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Gerstein Science Infonnation Centre
issary
v_'. '
A TREATISE ON THE ANALYTIC GEOMETRY
OF THREE DIMENSIONS.
CAMBRIDGE :
/ r. -^ •^/y^yr
CT'^CT
A TREATISE
ON THE
ANALYTIC GEOMETRY
OF
THREE DIMENSIONS.
BY
GEORGE SALMON, D.D., D.C.L., LL.D., F.R.S.,
EEGIUS PROFESSOR OF DIVINITY IN THE UKIVERSITY OF DUBLIN.
FOURTH edition:
(HODGES, FIGGIS, & CO., GEAFTON STEEET,
BOOKSELLERS TO TEE VNIVJEf.SnY.
MDCCCLXXXII. i
i
C r'
'■■■tf^M-^ ,.„ ,
0(\
SS3
PREFACE TO THE THIRD EDITION.
In the preface to the second edition of my Higher
Plane Curves , I have explained the circumstances
under which I obtained Professor Cayley's valuable
help in the preparation of that volume. I have
now very gratefully to acknowledge that the same
assistance has been continued to me in the re-editing
of the present work. The changes from the preceding
edition are not so numerous here as in the case of
the Higher Plane Curves, partly because the book
not having been so long out of print required less
alteration, partly because the size to which the
volume had already swelled made it necessary to be
sparing in the addition of new matter. Prof. Cayley
having read all the proof sheets, the changes made at
his suggestion are too numerous to be particularized ;
but the following are the parts which, on now looking
through the pages, strike me as calling for special
«^^knowledgement, as being entirely or in great
measure derived from him; Arts.* 51 — 53 on the six
coordinates of a line, the account of focal lines Art. 146,
xVrts-t 'H4 — 322 on Gauss's method of representing
the coordinates of a point on a surface by two
parametcTs. The discussion of Orthogonal Surfaces
is taken from a manuscript memoir of Prof. Cayley's,
* These articles have been altered in the present edition.
t Now Arts. 377-384.
VI PEEFACE.
Arts.* 332 — 337 nearly without alteration, and the
following articles with some modifications of my own.
Prof. Cayley has also contributed Arts.f 347 and 359
on Curves, Art.:]: 468 on Complexes, Arts. 567 to the
end of the chapter on Quartics, and Arts.§ 600 to
the end. Prof. Casey and Prof. Cayley had each
supplied me with a short note on Cy elides, but I
found the subject so interesting that I wished to
give it fuller treatment, and had recourse to the
original memoirs.
I have omitted the appendix on Quaternions
which was given in the former editions, the work of
Professors Kelland and Tait having now made
information on this subject very easy to be obtained.
I have also omitted the appendix on the order of
Systems of Equations, which has been transferred to
the Treatise on Higher Algebra.
I have, as on several former occasions, to acknow-
ledge help given me, in reading the proof sheets, by
my friends Dr. Hart, Mr. Cathcart and Dr. Fiedler.
* Now Arts. 476-479. J Now Art. 453.
t Now Arts. 316 and 328. § Now Art. 620.
Owing to the continued pressure of other en-
gagements I have been able to take scarcely any
part in the revision of this fourth edition. My friend,
Mr. Cathcart, has laid me under the great obligation
of taking the work almost entirely off my hands,
and it is at his suggestion that some few changes
have been made from the last edition.
Teinity College Dublin,
Sept., 1882.
CONTEiNTS.
Thefolhiving selected course is recommended to Junior Readers : The Theory 0/ Surfaces
of the Second Order, pp. 1 — 125, omitting articles specially indicated in footnotes.
Confocal Surfaces, Arts. 157-170. The Curvature of Quadrics, pp. 1G7— 172,
The General Theory of Surfaces, Chap. xi. The Theory of Curves, Arts. 314—323,
358 — 360, 364 — 366, 373 — 375. And the Chapter on Families of Surfaces,
Arts. 422—445.
CHAPTER I.
THE POIMT.
Method of coordinates , . . . . .1
Properties of projections . . ' . . . - . 3
Coordinates of point cutting in a given ratio the distance between two points . 6
Coordinates of centre of a tetrahedron .... 6
Distance between two points (rectangular coordinates) . . .6
Dii-ection-cosines of a line ...... 7
Area of a figure in terms of areas of its projections . , .7
Angle between two lines in tei-ms of their direction-cosines . . 8
Perpenchcular distance of a point from a line . . . .8
Direction-cosines of the perpendicidar to the plane of two lines , , 9
Transformation of Coordinates . . . . .9
Distance between two points (obUque coordinates) ... 11
Degree of an equation unaltered by transformation . . . .11
CHAPTER II.
