = Q we see that the symmetroid with
fourteen nodes, may be written Jfl+ Jgm + *Jhn = Q, and
corresponds to the Jacobian of four quadrics through four
points.
If we make these quadrics have one, or two, more common
points, we obtain symmetroids with 15 or 16 nodes. In these
cases the planes /, g, &c., must be connected by one or two
equations of the form
246 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
obtained by adding the four equations, which express
that a point f , tj, , &, lies on the quadrics u lt u 2 , u 3 , and w 4 ,
these equations being first multiplied by a, , 7, and S re
spectively. We thus obtain the important result proved
directly by Jessop * that the 14, 15, and 16 nodal quartics
are all included in the form
subject in the case of 15 and 16 nodal quartics to one or two
conditions of the form
[573. The 16nodal or Kummer's Quartic was discussed
in Art 455a in connexion with quadratic complexes, but it
will be instructive to consider this surface from an indepen
dent point of view.f We have seen in Art. 572 that the
enveloping cone from each node breaks up into six planes
intersecting on the fifteen lines joining the node in question
to the others. It is easy to see that these planes touch the
tangent cone at the node along the six lines of closest con
tact. Each of these planes contains six nodes, namely, the
original one and one in each of the other five planes, and as
six planes pass through each node there are sixteen such
planes altogether. Further, these planes are tropes ; for the
section of the quadric by any one of them has the six nodes
lying on it as double points and must accordingly be a repeated
conic through the nodes, for the only other plane section of
a quartic having six double points is made up of four right
lines, and this hypothesis is inadmissible as no three nodes
are collinear. Thus, then, the sixteen nodes lie by sixes on
conies in sixteen tropes, and the tropes touch in sixes the
quadric tangent cones at the nodes. This arrangement of
points and planes is known as a 16 6 configuration.
" Quarterly Journal, vol. 81.
fThe subject matterrof this and the next three Articles is largely taken
from Hudson, op. cit. Art. 545.
SURFACES OF THE FOURTH DEGREE. 247
Denoting one node by 0, the tropes through it by the
numbers I to 6, and the other nodes by the binary symbols
12, 13, &c., we can now see how these latter are distributed
on the ten remaining tropes. For two triads of tropes such
as 1, 2, 3 and 4, 5, 6 cut any plane in two triangles circum
scribed to the section by that plane of the nodal cone at 0,
and therefore the six vertices of these triangles lie on a conic.
In other words, the lines joining to 12, 23, 31, 45, 56, 64
lie on a quadric cone. There are ten such cones, and only
ten in general, corresponding to the ten partitions of the six
tropes into a pair of triads. But the six nodes lying on any
other trope connect to by the edges of a quadric cone, which
cone must be identical with one of the ten just mentioned ;
and hence the six nodes 12, 23, 31, 45, 56, 64 lie on a trope.
123
We may call this the trope j^ and can now at once
specify the tropes through any node; for instance, those
, 123 124 125 126
through 12 are 1, 2,
573a. As the general quartic surface contains thirtyfour
constants in its equation, and as the existence of a node im
plies one relation between these, we would expect that the
equation of Rummer's Quartic should contain eighteen inde
pendent constants, and that thus six arbitrary points can be
selected as nodes.
The notation of the preceding article enables us to verify
this and to see that six arbitrary points, no four of which are
coplanar, determine a 16 6 configuration in twelve different
ways. For let us call one definite point and a second 12,
which is clearly permissible since no two nodes have any
special relation to each other. If to the other four points
are attached the symbols 23, 34, 45, 51, nothing is implied
except that no four of the six points are coplanar ; and this
allocation of symbols can be made in twentyfour ways.
This being done, the ten planes
123 234 345 115
1, A d, 4, 5,
248 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
are determined, for three points in each are known, and the
ten other points will be found to be determined each as the
intersection of some three of these ten planes. The reduction
to twelve distinct ways depends on the fact that to call the
four points 51, 45", 34, 23 instead of 23, 34, 45, 51 will not
lead to a different configuration ; for the same ten additional
points will be determined though their symbols will be changed
by the interchange of 1 with 2 and 3 with 5, e.g. 25 becomes
13 and so on.
5736. Analytical treatment of the 16 6 configuration. Let
the symbol a denote the operation of interchanging y with z
and x with w, and let b and c have similar meanings. Further,
let A, B, C denote the operations of changing the signs of
y and z, of z and x, and of x and y respectively, while d
denotes that no interchange, and D that no change of signs,
is made. From the equation
ax + fty + yz + Sw? = 0,
which we call dD, fifteen others are derived by combining
some one of the operations a, b, c, d, with A, B, C or D, and
we will call these equations by the symbol of the operation
whereby they are derived. These sixteen equations may be
taken as representing at the same time the planes or points
of a 16 6 configuration ; for instance the point bC represented
by aw + &z  yy  Bx = is the point (  B,  7, , a) which lies
on the plane ax+ 0y+ yz+ Bw = 0, that is the plane dD ; thus
the six points in the plane dD are aB, aC, bC, bA, cA, cB.
The six points that lie in the plane aB are dD, dA, cA, cC,
bC, bD, a result that can be obtained immediately by operat
ing on the previous six points with aB and remembering that
a 1 = d, ab = c, and so on.
The relations between the points and planes are given by
the following diagram; the six planes (or points) passing
through (or lying on) a given point (or plane) are those whose
symbols appear in the same row or column as the symbol of
the given point (or plane) :
SURFACES OF THE FOURTH DEGREE. 249
dD aB bC cA
aC dA cD bB
bA cC dB aD
cB bD aA dC
Eighteen constants are involved in the foregoing equa
tions, namely, three to fix each plane of reference, three to
fix the ratios of the coordinates, and the three ratios
a:j3:y:8. They are accordingly sufficiently general to
represent any 16 6 configuration.
573c. The equation of Kummer's Quartic which has the
points and planes of last article for nodes and tropes can now
be written down. For it must be unchanged by any of the
operations a, b, c, A, B, C, and is therefore of the form
x* + y* + z* + tv* + L (y*z* + x 9 w z ) + M (z*x* + y*w*)
+ N (x'y* + * 2 ^ 2 ) + %Kxyzw = 0.
The condition that (a, y9, 7, 8) is a node gives the equations
, / 8 4 +7 4 a 4 5 4
Ll =  5 
a 2
with two similar ones, and
a/
If a, /3, 7, 8, be eliminated from these we obtain the con
dition already found in Art. 4550, viz.
4  L 2  M z  N' 2 + LMN+ D* = 0.
If B be put equal to zero the equation of the surface takes
the particular form
x 4 + y* +z 4 + w*  2p (ifz* + x*w 2 )  1q (z*x 2 + y*w z )
2r (x*y*+z*w*) = Q
where p, q, and r are the cosines of the angles of a plane tri
angle whose sides are a 2 , /S 2 , and 7 2 . In this case the sixteen
nodes lie in fours in the coordinate planes, and the section of
the surface by any one of these planes is a pair of conies
intersecting in the nodes and having as the sides of their
common self con jugate triangle the traces of the other co
ordinate planes.
250 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
Such a surface is called a Tetrahedroid, and includes the
Wavesurface as a particular case, as can be at once seen by
writing in the equation just given when multiplied by aWc?
x 3 , if, **,!, (fr'+c 2 )
in place of
bcx*, cay*, abz*, w* t 2bcp,
respectively, when we obtain the ordinary equation of the
Wavesurface.
Ex. 1. The coordinates of a point of Rummer's Surface are expressible in
terms of two parameters as follows :
x: y : z : w : : 1 : \ (u + v) : uv : (u  w) 2 (
where i( r = u  lt r and v r =v  k r .
Let 0, 0, 0, 1 be a node, xz  i/ 2 = 0, the nodal cone at it and the six planes
of the type
r = fcr 2 .c  2k r y + z =
123
be the six tropes through it. Let to be the trope .
Writing the equation of the surface in the form
to 3 (xz  y 2 ) + ivu 3 + ^ = . . . . (i)
we require
(xz  y) $*  M 3 2 = Air 1 ir a ir 3 ir 4 ir, ) ir 6 . . . (ii)
Any tangent to the section of xz  J/ 3 by w has as its equation
(Px  20y + z = 0, and if u and v be the values of corresponding to the tan
gents from any point we have
x : y : z : : 1 : \ (u + v) : uv.
The equation (u^MyV^v^  tt 4 tt 5 ?t 6 t' 1 v ! , 3 ) (u  t>) J = is satisfied for the
six nodes in w, and as it is symmetrical in u and v and u?v y is the highest
term, it represents the conic through them, <f> = 0. Substituting in (ii) we
get
whence A = 4, for u 3 must be a symmetrical function of and v. Solve finally
for to : x from (i).
Ex. 2. Six of the sixteen planes of Art. 5736 are the planes corresponding
to the point (a, /i, 7, 8) with reference to the six mutually apolar * linear
complexes p = 0, q t = 0, rs = 0. They are the six called in that
Article oB, aC, bC, bA, cA, cB.
Two linear complexes Ap + Bq + &c. = 0, A'p + Rq + &c. = are
said to be ajvlur when their coefficients satisfy the relation
AD' + A'D + BE' + B'E + CF' + C'F = 0.
In this case the conjugate line with respect to ope complex of any ray of
the second is also a ray of the second.
SURFACES OF THE FOURTH DEGREE. 251
Ex. 3. The ten other planes are the polar planes of (a, 0, y, 5) with re
spect to ten quadrics thus denned : take any three of the linear complexes of
Ex. 2, and their common rays will be one system of generators of one of the
quadrics while the common rays of the three remaining complexes will be
the other system of generators of the same quadric. The equations of the
quadrics are x* + y 3 + z* + tc* = ; y 2 + z*  x*  w~ = and two similar ;
yz + xw = and four similar; the results can be easily verified by means of
these equations.
Ex. 4. The preceding examples, together with the fact that six mutually
apolar linear complexes can always be reduced to the forms of Ex. 2 by
choosing as edges of the tetrahedron of reference the directing lines (Art.
455) of the three congruences obtained by combining the complexes into
three pairs, lead to the following theorem: Six mutually apolar linear com
plexes and one arbitrary point determine uniquely a 16g configuration.
574. Very few investigations concerning non singular
quartic surfaces have been published. The following is a
slight sketch of a method by which Rohn * has proved that
the maximum number of ovals which a nonsingular quartic
surface may possess is ten. By an oval portion of a surface
he means one without singularities and closed. That quartics
may actually possess as many as ten ovals appears from the
equation
XYZ W  k*(x 2 +if+ z  r')  c 2 =
in which X, Y, Z, and W are, when equated to zero, the
Cartesian equations, in the standard form of Art. 27, of the
faces of a regular tetrahedron whose centre is the origin and
the distance of whose corners from the origin exceeds r. If
c were zero this equation would represent a 12nodal quartic
having six identical portions lying outside the tetrahedron
in the acute angle between each pair of faces and four
identical portions lying inside the tetrahedron and near the
corners, these ten portions being joined at the nodes but
otherwise distinct. Now suppose a small value to be given
to c, and we obtain ten oval portions lying within the former
portions.
The proof that ten is the maximum number depends on
somewhat intricate considerations of the tangent cone to a
10nodal quartic from one of the nodes and cannot be given
here. But it is easy to see that this number cannot exceed
* Lupzigtr Berichte, LXIII. (1911).
252 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
twelve. For if possible let S = Q be a quartic with thirteen
ovals, and suppose that the sign of the function S is positive
for points inside the ovals. Let F=0 be a quadric which
does not meet all the ovals in real curves and consider the
pencil of quartics S  k 7 s = 0, as k grows from zero.
The members of this pencil are quartics with thirteen
ovals lying within those of S, up to such a value of k as
makes one of the ovals shrivel up into a point, A, which will
be a conjugate point on that particular quartic. Consider
the tangent cone from A to this quartic. It will be of the
sixth order and be such that any plane cuts it in twelve ovals.
But this is impossible ; for if a plane sextic had twelve
ovals we could choose one point on eleven of them and three
on the twelfth and through these fourteen points describe a
plane quartic which would meet the eleven ovals in two
points each and the twelfth in four and therefore the sextic
in twentysix. Hence the quartic surface can only have at
most twelve ovals.
The process of replacing ovals by conjugate points is
called a Schrumpfungsprozess (shrivelling) by Rohn, and he
shows that any number of ovals up to ten can be thus
replaced one by one, the remaining ovals lying within the
original ones. In fact the only difference is that the quadric
V is made to pass through the conjugate points already
obtained, so that S  kV' 2 preserves these singularities and is
made to acquire one more conjugate point at each step.]
CHAPTER XVII.
GENERAL THEORY OF SURFACES.
SECTION I. SYSTEMS OF SURFACES.
575. WE shall in this chapter proceed, in continuation of
Art. 287, with the general theory of surfaces, and shall first
give for surfaces in general a few theorems proved for quadrics
(Art. 233, &c.).
The locus of the points whose polar planes with regard to
four surfaces U, V, W, T (whose degrees are m, n, p, q) meet
in a point, is a surface of the degree m + n +p + q  4 ;
the Jacobian of the system, which is also the locus of all
double points of the system \U+fj,V+ vW+pT. For its
equation is evidently got by equating to nothing the deter
minant whose constituents are the four differential coefficients
of each of the four surfaces. If a surface of the form
\U+fjiV + i'W touch T, the point of contact is evidently a
point on the Jacobian, and must lie somewhere on the curve
of the degree q (m + n +p + q  4) where the Jacobian meets
T. In like manner, pq (m + n+p + q4} surfaces of the form
\U + fj,V can be drawn so as to touch the curve of intersection
of T, W ', for the point of contact must be some one of the
points where the curve TW meets the Jacobian.
It follows hence, that the tactinvariant of a system of three
surfaces U, V, W (that is to say, the condition that two of the
mnp points of intersection may coincide), contains the coeffici
ents of the first in the degree np (2m + n + p  4) ; and in like
manner for the other two surfaces. For, if in this condition
we substitute for each coefficient a of U, a + \a, where a is
the corresponding coefficient of another surface U' of the same
253
254 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
degree as U, it is evident that the degree of the result in X is
the same as the number of surfaces of the form U + \U\ which
can be drawn to touch the curve of intersection of V, W*
I had arrived at the same result otherwise thus : (see
Quarterly Journal, voL i. p. 339). Two of the points of inter
section coincide if the curve of intersection UV touch the
curve U W. At the point of contact then the tangent planes
to the three surfaces have a line in common ; and these planes
therefore have a point in common with any arbitrary plane
ax + fty + yz + Siv. Thus the point of contact annuls the
determinant, which has for one row, a, /?, 7, 8 ; and for the
other three, the four differentials of each of the three surfaces.
The condition that this determinant may vanish for a point
common to the three surfaces is got by eliminating between the
determinant and U, V, W. The result will contain a, /?, 7, 8
in the degree mnp ; and the coefficients of U in the degree
np (m + n+p3) + mnp. But this result of elimination con
tains as a factor the condition that the plane ax + fty + yz + Bw
may pass through one of the points of intersection of U, V, W.
And this latter condition contains a, /9, 7, B in the degree mnp,
and the coefficients of U in the degree np. Dividing out this
factor, the quotient, as already seen, contains the coefficients of
U in the degree
np (2m + n+p 4).
576. The locus of points whose polar planes with regard
to three surfaces have a right line common is, as may be
inferred from the last article, the Jacobian curve denoted by
the system of determinants
PI, CT it Z7 if U,
V lt V,, V 3 , F 4
W lt W 9t W 3 , W t = 0.
But this curve (see Higher Algebra, Art. 271) is of the degree
(m* + n 2 +p'* + m'n + np' + p'm'},
where m' is the degree of U lt &c., that is to say, m' = ml, &c.
' Moutard, Terquem's Nouvelles Annales, xix. p. 68.
SYSTEMS OF SURFACES. 255
If a surface of the form \U + pV touch W, the point of contact
is evidently a point on the Jacobian curve, and therefore the
number of such surfaces which can be drawn to touch W is
equal to the number of points in which this curve meets W,
that is to say, is p times the degree of that curve. Reasoning
then, as in the last article, we see that the tactinvariant of two
surfaces U, V, that is to say, the condition that they should
touch, contains the coefficients of U in the degree
n (n' 2 + 2m' n' + 3m'),
or n (n + Imn + 3w' 2  4n Sm + 6).
This number may be otherwise expressed as follows : if the
degree and class of V be M and N, and the degree of the tangent
cone from any point be R, then the degree in which the coeffi
cients of U enter into the tactinvariant is
We add, in the form of examples, a few theorems to which
it does not seem worth while to devote a separate article.
Ex. 1. Two surfaces, U, V of degrees m, n intersect ; the number of
tangents to their curve of intersection, which are also inflexional tangents of
the first surface, is inn (3m + 2n  8).
The inflexional tangents at any point on a surface are generating lines of
the polar quadric of that point ; any plane therefore through either tangent
touches that polar quadric. If then we form the condition that the tangent
plane to V may touch the polar quadric of U, which condition involves the
second differentials of U in the third degree, and the first differentials of V in
the second degree, we have the equation of a surface of degree (3m + 2n  8)
which meets the curve of intersection in the points, the tangents at which are
inflexional tangents on U.
Ex. 2. In the same case to find the degree of the surface generated by the
inflexional tangents to U at the several points of the curve UV.
This is got by eliminating x'y'z'w' between the equations
U' = 0, V = 0, A U' = 0, A 2 tP = 0,
which are in x'y'z'w' of degrees respectively m, n, m  1, m  2, and in xyzw
of degrees 0, 0, 1, 2. The result is therefore of degree mn (3m  4).
Ex. 3. To find the degree of the developable which touches a surface along
its intersection with its Hessian. The tangent planes at two consecutive points
on the parabolic curve intersect in an inflexional tangent (Art. 269) ; and, by
the last example, since n 4 (m  2), the degree of the surface generated by
these inflexional tangents is 4w (m  2) (3m  4). But since at every point of
the parabolic curve the two inflexional tangents coincide, and therefore the
surfaces generated by each of these tangents coincide, the number just found
must be divided by two, and the degree required is 2m (m  2) (3m  4).
256 ANALYTIC GEOMETRY OP THREE DIMENSIONS.
Ex. 4. To find the characteristics, as in Art. 330, of the developable circum
scribed along any plane section to a surface whose degree is m. The section
of the developable by the given plane is the section of the given surface, together
with the tangents at its 3m (m  2) points of inflexion. Hence we easily find
degree = 6m(m2), class = m(m  1), r=m(3m5), a=0, /3 = 2m(5mll), &c.
Ex. 5. To find the characteristics of the developable which touches a sur
face of degree m along its intersection with a surface of degree n. We find
class = mn(m  1), a = 0, r = mn (3m + n  6), whence the other
singularities are found as in Art. 330.
Ex. 6. To find the characteristics of the developable touching two given
surfaces, neither of which has multiple lines. We find
class = mn (m  I) 8 (n  l) s , a = 0, r = mn (m  1) (n  1) (m + n  2).
Ex. 7. To find the characteristics of the curve of intersection of two de
velopables.
The surfaces are of degrees r and r', and since each has a nodal and
cuspidal curve of degrees respectively x and m, x' and m', therefore the curve
of intersection has rx' + r'x and rm' + r'm actual nodal and cuspidal points.
The cone therefore which stands on the curve, and whose vertex is any point,
has nodal and cuspidal edges in addition to those considered at Art. 343 ; and
the formulae there given must then be modified. As there the degree is r;' ;
but the degree of the reciprocal of this cone is
rr' (r + r'  2)  r (2x r + 3m')  r' (2s + 3m),
or, by the formulae of Art. 327, rank = rn' + r'n. In like manner
class = or' + a'r + 3rr'.
Ex. 8. To find the characteristics of the developable generated by a line
meeting two given curves. This is the reciprocal of the last example. We
have therefore class = rr', rank = rm' + r'm, degree = /3r' + jS'r + 3rr'.
Ex. 9. To find the characteristics of the curve of intersection of a surface
and a developable. The letters M, N, R relate to the surface as in the present
article ; TO, n, r to the developable. We find degree = Mr, rank = rR + nM,
class = aM + 3rR.
Ex. 10. To find the characteristics of a developable touching a surface
and also a given curve. We find degree = &N + 3rR, rank = rR + mN,
class = Nr.
577. The theory of systems of curves given in Higher Plane
Curves, p. 372, obviously admits of extension to surfaces. Let
it be supposed that we are given one less than the number
of conditions necessary to determine a surface of degree // :
the surfaces satisfying these conditions form a system whose
characteristics are /*, v, p; where p. is the number of sur
faces of the system which pass through any point, v is
the number which touch any plane, and p the number
which touch any line. It is obvious that the sections of the
SYSTEMS OF SURFACES. 257
system of surfaces by any plane form a system of curves
whose characteristics are p, p ; and the tangent cones drawn
from any point form a system whose characteristics are p, v.
