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Full text of "A treatise on the analytic geometry of three dimensions. Rev. by Reginald A.P. Rogers"

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 tri-dimensional 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 
twenty-two 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 co-ordinates, Lame's equations and the 
connection between those and Cayley's differential 
equation (486). In 486ft and c, the Dupin-Darboux 
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 twenty-seven right lines on a cubic 
surface and the twenty-eight 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 
sub-divided 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. 588-608) 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. 620-630). 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 
re-written 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 order-class 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 sixteen-nodal 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 (Malus-Dupin), "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 

" Thread-construction " for normal congruence 71 

Congruence of tangent lines to two confocal quadrics .... 71 

Directed normal congruences 71 

The directing curves of a doubly-directed normal congruence are focal 

conies ............ 72 

The surfaces normal to a doubly-directed congruence are cyclides of 

Dupin, having circular lines of curvature 73 

Cyclides of Dupin ; equations in elliptic and Cartesian coordinates, 

thread-construction, 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 one-sheeted 

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 bi-circular quartics 11:! 

A class of systems which are envelopes of a one-parameter 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 Dupin-Darboux 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 CENTRO-SURFACE, 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 centre-surface of quadric 141 

Section by principal planes 144 

Its cuspidal curves 145 

Its nodal curves 146 

Centro-surface 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 centre-surface . . . 150 

Grouping of the 28 bitangents of centro-surface 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 

Twenty-three 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 double-sixes .... 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 twenty-seven 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 non-plane 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 

Five-fold generation of cyclides 227, 229 

Cases where centre-locus is specially related to Jacobian sphere . 227, 237 

The five Jacobian spheres are mutually orthogonal 230 

Confocal cyclides 232 

Sphero-quartics 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 octo-nodal 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 non-singular quartics 251 



CHAPTER XVII. 

GENERAL THEORY OP SURFACES. 

SECTION I. SYSTEMS OP SURFACES 253 

Jacobian of four surfaces 253 

Degree of tact-invariant 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 

Quadro-quadric 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 

Node-couple 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 
n-1 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 direction-angles of the generating line. Remember- 
ing that U lt U 2 , U 3 are proportional to the direction-cosines 
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 direction-cosinea 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'=x-l0,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 direction-cosines 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 direction-cosines 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 direction-cosines 
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 , 

z-y, 0, - (x - a) 
0, z-y, -(y-ft) =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 left-hand 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 cono-cuneus 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 



x-a,y - 



= 0, 



or {m(z-i)-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 



x-a 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, (x-a)* + 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 

M-iy 'i/ 
1 . M/rt. tt-o 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 n-I, 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) (a-x) + (sdx + tdy) (ft-y) = 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 r-2 (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 (a-y}, 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 = c-p + 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 
direction-cosines of a tangent plane, while dx, dy, dz are 
proportional to the direction-cosines 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 z-px- 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 

(x-a)' 2 + (y-p)* + 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 

(x-a)*+{y-4>(a)}* + z* = v*, (x-a) + {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=*-^(x-a) 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 , 

Ce-a)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 

x-a=f(p, q}, y-P=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 
(r-F ll )(t-F,. 2 ) = (s-F 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, 

rt-s 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 Pro-Hessian. 

In order to find in what points the developable is met by 
the Pro-Hessian, 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 Pro-Hessian 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, 

(x-a?+(y-p)*+(z-^ = R\ 
x-a+p(z-j)=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* + (z-y)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 Pro-Hessian (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 Pro-Hessian 
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 Pro-Hessian, 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 a-F m = A, 0-F in = B, y-F^ = C, 8-F 22!1 = D; 
then, differentiating pr = a\ we have 

(A + Bm) r + (C + Dm) p-2(B + Cm) a = 0. 
Substituting for m from the equation <r + rm = 0, and remem- 
bering that pr = <r 2 , we have 

AT* - 3-Bo-r 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 
direction-cosines 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 direction-cosines 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 

(x-a^ + (y-^ + z^ = r\ x- a + m(y - ffi-0. 

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 direction-cosines 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*)s-pqr} = 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. 
1-120; 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 Line-Complex (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 one-fold, two-fold, three-fold 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 one-fold, two-fold, and three-fold relation, or may be called 
those of a triple, double, or single system of right lines. 