INTERPRETATION OF EQUATIONS.
Meaning of a single equation ; of a system of two or three equations . 12
Every plane section of a surface of the «"" degree is a curve of the n^^ degree . 14
Every right Hue meets a surface of the n^^ degree in n points . . 14
Order of a curve in space defined . . . . ,14
Three surfaces of degrees m, n,p intersect in mnp points . . 15
Cylindi-ical sm-faces defined ...... 15
CHAPTER III.
THE PLANE AND RIGHT LINE.
Every equation of the first degree represents a plane . . 16
Equation of a plane in terms of its direction-cosines and perpendicular from origin 16
Angle between two planes ...... 17
Condition that two planes may be mutually pei-pendicular . . .17
Equation of plane in terms of intercepts made on axes ... 17
VIU
CONTENTS.
Equation of plane through three points
Interpretation of terms in this equation
Value of determinant of the direction-cosines of three right lines
Length of perijendicular from a given point on a given plane
Coordinates of intersection of three planes .
Condition that four planes may meet in a point
Volume of tetrahedron in terms of coordinates of its vertices
Volume of tetrahedron, the equations of whose faces are given
Equations of surfaces passing through intersection of given surfaces
The equatipn of any plane can be expressed in terms of those of four given planes
QUADRIPLANAR COORDINATES, ANp THEIR DpAL lUTERPRKTATION
Anharmonic ratio of four planes ....
The Right Line .....
Equations of a right line include four constants
Condition that two lines may intersect
Direction-cosines of a line whose equations are given
Equations of perpendicular from a given point on a given plane
Direction -cosines of the bisectors of the angle between t>yo lines ,
Equation of plane bisecting the angle between two given planes
Angle between two lines .....
Conditions that a line may lie in a given plane . .
Number of conditions that a Hne may lie in a given surface .
Existence of right lines on surfaces of the second and third degrees
Equation of plane drawn through a given line perpendicular to a given plane
Equation of plane parallel to two given lines
Equations and length of shortest distance bietween two given lines
The Six Coordinates of a Line
Identical relation connecting them ....
Moment of two right lines ....
Relation connecting the mutual distances of four points in a plane
Volume of a tetrahedron in terms of its edges
Relation connecting mutual distances of four points on a sphere .
Radius of sphere cu-cumscribing a tetrahedron
Shortest distance between two opposite sides . .
Angle of inclination of two opposite sides
General formulfe for linear transformation of quadriplanar coordinates
Volume of tetrahedron in terms of homogeneous coordinates of vertices
Homogeneous coordinates of right line as ray and as axis
Passage from one system to the other
Condition of intersection of right lines . , ,
Lines meeting four right Unes ....
Transformation of homogeneous coordinates of line .
PAOR
18
19
19
20
21
21
21
22
22
23
23
24
24
24
25
26
27
27
28
28
29
29
29
29
30
31
33
33
34
34
35
35
37
37
37
37
39
40
42
43
43
41
CHAPTER IV.
properties op quadrics in general.
Number of conditions necessary to determine a quadric .
Result of transformation to parallel axes ,
Equation of tangent plane at any point . . ,
Equation of polar plane ....
Cones defined — tangent cone , . , .
45
46
47
48
48
CONTENTS.
IX
the origin
the rectangles
's method
Cylinder the limiting case of a cone ....
Locus of extremities of harmonic means of radii through a point
Every homogeneous equation in x, _?/, z represents a cone whose vertex is
Polar plane of a point with regard to a cone . .
Discriminant of a quadric ....
Coordinates of centre .....
Conditions that a quadric may have an infinity of centres
Equation of diametral plane . . . .
Conjugate diameters ....
A quadric has three principal diametral planes
Formation of equation representing the three principal planes
Rectangles under segments of intersecting chords proportional to
imder the segments of a pair of parallel chords
Equations of tangent plane and cone, &c., derived by Joachimsthal'
Condition that a plane may touch the surface . ,
Coordinates of the pole of a given plane .
Condition that a line may touch the si^rface
This condition a quadratic function of the six coordinates of a line
This condition derived from the former condition .
Coordinates of polar line of a given Une ....
Determination of points of contact of tangent planes through a given right line
Complex of lines ......
Conditions that a right line lie in a quadric ....
CHAPTER V.
CLASSIFICATION OF QUADRICS.
Functions of coefficients which are unaltered by rectangular transformation
Discriminating cubic ......
Cauchy's proof that its roots are real ....