Several of the following theorems given by De Jonquieres
(Comptes Eendus, LVIII., p. 567), answer to theorems
already proved for curves.
(1) The locus of the poles of a fixed plane with regard to
surfaces of the system is a curve of double curvature of de
gree v. The locus is a curve, since the plane itself can only
be met by the locus in a finite number of points v. Taking
the plane at infinity, we find, as a particular case of the above,
the locus of the centre of a quadric satisfying eight condi
tions. Thus, when eight points are given, the locus is a curve
of the third degree ; when eight planes, it is a right line.
(2) The envelope of the polar planes of a fixed point, with
regard to all the surfaces of the system, is a developable of
class /*.
(3) The locus of the poles ivith regard to surfaces of the
system, of all the planes which pass through a fixed right
line, is a surface of degree p. There are evidently p and only
p points of the locus, which lie on the assumed line. The
theorem may otherwise be stated thus : understanding by the
polar curve of a line with respect to a surface, the curve
common to the first polars of all the points of the line ; then,
the polar curves of a fixed line with regard to all the surfaces
of the system lie on a surface of degree p.
(4) Reciprocally, The polar planes of all the points of a
line, with respect to surfaces of tlie system, envelope a surface
of class p.
(5) The locus of the points of contact of lines drawn from
a fixed point to surfaces of the system is a surface of degree
fjk + p, having the fixed paint as a multiple point of order p.
This is proved as for curves. The problem may otherwise be
stated : " To find the locus of a point such that the tangent
plane at that point to one of the surfaces of the system which
passes through it shall pass through a fixed point." Hence
we may infer the locus of points, where a given plane is cut
VOL. II. 17
258 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
orthogonally by surfaces of the system. It is the curve in
which the plane is cut by the locus surface n + p, answering
to the point at infinity on a perpendicular to the given plane.
(6) The locus of points of contact, with surfaces of the
system, of planes passing through a fixed line, is a curve of
degree v + p meeting the fixed line in p points. This also may
be stated as the locus of a point, the tangent plane at which
to one of the surfaces of the system passing through it con
tains a given line.
(7) The locus of a point such that its polar plane with re
gard to a given surf ace of degree m, and the tangent plane at
that point to one of the surfaces of the system passing through
it, intersect in a line which meets a fixed right line, is a sur
face of degree m/j, + p. The locus evidently meets the fixed
line in the p points where it touches the surfaces of the
system, and in the m points where it meets the fixed surface,
these last being multiple points on the locus of order p.
(8) If in the preceding case the line of intersection is
to lie in a given plane, the locus will be a curve of degree
m(m  l)fji+mp + v. The v points where the fixed plane is
touched by surfaces of the system are points on the locus ;
and also the points where the section of the fixed surface by
the fixed plane is touched by the sections of the surfaces
of the system. But the number of these last points is
m (m  l)/i + mp.
The locus just considered meets the fixed surface in
m\m(mY)n + mp + v} points. But it is plain that these
must either be the m (m l)/tt+ mp points just mentioned, or
else points where surfaces of the system touch the fixed sur
face. Subtracting, then, from the total number the number
just written, we find that
(9) The number of surfaces of the system which touch a
fixed surface is m(m  1)V + m(m  l)p + mv ; or, more gener
ally, if n be the class of the surface, and r the degree of
the tangent cone from any point, the number is np + rp + mv.
We can hence determine the number of surfaces of the
form \U + V which can touch a given surface. For if U and
SYSTEMS OF SURFACES. 259
Fare of degree m, these surfaces form a system for which
/*=!, v = 3(ml) 2 , p = 2(wl). If, then, n be the degree
of the touched surface, the value is
n (n  I) 2 + 2n (  1) (m  1) + 3n (m  I) 2 ,
the same value as that given, Art. 576. This conclusion may
otherwise be arrived at by the following process.
578. If there be in a plane two systems of points having a
(n, m) correspondence, that is, such that to any point of the
first system correspond m of the second, and to any point of
the second correspond n of the first : and, moreover, if any right
line contains r pairs of corresponding points, then the number
of points of either system which coincide with points corre
sponding to them is m + n + r. Let us suppose that the co
ordinates of two corresponding points xy, x'y , are connected
by a relation of the degrees fj,, /*' in xy, x'y' respectively ; and
by another relation of the degrees v, v ; then if x'y' be given,
there are evidently /JLV values of xy, hence n = nv. In like
manner m = fiv'. If we eliminate x, y between the two equa
tions, and an arbitrary equation ax + by + c = Q, we obtain a
result of the degree fiv' + p,'v in x'y' : showing that if one point
describe a right line, the other will describe a curve of the
degree \iv + n'v, which will, of course, intersect the right line
in the same number of points, hence r = /*i/ + pv. But if we
suppose x' and y' respectively equal to x and y, we have
(fi + //) (y + 1/) values of x and y ; a number obviously equal
to m + n + r. [This proof assumes that the sets of m and n
points are complete intersections, but a general proof can be
given by means of the principle of correspondence.]
579. Let us now proceed to investigate the nature of the
locus of points, whose polar planes with respect to surfaces of
the system coincide with their polars with respect to a fixed
surface ; and let us examine how many points of this locus
can lie in an assumed plane. Let there be two points A and a
in the plane, such that the polar plane of A with respect to
the fixed surface coincides with the polar plane of a with
17*
260 ANALYTIC GEOMETRY OF THEEE DIMENSIONS.
respect to surfaces of the system. Now, first, if A be given,
its polar plane with regard to the fixed surface is given ; and
the poles of that plane with respect to surfaces of the system
lie, by theorem (1), on a curve of degree v. This curve
will meet the assumed plane in the points a which correspond
to A, whose number therefore is v. On the other hand, if
a be given, its polar planes with respect to surfaces of the
system envelope, by theorem (2), a developable whose class is
fi ; but the polar planes of the points of the given plane with
regard to the fixed surface envelope a surface whose class is
(m  1) 2 ; * this surface and the developable have common
p. (m  I) 2 tangent planes, which will be the number of points
A corresponding to a. Lastly, let A describe a right line,
then its polar planes with respect to the fixed surface envelope
a developable of the class m  1 ; but with respect to the
surfaces of the system, by theorem (4), envelope a surface
of the class p. There may, therefore, be p (m  1) planes
whose poles on either hypothesis lie on the assumed line.
Hence, last article, the number of points A which coincide
with points a is /u, (m  1) 2 + p (m  1) + v. The locus, then,
of points whose polar planes with respect to the system,
and with respect to a fixed surface, coincide, will be a curve
of the degree just written, and it will meet the fixed surface
in the points where it can be touched by surfaces of the
system.
580. We add a few more theorems given by De Jonquieres.
(10) The locus of a point such that the line joining it to a
fixed point, and the tangent plane at it to one of the surfaces
of the system which pass through it, meet the plane of a
fixed curve of degree m in a point and line which are pole
and polar with respect to that curve, is a curve of degree
urn (m  1) + pm + v.
This is proved as theorem (8). Let the fixed curve be the
imaginary circle at infinity, and the theorem becomes the
* It was mentioned (Art. 524) that if the equation of a plane contain two
parameters in the degree n, its envelope will be of the class n a .
SYSTEMS OF SURFACES. 261
locus of the feet of the normals drawn from a fixed point to
tin' xurftirix i if tin xii stem is a curve of degree 2//.+ 2p + v.
(11) If there be a system of plane curves, whose character
istics are /*, v, the locus of a point such that its polar with re
gard to a fixed curve of degree m, and the tangent at it to one
of the curves of the system which pass through it, cut a given
finite line harmonically, is a curve whose degree is mp + v.
Consider in how many points the given line meets the locus,
and evidently its v points of contact with curves of the system
are points on the locus. But, reasoning as in other cases, we
find that there will be m points on the line, which together
with their polars with respect to the fixed curve divide the
given line harmonically. And since these are points on the
iocus for each of the /* curves which pass through them, the
degree of the locus is mp+v. Taking for the finite line the
line joining the two imaginary circular points at infinity, it
follows that there are m (m/j. + v} curves of the system which
cut a given curve orthogonally. De Jonquieres finds that in
like manner the locus of a point such that its polar plane with
regard to a fixed surface, and the tangent plane at that point
to one of the surfaces of the system, meet the plane of a fixed
conic in two lines conjugate with respect to the conic, is a sur
face of degree mp + p. And consequently that a surface
of this degree meets the fixed surface in points where it is cut
orthogonally by surfaces of the system.
(12) If from each of two fixed points Q, Q' tangents be
drawn to a system of plane curves of class n, the locus of
the intersections of the tangents of one system with those of
the other is a curve of degree v (2w  1). For consider any
curve touching the line QQ', then one point of the locus will
be the point of contact, and n  1 of the others will coincide
with each of the points Q, Q'. And since there may be v such
curves, each of the points Q, Q', is a multiple point of order
v (n  1), and the line QQ' meets the locus in v (2n  1) points.
Let the points QQ' be the two circular points at infinity, and
it follows that the locus of foci of curves of the system is a
curve of degree v (2w  1). If we investigate, in like manner,
2<'<2 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
the locus of the intersection of cones drawn to a system of sur
faces from tiro fixed points QQ', it is evident, from what has
been said, that any plane through QQ' meets the locus in a
curve whose degree is p (2  1) ; but the line QQ' is a mul
tiple line of order p, being common to both cones in every
case where the line QQ' touches a surface of the system.
The degree of the locus therefore is 2np : and accordingly, 4p
is the degree of the locus of foci of sections of a system of
quadrics by planes parallel to a fixed plane.
Chasles has given the theorem that if there be a system
of conies whose characteristics are p, v, then 2i>  p. conies of
the system reduce to a pair of lines, and 2/u.  v to a pair of
points. It immediately follows hence, as Cremona has re
marked, that if there be a system of quadrics, whose char
acteristics are /*, v, p, of which <r reduce to cones and a to
plane conies, then considering the section of the system by
any plane, we have v = 2p  p, </ = 2/ip, and, reciprocally,
a = 2i>  p. These theorems, however, are obviously subject
to modifications if it can ever happen that a surface of the
system can reduce to a pair of planes or a pair of points.
Thus in the simple case of the system through six points and
touching two planes, the ten pairs of planes through the six
points are to be regarded as surfaces of the system, since a
pair of planes is a quadric which touches every plane. For
the same reason the problem to describe a quadric through six
points to touch three planes does not, as might be thought,
admit of 27 but only of 17 solutions, the ten pairs of planes
counting among the apparent solutions.
I have attempted to enumerate the number of quadrics
which satisfy nine conditions, Quarterly Journal, vin. 1
(1866). The same problem has been more completely dealt
with by Chasles and Zeuthen (see Comptes Bendus, Feb. 1866,
p. 405).
SECTION II. TRANSFORMATIONS OF SURFACES.
581. The theory of the transformation of curves and of
the correspondence of points on curves (explained Higher
TRANSFORMATIONS OP SURFACES. 263
Plane Curves, Chap, vm.) is evidently capable of extension
to space of three dimensions, but only a very slight sketch
can here be given of what has been done on this subject.
It will be recollected that a unicursal curve is a curve, the
points of which have a (1, 1) correspondence with those of a
straight line ; or, analytically, we can express the coordinates
x, y, z of a point of it as proportional to homogeneous functions,
of the same degree m, of two parameters X, p. Similarly, a
unicursal surface is a surface, the points of which have a
(1, 1) correspondence with those of a plane ; or, analytically,
we can express the coordinates x, y, z, w of any of its points
as proportional to homogeneous functions, of the same degree
m, of three parameters \, fi, v. When the points of a surface
have thus a (1, 1) correspondence with those of a plane, it is
evident that every curve on the surface corresponds in the
same manner to a curve in the plane, which latter curve may,
therefore, be taken as a representation (Abbildung) of the
former curve.
582. It is geometrically evident that quadrics and cubics
are unicursal surfaces. If we project the points of a quadric
on a plane by means of lines passing through a fixed point
on the surface, we obtain at once a (1, 1) correspondence
between the points of the quadric and of the plane. In the
case of the cubic, taking any two of the right lines on the
surface, any point on the surface may be projected on a plane
by means of a line meeting the two assumed lines, and we
have in this case also a (1, 1) correspondence between the
points of the surface and of the plane. From the construc
tion in the case of the quadric can easily be derived analytical
expressions giving x, y, z, w as quadratic functions of three
parameters. And such expressions can be obtained in several
other ways : for instance, coordinate systems have been
formed by Pliicker and Chasles (see Art. 393) determining
each point on the surface by means of the two generators
which pass through it. And, indeed, the method by which
the generators are expressed by means of parameters (Art. 108)
264 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
at once suggests a similar expression for the coordinates
of a point (see Art. 421, Ex. 2) on the surface. Thus, on the
quadric xw = yz, the systems of generators are X# = py, pw = \z ;
\x = vz, vw = \y, whence the coordinates of any point on
the quadric may be taken //, \v, X^., X 2 . The construction
we have indicated in the case of a cubic may also be used to
furnish expressions for the coordinates in terms of parameters ;
but other methods effect the same object more simply. For
instance, Clebsch has used the theorem that any cubic may
be generated as the locus of the intersection of three corre
sponding planes, each of which passes through a fixed point.
If A, B, C ; A', B', C' ; A", B", C" represent planes, we evi
dently obtain the equation of a cubic by eliminating X, /*, v
between the equations
\A + pB + vC = Q, \A' + fiB' + vC' = 0, \A" + t*B" + vC" = Q ;
and if we take X, /*, v as parameters, we can evidently, by
solving these three equations for x, y, z, w, which they im
plicitly contain, obtain expressions for the coordinates of any
point on the cubic, as cubic functions of the three parameters.
583. It will be more simple, however, if we proceed by a
converse process. Let us suppose that we are given a system
of equations x :y: z:w = P : Q : R : S, where P, Q, E, S are
homogeneous functions, of degree m, of three parameters X, p, v.
This system of equations evidently represents a surface, the
equation of which can be found by eliminating X, /*, v from
the equations, when there results a single equation in x, y, z, w.
If X, /*, v be taken as the coordinates of a point in a plane, the
given system of equations establishes a (1, 1) correspondence
between the points of the surface and of the plane. P = 0,
&c., denote curves of degree m in that plane. Let us first
examine the degree of the surface represented by the system
of equations, or the number of points in which it is met by
an arbitrary line ax + by+cz + dw, a'x+b'y+c'z + d'ic. To
these points evidently correspond in the plane the intersec
tions of the two curves
TRANSFORMATIONS OF SURFACES. 265
whence it follows that the degree of the surface is in general
M. If, however, the curves P, Q, R, S have a common points,*
the two curves have besides these only m  a other points of
intersection, and accordingly this is the degree of the surface.
Then to any plane section of the surface will correspond in
the plane a curve aP + bQ + cR + dS passing through the a
points : these two curves will have the same deficiency, and
we are thus in each case enabled to determine whether a plane
section of the surface contains multiple points, that is to say,
whether the surface contains multiple lines. To the section
of the surface, by a surface of degree k t ax* + &c. = cor
responds in the plane a curve aP*+ &c. =0 of degree mk, and
on this each of the a points is a multiple point of order k.
Again, the given system of equations determines a point on the
surface corresponding to each point of the plane, except in the
case of any of the a points. For each of these, the expressions
for x, y, z, w vanish, and their mutual ratios become indeter
minate : to one of these points then corresponds on the surface
not a point, but a locus, which will ordinarily be a right line
on the surface. To a curve of degree p on the plane will
correspond on the surface a curve the degree of which (that
is to say, the number of points in which it is met by an arbitrary
plane) is the same as the number of points in which the given
plane curve is met by a curve aP + bQ + cR + dS. This
number will be, in general, mp, but it will be reduced by one
for each passage of the given curve through one of the a
points.
584. In conformity, then, with the theory thus explained, let
P, Q, R, S be quadratic functions of X, /*, v ; then P = 0, &c.
represent conies ; and in order that the corresponding surface
should be a quadric, it is necessary and sufficient that the conies
P, Q, R, S should have two common points A, B. Then to
any point in the plane ordinarily corresponds a point on the
surface, except that to the points A , B correspond right lines
* For simplicity, we only notice the case where the common points are
ordinary points, but of course some of them may be multiple points.
266 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
on the surface. To a plane section of the quadric corresponds
in general a conic passing through AB ; but this conic may
in some cases break up into the line AB, together with another
line ; and in fact the previous theory shows that to every right
line in the plane thus corresponds in general a conic on the
quadric. If, however, the line in the plane pass through either
of the points A, B, the corresponding locus on the quadric is
only of the first degree, and we are thus by this method led
to see the existence of two systems of lines on the surface,
the lines of one system all meeting a fixed line A, those of
the other a fixed line B.
585. If the conies P, Q, R, S have but one common point A ,
the surface is a cubic ; but as each plane section of the cubic
corresponds to a conic, and is therefore unicursal, it must have
a double point, and the cubic surface has a double line. And
since to every line through the point A corresponds a line on
the surface, we see that the cubic is a ruled surface. In like
manner, if P, Q, R, S have no common point, the surface is
a quartic ; but every plane section being unicursal, the quartic
has a double curve of the third degree ; this is either Steiner's
surface already referred to or a ruled surface.
586. Again, let P, Q, R, S be cubic functions of X, /*, v; in
order that the surface represented should be a cubic, the curves
P, Q, R, S must have six common points. Then the deficiency
of the curve aP + &c. being unity, this is also the deficiency
of a plane section of the cubic ; that is to say, the surface has
no double line. To the six points will correspond six non
intersecting lines on the surface ; these will be one set of the
lines of a Schlafli's doublesix.
To a line in the plane corresponds on the surface a skew
cubic curve, but if the line pass through one of the six
points, the corresponding curve will be a conic, and if the line
join two of the six points, the corresponding curve will be a
right line. We thus see that there are on the surface, in
addition to the six lines with which we started, fifteen others,
TKANSFORMATIONS OF SURFACES. 267
each meeting two of the six lines. Again, to a conic in the
plane corresponds in general a sextic curve on the surface, but
this will reduce to a line if the conic pass through five of the
six points. We have thus six other lines on the surface,
each meeting five of the original six ; and thus the entire
number is made up of 27 = 6 + 15 + 6.
Suppose, however, P, Q, B, S to be still cubic functions,
but that the curves represented by them have only five common
points, then, by the previous theory, the surface represented
is a quartic, but the deficiency of a plane section being unity,
the quartic must have a double conic. There will be on the
quartic right lines, viz. five corresponding to the five common
points, one corresponding to the conic through these points,
and ten to the lines joining each pair of the points ; or sixteen
in all (see Art. 559). This is the method in which Clebsch
arrived at this theory (Crelle, LXIX. p. 142).
587. The deficiency or genus of a plane curve of degree
n with 8 double points and K cusps is \(n  Y)(n  2)  8  K ;
it is equal to the number of arbitrary constants contained
(homogeneously) in the equation of a curve of degree n  3,
which passes through the S + K double points and cusps ; and
it was found by Clebsch that there is a like expression for the
" deficiency " of a surface of degree n having a double and
a cuspidal curve ; it is equal to the number of arbitrary
constants contained (homogeneously) in the equation of a
surface of degree n  4, which passes through the double and
cuspidal curves of the given surface.* Prof. Cayley thence
deduced the expression
D = J (n  !)(  2)(n  3)  (n  3)(6 + c)
* More generally, if the surface has an tple curve and also ./pie points,
then it is found by Dr. Noether (An. di Mat. (2), v. p. 163) that the deficiency
is equal to the number of constants, as above, in the equation of a surface
of degree n4, which passes (t1) times through the tple curve (has this
for an (i l)ple line), and (;'  2) times through each;ple point (has this fora
(;2)ple point).
268 ANALYTIC GEOMETRY OF THKEE DIMENSIONS.
where b, q are the degree and class of the double curve, c, r
those of the cuspidal curve, t the number of triple points
on the double curve, ft, 7, * the number of intersections of
the two curves (ft of those which are stationary points on
the double curve, 7 stationary points on the cuspidal curve, i
not stationary on either curve), and 6 the number of singula
rities of a certain other kind. In the case where there is only
a double curve without triple points the formula is
D = \(n  l).(n  2) (  3)  (n  3) b + q.