A. The parameters have a one-fold 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 two-fold relation. We have 
here a congruence of lines. A well-known 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 
centre-surface or locus of the centres of curvature of the 
given surface, and the normals are thus bitangents of the 
centre-surface. 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 order-class 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 order-class 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 three-fold. We have 
here a regulus of lines or ruled-surface, 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 direction-cosines of all dia- 
meters are in tlu ratio P : Q : R. 

Since the direction-cosines of the perpendicular to the 
polar plane of any point xyz are proportional to 



THE LINEAR COMPLEX. 41 

- Qz + Ey + S, Pz-Ex + 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 direction-cosines 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 direction-cosines of the tangent line, I, m, n those of the 
principal normal, and A, p, v those of the bi-normal, then 

ya, - xfi - hy = 0, 
and by differentiating with regard to s, and using the first of the Frenet-Serret 

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'fa-ijfa + 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. 455a-o 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 equi-inharmonic 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 quadri-quadric 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 equi-anhar- 
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 
dp-i 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 thirty-two 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 y-z' + 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 (Z-PX)(Z-PX)=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 -P-v/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 (v-kj . . . 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 direction-cosines (/, 
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 direction-cosines 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 direction-cosines 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 " doubly-infinite " 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 limit-points 
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 direction-cosines 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 direction-cosines 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 half-way between the limit points 
coincides with tliat half-way 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 direction-cosines 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 two-sheeted locus of the foci or envelope of the focal 
planes ; the middle surface is the locus of the " middle point " 
which lies half-way between the limit points and as we have 
seen (Art. 457) coincides with the point half-way between the 
foci. The envelope of the plane drawn through the middle 
point perpendicular to the ray is called the middle envelope ; 
and the two-sheeted 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 - a-x- - b-y- 

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 direction-cosines 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 direction-cosines 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 direction-cosines 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 singly-directed 
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 doubly-directed 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 C-L. 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 
doubly-directed 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 doubly-directed 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 doubly-directed 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 * = te-ag* + (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 Dupin-cyclide, intersect two fixed sphero-curves.] 

[457e. Ribauco-ur'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 Couronnf-s 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 doubly-directed 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 direction-cosines 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' = Zjl-j + 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 direction-cosines 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 direction-cosines 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 SjL-j - 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 non-consecutive 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 7i-2 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 + w-yy.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=p-l, n = q-l, 
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=p-l, 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 4-m 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 one-parameter 
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 direction-cosines 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 left-hand 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 direction-cosines 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'f-cole 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 r-f(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- + N-T. 
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 (SU-II) = 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 right-hand 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) \-Lh-Nf, 



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 bc-f 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 fiN-vM, &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 bc-f* = 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") 

4-2(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} (2d-p) (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 
above-mentioned 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 direction-cosines 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 direction-cosines 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 direction-cosines (/, 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 Frenet-Serret 
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 S-curves 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 direction-cosines 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 direction-cosines 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 Frenet-Serret 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+i-a 

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 Frenet-Serret 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- = a-j- + /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 cos-0 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 one-parameter 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+ Rdr-Q ... (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 

l-z + 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 direction-cosines 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 Systi-mes 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 CENTBO-SURFACE, 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 sphero-conic. 

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' + b-yy' - 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 direction-cosines 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 thirty-sixth 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 " bi-circular 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. 1-44 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 (0-ff) 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 hero-conies. 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 - n-ri - - 

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 
sphero-quartics, 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 direction-cosines are 



shall now calculate the length and the 
a c 

direction-cosines 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 direction-cosines 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 direction-cosines 
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 + ~r-f 

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 direction-cosines are obtained from the con- 
sideration that it is perpendicular to the two lines whose 
direction-cosines are respectively 

P X P"l/' P Z y-, P X T> P I/ T> P % 

Forming, by Art. 15, the direction-cosines 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 
direction-cosines 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 CENTRO-SURFACE 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 right-hand 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 CENTRO-SUBFACE 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) (a-c) (a-d) = -4 s , (6 -a) (6 - c) (6-d)= - B s , 
(c-a) (c-6) (c-d) = -<7 2 , (d-a) (<!-&) (d-c) = -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 centre-surface will arise every time 
that a surface contains a right line which meets the circle at infinity. 



GENERALIZED CENTRO-SURPACE 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) + (aC-cA)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 + a-D, 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 twenty-fourth. 