EUipsoids .......
Hyperboloids of one and of two sheets ....
Asymptotic cones ......
Paraboloids .......
Actual reduction of equation of a paraboloid . . , .
CHAPTER VI.
PAOB
48
48
49
50
51
52
52
52
54
55
55
56
56
58
58
59
59
59
60
62
63
64
66
66
67
68
69
69
72
73
properties of quadeics deduced from special forms of their equations,
Central Surfaces . . . . . . .75
Equation referred to axes ...... 75
Length of normal . . . . . . .76
Sum of squares of reciprocals of three rectangular diameters is constant . 76
Locus of intersection of three tangent planes which cut at right angles . 77
Conjugate Diameters ...... 77
Sum of squares of three conjugate diameters is constant . . .78
Parallelepiped constant whose edges are conjugate diameters . . 78
Sum of squares of projections of conjugate diameters constant on any line or plane 79
Locus of inrersection of tangent planes at extremities of conjugate diameters 80
Quadratic which determines lengths of axes of a central section . . 81
One section through any semi-diameter has it as an axis ... 81
Axes found when the quadric is given by the general equation . . 82
CONTENTS.
PAOB
CiBCTJLAti Sections ...... 82
Form of equations of concyclic surfaces . . . . .83
Two circular sections of opposite systems lie on the same sphere . . 84
Umbilics defined . . . . . . .85
Eectilinear Generators . . ... 85
Two lines of opposite systems must intersect . . . .87
No two lines of the same system intersect . ... 87
Ruled surfaces defined . . . . . . .87
Distinction between developable and skew surfaces ... 89
A right line whose motion is regulated by three conditions generates a surface 89
Surface generated by a line meetuig three director lines ... 90
Ditto, when lines are given by their six coordinates . . .92
Tour generators of one system cut any generator of the other in a constant
anharmonic ratio ...... 92
Surface generated by lines joining corresponding points on two homographically
divided lines . . . . . . .92
Non-central Surfaces ...... 92
Functions unaltered by transformation of coordinates . . .93
Circular sections of paraboloids . . . . . 93
Eight lines on hyperbolic paraboloid . , . . .94
Method of constructing hyperboUc paraboloid ... 95
Conditions for Surfaces of Revolution . . . .96
Examples of Loci . ., . . . .99
Locus of intersection of three rectangular tangent lines to a quadric . . 100
Method of finding equation of cone, given its vertex, and a curve through which
it passes ....... 101
Reciprocal cones . . . . . . .101
Orthogonal hyperboloids, equilateral hyperboloida . . . 100, 102
CHAPTER VII.
methods op abridged notation.
Reciprocal Suefaces ....
Degree of the reciprocal of a surface, how measured
Reciprocal of a curve in space ....
Osculating plane of a curve defined .
Plane section of one sui-face answers to tangent cone of reciprocal
Reciprocal of quadric, when ellipsoid, hyperboloid, or paraboloid
Reciprocal of i-uled surface is ruled surface of same degree
Reciprocal cones defined ....
Sections by any plane of reciprocal cones are reciprocal .
Focal lines of a cone defined
Reciprocal of sphere .....
Properties of surfaces of revolution obtained by reciprocation
Equation of reciprocal of a quadric
Tangential Equations ....
Tangential equation of a quadric . .
Equation of system of quadrics having common curve
All quadrics through seven points pass tlirough an eighth
Locus of centres of quadrics touching eight planes
„ passing through eight points
103
103
104
104
104
105
105
105
105
106
106
107
108
109
109
ill
111
112
113
CONTENTS. XI
PAOB
Four cones pass through the intersection of two quadrics , . 113
Properties of systems of quadrics having double contact . . .115
Propeilies of three quadrics having one plane curve common . . 115
Similar quadrics . . . . . , ,115
Geometrical solution of problem of circular sections . . . 116
Twelve umbilics he three by three on eight lines . . . .116
Quadrics touching along a plane curve . . . .117
Properties of their sections . . . . , .117
Two quadrics enveloped by the same third intersect in plane curves . 117
Form of equation referred to self -conjugate tetrahedi-on . . .117
Lines joining vertices of tetrahedron to corresponding vertices of polar tetrahedron
all generators of same hyperboloid . . . . .118
Analogues of Pascal's theorem ..... 122
Hexagon of generators of a hyperboloid ..... 123
CHAPTER VIII.
FOCI AND CONFOCAL SURFACES.