Thus in the several cases,
Quadric surface n = 2, 6 = 0, q =
General cubic surface n = 3, 6 = 0, g =
Quartic with double right line n = 4, 6 = 1, q = Q
conic n = 4,6 = 2,g = 2
Quintic with a pair of
nonintersecting double right lines n = 5, 6 = 2, q =
Quintic with a double skew cubic rc = 5, 6 = 3, <? = 4
and in all these cases we find D = or the surface is unicursal.
CREMONA TRANSFORMATIONS.
[58?a. As in two dimensions (see Higher Plane Currex,
ch. vin.) the most powerful method of studying algebraic
surfaces of high degree is to establish a (1, 1) correspondence
between their points and those of a simpler surface. Such a
correspondence is called birational, since when the coordinates
of a point on either surface are given, those of the correspond
ing point are determined algebraically and uniquely, and
therefore rationally ; the surfaces are said to be transformed
into or represented upon one another. The representation of
a unicursal surface upon a plane is an important example ;
it follows from the definitions that any surface which can be
birationally transformed into a unicursal surface is itself
unicursal.
Let the two surfaces / (rr, y, z, w), F (X, 7, Z, W), sup
posed to exist in unconnected spaces, be in birational corre
spondence. Then if a (ar, y, z, w) and A (X, Y, Z, W} are
TRANSFORMATIONS OF SURFACES. 269
corresponding points, the coordinates of A are determined,
when those of a are given, by equations of the form
X:y:Z:TP=fc:*,:&:*4, , (1)
where fa, < 2 > $ 3 , fa are homogeneous functions of x, y, z, w
of degree n say, which is the degree of the transformation (1).
Then if we eliminate x, y,z, whom the equation /=0 by
means of the equations (1) we obtain the equation F=0 of
the second surface.
If A is an arbitrary point in the second space, not lying on
F, then (1) represents three surfaces of degree n in the first
space, which have in general n 3 common points ; so that, re
garded as a correspondence between the whole of the two
spaces, the transformation is not birational, but of order
n 3 . But if A lies on F, then one and in general only one
of the n 3 points lies on /, and this one is the point a, uniquely
determined. That is to say, the equations (1) are not ration
ally reversible by themselves, but if we join to them the equa
tion /=0, we can express the coordinates of a rationally in
terms of those of A in the form
a; : y : * : to = *! : * a : * 3 : * 4 , . (2)
where fa, $ 2 , $ 3 , $ 4 are homogeneous functions of X, Y, Z, W
of degree N say. The transformation (2) is the inverse of (1).
587&. But if, for all positions of A, the three surfaces
given by (1) have fixed points or curves in common, the num
ber of variable points in the first space corresponding to A is
less than n 3 , and if the common elements are such as to ab
sorb all but one of the variable points of intersection, we have
a (1, 1) correspondence between the whole of the two spaces ;
this is called a Cremona transformation. Then the equations
(1) can be reversed without the aid of the equation of the sur
face, and are exactly equivalent to (2), and to every surface
/ in one space there corresponds a surface F in the other.
The family of surfaces (<) whose general member is
<j) = ^fa + a^fa + a 3 fa + a t fa
corresponds to all the planes in the second space ; this family
is linear and triply infinite, and the necessary and sufficient
270 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
condition for a Cremona transformation is that any three
members shall have one and only one point of intersection
which varies with the parameters %, # 2 , a 3 , a. A family of
surfaces which satisfies these conditions is called homaloidal ;
every homaloidal family gives rise to a Cremona transforma
tion, in which it corresponds to the family of planes in the
other space, and therefore every homaloid, or member of a
homaloidal family, is unicursal. The same remarks apply to
the family ($) in the second space.
Any point common to all the surfaces <, whether an iso
lated point or lying on a common curve, is an exception to
the (1, 1) correspondence. The equations to determine the
coordinates of the corresponding point cease to have meaning
in the form (1), and in the form (2) they cease to be indepen
dent, and give a locus, either a curve or a surface, instead of a
single point.
Now let there be a curve c of degree m common to all the
<'s ; then any two <'s have a variable curve of intersection
7 of degree n?  m, which corresponds to a straight line in the
second space. To the n?  m intersections of 7 with any plane
correspond the intersections of a straight line with any #.
whose number is N. Therefore the degree of the inverse
transformation is
N=n z m.
Thus if n = 1, N must be 1 ; if n  2, then N can be 2, 3,
or 4, etc.
There are (n + l)(w + 2)(n + 3) terms in the general homo
geneous function of degree n in four variables, and therefore
as many linearly independent surfaces of degree n, and any
three of these meet in n 3 points. In order to select a homa
loidal family, we must impose on the general surface con
ditions whose postulation, or the number of independent
relations between the coefficients to which they give rise,
is $(n + l)(n + 2Xw + 3)  4, and whose equivalence, or the
number of variable points of intersection which they absorb,
is n 8  1. Guccia (Rend. C.M. Palermo, I. p. 339) has shown
that the second of these statements includes the first ; it follows
TRANSFORMATIONS OF SURFACES. 271
that none of the conditions can be such as to increase the
postulation without increasing the equivalence (e.g. the con
dition of dividing a given segment harmonically), for if it were
so, by releasing this condition we should have a family which
satisfied the condition of equivalence, but whose postulation
was too low. Thus the conditions all impose upon the family
common elements, whose aggregate is called the fundamental
system. These fundamental elements may be either points
or curves, either simple or of given multiplicity ; and the
surfaces may be required to pass through the element, or to
have contact of a given order with a given surface at the
point or along the curve, or to have several sheets passing
through the element, with different conditions of contact for
the different sheets. Further complications arise from the
intersections of different fundamental curves, or the passage
of these curves through points of higher singularity. It is
impossible at present to give any general formulae for either
postulation or equivalence, but some partial results have been
obtained (Cayley, Coll. Math. Papers, vn. p. 189 ; Noether,
An. di Mat. (2) v. p. 163 ; Hudson, Proc. Lond. Math. Soc.
(2), xi. p. 398).
587c. Before going further with the general theory, let us
discuss a simple example. There is a quadratic transforma
tion whose inverse is also quadratic, n = N=%, and the degree
of the fundamental curve is ri 2  N=2. This cannot be two
skew straight lines, for their equivalence is found to be 8,
which is too great ; it is therefore a conic c. Then the
postulation is 5 and the equivalence 6, and each is too low by
1 ; therefore there is also one simple fundamental point a.
Let the equations of c be w = 0, q(x, y, z) = 0, and let the
coordinates of a be (0, 0, 0, 1). Then the quadric homaloidal
family is
< = (a^x + a z y + a 3 z)w + a t q,
and the equations of transformation are
X : Y:Z: W=xw : yw : zw : q,
and of the inverse,
272 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
where Q is the same function of X, Y, Z that q is of x, y, z,
so that the second homaloidal family is of the same nature as
the first, with a fundamental conic C, W=Q, Q = Q, and a
fundamental point A (0, 0, 0, 1).
To a general surface/ (x, y, z, 10} of degree n there corre
sponds the surface/ (XW, YW, ZW, Q}, or say F (X, Y, Z, W],
of degree 2/i. Every term of F has a factor of the form
W a Q" ~ "., and therefore the surface has C as an ;ifold conic,
and similarly it has A as an rafold point. But if / passes
through a, then every term of /contains either x, y, or z, and
after substitution W is a factor, and the remaining factor F
is of degree 2;t  1. Now WF has C, A as nfold elements,
and as W passes once through G and not through A, there
fore F passes n  1 times through C and n times through A ;
the passage of / through a lowers by 1 the degree of the
transformed surface and also the multiplicity of C on it. The
whole plane W corresponds to the point a, and F corresponds
to the rest of /. F meets W in the conic C counted n I
times and in a residual straight line ; the part of F near the
residual corresponds to the part of / near a, or we may say
that in the limit, the residual corresponds to the tangent
plane of / at a. If / passes through c, the factor Q is to be
dropped from the equation of the corresponding surface.
This factor represents the cone vertex A standing on C ; each
of its generators corresponds to a single point of c. For
example, let q = yz + zx + xy, then the point 6 (0, 1, 0, 0) lies
on c. If /passes through 6, each term of / contains x, z, or
w as a factor, and each term of F contains XW, ZW, or
YZ + ZX + XY, and therefore the surface F contains the whole
of the straight line X=Z = Q, which is a generator of the
cone Q and corresponds to b. Similarly, if / passes s times
through a and t times through c, then the factor W'Q* is
dropped, the transformed surface is of degree 2n  s  22, and
on it A is an (n  2^fold point and C is an (n  s  )fold
curve ; the residual intersection of F with W is a curve of
degree s corresponding to the tangent cone of /at a.
Inversely, every point of the plane w corresponds to A.
TRANSFORMATIONS OP SURFACES. 273
Since the quadrics (<) meet w in the conic c, a </> can contain
no other point of w unless it degenerates and contains every
point ; hence every point of w presents the same condition
i'/, = 0) to the family (</>) and corresponds to the same point
A in the second space, which is a fundamental point, since
there is not one definite point corresponding to it. Since
every straight line meets w in one point, the variable conic
P of intersection of any two $'s passes once through A,
which is a simple fundamental point. In the same way, a
generator / of the cone q meets < in a and one point of c and
in no other point, unless it lies altogether on (f> ; every point
of I presents the same condition to <f> and corresponds to the
same point jB in the second space. Now </> meets the cone
q in c and two generators of the cone ; i.e. every </> contains
two positions of I, and every plane in the second space con
tains two positions of B, whose locus is therefore a funda
mental conic, in fact C.
These surfaces w, q are the principal elements, corre
sponding to the fundamental elements in the other space.
The Jacobian of the homaloidal family is given by a numerical
multiple of w' 2 q, and is entirely composed of principal surfaces.
By means of this transformation we can study a variety
of surfaces. For example, consider a quartic surface / having
a double conic c. Take c and any point a of the surface as
the fundamental system of a quadroquadric transformation ;
then /is transformed into a cubic surface F. All curves on
/ are transformed into curves on F, except that if / contains
a straight line I through a, since / meets the plane of c in c
counted twice and in no other point, therefore I also meets c
and is transformed into a point ; but all the curves upon a
cubic surface are known, and hence we can discover all those
that exist upon/, as well as other properties of the surface.
The quadroquadric transformation can be specialised in
two ways : (1) the conic degenerates, (2) the point lies on the
conic.
(1) If c is a pair of lines, we may assume q = yz\ then the
cone Q also degenerates into the pair of planes YZ, one cor
VOL. II. 18
274 ANALYTIC GEOMETKY OF THREE DIMENSIONS.
responding to each of the lines of c. If any surf ace /contains
only one of the lines, the degree of the transformed surface
is lowered by 1. The inverse transformation is specialised in
just the same way.
(2) The fundamental point a cannot lie in the plane of
the conic c without lying on c, for otherwise the whole
hoinaloidal family would degenerate and consist not of
quadrics but of planes. But if a moves up to a point of c, in
the limit </> touches a fixed plane at a ; this condition replaces
that of passing through the isolated point. If c does not
degenerate, let its equations now be x = 0, yw + z 2 = ; let the
coordinates of a be (0, 0, 0, 1), and let the fixed tangent plane
at a be y = 0. Then </> has the form
(aiX + a^y + a 3 z)x + a 4 (yw + z*)
and the equations of transformation are
X: Y:Z : W=x* : xy : xz : yw + z z
x:y:z:w = XY:Y 2 :YZ: XWZ 2 .
The cone q degenerates into the plane of c and the fixed
tangent plane at a. If a surface / passes through a, the
degree of the corresponding surface F is lowered by 1 ; if /
also touches y at a, the factor Y~ is dropped, and the degree
of F is lowered by 2.
587d. The only other homaloidal families of quadrics are
found to be those with a fundamental straight line and three
points, and those with three simple points and a point of con
tact. The inverse transformations are of degrees 3 and 4
respectively.
Of higher transformations, we may briefly refer to the
cubo cubic transformation given by three equations linear in
each set of coordinates, which has occupied a large space in
the literature of the subject. The fundamental system in
each space consists of a sextic curve of genus 3 ; ,to the
points of each of these curves correspond the trisecants of the
other, which generate a scroll of degree 8, the Jacobian of
the family. The sextic may degenerate in a great variety of
ways, which are usually different in the two spaces. If it
TRANSFORMATIONS OF SURFACES. 275
consists of the six edges of a tetrahedron, the cubic homaloids
all have four double points, and the equations take the ele
gant form
Xx=Yy = Zz= Ww.
The simplest transformation of degree n is monoidal (Art.
316) when the homaloids have a common multiple point of
order n 1. In particular, let w^> + ^r be any monoid, where
<f>, "fy are functions of x, y, z only ; this forms part of a homa
loidal family, of which the other independent members are
x&, y0, zd. For the equations
X : Y : Z : W = x& : y0 : zd : w<f> + ^
are rationally reversible in the form
x : y : z : w = X$ : Y<P : Z$ : W9  V.
Since we can always find a monoid through any given curve,
this gives a Cremona transformation that changes any given
curve into a plane curve ; the usual method of projection is
a birational transformation of the curve alone, and not of the
whole space.
587e. Now let us return to the general theory, and con
sider what is the principal system corresponding to the funda
mental system. To a point a lying on all the surfaces (<)
there corresponds either a surface or a curve. If it is a sur
face J', which meets every straight line, then a lies on the
variable intersection 7 of every pair of </>'s, i.e. a is either an
isolated fundamental point, or a multiple point lying on a
fundamental curve of lower multiplicity, and the degree of J'
is equal to the number of branches of 7 through a. But if a
is an ordinary point of a fundamental curve c, then 7 does not
in general pass through a, and corresponding to a we have
not a surface but a curve K. The degree of K, or the number
of its intersections with any plane, is equal to the multiplicity i
of c on (<). As a moves on c, this curve K generally moves
and describes a surface J", whose degree is equal to the num
ber of intersections of c, 7. If m is the degree of c, or the
number of intersections of c with any plane, then any $ meets
J" in m curves such as K, and the remaining part of the inter
18*
276 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
section of $, J" consists of fundamental curves only. But if
7 does not meet c, then to every point a of c there corresponds
"the same curve (7, which does not describe a surface, but is
fixed. Then every point of C corresponds to every point of c,
and C is also a fundamental curve, of degree and multiplicity
equal respectively to the multiplicity and degree of c.
It can be shown that /' is a unicursal surface, and that
its different points are in (1, 1) correspondence with the dif
ferent directions issuing from the point to which J' corre
sponds. The surfaces such as /', J", taken with proper
multiplicities, make up the Jacobian of ($), which can be
defined as the locus of double points of $'s ; this can be proved
by showing that a double point, if it lies upon /' or J", pre
sents less than four conditions to ($), and is therefore possible,
and conversely that a possible position for a double point must
correspond to a fundamental point in the other space.
All the points of /' (or of K) present the same condition
to ($), and any # which contains one such point contains
them all. If a $ degenerates, one component corresponds to
the plane in the other space, and the other component does
not correspond to a surface, but to a curve or point, and there
fore is part of the Jacobian.
To a general surface / (x, y, z, w) of degree t there corre
sponds the surface / (^, $ 2 , $ 3) $ 4 ), or say F (X, Y, Z, W),
of degree tN, and containing the second fundamental system
repeated t times ; but if/ passes through a fundamental ele
ment, the corresponding factor J is dropped from the trans
formed equation, and the degree of F is lowered. To a
general curve of degree t there corresponds a curve of degree
tN, which again is lowered if the first curve meets a funda
mental curve or passes through a fundamental point.
587/. Segre (An. di Mat. (2), xxv. p. 2) has used the quadric
transformation to analyse the higher singularities of surfaces
into their constituents. We have seen that if a is an sfold
point on/, a quadric transformation with a as fundamental
point transforms / into a surface having, corresponding to a,
CONTACT OF LINES WITH SURFACES. 277
a plane curve of degree s. If the tangent cone at a has
multiple edges, the curve has multiple points, which may be
multiple points of the surface. The order of these points is
in general less than s, but it may be as high as s, if the tangent
cone at a consists of s planes meeting in a line ; in every case,
a finite number of quadric transformations reduces the singu
larity to a set of multiple points of lower order.]
SECTION III. CONTACT OF LINES WITH SURFACES.
588. We now return to the class of problems proposed in
Art. 272, viz. to find the degrees of the curves traced on a
surface by the points of contact of lines which satisfy three
conditions, and of the scrolls generated by such lines. The
cases we shall consider are :
(A) flecnodal lines, which meet the surface in four con
secutive points ;
(B) lines which are inflexional tangents at one point and
ordinary tangents at another ;
(C) triple tangent lines, which are ordinary tangents at
three points.
Now to commence with problem A ; if a line meet a sur
face U in four consecutive points, the tangent plane meets U
in a curve having at the point a flecnode, or node having there
an inflexion on one branch ; the tangent to this branch is the
flecnodal line. We must at the point of contact not only
have C7' = 0, but also AU' = Q, A*U'=0, J 3 C7' = 0. The tan
gent line must then be common to the surfaces denoted by
the last three equations.
But since the six points of intersection of these surfaces
are all coincident with x'y'z'w', the problem is a case of that
treated in Art. 473. Since then, by that article, the condition
II = 0, that the three surfaces should have a common line, is
of degree
\'\"fj. + X"X/u/ + XX'//  XX 'X" ;
substituting
X = 1, X' = 2, X" = 3 ; /j, = n  1 , // = n  2, //' = n  3 ;
278 ANALYTIC GEOMETRY OF THEEE DIMENSIONS.
we find that II is of degree lln  24. The points of contact
then of lines which meet the surface in four consecutive points
lie on the intersection of the surface with a derived surface S
of degree lln  24.*
The intersection of this surface S with the given surface
U is a curve of degree 4 = n (lln  24), " the flecnodal curve "
of U.
589. We proceed to give Clebsch's calculation, deter
mining the equation of this surface S which meets the
given surface at the points of contact of lines which meet
it in four consecutive points. It was proved, in the last
article, that in order to obtain this equation it is necessary to
eliminate between the equations of an arbitrary plane and
of the surfaces AU', A' 2 U', A 3 U'. This elimination is per
formed by solving for the coordinates of the two points of
intersection of the arbitrary plane, the tangent plane AU',
and the polar quadric A' 2 U' ; substituting these coordinates
successively in A 3 U', and multiplying the results together.
Let the four coordinates of the point of contact be x l} x 2 , x 3 , x 4 ;
the running coordinates y lt y^, y 3 , y ; the differential coeffici
ents u lt w 2 , u 3 , u ; the second and third differential coefficients
being denoted in like manner by suffixes, as w 12 , w 123 . Through
each of the lines of intersection of AU' , A*U', we can draw
a plane, so that by suitably determining t lt t z , t 3 , 4 , we can, in
an infinity of ways, form an equation identically satisfied
J 8 V + (t iyi + t^ + t 3 y 3 + t t yj A U'
 (i).
* I gave this theorem in 1849 (Cambridge and Dublin Journal, iv. p. 260).
I obtained the equation in an inconvenient form (Quarterly Journal, i. p. 336) ;
and in one more convenient ((Philosophical Transactions, CL. 1860, p. 229)
which I shall presently give. But I substitute for my own investigation the
very beautiful piece of analysis by which Prof. Clebsch performed the eli
mination indicated in the text, Crelle, LVIII. p. 93. Prof. Cayley has observed
that exactly in the same manner as the equation of the Hessian is the trans
formation of the equation rt  s* which is satisfied for every point of a
developable, so the equation S = is the transformation of the equation
(Art. 437) which is satisfied for every point on a ruled surface.
CONTACT OF LINES WITH SURFACES.
279
We shall suppose this transformation effected ; but it is not
necessary to determine the actual values of t lt &c., for it will
be found that these quantities disappear from the result.
Let the arbitrary plane be c^ + C 2 y 2 + crf/a + c 4 y 4 , then it is
evident that the coordinates of the intersections of the arbi
trary plane, the tangent plane u l y l + u. 2 y 2 + u t y 3 + w 4 y 4 , and
J U', are the four determinants of the two systems
Pi' P2> Ps, Pi
"l.
gli
W
These coordinates have now to be substituted in A*U\ which
we write in the symbolical form (o i y l + a. 2 y 2 + a 3 y 3 + a 4 y 4 ) 3 ;
where j means = , &c., so that, after expansion, we may,
CLX^
for any term ^a^y^^j^ substitute w 123 T/ 1 ?/ 2 y 3 , &c. It is
evident then that the result of substituting the coordinates
of the first point in A A U' may be written as the cube of the
symbolical determinant ^a 1 c 2 M a p 4 , where, after cubing, we
are to substitute third differential coefficients for the powers
of the a's as has been just explained. In like manner, we
write the result of substituting the coordinates of the second
point C5'& 1 c 2 / 3 <7 4 ) 3 , where 6 : is a symbol used in the same
manner as a v The eliminant required may therefore be
written
(5'a 1 c 2 ^^ 4 ) 3 (5V 2 W;# 4 ) 3 = 0.*
The above result may be written in the more symmetrical
form
For, since the quantities a, b are after expansion replaced by
differentials, it is immaterial whether the symbol used origin
ally were a or b ; and the lefthand side of this equation when
*The reason why we use a different symbol for j, <fec., in the second
determinant is because if we employed the same symbol, the expanded result
would evidently contain sixth powers of a, that is to say, sixth differential co
efficients. We avoid this by the employment of different symbols, as in Prof.