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 centro-surface 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 centro-surface, 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 
- /3-yax = (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 CENTRO-SURFACE. 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 centro-sur- 
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 centre-surface therefore is 2w (w 2 -m- 1). 

510. Darboux * investigates as follows the degree of the 
centre-surface. 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 
centro-surface. 

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(p-l) 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 
centro-surface, 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 M-N=2(p-v) 

and putting in the values already obtained for /*, v, N, we have 
M=2m (w-1) (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 CENTRO-SURFACE. 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 centre-surface 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 
centro-surface 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(m-l), 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 twenty-fourth order, being the complete intersection 
of a quartic with a sextic surface. The nodal curve is of the 
twenty-sixth 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 Sphero-Quartics, 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* + c-z* = (*- + 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 twenty-four stationary, 
and twenty-seven 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 thirty-sixth 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 cnic-node, 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 = C-2 + 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 s-ple 
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 s-ple 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 Ar-ple 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 twenty-three 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 twenty-three 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 + (n-y)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 twenty-seven 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. 231-326. [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 Twenty-seven 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 + c-y 2 + 2J07 + 2m-ya + 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 (n-l) 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 3-nodal 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 co-ordin- 
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 before-mentioned 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 C-j, 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 twenty-four lines on the cubic besides the 
three in the plane ; that is to say, twenty-seven 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 twenty-seven 



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 forty-five 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, twenty-seven 
in all. 

If we denote each of the eighteen lines by the three which 
it meets, the twenty-seven 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 
non-intersecting 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 "double-six". 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 twenty-seven lines 
can be constructed thirty-six " double-sixes ". [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 
twenty-seven 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 non-intersecting 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 twenty-seven 
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 forty-five 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. 1-3 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 non-vanishing 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 -j-u\ 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 double-six 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-, -9-5, &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, 7-6. 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(a-f)* + 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 one-hundredth 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*2a-x, 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 one-hundredth 
degree. 

There are linear contravariants, the simplest of which, of 
the thirteenth degree, has been already given ; the next being 
of the twenty-first, 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, 
twenty-second, &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 twenty-seven 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 twenty-seven 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 mem-jir, 
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 twenty-seven 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 co-extensive 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 six-nodal 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 sixteen-nodal 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 triple-point 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 + bx-y) + 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 non-intersecting 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 non-plane 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 pinch-points 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 pinch-points 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 pinch-points 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 pinch-points 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 pinch-point 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 pinch-point 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 pinch-points 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 non-intersecting 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 pinch-points 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 four-nodal 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 line-pairs of C, 
and to coincident line-pairs 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 x-y-z+w=Q 
y-z-x+w=Q z-x-y + 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 

ay-z 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 non-proper 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 
one-one 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), x-z 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 pinch-points ; in (2) these have coincided in pairs ; 
in (3) there are two pinch-points and a triple point, while in 
(4) the pinch-points 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. 255-264, 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 two-fold 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 two-fold line, and on each 
such two-fold 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 point-pair. 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 point-equation corresponding to 
the point-pair 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, 



'2-20 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 left-hand 
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 : : m-n: n-l: 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(l-m) =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 three-dimensional 
space of the " surface " common to two quadratic ' varieties " 
in four-dimensional space. 

He obtains seventy-six 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 
fifty-five 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 twenty-seven 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 twenty-seven 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 non-singular 
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 twenty-eight 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 X-2\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 point-spheres belonging to the 
system aX+ ftY+yZ + SW; and therefore, from the mode of 
generation, every point of the curve FJ is a point-sphere 
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 point-sphere 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 point-sphere 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 point-sphere 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 five-fold 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 self-conjugate 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 d-b 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 

X-a X-6 X-c \-d X-e 

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 b-e 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 anti-points 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 sphero-quartic. 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 
aphero-quartic curve. The properties of these sphero-quartics 
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 last-mentioned, 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 
tangent-spheres 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 self-conjugate 
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 
point-sphere 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 point-sphere Z, is a node, for 
being orthogonal to X and to Y it lies on these spheres. 
Similarly the point-spheres X+tY, X-iY are nodes on the 
surface. We also see that from Z the cyclide inverts into 
a quadric of revolution, and that the spheres of self-inversion 
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 point-spheres. 