Cayley's " Hypcrdeterminant Calculus " (Lessons on Higtier Algebra, Lesson
Xiv.), with which the method here used is substantially identical.
280 ANALYTIC GEOMETEY OF THREE DIMENSIONS.
expanded is merely the double of the last expression. We
have now to perform the expansion, and to get rid of p and
q by means of equation (I). We shall commence by thus
banishing^) and q.
590. Let us write
F = (Sa&UspJ (SbjCfaqJ, G = (2V 2 w 3 jp 4 ) (Sa^utfJ.
The eliminant is F 3 +G 3 = 0, or (F+G)  3FG(F+G) =0.
We shall separately examine F+G, and FG, in order to get
rid of p and q. If the determinants in F were so far ex
panded as to separate the p and q which they contain we
should have
F= (m t p l + m. 2 p 2 + m&i + m^ (n^ + w 2 g 2 + n 3 q 3 + 4 # 4 ),
G = (n&i + ntfz + nsp 3 + )i>^ (m^ + m 2 q 2 + m 3 q z + w 4 ? 4 ) ,
where, for example, w 4 is the determinant Ha^u^ and n is
Sb^Uz. If then i, j be any two suffixes, the coefficient of
m ( nj in F + G is (p#> +pjqi) . And we may write
F+G = SSmtHj (#& +p&^ ,
where both i and j are to be given every value from 1 to 4.
But, by comparing coefficients in equation (I), we have
whence F+G = %SSm#ijti v +
Now it is plain that if for every term of the form pq^ +pjq t we
substitute tiUj + tjU h the result is the same as if in F and G
we everywhere altered p and q into t and u. But, if in the
determinants 5"a 1 c 2 w 3 g r 4 , J6 1 c 2 w 3 g' 4 we alter q into u, the deter
minants would vanish as having two columns the same. The
latter set of terms therefore in F + G disappears, and we have
Now, if we remember what is meant by m it n,, this double
sum may be written in the form of a determinant
n , u n , u l3 , u u , !, Cj,
21 , ?<5j 2 , ?^ 3 , w 24 , rt 2 , c 2 , w
CONTACT OF LINES WITH SURFACES. 281
For since this determinant must contain a constituent from
each of the last three rows and columns it is of the first degree
in w n , &c., and the coefficient of any term u u is
 {Sa. 2 c 3 u 4 2 f & 1 c 2 w 3 + 5 > a 1 c 2 w 3 5 I 6 2 c 3 w 4 [ or (w x n 4 + w^).
In the determinant just written the matrix of the Hessian
is bordered vertically with a, c, u ; and horizontally with b, c, u.
As we shall frequently have occasion to use determinants of
this kind we shall find it convenient to denote them by an
abbreviation, and shall write the result that we have just
arrived at,
F + G=2(' C > W Y
\o, c, uj
591. The quantity FG is transformed in like manner. It
is evidently the product of
(m l p l + m. 2 p 2 + m.jp 3 + m^^ (m l q l + m 2 q 2 + m 3 q 3 + m&^
and (w^ + n. 2 p 2 + n 3 p 3 + n t pj (n l q l + n 2 q 2 + n 3 q 3 + nflj .
Now if the first line be multiplied out, and for every term
(Pil2 + Ptf[i) we substitute its value derived from equation (I),
it appears, as before, that the terms including t vanish, and it
becomes ^Sn^m/ttg, which, as before, is equivalent to ( ' c> u ),
\d, c, u/
where the notation indicates the determinant formed by
bordering the matrix of the Hessian both vertically and hori
zontally with a, c, u. The second line is transformed in like
manner; and we thus find that (F+ G) 3  3FG(F+G) =0
transforms into
/a, c, u\ f 4 fa, c, u\ a _ g (a, c, u\ (b, c, u\] =Q
\b, c, u) \ \b, c, u) \a, c, u) \b, c, u))
It remains to complete the expansion of this symbolical
expression, and to throw it into such a form that we may be
able to divide out c^ + c a x 2 + c 3 x 3 + c 4 # 4 . We shall for short
ness write a, 6, c, instead of a l x l + a z x. z + a 3 x 3 + 4 x 4 , b^ + &c.,
c^^ + &c.
592. On inspection of the determinant, Art. 590, which
(CL C U\
, ' ), it appears that since
0, C, U/
u ll x l + u 12 x 2 + u l3 x 3 + Wj 4 x 4 = (n  1) u lt &c.,
282
ANALYTIC GEOMETBY OF THREE DIMENSIONS.
this determinant may be reduced by multiplying the first four
columns by x lt a? 2 , x a , x v and subtracting their sum from the
last column multiplied by (n  1), and similarly for the rows ;
when it becomes
n f n
6 4 T O, 4 ' 0?' 6
c 4 , 0, 0,  c
0,  a,  c,
 I) 2
0,
c 2 , C 3 ,
0, 0,
which partially expanded is
where ( f ) denotes the matrix of the Hessian bordered with
\o/
a single line, vertically of a's and horizontally of 6's.
In like manner we have
a, c, u\ _ _
"
2 a _ 9 (a
c '
c ' w ) =  _1 f ci
, c, w7 (w  \Y [
Now as it will be our first object to get rid of the letter a, we
may make these expressions a little more compact by writing
c^  6^ = d lt &c., when it is easy to see that
Thus
/*. c, w\ = _J /d\ (a, c, u\ _ 1 f (a\_ n (c \\
\b, c, u)' ( n  1)2 U> \b, c, u) ~ (^TiyH W W J '
and the equation of the surface, as given at the end of the
last article, may be altered into
CONTACT OF LINES WITH SURFACES. 283
593. We proceed now to expand and substitute for each
term a^a^, &c., the corresponding differential coefficient.
Then, in the first place, it is evident that
a 3 = n (n  1) (n  2) u = ; a 2 ^ = (n  1) (n  2) u v &c.
Hence a 2 (")  (n  1) (n  2)
But the last determinant is reduced, as in many similar
cases, by subtracting the 'first four columns multiplied re
spectively by x lt x%, x 3 , 4 from the fifth column, and so
causing it to vanish, except the last row. Thus we have
a a (j?)= (n2)Hc.
Again (see Lessons on Higher Algebra, Art. 36),
We have therefore
(*)= (n  2) 2 jr~S. =  4 C  '2
Lastly, it is necessary to calculate a ( a j (**,}. Now if U mn
denote the minor obtained from the matrix of the Hessian
by erasing the line and column which contain &,, it is easy
to see that a () (^) =  (n ^)SU mil U < . n u mn c. i d Q , where the
\c / \a/
numbers m, n, p, q are each to receive in turn all the values
1, 2, 3, 4. But (see Lessons on Higher Algebra, Art. 33)
U^U^^U^Ur,
Substituting this, and remembering that SU^u^^^H, we
have
Making then these substitutions we have
*
(5) () + 4 ( "  >2 > Hci 6)  (re  2)ffc<i 0
284 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
But attending to the meaning of the symbols d^ &c., we see
that d or d^ + d 2 x 2 + d z x z + dx vanishes identically. If then
we substitute in the equation which we are reducing the
values just obtained, it becomes divisible by c 3 , and is then
brought to the form
d
594. To simplify this further we put for d its value,
when it becomes
J i b \ ;/ c M 3 oAA f fb\ ,/c\) f 2 /6\ , ib
4 K)  %)) ' 3 U) {< a)  b (a)l { c \b) ~ 2 H
Now this is exactly the form reduced in the last article,
except that we have 6 instead of a, and a instead of d. We
can then write down
= 4 {' 3 ()' + S (  2 ^ () (a)  3 <  2 ^ a
while the remaining part of the equation becomes
But (last article) the last term in both these can be reduced
to 12 (n  2) 2 H*c ( c \ Subtracting, then, the factor c 8 divides
out again, and we have the final result cleared of irrelevant
factors, expressed in the symbolical form
595. It remains to show how to express this result in the
ordinary notation. In the first place we may transform it by
the identity (see Lessons on Higher Algebra, Art. 33)
a > b \  a \ b

whereby the equation becomes
CONTACT OF LINES WITH SURFA<
Now ( ) ( ) ( 7 ) expresses the covariant which we have
\a/ \aj \bj
before called S. For giving to U mn the same meaning as
before, the symbolical expression expanded may be written
2U mH U M U rl u mHr u Ht , where each of the suffixes is to receive
every value from 1 to 4. But the differential coefficient of H
with respect to x r can easily be seen to be 2U, nn u mnr , so that S
is 5U n   , which is, in another notation, what we have
dx r dx,
called 0, Art. 544. The covariant S is then reduced to the
form S  4H$, where
where U Mi denotes a second minor formed by erasing two
rows and two columns from the matrix of the Hessian, a form
scarcely so convenient for calculation as that in which I
had written the equation, Philosophical Transactions, CL.
1860, p. 239. For surfaces of the third degree Clebsch
has observed that $ reduces, as was mentioned before, to
2U mn H mn , where H mn denotes a second differential coefficient
of H.
596. The surface S touches the surface H along a certain
curve. Since the equation S is of the form  4H$ = 0,
it is sufficient to prove that touches H. But since is got
by bordering the matrix of the Hessian with the differentials
of the Hessian, = is equivalent to the symbolical expression
TT\
O. But, by an identical equation already made use of,
we have
H (c t H\_(H\(c\_(H
*Uw wW
where c is arbitrary. Hence touches H along its intersection
(H\
] of degree In  15. It is proved then
c /
that S touches H, and that through the curve of contact
an infinity of surfaces can pass of degree 7n  15.
286 ANALYTIC GEOMETEY OF THEEE DIMENSIONS.
597. The equation of the surface generated by the flecnodal
tangents is got by eliminating xy'z'w' between U' = 0, A U' = 0,
J7' = 0, J 3 tT0; which result, by the ordinary rule, is of
degree,
n (n  2) (n  3) + 2/i (n  1) (n  3) + Sn (n  1) (n  2)
= 6w 3  22n 2 + 18n.
Now this result expresses the locus of points, whose first,
second, and third polars intersect on the surface ; and, since if
a point be anywhere on the surface, its first, second, and third
polars intersect in six points on the surface, we infer that
the result of elimination must be of the form U 6 M = 0. The
degree of M, which is the scroll (A), is therefore
a = 2w(w3) (3n2).
598. We can in like manner solve problem (B) of Art. 588.
For the point of contact of an inflexional tangent we have
U' = 0, A U' = 0, J' 2 CT = 0; and if it touch the surface again,
we have also TF" = 0, where W is the discriminant of the
equation of degree n3 in X : //,, which remains when the
first three terms of the equation of Art. 272 vanish. For
W then we have X" = (n + 3) (n  4) , /u," = (n  3) (n  4) ; and
having, as in Art. 588 and last article, X = 1, /* = n  1 ; X' = 2,
ft = n  2, we find for the degree of the resultant
2 (n  3) (n  4) +(n  2) ( + 3) (n  4)
+ 2 (n  ] ) (n + 3) (n  4)  2 (n + 3) (n  4)
= (n4)(3n* + 5n24).
The locus of the points of threepoint contact of in
flexional tangents which touch the surface elsewhere is there
fore of degree
6 8 = n(n4) (3w 2 + 5w24).
In order that a tangent at the point xy'z'w may else
where be an inflexional tangent, we must have JCT =
(an equation for which X= 1, /* = n  1), and, besides, we must
have satisfied the system of two conditions, that the equation
of degree n 2 in X : /*, which remains when the first two
terms vanish of the equation of Art. 272, may have three
roots all equal to each other. If then X', /*' ; X", p" be the
CONTACT OF LINES WITH SURFACES. 287
degrees in which the variables enter into these two condi
tions, the degree of the surface which passes through the
points of simple contact is, by Art. 473, X'/z" + x."// + (  2)X'X".
But (see Higher Algebra, Lesson 19)
X'X" = (n  4) O 2 + n + 6), X>" + X>' = (n  2) (n  4) (n + 6) ;
the degree of the resultant is (n  2) (n  4) (n 2 + 2w + 12), and
the locus of the points of simple contact of tangents which
are inflexional elsewhere is of degree
I.,  n (w  2) (n  4) (n a + 2n  12).
The equation of the scroll (B) generated by the tangents
is found by eliminating x'y'z'w between the four equations
U' = 0, JCT = 0, J 2 ET = 0, TF = 0; and from what has been
stated as to the degree of the variables in each of these equa
tions the degree of the resultant is
n (n  2) (n  3) (n  4) + 2n (n  1) (n  3) (n  4)
+ n (  l)(n  2) (n + 3) (n  4) = n (n  4) (n 3 + 3n a  20 + 18).
But it appears, as in the last article, that this resultant con
tains as a factor, U in the power 2 (n+ 3) (n  4). Dividing
out this factor the degree of the scroll (B) remains
b = n (n  3) (  4) (w 2 + 6  4).
599. Next we have the problem (C). The locus of the
points of contact of triple tangent lines is investigated in like
manner, except that for the conditions that the equation con
sidered above should have three roots all equal, we substitute
the conditions that the same equation should have two distinct
pairs of equal roots. But (see Higher Algebra, Lesson 19)
for this system of conditions we have
X'X" =  (n  4) (n  5) (n* + 3n + 6),
X>" + X V  (  2) (  4) (n  5) (n + 3) .
The degree of the resultant is, therefore,
i (n  2) (n  4) (n  5) (n s + 5n + 12),
and the degree of the locus of the points of contact of triple
tangents is
c 2 = \ n (n  2) (n  4) (n  5) (n 2 + 5n + 12).
To find the scroll (C) generated by the triple tangents, we
are to eliminate x'y'z'w' between C7' = 0, JC7' = 0, and the two
288 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
conditions, the degree of the resultant being
nfjL/i' + n (n  1) (X'/i" + ^'V) '>
but since this resultant contains as a factor 7*'*", we have to
subtract n\'\" from the number just written. Substituting
i (n  2) (n  3) (n  4) (n  5) for ///u", and the values last
given for X'X", X'/t" + X'V, we get, for the degree of the scroll
(C), after dividing by three,
c = f (n  3) (  4) (n  5) (w 2 + 3n  2).
The following examples are solved by the numbers found
in Art. 588 and the last three articles :
Ex. 1. To find the degree of the curve formed by the points of simple
intersection of the fourpoint tangents.
The complete curve of intersection with U of the ruled surface M whose
degree is a consists of the curve of points of simple intersection, whose degree
we call Oj, and of the curve of fourfold points, of degree a 4 . We have
manifestly 4o 4 + o 1 = na. Putting in the values of a, o 4 , we find
a, = 2 (n  4) (3n 2 + n  12).
Ex. 2. To find the degree of the curve formed by the points of simple
intersection of inflexional tangents which touch the surface again.
The complete curve of intersection of the scroll (B) with U consists of the
curve of points at which the tangents are inflexional, of degree b 3 ; of that of
the ordinary contacts, of degree 6 2 ; and of that of the simple intersections, of
degree 6 r Among these we have the obvious relation nb = 36 3 + 26 3 + &i ;
putting in the values of b, b s , fe a we find
6j = n(n 4) (n  5) (n 3 + 6n 2  n  24).
Ex. 3. To find the degree of the curve formed by the points of simple
intersection of triple ordinary tangent lines.
Here with a similar notation nc = 2c a + c lt whence we have
Cl = n (n  4) (n  5) (n  6) (n 3 + 3?i 3  2n  12).
COO. There remains to be considered another class of
problems, the determination of the number of tangents which
satisfy four conditions. The following is an enumeration
of these problems. To determine the number of lines which
are :
Q9) fivepoint tangents ;
(7) flecnodal (fourpoint) tangents in one place and ordinary
tangents in another ;
(&) inflexional (threepoint) in two places ;
CONTACT OF LINES WITH SURFACES. 289
(e) inflexional in one place and ordinary tangents in two
others ;
(f) ordinary tangents in four places.
601. To find the points on a surface where a line can be
drawn to meet the surface in five consecutive points, we have
to form the condition that the intersection of A U' t J 2 U', and
an arbitrary plane should satisfy A* U' as well as A 3 U'. Clebsch
applied to A*U f the same symbolical method of elimination
which has been already applied to A 3 U'. He succeeded in
dividing out the factor c 6 from this result ; but in the final
form which he found, and for which I refer to his memoir,
there remain c symbols in the second degree, and the result
being of degree 14/i  30 in the variables, all that can be con
cluded from it is that through the points which I have called
ft (Art. 600) an infinity of surfaces can be drawn of degree
14/i  30. We can say, therefore, that the number of such
points does not exceed n (Lin  24) (14n,  30).
602. The numerical solutions of the problems proposed in
Art. 600 accomplished by Dr. Schubert * are derived from the
principle of correspondence, which may be stated as follows :
Take any line and consider the correspondence between
two planes through it, such that when the first passes through
a given point there are p points which determine the second,
and when the second passes through a given point, q points
determine the first, and, moreover, such that there are g pairs
of corresponding points whose connecting lines meet an arbi
trary right line, then the number of planes of the system
which contain a pair of corresponding points is p + q ; but
since of these there are g whose connecting lines meet the
arbitrary line, the remaining p + q  g contain coinciding pairs
of points of the systems.
603. Now to determine y3. A fivepoint contact arises
from a fourpoint contact by the coincidence of one additional
* Gijtt. Nachr., Feb. 1876 ; Math. Ann., x. p. 102, xi. pp. 34878. See also
his Kalkill der abtahltnden Geometric (1879), pp. 2367, 216.
VOL. II. 19
290 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
simple point of intersection. To each of the 4 points in
a plane correspond n  4 simple intersections of the osculating
tangents at them with U ; and to each of the points % in
the plane corresponds a single fourfold point. Hence the
number p + q for these two systems is (n  4) 4 + %. But
the surface M meets any right line in a points through each
of which passes a line connecting the n  4 points a a to the
corresponding d 4 ; hence in this case g is (n  4) a. Accord
ingly, the number of coincidences of a point cz 1 with a point
a^ is
ft = (n  4) 4 + #!  (n  4) a = (n  8) 4 + 4a
= 5n(n4)(7n12).
The same number is found from the analogous relation
yS = 6 2 + Z> 3  b,
since the union of a threepoint with an ordinary contact also
leads to a fivepoint one.
Again, fourpoint tangents having another ordinary con
tact may arise either through coincidence of two simple
intersections on a fourpoint tangent, giving in a similar
manner by the principle of correspondence
7 = 2 (n  5) %  (n  5) (n  4) a ;
or, through the coincidence of a simple intersection with the
threepoint contact of an inflexional tangent which touches
elsewhere, giving
7 = (n  5) 6 3 + b :  (n  5) b ;
or, lastly, by the coincidence of two contacts of a triple ordin
ary tangent, giving
7 = 4c 2  6c.
Each method leads to
7 = 2 (n  4) (n  5) (3n  5) (n + 6).
Tangents inflexional in two places arise from the coinci
dences of an ordinary intersection with an ordinary contact
on an inflexional tangent, thus
(n  5) 6 2 + ^  (n  5) b = 2S,
which gives
S = *n(n4) (n5) (w 3 + 3w 2 + 29n  60).
Inflexional tangents having two further ordinary contacts
CONTACT OF PLANES WITH SURFACES. 291
arise from coincidences of two simple intersections among
those on inflexional tangents having one other ordinary con
tact, thus
2e = 2 (  6) 6,  (n  5) (n  6) b ;
or, from coincidence of a simple intersection with one of the
ordinary contacts among those on tangents having three such,
whence
= ( n  6) c a + 3^  3 (n  6) c
= in (n  4) (n  5) (n  6) (n 8 + 9rc 2 + 20w  60).
Finally, four ordinary contacts arise from coincidence of
two simple intersections in the case of a tangent line having
three ordinary contacts. Whence
4= 2 (n  7) Cj  (n  6) (n  7) c ;
t= T Vw(w4)O5) (n6) (n  7) (n 3 + 6w 2 + In  30).
SECTION IV. CONTACT OF PLANES WITH SURFACES.
604. We can discuss the cases of planes which touch a
surface in the same algebraic manner as we have done those
of touching lines. Every plane which touches a surface meets
it in a section having a double point ; but since the equation
of a plane includes three constants, a determinate number of
tangent planes can be found which will fulfil two additional
conditions. And if but one additional condition be given, an
infinite series of tangent planes can be found which will satisfy
it, those planes enveloping a developable, and their points of
contact tracing out a curve on the surface. It may be re
quired either to determine the number of solutions when two
additional conditions are given, or to determine the nature of
the curves and developables just mentioned, when one addi
tional condition is given.
[The former class of problem determines the six types of
singular tangent planes ; the following are the singularities
of the sections and the numbers of the planes : *
[*For a full discussion of these the reader is referred to the original
memoirs : Salmon, Trans. R. Jr. Ac., xxm. p. 461, 1855 ; Clebsch, Crelle, LXIII.
p. 14, 1863 ; Cayley, Coll. Math. Papers, vi. p. 859, 1869 ; Schubert, Math.
Ann., xi. p. 375, 1877 ; Basset, Quart. J., XL. p. 210, 1908 ; XLI. p. 21, 1909.]
19*
292 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
(i) a biflecnode,
o> 4 = a = 5n (7w 2  28n + 30) ;
(ii) a node with an undulation on one branch,
<M 6 = ft = 5n (n  4) (7n  12) ;
(iii) a tacnode,
5 = 2n (n2) (lln 24);
(iv) a flecnode and another node,
w 2 = n (n  2) (lln  24) (n 3  n 2 + n  16) ;
(v) a cusp and a node,
! = 4n (n  2) (n  3) (n 3 + 3n  16) ;
(vi) three nodes,
<S 3 = n (n  2) (n 7  4n 6 + 7n 5  45n 4
+ U4n 3  llln 2 + 548n  960).]
The first of these problems has been solved, as follows,
by Clebsch, but with an erroneous result, as has been shown
by Schubert. It was proved (Art. 537) that the points of
inflexion of the section by the tangent plane at any point on
a surface, of the polar cubic of that point, lie on the plane
xH^ + yH 2 + zH 3 + wH 4 . Let it be required now to find the
locus of points x'y'z'w on a surface such that the line joining
x'y'z'w to one of these points of inflexion may meet any
assumed line : this is, in other words, to find the condition
that coordinates of the form \X' + JJLX, \y' + py, &c. (where
xyzw is the intersection of the assumed line with the tangent
plane) may satisfy the equation of the polar with respect to
the Hessian AH', and also of the polar cubic A 3 U'. Now
the result of substitution in AH' is 4 (n  2) \H' + fiAH'.
When we substitute in A 3 U', the coefficient of X 3 vanishes
because x'y'z'w' is on the surface, and that of X vanishes
because xyzw is in the tangent plane. The result is then
3 (n2) XJ 2 U' + fiA*U' = 0. Eliminating X : yt* between these
two equations, we have 4:H'A 3 U' = 3AH'A' 2 U', where in A 3 U',
&c., we are to substitute the coordinates of the intersection
of an arbitrary line with the tangent plane ; that is to say,
the several determinants of the system
tt lf W 2 , M 3 , M 4
a, /3, 7, 8
CONTACT OF BLANKS WITH SURFACES. 293
By this substitution J 3 CT becomes in x'y'z'w' of degree
n 3 + 3 (n  1) = 4n  6, and H' being of degree 4 (n  2), the
equation is of degree 8  14. This, then, is the degree of
the locus required.
Now the points at which two fourpoint tangents can be
drawn belong to this locus. At any one of these points the four
point tangents evidently both lie on the polar cubic of that point,
and their plane will therefore intersect that cubic in a third
line which, as we saw (Art. 537), lies in the plane AH . Every
point on that line is to be considered as a point of inflexion
of the polar cubic ; and therefore the plane through the point
x'y'z'w' and any arbitrary line must pass through a point of
inflexion. The points then, whose number we are investi
gating, and which are evidently double points on the curve
US, are counted doubly among the n (lln  24) (Sn  14) in
tersections of the curve US with the locus determined in this
article. Let us examine now what other points of the curve
US can belong to the locus. At any point on this curve the
fourpoint tangent lies in the polar cubic, the section of which
by the tangent plane consists of this line and a conic ; and
since all the points of inflexion of such a system lie in the
line, the fourpoint tangent itself is, in this case, the only line
joining x'y'z'w to a point of inflexion. And we have seen
(Art. 597) that the number of such tangents which can meet
an assumed line is In (n  3) (3n  2). Now Schubert first
pointed out in applying his method of enumeration to the
present problem, as we shall immediately show, that these
lines must be counted three times. We have, then, the
equation
2a + Qn (n  3) (Sn  2) = n (lln  24) (Sn  14),
whence a = 5n (In 2  28n + 30),
which is the solution of the problem proposed. Schubert
also deduces this result from the principle of correspondence.
The points of contact of the inflexional tangents which
meet an arbitrary given right line I are easily shown as in
Art. 576, Ex. 2, to lie on the intersection of U with a surface
of degree 3n  4. This surface meets the flecnodal curve
'294 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
(see notation in Examples, Art. 599) in (3n  4) a 4 points,
which consist of the a points of contact of fourpoint tangents
which meet the line I, and the d = (3  4) a 4  a flecnodes,
whose ordinary inflexional tangent meets I.
Accordingly, we may suppose a pencil of rays in a plane
such that to each ray which meets a fourpoint tangent corre
sponds one which meets the other inflexional tangent at the
same flecnode. In such a pencil there will be a + d = (3n  4) a^
rays meeting as well a fourpoint tangent as also the other
inflexional tangent at its flecnode. But these rays include
the a 4 rays to the points of the flecnodal curve in the plane
of the pencil and (n  1) 4 which lie in the tangent planes
through the vertex of the pencil to U at flecnodes. Thus
there remain
a + d  0,4  (n  1) 4 = 2 (n  2) 4
rays having the above property. These must be the rays
which intersect tangents which have fourfold contact at
parabolic points. It is not difficult to show otherwise from
Art. 596 by the usual algebraical methods that there are
2(n2) (llw24)
points on a surface of the degree n in which coincident
inflexional tangents have a fourpoint contact.
The d tangent lines generate a ruled surface intersecting
U in a curve of degree nd which consists of the curve of
threefold points whose degree is 4 and of that of ordinary
intersections of degree a^. These give
%' + 3a 4 = nd.
Now applying the principle of correspondence, to each of
the a 4 points in a plane correspond n  3 simple intersections
of the tangents at them with U and to each of the points a/
corresponds a single flecnode. But the surface generated by
d lines meets any right line in d points through each of which
pass n  3 lines connecting a point a/ with a point a 4 . Hence
putting (n  3) d for g,
a/+(w3) a 4 (n3) d
is the number of coincidences of a flecnode and one of the
simple points on the ordinary inflexional tangent. Now we
CONTACT OP PLANES WITH SURFACES. 295
saw that in 2 (n  2) 4 fourfold points the two osculating
tangents coincide, hence the difference
/+ (n  3) 4  (n  3) d  2 (n  2) a 4 = (Sn  14) a 4  3a
is double the number of biflecnodal points, as above.
605. The second class of problems, referred to at the be
ginning of Art. 604, determines the curves traced out by the
points of contact of, and the torses enveloped by, the tangent
planes which meet the surfaces in sections having (i) a flec
node, (ii) a cusp, (iii) two nodes. The first has been partly
considered by anticipation in Art. 588 ; the other two we shall
now consider.
Let the coordinates of three points be x'y'z'w', x'y'z'w" ,
xyzw\ then those of any point on the plane through the
points will be \x' + px" + vx, \y' + py" + vy , &c. ; and if we
substitute these values for xyzw in the equation of the surface,
we shall have the relation which must be satisfied for every
point where this plane meets the surface. Let the result of
substitution be [7] = 0, then [U] may be written
x" ir + \ a  l nA it u' + \ n ~ i vA u f + i\ n  2 oj,, + vA?u + &c. = o,
where J,, = *"^ + y + "& + w"^ ,
d d d d
A = xj, + y r, . + Zj, + Wj, .
ax dy dz aw
The plane will touch the surface if the discriminant of this
equation in X, /*, v vanish. If we suppose two of the points
fixed and the third to be variable, then this discriminant will
represent all the tangent planes to the surface which can be
drawn through the line joining the two fixed points.
We shall suppose the point x'y'z'w' to be on the surface,
and the point x"y"z"w" to be taken anywhere on the tangent
plane at that point ; then we shall have U' = 0, A tl U' = 0,
and the discriminant will become divisible by the square of
AU'. For of the tangent planes which can be drawn to a
surface through any tangent line to that surface, two will
coincide with the tangent plane at the point of contact of
296 ANALYTIC GEOMETRY OF THREE DIMENSIuNS.
that line. If the tangent plane at x'y'z'w be a double tan
gent plane, then the discriminant we are considering, instead
of being, as in other cases, only divisible by the square of
the equation of the tangent plane, will contain its cube as a
factor. In order to examine the condition that this may be
so, let us, for brevity, write the equation [U] as follows, the
coefficients of X", X"~y being supposed to vanish,
T\ n  l v + iX"V/* 2 + 25/w + (V) + &c. = 0.
T represents the tangent plane at the point we are considering,
C its polar quadric, while A = is the condition that x'y'z'w"
should lie on that polar quadric. Now it will be found that
the discriminant of [ U] is of the form
T 2 A (B 2 AC)^+T 3 ^ = 0,
where <j> is the discriminant when T vanishes as well as U'
and A^U'. In order that the discriminant may be divisible
by T 3 , some one of the factors which multiply T 2 must either
vanish or be divisible by T.
606. First, then, let A vanish. This only denotes that the
point x"y"z"w" lies on the polar quadric of xy'z'w ; or, since
it also lies in the tangent plane, that the point x"y"z"w" lies
on one of the inflexional tangents at xy'z'w. Thus we learn
that if the class of a surface be p, then of the p tangent
planes which can be drawn through an ordinary tangent line,
two coincide with the tangent plane at its point of contact,
and there can be drawn p  2 distinct from that plane ; but
that if the line be an inflexional tangent, three will coincide
with that tangent plane, and there can be drawn only p  3
distinct from it. If we suppose that x"y"z"w" has not been
taken on an inflexional tangent, A will not vanish, and we
may set this factor aside as irrelevant to the present discus
sion.
We may examine, at the same time, the conditions that T
should be a factor in B*AC, and in <.
The problem which arises in both these cases is the fol
lowing: Suppose that we are given a function V, whose
degrees in xy'z'w', in x"y"z"w" and in xyzw are respectively
CONTACT OP PLANES WITH SURFACES. 297
\, /z, fj,. Suppose that this represents a surface, having
as a multiple line of order p, the line joining the first two
points ; or, in other words, that it represents a series of planes
through that line ; to find the condition that one of these
planes should be the tangent plane T, whose degrees are
n  1, 0, 1. If so, any arbitrary line which meets T will
meet V, and therefore if we eliminate between the equations
T=0, F = 0, and the equations of an arbitrary line
ax + by + cz + dw = 0, a'x + b'y + cz + d'w = 0,
the resultant E must vanish. This is of degree p in abed,
in a'b'c'd', and in x'y'z'w" , and of degree p.(n  1) + X in
x'y'z'w'. But evidently if the assumed right line met the
line joining x'y'z'w ', x"y"z"w", R would vanish even though T
were not a factor in V. The condition (M" = 0), that the two
lines should meet, is of the first degree in all the quantities
we are considering ; and we see now that R is of the form
M^R. R' remains a function of x'y'z'w alone, and is of
degree /z. (n  2) + \.
607. To apply this to the case we are considering, since
the discriminant of [U] represents a series of planes through
x'y'z'w', x'y"z"w", it follows that B*  AC and <f> both represent
planes through the same line. The first is of the degrees
2(n2), 2, 2, while <f> is of the degrees (n  2) (n 2  6) ,
n 3  2n? + n  6, n 3  2/i 2 + n  6, as appears by subtracting the
sum of the degrees of T 2 , A, and (B  AC) 2 from the degrees
of the discriminant of [U], which is of degree n (n I) 2 in
all the variables. It follows then from the last article that
the condition Cff = 0) that T should be a factor in B*  AC
is of degree 4 (n  2), and the condition (K=0) that T should
be a factor in < is of degree (n  2) (n 3 n* + n 12) . At all
points then of the intersection of U and H the tangent plane
must be considered double. H is no other than the Hessian ;
the tangent plane at every point of the curve UH meets the
surface in a section having a cusp, and is to be counted as
double (Art. 269). The curve UK is the locus of points of
contact of planes which touch the surface in two distinct
298 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
points (Art. 286). It is called by Prof. Cayley the nodecouple
curve.
608. Let us consider next the series of tangent planes
which touch the surface along the curve UH. They form a
developable whose degree is 2w (n  2) (3n  4) (Art. 576, Ex. 3).
The class of the same developable, or the number of planes
of the system which can be drawn through an assigned
point, is An (n  1) (n  2). For the points of contact are
evidently the intersections of the curve UH with the first
polar of the assigned point. We can also determine the num
ber of stationary planes of the system. If the equation of U,
the plane z being the tangent plane at any point on the curve
UH, be z + y 2 + u 3 + &c. = 0, it is easy to show that the direction
of the tangent to UH is in the line ~ = 0. Now the tan
gent planes to U are the same at two consecutive points
proceeding along the inflexional tangent y. If then w 3 do
not contain any term a; 3 (that is to say, if the inflexional tan
gent meet the surface in four consecutive points), the direction
of the tangent to the curve UH is the same as that of the
inflexional tangent ; and the tangent planes at two consecutive
points on the curve UH will be the same. The number of
stationary tangent planes is then equal to the number of inter
sections of the curve UH with the surface S. But since the
curve touches the surface (Art. 596), we have
a = 2n(n2) (lln24).
From these data all the singularities of the developable
which touches the surface along UH can be determined as in
Art. 326 ; we have
degree = n (n  2) (28w  60), class = 4n (n  1) (n  2),
rank = 2 (n  2) (3  4),
a = 2rc (n  2) (lln  24), = n (n  2) (70w  160) ;
<2g = n (n  2) (16n 4  64n 3 + 80w 2  108w + 156),
2fc = n (n  2) (784w 4  4928w 3 + 10320w 2  7444n + 548).
The developable here considered answers to a cuspidal
line on the reciprocal surface, whose singularities are got by
CONTACT OP PLANES WITH SURFACES. 299
interchanging degree and class, a and /?, &c., in the above
formula.
The class of the developable touching the surface along
UK, which is the degree of a double curve on the reciprocal
surface, is seen as above to be n (n  1) (n  2) (n 3  ri* + n 12).
Its other singularities will be obtained in the next Chapter,
where we shall also determine the number of solutions in some
cases where a tangent plane is required to fulfil two other
conditions.
CHAPTER XVII0.
THEORY OF RECIPROCAL SURFACES.
609. [WE have seen that a surface without multiple points
has in general two sets of singular tangent planes, viz. the
nodecouple torse and the spinode torse, while the reciprocal
surface has a double curve and a cuspidal curve. These
singularities, which must occur either on the surface or on
its reciprocal, are called ordinary singularities; in dealing
with the general theory of reciprocals, we suppose the original
surface to possess them all.
In connexion with these, there present themselves finite
numbers of other singular points and planes, which will,
however, be different according as the surface is regarded as
a locus or an envelope. For example, on the double curve
there are points of transition between the parts where the
section of the surface has a node with real and imaginary
tangents respectively. For an envelope, these are points of
intersection of the double and cuspidal curves which are
stationary on the latter (see Art. 624) ; while for a locus, they
are the pinchpoints, at which the two tangent planes coincide.
Further, each of these singularities is in general accompanied
by others ; for example, at a pinchpoint, any section of the
surface has a cusp, but certain sections have higher singu
larities, and their planes are singular tangent planes.]
Let b, c be the degrees of the double and cuspidal curves.
The tangent cone, determined as in Art. 277, includes doubly
the cone standing on the double curve and trebly that stand
ing on the cuspidal curve, so that if the degree of the tangent
cone proper be a, we have
a + 26 + 3c = w (n  1).
300
THEORY OF RECIPROCAL SURFACES. 301
The class of the cone is the same as the degree of the reci
procal. Let the cone have 8 double and K cuspidal edges.
Let b have k apparent double points, and t triple points which
are also triple points on the surface ; and let c have h apparent
double points. Let the curves b and c intersect in 7 points
which are stationary points on the former, in ft which are
stationary points on the latter, and in i which are singular
points on neither. Let the curve of contact a meet 6 in p
points, and c in cr points. Let the same letters accented de
note singularities of the reciprocal surface.
610. We saw (Art. 279) that the points where the curve
of contact meets A 2 U, give rise to cuspidal edges on the
tangent cone. But when the line of contact consists of the
complex curve a + 26 + 3c, and when we want to determine
the number of cuspidal edges on the cone a, the points where
b and c meet A*U are plainly irrelevant to the question.
Neither shall we have cuspidal edges answering to all the
points where a meets A' 2 U, since a common edge of the cones
a and c is to be regarded as a cuspidal edge of the complex
cone, although not so on either cone considered separately.
The following formulae contain an analysis of the intersections
of each of the curves a, b, c, with the surface J U,
a (n  2) = K + p + 2<r )
6 O2) = p+2/3 + 37 + 3* ............. (A).
The reader can see without difficulty that the points indicated
in these formulas are included in the intersections of A 2 U
with a, b, c, respectively ; but it is not so easy to see the
reason for the numerical multipliers which are used in the
formulae. Although it is probably not impossible to account
for these constants by a priori reasoning, I prefer to explain
the method by which I was led to them inductively.*
* The first attempt to explain the effect of nodal and cuspidal linea
on the degree of the reciprocal surface was made in the year 1847, in two
papers which I contributed to the Cambridge and Dublin Mathematical
Journal, n. p. 65, and iv. p. 188. It was not till the close of the year 1849 N
302 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
611. We know that the reciprocal of a cubic is a surface
of the twelfth degree, which has a cuspidal edge of the twenty
fourth degree, since its equation is of the form 64S 3 = T' 2 ,
where S is of the fourth and T of the sixth degree (Art. 519).
Each of the twentyseven lines (Art. 530) on the surf ace answers
to a double line on the reciprocal. The proper tangent
cone, being the reciprocal of a plane section of the cubic, is
of the sixth degree, and has nine cuspidal edges. Thus we
have a' = 6, b' = 27, c = 24, ri = 12, a+ 2&' + 3c' = 12.11. The
intersections of the curves c and b' with the line of contact of
a cone a through any assumed point, answer to tangent planes
to the original cubic, whose points of contact are the inter
sections of an assumed plane with the parabolic curve UH,
and with the twentyseven lines. Consequently there are
twelve points a' and twentyseven points p ; one of the latter
points lying on each of the lines, of which the double line of
the reciprocal surface is made up.
Now the sixty points of intersection of the curve a with
the second polar, which is of the tenth degree, consist of
the nine points K', the twentyseven points p', and the twelve
points a'. It is manifest, then, that the last points must
count double, since we cannot satisfy an equation of the form
9\+ 27/ti +12v = 60, by any integer values of X, /u,, v, except
(1, 1, 2). Thus we are led to the first of the equations (^4).
Consider now the points where any of the twentyseven
lines 6 meet the same surface of the tenth degree. The points
ft answer to the points where the twentyseven right lines
touch the parabolic curve ; and there are two such points on
each of these lines (Art. 287). There are also five points t
on each of these lines (Art. 530), and we have just seen that
there is one point p. Now, since the equation \ + 2ft + 5i/ = 10,
can have only the systems of integer solutions (1, 2, 1) or
however, that the discovery of the twentyseven right lines on a cubic by en
abling me to form a clear conception of the nature of the reciprocal of a cubic,
led me to the theory in the form here explained. Some few additional details
will be found in a memoir which I contributed to the Transactions of the Royal
Irish Academy, xxm. p. 461.
THEORY OP RECIPROCAL SURFACES. 303
(3, 1, 1), the ten points of intersection of one of the lines
with the second polar must be made up either p + 2/9'+ t', or
Sp' + fi+t', and the latter form is manifestly to be rejected.
But, considering the curve b' as made up of the twentyseven
lines, the points t' occur each on three of these lines : we are
then led to the formula b' (ri  2) = p + 2/9' + 3t' .
The example we are considering does not enable us to
determine the coefficient of 7 in the second formula (A),
because there are no points 7 on the reciprocal of a cubic.