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 non-singular 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 twenty-four 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 thirty-four constants, that is, the surface may 
be made to satisfy thirty-four 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.Qltf-O, 

with eight nodes, the common intersection of three quadric 
surfaces ; this is the octadic 8-nodal 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, centro-surface 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 8-nodal surface, called 
the octo-dianome, 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 last-mentioned 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 
ennea-dianome ; 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 deca-dianome. 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 7-nodal 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 octo-dianomes and ennea-dianomes are 
respectively (P, Q) 2 + #v and P 2 + 8^.] 

A. kind of 10-nodal 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 16-nodal 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 15-nodal 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 14-nodal 

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 3-nodal 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 thirty-two points on the 
quartic lying in pairs on ninety-six 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 co-ordinates 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 plane-pairs 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 one-one 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 16-nodal 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. 

f-The 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 thirty-four 
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 twenty-four 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 
Wave-surface 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 
Wave-surface. 

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 non-singular 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 12-nodal 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 
10-nodal 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 twenty-six. 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 + q-4} 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 tact-invariant 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+p-3) + 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' = m-l, &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 tact-invariant 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 tact-invariant 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(m-2), class = m(m - 1), r=m(3m-5), a=0, /3 = 2m(5m-ll), &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(m-Y)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(m-l) 2 , p = 2(w-l). 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' xurftiri-x 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/i-p, 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 double-six. 

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 t-ple 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 n-4, which passes (t-1) times through the t-ple 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 

non-intersecting 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 ;i-fold conic, 
and similarly it has A as an ra-fold 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 n-fold 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 quadro-quadric 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 quadro-quadric 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 

(a-iX + 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: XW-Z 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 s-fold 
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' P-2> 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 left-hand 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 
(Pil-2 + 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?)= -(n-2)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, 
J-7' = 0, J 3 tT-0; 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(w-3) (3n-2). 

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 n-3 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) 
= (n-4)(3n* + 5n-24). 

The locus of the points of three-point contact of in- 
flexional tangents which touch the surface elsewhere is there- 
fore of degree 

6 8 = n(n-4) (3w 2 + 5w-24). 

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 four-point 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) five-point tangents ; 

(7) flecnodal (four-point) tangents in one place and ordinary 
tangents in another ; 

(&) inflexional (three-point) 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 five-point contact arises 
from a four-point contact by the coincidence of one additional 

* Gijtt. Nachr., Feb. 1876 ; Math. Ann., x. p. 102, xi. pp. 348-78. See also 
his Kalkill der abtahltnden Geometric (1879), pp. 236-7, 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(n-4)(7n-12). 
The same number is found from the analogous relation 

yS = 6 2 + Z> 3 - b, 

since the union of a three-point with an ordinary contact also 
leads to a five-point one. 

Again, four-point tangents having another ordinary con- 
tact may arise either through coincidence of two simple 
intersections on a four-point 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 
three-point 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(n-4) (n-5) (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(w-4)O-5) (n-6) (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 (n-2) (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 (n-2) 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 four-point 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 
four-point 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 four-point 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 four-point 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 four-point 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 four-point 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(n-2) (llw-24) 

points on a surface of the degree n in which coincident 
inflexional tangents have a four-point 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/+(w-3) a 4 -(n-3) 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 = x-j, + 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(n-2), 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 node-couple 
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(n-2) (lln-24). 

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 
node-couple 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 pinch-points, at which the two tangent planes coincide. 
Further, each of these singularities is in general accompanied 
by others ; for example, at a pinch-point, 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 O-2) = 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 twenty-seven 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 twenty-seven lines. Consequently there are 
twelve points a-' and twenty-seven 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 twenty-seven 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 twenty-seven 
lines 6 meet the same surface of the tenth degree. The points 
ft answer to the points where the twenty-seven 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 twenty-seven 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 twenty-seven 
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 fifty-four 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 + rc-12), 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 

+ 548w-960), 
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(rc-2)(w-3) = 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 (w-2) ( 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(rc-2) (n-3) (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 =(a-b-c) (w-2) + 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 = (a-26-3c)(n-2) (w-3) 

+ 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)(n-2)(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 -4/9-90- ) 

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 = 2p--3* 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(p-l) (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 (w-l), 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(r-2) = w (r-4 + 2), 
x (r - 2) = n (r - 4) + 2/8 + 87 + 3f , 
m (r-2) = 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 
non-consecutive 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, 
9-18 = 27-0-, < 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 non-inter- 
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 /it-fold 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 (/x-1) (/* -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 (m-Y) + 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. 