Lastly, the two hundred and forty points in which the curve
c meets the second polar are made up of the twelve points a',
and the fiftyfour points /9'. Now the equation 12X + 54/* = 240
only admits of the systems of integer solutions (11, 2), or (2, 4),
and the latter is manifestly to be preferred. In this way we
are led to assign all the coefficients of the equations (A) except
those of 7.
612. Let us now examine in the same way the reciprocal
of a surface of degree n, which has no multiple points. We
have then n = n ()i  I) 2 , ri  2 = (n  2) (n 2 + 1) , a = n (n  1) ;
and for the double and cuspidal curves we have (Art. 286)
b'=*n(n  l)(n  2) (n 3  w 2 + rc12), c' = 4n (n  l)(n 2).
The number of cuspidal edges on the tangent cone to the
reciprocal, answering to the number of points of inflexion on
a plane section of the original, gives us K=3n (n  2). The
points p and a' answer to the points of intersection of an
assumed plane with the curves UK and UH (Art. 607) ;
hence p' = n (n 2)(n 3  w 2 + n  12), <r' = 4n (n  2). Substi
tute these values in the formula a(ri  2) = K + p + 2cr', and it
is satisfied identically, thus verifying the first of formulae (A).
We shall next apply the same case to the third of the
formulae (A). It was proved (Art. 608) that the number of
points /S' is 2n(n 2)(lln 24). Now the intersections of
the double and cuspidal curves on the reciprocal surface
answer to the planes which touch the original surface at the
points of meeting of the curves UH and UK. If a plane
meet the surface in a section having an ordinary double
304 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
point and a cusp, since from the mere fact of its touching at
the latter point it is a double tangent plane, it belongs in two
ways to the system which touches along UK ; or, in other
words, it is a stationary plane of that system. And, since
evidently the points ft' are to be included in the intersections
of the double and cuspidal curves, the points U, H, K must
either answer to points ft' or points 7'. Assuming, as it is
natural to do, that the points ft count double among the inter
sections of UHK, we have
7' = n{4 (n  2) } {(n  2) (n 3  n 2 + n  12) }  4n(w  2) (lln  24)
= 4w (n  2) (n  3) (n 3 + 3n  16).
But if we substitute the values already found for c, n, a', ft,
the quantity c' (n  2)  2<r'  kft 1 becomes also equal to the
value just assigned for 7'. Thus the third of the formulae (^4)
is verified. It would have been sufficient to assume that the
points ft count /JL times and that the points 7 count X times
among the intersections of UHK, and to have written that
formula provisionally c (n  2) = 2<r + pft + X/y, when, proceed
ing as above, it would have been found that the formula
could not be satisfied unless \= 1, /* = 4.
It only remains to examine the second of the formulas (^l).
We have just assigned the values of all the quantities involved
in it except t'. Substituting then these values, we find that
the number of triple tangent planes to a surface of the nth
degree is given by the formula
W = n(n 2) (V  4/i + 7/i 5  45/t 4 + 114/i 3  lllw 2
+ 548w960),
which verifies, as it gives t' = 45 when n = 3.
613. It was proved (Art. 279) that the points of contact
of those edges of the tangent cone which touch the sur
face in two distinct points lie on a certain surface of degree
(n  2) (n  3). Now when the tangent cone is, as before, a
complex cone a + 26 + 3c, it is evident that among these double
tangents will be included those common edges of the cones ab,
which meet the curves a, b in distinct points ; and, similarly
for the other pairs of cones. If then we denote by [ab] the
THEORY OP RECIPROCAL SURFAt 305
number of the apparent intersections of the curves a and 6,
that is to say, the number of points in which these curves
seen from any point of space seem to intersect, though they
do not actually do so, the following formulae will contain an
analysis of the intersections of a, b, c, with the surface of the
degree (n  2) (n  3) :
a (n  2) (n  3) = 28 + 2 [ab] + 3 [ac],
b (n  2) (n  3) = 4k + [ab] + 3 [be],
c(rc2)(w3) = 6A+[ac] + 2 [be].
Now the number of apparent intersections of two curves is
at once deduced from that of their actual intersections. For
if cones be described having a common vertex and standing
on the two curves, their common edges must answer either
to apparent or actual intersections. Hence,
* [ab] = ab 2p, [ac] = ac  3<r, [be] = be  3  2y  i.
Substituting these values, we have
a (n  2) (n  3) = 2S + 2a6 + Sac  4p  9<r )
6 (n  2) (n  3) = 4A; + ab + 3bc  9  67  3*  2/> [ ... (B).
c (n  2) (n  3) = Qh + ac + 2bc  6/3  4<y  2i  3<r j
The first and third of these equations are satisfied identically
if we substitute for , 7, p, a, &c., the values used in the last
article, to which we are to add 28'<=w (w2) ( 2 9), i' = 0,
and the value of h' got from (Art. 608),
2A' = n (n  2) (16w 4  64/i 3 + 80n 2  108n f 156).
The second equation enables us to determine k' by the equa
tion
8&' = n (n  2) (n w  6/i 9 + 16/* 8  54w 7
+ 164n 8  288n 5 + 547 rc 4  1058w 3 + 1068n 2  1214w + 1464) ;
from this expression the rank of the developable, of which b'
is the cuspidal edge, can be calculated by the formula
R' = b'*b'2k'6t'3y.
Putting in the values already obtained for these quantities
we find
jR' = n(rc2) (n3) (n a
* If the surface have a double curve, but no cuspidal, there will still be a
determinate number i of cuspidal points on the double curve, and the above
equation receives the modification [aft] = aft  2p i. In determining, however,
the degree of the reciprocal surface the quantity [ab] is eliminated.
VOL. II. 20
306
This is then the rank of the developable formed by the planes
which have double contact with the given surface.
614. From formulae (^4) and (B) we can calculate the
diminution in the degree of the reciprocal caused by the
singularities on the original surface enumerated in Art. 609.
If the degree ttf a cone diminish from m to m  I, that of its
reciprocal diminishes from m (m  1) to (m  1) (m  I  1) ; that
is to say, is reduced by I (2tn  I  1). Now the tangent cone to
a surface is in general of degree n (n  1), and we have seen
that when the surface has double and cuspidal curves this
degree is reduced by 26 + 3c. There is a consequent diminu
tion in the degree of the reciprocal surface
D = (26 + 3c) (2w 2  2w  26  3c  1).
But the existence of double and cuspidal curves on the surface
causes also a diminution in the number of double and cuspidal
edges in the tangent cone. From the diminution in the
degree of the reciprocal surface just given must be subtracted
twice the diminution of the number of double edges, and
three times that of the cuspidal edges. Now, from formulae
(^4), we have
K =(abc) (w2) + 6 + 4 7 +3.
But, since if the surface had no multiple curves, the number of
cuspidal edges on the tangent cone would be (a + 26 + 3c)(w  2),
the diminution of the number of cuspidal edges is
K = (36 + 4c) (n  2)  6/3  4 7  3*.
Again, from the first system of equations in the last article,
we have
(a  26  3c) (n  2) (n  3) = 28  8k  ISA  12 [6c],
and putting for [6c] its value
28 = (a263c)(n2) (w3)
+ Sk + 187* + 126c  36  24 7  12z.
But if the surface had no multiple curves, 28 would
= (a + 26 + 3c) (n 2) (n  3).
The diminution then in the number of double edges is given
by the formula
(46 + 6c)(n2)(n 3)  8A  18A
TIIKORY OF RECIPROCAL SURFACES. S07
Thus the entire diminution in the degree of the reciprocal
D  3K  %H. is, when reduced,
n (76 + 12c)  46 2  9c 2  Sb  15c
+ 8k + ISh  18  127  12i + 9*.
615. The formulae (B), reduced by the formula
a + 26 + 3c = n (n 1),
become a (4w + 6) = 28 a 2 4/990 )
b (  4w + 6) = 4fc  2Z> 2  9/9  67  3i  2/> ... (0).
c (  4n + 6) = 6A  3c 2  6/3  4 7  2*  3o j
To each of these formulae we add now four times the
corresponding formula (A); and we simplify the results by
writing for a 2  a  28  3/t, the degree ri of the reciprocal sur
face, by giving E the same meaning as in Art. 613, and by
writing for c 2  c  2h  3/3, the degree S of the developable
generated by the curve c ; we thus obtain the formulse in the
more convenient shape,
n  a = K  <r \
2 = 2p3* f .................. (D).
From the first of equations (^4) and (D) we may also obtain
the equation
(n  1) a = ri + p + 3cr,
the truth of which may be seen from the consideration that
a, the curve of simple contact from any one point, intersects
the first polar of any other point, either in the n points of
contact of tangent planes passing through the line joining the
two points, or else in p points where a meets b, or the a points
where it meets c, since every first polar passes through the
curves b, c.
616. The effect of multiple curves in diminishing the
degree of the reciprocal may be otherwise investigated. The
points of contact of tangent planes, which can be drawn
through a given line, are the intersections with the surface of
the curve of degree (n  I) 2 , which is the intersection of the
first polars of any two points on the line. Now, let us first
consider the case when the surface has only an ordinary
double curve of degree b. The first polars of the two points
20*
308 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
pass each through this curve, so that their intersection breaks
up into this curve b and a complemental curve d. Now, in
looking for the points of contact of tangent planes through
the given line, in the first place, instead of taking the points
where the complex curve b + d meets the surface, we are
only to take those in which d meets it, which causes a reduc
tion bn in the degree of the reciprocal. But, further, we are
not to take all the points in which d meets the surface : those
in which it meets the curve b have to be rejected ; they are in
number 26 (n  2)  r (Art. 346) where r is the rank of the
system b. Now, these points consist of the r points on the
curve b, the tangents at which meet the line through which
we are seeking to draw tangent planes to the given
surface, and of 26 (n  2)  2r points at which the two polar
surfaces touch. These last are cuspidal points on the double
curve b ; that is to say, points at which the two tangent
planes coincide, and they count for three in the intersections
of the curve d with the given surface, since the three surfaces
touch at these points ; while the r points being ordinary
points on the double curve only count for two. The total
reduction then is
nb + 2r + 3 {26 (n  2)  2r} = b(ln 12)  4r,
which agrees with the preceding theory.
If the curve 6, instead of being merely a double curve,
were a multiple curve on the surface of order of multi
plicity p, I have found for the reduction of the degree of the
reciprocal (see Transactions of tJie Royal Irish Academy,
xxm. p. 485)
6 (p  1) (3p + 1) n  2bp (p*  l~)p* (p  l)r,
for the reduction in the number of cuspidal edges of the cone
of simple contact
ft{3 (p l) 2 n p(pl) (2p  1)}  p (p  1) (p  2>,
and for twice the reduction in the number of its double edges
26p (p  1) n 2  6 (p  1) (Up  8)n
+ bp (p  1) (Qp  2)  6y (p  I) 2 +p (p  1) (4p  6) r.
[For example, consider a surface of degree mp degenerat
ing into p surfaces of degree m meeting one another in a
THEORY OF RECIPROCAL SURFACES. 309
common curve of degree m? ; then r = 2/w 2 (wl), and the
reduction in class is
m\mp (p  1) (3p + 1)  1p (p*  1)  2p 2 (p  1) (TO  1) }
= mp (mp  I) 2  p . m (m  I) 2 ,
as it ought to be.]
The method of this article is not applied to the case
where the surface has a cuspidal curve in the Memoir from
which I cite, and I have not since attempted to repair the
omission.
617. The theory just explained ought to enable us to
account for the fact that the degree of the reciprocal of a
developable reduces to nothing. This application of the
theory both verifies the theory itself and enables us to deter
mine some singularities of developables not given, Arts. 325,
&c. We use the notation of the section referred to. The
tangent cone to a developable consists of n planes ; it has
therefore no cuspidal edges and \n (n  1) double edges. The
simple curve of contact a consists of n lines of the system
each of which meets the cuspidal edge m once, and the
double curve x in r  4 points (see Art. 330). The curves m
and x intersect at the a points of contact of the stationary
planes of the system ; for since there three consecutive lines
of the system are in the same plane, the intersection of the
first and third gives a point on the curve x. It is only on
account of their occurrence in this example that I was led to
include the points i in the theory.
We have then the following table :
Notation of this Chap., n, a,b, c ; p, <r, K, ft, h, i; n, S ;
Notation of Chap. XII., r, n, x, m ; n (r  4), n, 0, ft, h, a ; 0, r ;
and the quantities t, y, R, remain to be determined. On sub
stituting these values in formulae (^4) and (D), Arts. 610, 615,
we get the system of equations
w(r2) = w (r4 + 2),
x (r  2) = n (r  4) + 2/8 + 87 + 3f ,
m (r2) = 2w + 4/9 + 7, ,~
n=  n,
2# = 2n (r  4)   3a,
310 ANALYTIC GEOMETEY OF THREE DIMENSIONS.
The first and fourth of these equations are identically true,
and the sixth is verified by the equations of Arts. 326, 327.
The three remaining equations determine the three quantities,
whose values have not before been given, viz. t the number
of "points on three lines " of the system ; 7 the number of
points of the system through each of which passes another
nonconsecutive line of the system ; and R the rank of the
developable of which x is the cuspidal edge. These quantities
being determined, we can by an interchange of letters write
down the reciprocal singularities, the number of " planes
through three lines," &c.
Ex. 1. Let it be required to apply the preceding theory to the case
considered, Art. 329. Call fcj the number of apparent double points on b,
Art. 609, &c.
Am. y = 6 (k  3) (k  4), 3* = 4 (k  3) (k  4) (k  5),
fcj = (k  3) (2fc 3  18fe 2 + 57k  65), R = 2 (k  1) (k  3).
And for the reciprocal singularities
y' = 2 (k  2) (k  3), 3t' = 4 (k  2) (k  3) (k  4),
fc/ = (k  2) (k  3) (2fc 2  lOfc + 11), R' = 6 (k  3) 2 .
Ex. 2. Two surfaces intersect the sum of whose degrees is p and their
product q.
Ans. y = q (pq  2q  6p + 16).
This follows from the table, Art. 342, but can be proved directly by the method
used (Arts. 343, 471), see Transactions of the Royal Irish Academy, xxm. p. 469.
R = 3q (p  2) {q (p  3)  1}.
Ex. 3. To find the singularities of the developable generated by a line
resting twice on a given curve. The planes of this system are evidently " planes
through two lines " of the original system : the class of the system is therefore
y ; and the other singularities are the reciprocals of those of the system whose
cuspidal edge is x, calculated in this article. Thus the rank of the system, or
the degree of the developable, is given by the formula
2fl' = 2m (r  4)  a  30.
618. Since the degree of the reciprocal of a ruled surface
reduces always to the degree of the original surface (Art. 124)
the theory of reciprocal surfaces ought to account for this re
duction. I have not obtained this explanation for ruled sur
faces in general, but some particular cases are examined and
accounted for in the Memoir in the Transactions of the Royal
Irish Academy already cited. I give only one example here.
THEORY OF RECIPROCAL SURFACES. 311
Let the equation of the surface be derived, as in Art. 464, from
the elimination of t between the equations
where a, a, &c., are any linear functions of the coordinates.
Then if we write k + l = p, the degree of the surface is /z,
having a double curve of degree  (/A  1) (/z  2), on which are
(fi  2) (JM  3) (/it  4) triple points. For the apparent double
points of this double curve we have
2fc = i O  2) O  3) (^ 2 5/i + 8);
and the developable generated by that curve is of degree
2 (JJL  2) (fj,  3) . It will be found then that we have
a = 2(^1), & = (/*!)(/* 2),* = 3(/* 2), 8 = 2(/* 2)(/* 3),
values which agree with what was proved (Art. 614), that the
number of cuspidal edges in the tangent cone is diminished
by 36 (//.  2)  3t, while the double edges are diminished by
26 (fj,  2) (JJL  3)  4fr. In verifying the separate formulae (J5)
the remark, note, Art. 613, must be attended to.
I have also tried to apply this theory to the surface, which
is the envelope of the plane aa" + b/3" + cy n + &c.. where a, , 7
are arbitrary parameters, but have only succeeded when
n = 3. We have here (see Art. 523, Ex. 2) n = 12, ri = 9, a = 18 ;
b, being the number of cubics with two double points (that is,
of systems of conic and line) which can be drawn through
seven points, is 21 ; c is 24, since the cuspidal curve is the
intersection of the surfaces of the fourth and sixth degree re
presented by the two invariants of the given cubic equation ;
for the same reason h = 180 and S = c 2  c  2h  30 = 192  3/8 ;
t, being the number of cubics with three double points (that
is, of systems of three right lines) which can be drawn through
six points, is 15. The reciprocal of envelopes of the kind we
are considering can have no cuspidal curve. This considera
tion gives K = 27, 8 = 108. The formulae (^4) and (D) then give
180 = 27 + p + la, 210 = p + 2+37 + 45, 240 = 2<r + 4 + 7,
918 = 270, < 2B = 2p/3, 3 (192  3/9) + 24 = So + .
These six equations determine the five unknowns and give
one equation of verification. We have
p = 8l, <r = 36, = 42, 7 = 0, # = 60,
312 ANALYTIC GEOMETEY OF THREE DIMENSIONS.
[As a simpler example, consider the ruled quartic surface
generated by a straight line resting on two given straight
lines and on a given conic. Each of the given lines is double
on the surface, and there is also one double generator, joining
the traces of the two given lines on the plane of the conic.
Thus the double curve is of degree 3, with one apparent
double point, equivalent to a twisted cubic of rank 4. We
must put b = 3, r = 4, n = 4 in the first formula of Art. 616 ;
the reduction in class is therefore
3(7 x 4  12)  4 x 4 = 32 = 4 x 3 2  4,
a,s it ought to be.
This reduction is twice that due to one of the noninter
secting double lines taken alone, which would give 6 = 1,
r = 0, reduction = 7 x 4  12 = 16 ; and we may consider
that the double generator, meeting both the other double
straight lines, causes no further reduction in class. For this
double generator reduces the degree of the tangent cone by
2, but also reduces the number of its cuspidal edges by 6,
on account of the two points of intersection of the generator
with the second polar, each of which has been shown to
decrease the number of cuspidal edges by 3. Since the cone
has no stationary planes, i = 0, and Pliicker's equation
i  K = 3(v  fi)
shows that to decrease K by 3 and p by 1 has no effect upon
the class.
More generally, consider the ruled surface of degree 2/i
generated by a straight line resting on two given straight
lines and on a given curve of degree /x. Each of the given
lines is /itfold on the surface, and for the reduction in class
we must put p = //,, b = 1, r = 0, n = 2/* in the second
formula of Art. 616, giving
2{(fi  1) (3/1, + l)2/x  2X/* 2  1)} = 8/*V  1) = 2/*(2/*  I) 2  2/*,
as it ought to be. There are also a certain number of double
generators, each resting twice upon the given curve, but as
before each meets the second polar in two points not on the
double lines, and has no effect upon the reduction in class.]
THEORY OF RECIPROCAL SURFACES. 313
619. It may be mentioned here that the Hessian of a ruled
surface meets the surface only in its multiple curves, and in
the generators each of which is intersected by one consecutive.
For (Art. 463) if xy be any generator, that part of the equa
tion which is only of the first degree in x and y is of the form
(xz + yw) <f>. Then (Art. 287) the part of the Hessian which
does not contain x and y is
d<b\f, dd>\ d<b d(j>Y
^ I 6 + w^ }  icz+ = } ,
dz / v dw) dz die }
which reduces to < 4 . But xy intersects </> only in the points
where it meets multiple curves. But if the equation be of the
form ux + vy' z (Art. 287) the Hessian passes through xy. Thus
in the case considered in the last article, the number of lines
which meet one consecutive are easily seen to be 2 (p  2) ;
and the curve UH, whose degree is 4//, (/A  2), consists of these
lines, each counting for two and therefore equivalent to
4 (IJL  2) in the intersection, together with the double curve
equivalent to 4 (/x1) (/* 2). Again, if a surface have a
multiple curve whose degree is m, and order of multiplicity
p, it will be a curve of degree 4 (p  1) on the Hessian, and
will be equivalent to 4 mp (p  1) on the curve UH. Now the
ruled surface generated by a line resting on two right lines
and on a curve m (which is supposed to have no actual
multiple point) is of degree 2w, having the right lines as
multiples of order m, having %m (mY) + h double generators,
and 2r generators which meet a consecutive one. Comparing
then the degree of the curve UH with the sum of the degrees
of the curves of which it is made up, we have
16m (m  1) = 8m (m  1) + 4w (m  1) + Sh + 4r,
an equation which is identically true.
ADDITION ON THE THEORY OF RECIPROCAL SURFACES.
620. [The third and fourth editions of this work contained
a memoir by Cayley on reciprocal surfaces, which is reprinted
in his Collected Mathematical Papers (xi. p. 225, vi. p. 582).