Node-couple 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 node-couple 7 cuspidal on double curve 

torse 



THEORY OF RECIPROCAL SURFACES. 315 

ORDINARY SINGULARITIES OF A SURFACE ALREADY POSSESSING : 

a double curve. a node-couple torse, 

j pinch-points / pinch-planes 

a cuspidal curve. a spinode torse. 

X close-points %' close-planes 

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 node-couple 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 node-couple torse is the envelope of 
the bitangent planes of the surface, and the node-couple 
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 pinch-points 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 pinch-point is a point 
at which the two tangent planes are coincident. The section 
of the surface by a plane passing through a pinch-point has a 
cusp, and the curves of contact a of all tangent cones pass 
through the point and have the same tangent there. The 
/ pinch-planes have the reciprocal properties, and touch the 
surface along straight lines. 

623. The % close-points 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 close-point, 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 % close-planes, like the pinch-planes, 
touch the surface along straight lines. 

624. The 7' planes are stationary on the node-couple 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 half-sheets) 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 
node-couple 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 half-sheets of the 
surface acnodal, and on the side of the two half-sheets cru- 
nodal, viz. the two half-sheets 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 off-points, 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(n-l) = a + 26 + 3c 

(2) a(a-l) = n + 2S' + 3*' 

(3) c-*' = 3(n-a) 

(4) a (n - 2) = K + p + 2<r - B + 2 |> (M - 2) - n - 2(3 



OP RECIPROCAL SURFACE. 31'9 

b M-2) 



[</(/<- 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)0-3) 

(9) c (n - 2) (n - 3) = 6/i + 180' + (ac - 3<r - #) + 2 (ta - i) 

+ 5 [2r 0* - 2) (^ - 3) - ^] 

(10) b(b-l) = q + 2(k+f) + M 

(11) c(c-l) = r + 2(A-h^) + 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 twenty-two 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 (n-1), 

S =$n (n-1) (n-2) (n-3), 
K = n (n - 1) (n - 2), 
n'= n (n-1) 2 , 
a = n (n - 1) , 



b' = in(n-l) (n-2) (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(n-2) (3-4), 
<r' = 4n (w - 2), 
^ = 2n(n-2)(llw-24), 
y'=4n(w-2)(n-3)(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. 
Anchor-ring, 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. 

BI-CIRCULAR 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. 

cubo-cubic, 274. 
Bitangent lines, 

congruence formed by, 37, 62. 

to centro-surfaco of algebraic surface, 148. 

to centre-surface 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. 
Close-planes, 315, 317. 
Close-points, 314 sqq. 
Cnic-nodes, 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 order-class 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 semi-directed, 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. 
Conic-node, 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 Bt-rational) 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. 

twenty-three 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. 
Cubo-cubic transformation, 274. 
Cuno-cuneus, 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 centre-surface 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. 

DECA-DIANOME, 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 (node-couple 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, 39-40. 
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. 
Double-six, 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. 
Dupin-Darboux 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. 
Ennea-dianome, 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. 
Five-point 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. 
Four-point 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 centra-surface 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. 
Node-oouple 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. 

doubly-directed, 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. 
Octo-dianome, 242. 
Off-points, 318. 



328 ANALYTIC GEOMETEY OF THREE DIMENSIONS. 

Order. 

of algebraic complex, 37. 

of algebraic congruence, 38. 

Order-class 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 centro-surface, 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. 
Pinch-planes, 314 sqq. 

Pinch-points, 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. 
Pro-Hessian, 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. 
Quadro-quadric 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, 202-13. 

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 non-intersecting 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, 203-6. 

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, 202-13. 

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). 
Sphero-conics for wave surface, 135, 136. 
Sphero-quartics, 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. 

triply-orthogonal, 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. 
Tact-invariant, 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 centra-surface 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. 
Thread-construction, 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 Bi-rational), 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. 
Triply-orthogonal systems of surfaces, 98 sqq. 

differential equation expressing condition that r = / (x, y, z) may form 
a, 103-11, 118. 

special cases of, 111-15, 232. 

condition for in curvilinear coordinates, 115 sqq. 

Dupin-Darboux 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. 
Frenet-Serret, 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 



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