This was left in an imperfect state (Cayley, C.M.P. vi.
314 ANALYTIC GEOMETRY OF THBEE DIMENSIONS.
p. 595; Wolffling, Math.naturwiss. Mitteilungen Wurtt. I.
p. 22 (1899), n. p. 87, in. p. 55). The following articles,
while retaining Cayley's notation and arrangement, are
based on Zeuthen's memoir, " Revision et extension des
formules numeriques de la theorie des surfaces reciproques,"
Math. Ann. x. p. 446 (1876).]
It will be convenient to give the following complete list of
the quantities which present themselves. The definitions in
the first and second columns have the properties which are
the most general when the surface is regarded as a locus and
as an envelope respectively. The last set have the same
generality from either point of view.
NUMBERS NOT REFERRING TO SINGULARITIES.
Section by any plane. Tangent cone from any point,
n degree n class
a' class a degree
B' double tangents & double edges
K inflexions K cuspidal edges
ORDINARY SINGULARITIES.
Nodecouple torse. Double curve,
b' class b degree
q degree q class
k' apparent double planes k apparent double points
t' triple planes t triple points
p degree of curve of contact p class of nodal torse
Spinode torse. Cuspidal curve,
c class c degree
r' degree r class
h' apparent double planes h apparent double points
a' degree of curve of contact tr class of cuspidal torse
Common planes of the two Intersections of the two curves.
torses.
ft stationary on spinode torse /8 cuspidal on cuspidal curve
7' stationary on nodecouple 7 cuspidal on double curve
torse
THEORY OF RECIPROCAL SURFACES. 315
ORDINARY SINGULARITIES OF A SURFACE ALREADY POSSESSING :
a double curve. a nodecouple torse,
j pinchpoints / pinchplanes
a cuspidal curve. a spinode torse.
X closepoints %' closeplanes
EXTRAORDINARY SINGULARITIES.
Special multiple points. Special singular planes.
B binodes B' bitropes
U unodes U' unitropes
osculatory planes 0' oscillatory points
Tangent cone at a general Curve of contact of a general
multiple point. singular tangent plane,
fj, degree // class .
i> class v degree
77+77 double edges, of which y' + y' double tangents, of
y touch the double curve which
y lie on the nodecouple torse
%+z cuspidal edges, of which ' + ? stationary tangents, of
z touch the cuspidal curve which
z' lie on the spinode torse
u double planes u nodes
v stationary planes v' cusps
we shall also write
x for v+fy + 3 x for v' + fy' + Sf
( denotes a sum extended (5" denotes a sum extended
to these multiple points.) to these singular planes.)
Tacnodes.
f nodes on double curve
d cusps on double curve
g nodes on cuspidal curve
e cusps on cuspidal curve
i intersections of the two curves.
Zeuthen has shown (Math. Ann., ix. p. 321) that at these
tacnodes the tangent planes have the reciprocal properties, so
that the same numbers are given by the reciprocal definitions.
316 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
621. We evidently have
a = a.
The definitions of p and a agree with those given, Art.
609 : the nodal torse is the torse enveloped by the tangent
planes along the double curve ; if the double curve meets
the curve of contact a at an ordinary point, then a tangent
plane of the nodal torse passes through the arbitrary point,
that is, p will be the number of these planes which pass
through the arbitrary point, viz. the class of the torse.
So also the cuspidal torse is the torse enveloped by the
tangent planes along the cuspidal curve ; and <r will be
the number of these tangent planes which pass through the
arbitrary point, viz. it will be the class of the torse. Again,
as regards p' and <r'.: the nodecouple torse is the envelope of
the bitangent planes of the surface, and the nodecouple
curve is the locus of the points of contact of these planes ;
similarly, the spinode torse is the envelope of the parabolic
planes of the surface, and the spinode curve is the locus of
the points of contact of these planes ; viz. it is the curve UH
of simple intersection of the surface and its Hessian; the
two curves are the reciprocals of the nodal and cuspidal
torses respectively, and the definitions of p', tr' correspond to
those of p and <r.
622. The j pinchpoints are not singular points of the
double curve per se, but are singular in regard to the curve
as double curve of the surface ; viz. a pinchpoint is a point
at which the two tangent planes are coincident. The section
of the surface by a plane passing through a pinchpoint has a
cusp, and the curves of contact a of all tangent cones pass
through the point and have the same tangent there. The
/ pinchplanes have the reciprocal properties, and touch the
surface along straight lines.
623. The % closepoints are points of the cuspidal curve
at which the section of the surface has a tacnode instead of
a cusp. In the neighbourhood of a closepoint, the surface
THEORY OF RECIPROCAL SURFACES. 317
resembles a flattened cone, limited by the cuspidal curve and
the contour of the surface. The curves of contact a of all
tangent cones pass through the point and have the same
tangent there. The % closeplanes, like the pinchplanes,
touch the surface along straight lines.
624. The 7' planes are stationary on the nodecouple torse,
and cut the surface in sections having a node and a cusp
(which are not multiple points of the surface). It follows
that the points 7 are points where the cuspidal curve with
the two sheets (or say rather halfsheets) belonging to it are
intersected by another sheet of the surface ; the curve of
intersection with such other sheet belonging to the double
curve of the surface has evidently a stationary (cuspidal) point
at the point of intersection.
The /?' planes are stationary on the spinode torse, and cut
the surface in sections having a tacnode (which is not a
multiple point of the surface) instead of a cusp. The tangent
at one of these points to the section is a generator both of the
nodecouple torse and of the spinode torse. Similarly, the
tangent to the cuspidal curve at one of its ft cusps is also the
tangent at that point to the double curve ; then intersecting
the two curves by a series of parallel planes, any plane which
is, say, above the cusp, meets the cuspidal curve in two real
points and the double curve in one real point, and the section
of the surface is a curve with two real cusps and a real node ;
as the plane approaches the cusp, these approach together,
and, when the plane passes through the cusp, unite into a
singular point in the nature of a triple point ( = node + two
cusps) ; and when the plane passes below the cusp, the two
cusps of the section become imaginary, and the double curve
changes from crunodal to acnodal.
625. At a point i the double curve crosses the cuspidal
curve, being on the side away from the two halfsheets of the
surface acnodal, and on the side of the two halfsheets cru
nodal, viz. the two halfsheets intersect each other along this
318 ANALYTIC GEOMETBY OP THKEE DIMENSIONS.
portion of the double curve. There is at the point a single
tangent plane, which is a plane i'.
The equivalent numbers of double points and cusps on
the double curve are
'  3)
and
d = d + y + S' \if(v 
and on the cuspidal curve,
and
e = e
The equivalent number of intersections of the two curves
is
t = i + 3/8 + 27 + 120' + 2 [/*] + T [4V + 4' + v'J .
(Zeuthen's symbols h, k stand for h + g, k +f respectively).
626. The oscillatory point 0' is understood rather more
easily by means of the reciprocal singularity of the osculatory
plane O ; this is a tangent plane meeting the surface in a
curve having the point of contact for a triple point.
[Cayley also considered C isolated cnicnodes, for which
/*=!/ = 2, y = r) = z = =u = v = Q; and o offpoints, or triple
points lying on the cuspidal curve, for which p, =v = 3,
z = v = l t y = i/=^=w = 0; with the reciprocal singularities.
All these sets are included in 5" and S'. It is assumed that
if at a general multiple point the tangent cone breaks up,
then among its parts there are no planes, and no repeated
sheets.
A full description of all the singularities is given in
Zeuthen's memoir cited in Art. 620.]
627. These quantities satisfy the following equations :
(1) n(nl) = a + 26 + 3c
(2) a(al) = n + 2S' + 3*'
(3) c*' = 3(na)
(4) a (n  2) = K + p + 2<r  B + 2 > (M  2)  n  2(3
OP RECIPROCAL SURFACE. 31'9
b M2)
[</(/< 2)]
(6) c (n  2) = 2<r + 4/3 + y + 8*' + 16B' + 120'
+ S[*G*2)]
(7) a (rc  2) (/i  3) = 28  GU + 3 (ac  3o  x ) + 2 (a&
+ 2[>(4^ +
(8) 6 (w  2) (n  3) = 4& + 90' + (ab  1p j) +3 (be I)
+ Z[y (^2)03)
(9) c (n  2) (n  3) = 6/i + 180' + (ac  3<r  #) + 2 (ta  i)
+ 5 [2r 0*  2) (^  3)  ^]
(10) b(bl) = q + 2(k+f) + M
(11) c(cl) = r + 2(Ah^) + 3e
and eleven other equations formed from these by exchanging
accented and unaccented letters (except a, f, d, g, e, i), to
gether with one other independent relation.
628. This new relation may be presented under several
different forms, equivalent to one another in virtue of the fore
going twentytwo relations ; one of these is
a + 2r  3c  4/  3*' + 2O'  14 U'  S' [2/z' + x + 677' + 8f]
= the same expression with accented and unaccented letters
interchanged.
Another form expresses that the surface and its reciprocal
have the same deficiency p, for which Zeuthen gives the
equation
24 (p + 1) = 24w  12a  15c + c + 3r + Gg + 9e + 2<r +
+ 6 X + 12 X ' + SB + 245' + 18*7+ 6C7'+ 60'
+ 5" [3x
629. From the equations of Art. 627 we deduce
n = n (n  I) 2  n (Ib + 12c) + 46 2 + Sb + 9c 2 + 15c  8A:  18/i  9*
+ 18/8 4 12y + 12i  24^'  3B  48B'  6t7 + 90'
+ 2 \Zxy + 3xz+12yz  x (ft  1)  3 (y + *)(/* 2)
 2) (/*  3)  (i, + 2?)]
which shows the effect of each singularity in reducing the
class.
320 ANALYTIC GEOMETRY OF THKEE DIMENSIONS.
630. In the case of the general surface of degree n without
singularities, b, q, k,f, d, t, p : j; c, r, h, g, e,<r,x; /3, 7, i ;
X, B, B', U, U', 0, 0' ; X S' all vanish and we have :
n = n,
a = n (n1),
S =$n (n1) (n2) (n3),
K = n (n  1) (n  2),
n'= n (n1) 2 ,
a = n (n  1) ,
b' = in(nl) (n2) (n 3  w 2 + n  12),
fc' = \n(n 2) (n 10  6/i 9 + I6n s  54n 7 + 164w 6  288w 5
( 547w 4  1058w 3 + 1068w 2  1214n + 1464),
' = in (n  2) (n 7  4/t 6 + 7w 5  45n 4
+ 114w 3  lllw 2 + 548n  960),
q' = n (n  2) (n  3) (w 2 + 2  4),
p = n (n  2) (n 3  n 2 + n  12),
c'  4w (n  1) (n  2),
A' = ^ (n  2) (16n*  64w 3 + 80w 2  108n + 156),
r' = 2n(n2) (34),
<r' = 4n (w  2),
^ = 2n(n2)(llw24),
y'=4n(w2)(n3)(w 3 3n + 16).
THE END.
INDEX OF SUBJECTS. VOLUME II.
(For Index of Authors cited, see p. 333.)
ABBILDUNG, 263.
Acnodal double curve, 317.
Anallagmatic surfaces, 159 n., 200, 22G.
Anchorring, 10, 11.
elliptical, 23.
Anharmonic ratio of four tangent planes through generator of ruled surface,
81, 82.
Apolar linear complexes, 250.
Applicable surfaces used to construct isotropic congruence, 75.
Apsidal surfaces, 130 sqg.
Asymptotic lines,
of ruled surface, 82.
congruence generated by tangents to, 65.
Axis of linear complex, 39 sqq.
BICIRCULAR quartic curves, 226, 235.
Biflecnode, 292.
Binodes on cubic, 1G6 and n.
general, 314 sqq.
Biplanes on cubic, 166.
Birational transformation
between points in plane and quartic with nodal line, 216.
general, 268 sqq.
Cremona, between two spaces, 269 sqq.
quadratic, 271 sqq.
cubocubic, 274.
Bitangent lines,
congruence formed by, 37, 62.
to centrosurfaco of algebraic surface, 148.
to centresurface of quadric, 151.
of plane quartic and lines on cubic surface, 190.
of cyclide, 227, 228.
Bitangent planes of cone of contact of quartic from point on nodal conic, 224.
Bitropes, 314.
CASSINIANS, 11.
Cayleyan, analogue of, 178.
Central plane through ray of rifled surface, 84.
Central points on ray of ruled surface, 84.
Centres, surface of (centre surf ace), 37, 68.
of quadric Clebsch's generalised form, 141 sqq.
of surface of rath degree, characteristics of, 148 sqq.
Characteristic, of envelopes, 20, 29 sqq., 33.
of families of surfaces, differoutial equation of, 29 8*77.
VOL. II. 321 21
322 ANALYTIC GEOMETRY OF THREE DIMENSIONS.
Circles, normal congruence of, 125.
forming lines of curvature, 73.
lying on cyclides, 229.
Class of algebraic congruences, 38.
Closeplanes, 315, 317.
Closepoints, 314 sqq.
Cnicnodes, 166 n., 167, 318.
Complex surface, Pliicker's, 42, 218, 220.
Complexes, rectilinear, 36 n., 37, 38.
general treatment of, 39 sqq.
linear, 39 sqq., 208.
special linear, 40.
quadratic, 42, 45 sqq.
algebraic of any order, general treatment, 43 sqq.
Complexes, of curves, 120 sqq.
of geodesies, 121.
Cones, Rummer's, 224.
bitangent to cyclide, 228.
tangent to general surface, 300 sqq., 314, 315.
developable, 309.
ruled surface, 310.
Confocal quadrics.
congruence of tangent lines to two, 63, 71.
congruence of generators of a system of, 78.
Congruences, of right lines, 36 n., 37, 38.
order, class, and orderclass of algebraic, 38.
of rays common to quadratic and linear complex, 56.
general treatment of, 56 sqq.
normal, 37, 59, 66 sqq. (see Normal).
of first order, 64.
of second order, 56 sqq.
three ways of defining, 58, 59.
as bitangents to surface, 62.
reciprocal of, 64.
of rays meeting twisted cubic, 64.
surfaces associated with, 64 sqq.
parabolic, hyperbolic, and elliptic, 65, 86.
formed by common tangents to two confocals, 68, 71.
directed and semidirected, 71.
Ribaucour's isotropic, 74 sqq. (see Isotropic).
formed by generators of confocal hyperboloids, 78.
of lines joining corresponding points on Hessian of cubic, 178.
Congruences of curves, 58, 120 n., 123.
normal, 123 sqq.
plane normal, 124 sqq.
circular normal (cyclic systems), 125 sqq.
Congruences of spheres, 237.
Conicnode, 166 n.
Conical surfaces (cones), 5, 7, 36.
Conjugate lines of linear complex, 39.
Conjugate lines in complex, 44, 51.
complex of quadratic complex, 51.
Conoidal surfaces (conoids), 7 sqq., 36.
Contact of lines with surfaces, 277 sqq.
of planes with surfaces, 291 sqq.
Coordinates, curvilinear, 104, 115 sqq.
Coordinates, line, 37 sqq., 190, 208.
Correspondence (see also Btrational) between points in a plane, 259.
of points on two surfaces, 262 sqq.
INDEX OF SUBJECTS. 323
Cosiugular complexes, 52.
Covariauts and invariants of cubic, 191 sqq.
I'runodal double curve, 317.
Cubic surfaces, 1G2 sqq. (see also Contents).
tangent cone of, 162, 302.
reciprocal of, 1G2, 302.
with double line, 91, 1G3 sqq.
ruled, 91, 163 K<J<I.
nodes on, limit to number of, 105.
nodes on, different kinds of, 166.
twentythree forms of, 170.
canonical form, 173 sqq., 177.
Hessian of, 174 sqq.
Steinerian of, 174.
circumscribing developable along parabolic curve, 175.
polar quadrics with regard to, reducing to planes, 176.
polar cubic of plane with regard to, 179 sqq.
right lines on, 183 sqq.
triple tangent planes of, 185.
invariants and covariants of, 191 sqq.
as unicursal surface, 26 i.
Cubic, twisted, as triple line on reciprocal of quartic, 204.
as double line on quartic, 207.
Cubocubic transformation, 274.
Cunocuneus, 8.
Curvature, circular lines of, 73.
of curves of a curvilinear complex, 121, 122.
lines of, preserved in inversion, 158, 159.
on surface of elasticity, 160.
on cyclides, 73, 233.
Curvilinear coordinates, 104, 115 sqq.
Cusp on plane section, 292, 295 sqq., 315.
Cuspidal curves, of centresurface of quadric, 144 sqq., 146 n.
of parallel surface, 154.
of negative pedal of quadric, 161.
on general surface, 300, 314.
Cuspidal edges, of surfaces of family, differential equation of, 31 sqq.
of developable enveloping surface, 33.
of developables of rectilinear congruence, 63.
Cyclides,
Dupin's, 72 sqq., 115, 127, 235.
general, 200, 225 sqq.
different forms of, 234 sqq.
Loria's classification of, 237.
Cyclic systems, 125 sqq.
congruences, 126.
Cylindrical surfaces, 4, 5, 36.
Cylindroid, 8.
DECADIANOME, 242.
Deficiency of surfaces, 267 and n., 268, 319.
Deformation.
of certain surfaces into surface of revolution, 10.
of rectilineal congruence with surface, 69.
of plane normal congruence of curves, 124, 125.
Degenerate focal surface, 64.
Degree of algebraic rectilinear complex, 39.
Developable surfaces.
partial differential equation of, 26 sqq.
21
324 ANALYTIC GEOMETEY OF THEEE DIMENSIONS.
Developable surfaces cont.
of rectilinear congruence, 63.
isotropic, 78.
circumscribing cubic along parabolic curve, 175.
touching surface along intersection with any surface, 256.
touching surface and curve, 256.
touching two surfaces, 256.
curve of intersection of two, 256.
surface with, 256.
generated by line meeting two given curves, 256.
generated by line meeting same curve twice, 310.
generated by double tangent planes (nodecouple torse), 299. 304 sqq.,
314 sqq.
generated by planes of cuspidal contact (spinode torse), 175, 255, 298,
314 sqq.
singularities of, 309 sqq.
Diameter of linear complex, 3940.
Dianodal surface, 242.
Differential equations, partial.
of families of surfaces, 1 sqq.
of cylinders, 4.
of lines, 5.
of conoidal surfaces, 7.
of surfaces of revolution, 9.
generated by lines parallel to fixe I place, 15.
of ruled surfaces, 19.
of envelopes, 20 sqq.
of developables, 26.
of tubular surfaces, 28.
of characteristics of families of surfaces, 29 sqq.
of cuspidal edges of surfaces of family, 32.
of first order, 29.
of second order, 33.
satisfied by parameter triply in orthogonal system, 103 sqq.
Directed congruences, rectilinear, 71.
determined by two right lines, 57.
Directing curves, 1, 8, 18, 33.
Director surface, 59.
Directrix of special linear complex, 40.
Doublesix, 187 sqq., 266.
fours, 222.
Double tangent lines of Kummer's quartic, 52.
points (see Nodal Points).
generators, 93 sqq.
lines (see Nodal Lines).
tangent planes, locus of points of contact, 297.
DupinDarboux theorem, 118.
generalised in two ways, 120, 122.
Dupin's theorem on triply orthogonal surfaces, 98 sqq., 104, 118.
ELASTICITY, Fresnel's surface of, 160.
Elliptic congruences, 65.
Slliptic coordinates, 71, 73, 74, 136.
Elliptic functions, coordinate of point of Wave Surface in, 135.
Enneadianome, 242.
Ennead, 243.
Envelopes.
general discussion of, 20 sqq.
of surface moving without rotation, 24.
INDEX OF SUBJECTS. 325
Envelopes cont.
of spheres, 21, 25, 28, 74, 226, 233 sqq.
of spheres touching three fixed spheres, 74.
of sphere cutting fixed sphere orthogonally, and having centre on fixed
quadric, 226, 234 sqq.
of certain points associated with system of surfaces involving one vari
able parameter, 257.
Equatorial surface lof complex, 42.
Equivalence, 270.
FAMILIES of surfaces (see also Differential Equations), 1 sqq.
Fivepoint tangent line, 288 sqq.
Flecnodal lines, 277, 293.
curve, 278, 293 sqq.
tangents, surface generated by, 286.
points of simple intersection of, 288.
touching surface elsewhere, 288, 290.
Flecnode on plane section, 292 sqq., 295.
Focal conies as directing curves of congruence, 72.
as generating Dupin cyclide, 72 sqq., 236.
Focal curve of cyclide, 226, 232.
Focal plane, of rectilinear congruence, 55, 62 sqq., 68.
of normal congruence, 68.
of isotropic congruence, 77.
of rectilinear congruence, 62.
Focal points of rectilinear congruence, 55, 62 sqq.
of normal congruence, 68.
of isotropic congruence, 77.
Focal surface of rectilinear congruence, 55, 62 sqq.
degenerate, 64, 71.
of normal rectilinear congruence, 68.
of isotropic congruence, 77, 78.
of algebraic congruence, 150 n.
of congruence of lines joining corresponding points on Hessian of cubic,
178.
Foci, of rectilinear congruence, 62.
Hamilton's virtual, 60.
Fourpoint contact (see Flecnodal).
Fundamental system of elements, 271.
GEODESICS, on cone, 33.
how connected with normal congruense, 70.
on single infinite family of surfaces, 119.
HELIX, generating conoid, 8.
Hessian, of developable, 27.
of cubic, 174 sqq.
double points on, 176 n.
developable touching surface along intersection with, 255, 298.
intersection of surface with its, 297.
of general surface, 174, 281 sqq., 285, 304.
of ruled surface, 313.
Homaloid, 270 sq.
Hyperbolic congruences, 65.
Hyperboloid of one sheet, 78, 86.
INFLEXIONAL tangents.
on ruled surface, 80.
surface generated by those on U along UV, 255.
326 ANALYTIC GEOMETBY OF' THREE DIMENSIONS.
Inflexional tangents cont.
touching surface elsewhere, 277, 286, 288.
common to two points, 288, 290.
touching surface twice again, 289 sqq.
Inflexions on plane section, 292, 315.
Invariants and covariants of cubic, 191.
Inversion, of Dupin's cyclide.
of polar reciprocal, 156.
of surfaces, general treatment, 156 sqq.
of lines of curvature, 158.
of cyclide, 220, 226.
of complex of curves, 158 sq.
effect of, on geodesic torsion, 159.
of confocal quadrics, 160.
Involution, of six lines, 57 n.
of points on generator of ruled surface, 82.
of tangent planes on double line of cubic, 163.
of points of contact of tangent planes through right line in cubic, 185.
Isotropic congruence, Ribaucour's, 74 sqq.
generated from sphere, 76.
spherical representation of, 76.
focal surface of, 77.
middle envelope of, 78.
ruled surfaces of, 86.
Isotropic developable, 78.
JACOBIAN, of four quadrics, 225, 245.
of four spheres, 226.
of four surfaces, 253.
curve, of four surfaces, 251.
of homaloidal family, 273.
KUMMEB'S quartic, 50, 201, 216.
LEMNISCATE, Bernouilli's, 11.
Level, lines of.
on conoids, 8.
Limit envelope of congruence, 65.
of normal congruence, 68.
Limit points of congruence, 5'J sqq., 64.
of normal congruence, 68.
Limit surface of congruence, 64.
of normal congruence, 68.
of isotropic congruence, 75.
Linear complex, 39 sqq.
principal, of quadratic complex, 53.
Locus of vertices of quadric cones through six points, 241.
of points whose polar planes to four surfaces are concurrent, 253.
of points whose polar planes to three surfaces are collinear, 24.
of various points associated with system of surfaces involving one
variable parameter, 257 sqq.
of points of contact of flecnodal tangents, 278.
of points of contact of inflexional double tangents, 286, 287.
of points of contact of triple tangents, 287.
of points of contact of double tangent planes, 297.
of points of simple intersection of flecnodal tangents, 238.
of points of simple intersection of double inflexional tangents, 288.
of points of simple intersection of triple tangents, 288.
INDEX OF SUBJECTS. 327
MAGNIFICATION in inversion, 159.
Middle envelope, of congruence, 65.
of isotropic congruence is minimal, 78.
Middle points, on rectilinear congruence, 63.
on normal congruence, 68.
on isotropic congruence, 75.
Middle surface, of congruence, 65, 66.
of normal congruence, 68.
isotropic congruence, 75.
Minimal surface, condition for, 78, 79.
Models, of wave surface, 128 n.
of cubic surfaces, 171 n.
Monoid, 238.
Multiple generators, 94.
Multiple lines, 163.
effect of, on degree of reciprocal, 303 sqq., 307 sjq.
Multiple points, 163.
Segre's m3thoi of decomposing, 168, 276.
NODAL lines (also Double Lines), 163, 206.
on certain ruled surfaces, 96 sq.
on centrasurface of quadric, 146.
on parallel surface of surface of ?ith degree, 15i.
on surface of elasticity, 160.
on negative pedal of quadric, 161.
on cubic (see Cubic).
on quartic (see Quartic).
on general surface, 300 sqq., 314.
Nodal points, nodes (also Double Points).
of Kummer's quartic, 50, 55.
of wave surface, 130.
of surface of elasticity, 160.
of negative pedal of quadric, 161.
of cubic, 165 sqq.
of quartic with nodal right line, 218.
on cyclides, 235 sq.
on quartics, 238 sqq.
on plane sections of surface, 292 sqq., 2)5 sqq.
Nodeoouple curve, 298.
torse, 293, 300, 314 sqq.
Normal rectilinear congruences, 66 sqq.
refracted, 69.
deformed with surface, 69, 70.
defined by geodesies, 70.
mechanical construction for, 71, 74.
directed, 71, 72.
doublydirected, 71, 72, 73.
orthogonal surfaces of, 72 sqq.
of normals to algebraic surface, 149.
Normal to algebraic surface, through given point, 148.
to algebraic surface in given plane, 149.
to algebraic surface meeting a given line, 149.
Normopolar surface, 152.
OCTADIC quartic surface, 211.
Octodianome, 242.
Offpoints, 318.
328 ANALYTIC GEOMETEY OF THREE DIMENSIONS.
Order.
of algebraic complex, 37.
of algebraic congruence, 38.
Orderclass of algebraic rectilinear congruence, 38.
Oscnodal edge, 16d, 214.
Oscular edge, 167 n.
on cubic, 167.
Osculatory plane, 315, 318.
point, 315, 318.
Ovals of quartic surfaces, 200.
PARABOLIC congruences, 65.
Parabolic curve.
on centrosurface, 151.
tangent planes to surface on, 175.
developable touching cubic along, 175.
developable touching surface along, 255, 238.
Paraboloid hyperbolic, 82, 83. ^ t.
Parallel surface, of surface of ?ith degree, 152.
of quadric, 145, 154, 242.
Parameter of distribution, 85, 122 n.
Parameters, defining families of surfaces, 1.
defining systems of right lines, 36 sqq.
defining unicursal surfaces, 263 sqq.
Parametric method, applied to congruences, 59.
applied to ruled surfaces, 84.
Pedal surfaces, 155 sq.
negative, 155, 160, 242.
of ellipsoid, 159.
Pinchplanes, 314 sqq.
Pinchpoints, 202, 205, 209, 211, 212, 218, 300, 315 sqq.
Podaire, 155.
Polar.
planes and lines in linear complex, 39, 40, 42.
quadrics, of cubic reducing to planes, 176, 181.
cubic of plane with regard to cubic, 179, 181.
plane of line giving corresponding points on Hessian of cubic, 180.
plane of point on cubic withTrespect to Hessian, 183.
Postulation, 270.
Principal.
linear complexes of quadratic complex, 53.
planes of congruence, 61, 64.
surfaces of congruence, 65.
elements and surfaces in Cremona transformation, 273.
ProHessian, 27.
Projection of lines on cubic into bitangents of plane quartic, 190.
QUADHICS.
system of, involving one variable parameter, 262.
satisfying nine conditions, 262.
as unicursal surfaces, 263 sqq.
homaloidal families of, 271 sqq., 274.
Quadroquadric transformation, 271 sqq.
Quartic surfaces (see also Contents).
Rummer's, 50, 201, 246.
Stoiner's, 171 n., 201, 207, 213 sq.
general treatment of, 200 s/</.
writers who have studied theory of, 200, 201.
with singular lines, 202 sqq.
INDEX OF SUBJECTS. 329
Quartic surfaces (see also Contents) cont.
classification of scrolls, 20213.
with triple linos, 202 sqq.
with double lines, 206 sqq.
with twisted cubic for double lines, 207 sqq.
with conic and right line for double lines, 210 sqq.
with three right lines for double lines, 211.
with two nonintersecting double right lines, 212.
with three concurrent double linos, 213.
with one double right line, 215, 217 sqq.
with eight nodei and a double right line, 218.
with double conic, 215, 221 sqq. 273.
with quadri quadric curves, 225 and n.
with circle at infinity for nodal curve (cyclides), 225 sqq.
triply orthogonal system of do., 113, 115, 232.
with cuspidal conic, 238.
without singular lines, 238.
with nodes, 23J sqq.
Weddle's, 241, 244.
Quintic surfaces, some unicursal, 263.
HADII, principal, parametric equation for, 79.
Rays, 36 n. (see also Right Lines).
Reciprocal, of rectilinear congruence, 64.
polar, of wave surface, 132 sq., 141.
of surface of centres of quadric, 146 n.
of cubic surface, 162, 302.
of cubic with double line, 161, 165.
of cubic with four double points, 171, 213.
of quartic scrolls with triple lines, 2036.
of quartic scrolls with double lines, 203, 210, 211.
of quartic with three concurrent double lines, 171, 213.
of octadic quartic, 241.
surfaces, general theory of, 300 sqq., 312 sqq.
of surface without multiple points, 303 sqq.
of developable, 309.
of ruled surface, 310 sqq.
Reference, surface of, 59.
Reflexion, of normal congruence, 69.
of isotropic congruence, 78.
Refraction of normal congruence, 69.
Regulus, 38.
Revolution, surfaces of.
partial differential equation, 9 sqq.
surfaces deformable into, 10.
Right lines, systems of, 36 sqq.
on cubic, 183 sqq., 302.
on quartic with nodal right line, 216, 217.
on quartic with nodal conic, 222 sqq.
Ring, parabolic, 242.
elliptic, 242.
Ruled surfaces (see also Scrolls), 36, 36 n., 38.
of congruence, 38, 65, 86.
general treatment of, 80.
normals along generator of, 82.
parametric treatment of, 84.
double curve on skew, 88.
multiple curve on a certain typo of, 89.
characteristics of tangent cone to, 89.
330 ANALYTIC GEOMETRY OF THEEE DIMENSIONS.
Ruled surfaces (see also Scrolls) cont.
generated by directing curves, 90 sqq.
of third degree, 163.
reciprocal of, 310 sqq.
Hessian of, 313.
SCHOLAR line, 167 n.
Scrolls, quartic, classification of, 20213.
generated by lines satisfying three conditions, 277 sqq.
generated by flecnodal tangents, 283.
generated by double inflexional tangents, 287.
generated by triple tangents, 237.
Singular.
lines, points, planes, and surfaces of complex of any degree, 43 sqq.
lines, points, planes, and surfaces of quadratic complex, 4:5, 45 sqq.
tangent planes, 291 sqq., 300 sqq.
Singularities of sections by tangent planes, 291 sqq.
of developables, 309.
ordinary, 300, 314.
extraordinary, 315.
Slope, lines of greatest, on conoids, 8.
Special linear complex, 40.
Spheres, coordinates consisting of five, 231 sqq., 237, 238.
envelopes of (see Envelopes).
Spheroconics for wave surface, 135, 136.
Spheroquartics, 136, 234.
Spinode torse, 298, 300, 314 sqq.
Staircase, spiral, 8.
Steinerian of cubic, 174.
Steiner's quartic, 171 n., 201, 207, 213 sq.
Striction, line of, 83.
of hyperbolic paraboloid, 83.
of hyperboloid, 84.
Surfaces (see also under Differential Equations).
ruled (see Ruled).
Pliicker's equatorial, 42.
Pliicker's complex, 42.
singular, of complexes, 43 sqq.
associated with rectilinear congruence, 64.
triplyorthogonal, systems of, 98 sqq.
apsidal, 130 sqq.
of third degree, 162 sqq. (see Cubic).
of fourth degree, 200 sqq. (see Quartic).
general theory of, 253 sqq.
systems of, 253 sqq.
of nth degree satisfying one less than number of conditions required t
determine, 256 sqq.
Symmetroid, 242, 245.
Syn normals, 152 and n.
System, fundamental, 271.
Syzygy of quadrics, 245.
TACSODB, 212, 218, 292, 315, 317.
Tactinvariant, of three surfaces, 253.
of two surfaces, 255.
Tangent lines to surface, satisfying three conditions, 277 sqq., 286.
to surface, satisfying four conditions, 277 sqq., 286.
to surface , touching four times, 289, 291.
I
INDEX OF SUBJECTS. 331
Tangent planes, having conic contact with Rummer's quartic, 50.
having circular contact with wavo surface, 133.
having conic contact with centrasurface of quadric, 146 n.
triple, of cubic, 185.
double, of cubic, 185.
singular, 291 sqq., 295 sqq.
through tangent line or inflexional tangent, 290.
double locus of points of contact, 297.
triple, 304.
osculatory, 318.
Tetrahedroid, 254.
Tetranodal cyclide, 235.
Threadconstruction, for normal congruence, 71.
for Dupin's cyclide, 74.
Torsal line, 167 n.
Torse (see Developable).
Torsion, of curves of a linear rectilinear complex, 42.
of curves of a curvilinear complex, 119 sqq.
geodesic, effect of inversion on, 159.
Torus, 242.
Transformation of surfaces (see also Birational), 262 sqq.
Triple lines on quartic surface (see Quartic), 202 sqq.
Triple point.
on quartic, 238 sqq.
Triple tangent lines, 277.
locus of points of contact of, 237.
scroll generated by, 2?7 sq.
Triple tangent planes, 304.
Triplyorthogonal systems of surfaces, 98 sqq.
differential equation expressing condition that r = / (x, y, z) may form
a, 10311, 118.
special cases of, 11115, 232.
condition for in curvilinear coordinates, 115 sqq.
DupinDarboux theorem on, 118 sqq.
corresponding to a cyclic system, 127.
Tropes (see also Tangent Planes), 213, 238, 246.
Tubular surfaces, 21 sqq., 28.
UNICUKSAL surfaces, 263 sqq.
Uniplane, 167.
Unitropes, 315.
Unode, 167, 315.
WAVE surface, 128 sqq.
sixteen nodal points of, 129, 130, 250.
sixteen tangent planes of circular contact of, 133.
expressed by elliptic functions of two parameters, 135.
equation of, in elliptic coordinates, 136.
two real sheets of, 137.
reciprocal of, 132, 133, 141.
model of, 128 n.
as a tetrahedroid, 250.
\\V 1<1 lu's quartic, 241, 244.
INDEX OF AUTHORS CITED.
APPELL, 135 n.
Aronhold, 195.
BALL, 8, 8 n.
Baker, 177, 189, 201, 244.
Basset, 168, 218, 291 n.
Bateman, 201.
Beltrami, 70.
Bennett, 177, 190.
Bernouilli, 11.
Bertrand, 128, 200.
Bianchi, 36 n., 125 n., 127 n.
Blythe, 171 n.
Bonnet, 103 n.
Boole, 3 n., 26 n., 30 n.
Bouquet, 103 n., 111.
Brill, 128.
Brioschi, 128 n.
CASEY. 158 ?!., 200, 225, 229, 232 n.,
233, 234.
Cassini, 156 n.
Castelnuovo, 214.
Cauchy, 128.
Cayley, 28 n., 36 n., 57 n., 89 ., 90 n.,
92 n., 96 and n., 97, 102 sqq., 103 n.,
105, 106?i., 108, 110, 111, 118, 128,
148 ., 160, 164, 166 n., 167 n., 171 n.,
178, 183 n., 200,201,206?i., 208, 209 n.,
211 n., 212 sqq., 240, 211, 242, 243 n.,
267, 271, 278 n., 279 n., 291 n., 298,
313, 318.
Chasles, 36 n., 57 ., 200, 262, 263.
Chillemi, 201 n.
Glebsch, 141, 143 sqq., 148, 173 n., 182,
199 and n., 200, 201 n., 222, 264, 267,
278 and n., 285, 289, 291 n., 292.
Connor, 201.
Cotter, 128.
Cotty, 201 n.
Crelle, 199 n., 200 n., 201 n.
Cremona, 165 n ., 173 n., 200, 223 n.,
260, 268 sqq., 275.
DARBOCX, 103 n., 112 and n., 113, 117
7i., 118 and n., 119 and ., 121, 122,
123 n., 127, 128, 146 n., 148 n., 150,
158 n., 200, 214, 222, 225, 229, 232
n., 233, 234, 235, 238.
De Jonquierea, 257, 260 sqq.
Desboves, 152 n.
Dupin, 69, 72, 73 and n., 74, 98, 102,
104, 105, 114 n., 115, 118, 119, 120,
122, 127, 200 n., 235, 236.
EISKNHART, 36 n., 123 n.
FORSYTE, 3 n., 30 n., 112 n., 117 n.,
119 ?i.
Fraser, 229 n.
FrenetSerret, 42, 120, 121.
Fresnel, 123 and n., 160.
GARNIER, 201 n.
Gauss, 59, 84, 1C4.
Geiser, 222, 223 n.
Gerbaldi, 201 n.
Gordon, 173 n.
Guccia, 270.
Guischard, 125 n.
HAMILTON, 36 n., 58, 60, 62, 64, 67,
128, 133 7i.
Henderson, 171 n., 185 n., 189.
Herschel, 128.
Hirst, 156.
Hudson, 201, 246 n., 271.
Hudson, H. P., Preface.
JESSOP, 36 71., 243 n., 246.
Joachimsthal, 119, 215.
KLEIN, 36 n., 150 n., 171.
Korndorfer, 200.
Kummer, 36 n., 43, 50, 59, 128, 200,
201 71., 224, 243, 246, 247, 249, 250.
LACOCR, 135 n , 201 n.
Lacroix, 83.
Lam(5,98., lOiandn., 115 sqq., 128 n.
Lancret, 73.
Levy, 103 n., 127.
Lie, 41.
Lilienthal, 123 n.
Lloyd, H., 128 ., 133 n.
Loria, 36 n., 128, 200, 237.
I MACCULLAOH, 128 n., 131 n., 133 n.
333
334
ANALYTIC GEOMETEY OF THEEE DIMENSIONS.
MacWeeney, 42.
Malus, 69.
Mannheim, 128 n.
Marcks, L., 150 n.
Maroni, 201 n.
Meunier, 158.
Monge, 29, 33, 34.
Montesano, 201 n., 215.
Morley, 201.
Moutard, 200, 254 n.
NOETHER, 267 n., 271.
PITCHER, 36 n., 37, 39, 42, 128 n., 133
n., 168, 218, 220, 237, 263.
Poncelet, 189, 190.
Purser, F., 151, 152 n.
QUETELET, 36 n.
REMY, 201.
Ribaucour, 36 n., 70, 74, 75 n., 76, 78
w., 123 ?i., 124, 125 sqq., 127 and n.
Roberts, M., 33.
Roberts, S., 153 n., 154.
Roberts, W., 114, 128 ., 136, 155, 156
n.
Rogers, R. A. P., 121 n.
Rohn, 200 and n., 201,238 sqq., 243 n.,
244.
Russell, R., 43 n., 178, 189.
SALMON, 291 n.
Sannia, 66 n., 86 n.
Schlafli, 168, 171 n., 186 sqq., 188, 189,
190, 266.
Schmidt, 200.
Schrotcr, 201 n.
Schubert, 289, 291 ., 292, 293.
Schur, 190.
Schwarz, 200.
Segen, 200, 212.
Segre, 168, 200, 221, 237, 276.
Serret, 103 n., Ill sqq.
Sisam, 200, 225 n.
Steiner, 171 ., 173 n., 180, 181 n., 201,
213, 214, 215, 221, 266.
Sturm, 173 n., 215 n.
Sylvester, 57 ., 173, 176, 192.
ToHTOLINI, 155.
Traynard, 201 n.
VAHLEN, 201 n.
Valcntiner, 225 n.
Van der Vries, 201 and n.
WEBB, G. R., Preface.
Weber, 128 n.
Weddle, 201,241, 244.
Williams, 200.
Wolffling, 128 n., 313.
ZEDTHEN, 200, 217 n., 224, 314, 315,
318.
Zimmerman, 201.
ABERDKKN : THE UNIVERSITY PRSSfl
16 197)
QA Salmon, George
553 A treatise on the analytic
S25 geometry of three dinenaiona
1912 5th ed.
v.2
PLEASE DO NOT REMOVE
CARDS OR SLIPS FROM THIS POCKET
UNIVERSITY OF TORONTO LIBRARY