Focus and dii-ectrix of a quadrio defined .... 126
Focus defined as intersection of two generators of circumscribed developable
through circle at infinity . , . . . .127
Foci in general lie on a curve locus . . . , .128
Surfaces when confocal ...... 128
Two kinds of foci . , ' . . . . . 128
Focal conies ....... 129
Analysis of species of focal conies for each kind of quadric . . 130
Foci of sections normal to a focal conic ..... 132
Focal lines of a cone ...... 133
Focal conies of paraboloids ...... 134
Focus and directrix property of quadrics .... 135
Tangent cone, whose vertex is a focus, is a right cone . . , 136
Reciprocal of a quadric with regard to a focus is a surface of revolution . 137
Property of umbilicar foci obtained by reciprocation , , , 138
Focal properties of quadrics obtained by reciprocation . . . 138
Focal Conics and Confocal Surfaces .... 139
Three confocals through a point all real and of different species . , 141
Coordinates of intersection of three confocals .... 142
Coordinates of umbilics ...... 142
Two confocals cut at right angles ..... 143
Axes of central section in terms of axes of confocals through extremity of
conjugate diameter ...... 144
Expression for length of perpendicular on tangent plane . , 145
pD constant along the intersection of two confocals . , . 146
Locus of pole of fixed plane with regard to a system of confocals . , 146
Axes of tangent cone are the three normals through its vertex , . 147
Transformation of equation of tangent cone to the three normals as axes of
coordinates ....... 149
Cones circumscribing confocal surfaces are confocal . . . 151
The focal lines of these cones are the generators of the hyperboloid through
the vertex ....... 152
Reciprocals of confocals are concyclic ..... 152
Xll
CONTENTS.
Tangent planes through any line to the two confocals which it touches are
mutually perpendicular ...... 153
Two confocals seen from any point appear to cut at right angles . . 153
Normals to tangent planes through a given line generate a hyperbolic paraboloid 153
Chasles's method of obtaining equation of tangent cone . . . 155
Locus of intersection of three rectangular planes each touching a confocal . 155
Intercept on bifocal chord between tangent plane and parallel central plane . 157
Given three conjugate diameters of a quadric, to find the axes . . 157
Locus of vertices of right cones enveloping a quadric ; which stand on given conic 158
Either of two focal conies may be considered as a locus of foci of the other . 159
Locus of intersection of three mutually perpendicular tangent lines . .160
Corresponding points on confocals ..... 161
Elliptic coordinates . . . . . , ,162
Projections of curves on ellipsoid, on planes of circular section . . 163
Ivoiy's theorem as to the distance of two corresponding points . . 164
Jacobi's analogue to the plane theorem that the sum of focal distances is constant 164
Locus of points of contact of parallel planes touching a seiies of confocals . 166
Curvature of Quadrics ...... 167
Radii of curvature of a normal and of an oblique section . . .168
Line of curvature defined . . . , , .170
Construction for principal centres of curvature .... 170
Surface of centres ; its sections by principal planes . . . 171
Equation of its reciprocal . . . . . .172
CHAPTER IX.
INVARIANTS AND COVAEIANTS OF SYSTEMS OF QUADRICS.
Fundamental invariants of a system of two quadrics . . . 173
Condition that a tetrahedron can be inscribed in one quadric, and self-conjugate
with regard to another ..... 175
Hesse's theorem as to vertices of two self-conjngate tetrahedra . . 175
Paure's property of spheres circumscribing self-conjugate tetrahedra . 175
Condition that two quadrics should touch . . , . , 175
Equation of surface parallel to a quadric .... 176
Point of contact of two surfaces, a double point on their curve of intersection . 177
Stationary contact defined ...... 178
Equation of surface of centres formed ..... 179
Condition that a tetrahedron can be inscribed in one quadric having two pairs
of opposite edges on another ..... 180
Equation of quadric touching four planes . . , , 181
Invariants of a cone and a quadric ..... 182
Two kinds of equilateral hyperboloids .... 183
Orthogonal hyperboloids . . . . , .183
Tangential equation of imaginary circle at infinity . . , 184
Two planes at right angles conjugate with regard to imaginary circle . . 184
Tangential equation of curve in space . . . . • 184
How to form reciprocal of tangential equation of a conic . . . 185
Equation of cone touching a quadric along a given plane section . . 185
Contravariants of a system of two quadrics . . , .186
Two principal covariant quadrics of a system of two quadrics . . 187
Equation of developable circumscribing two quadrics , .188
CONTENTS. Xlll
PAOK
Its sections by the principal planes ..... 189
Complex of lines meeting curve of intersection of two quadrics . . 190
Condition that a line should be cut harmonically by two quadrics . 190
Equation of developable generated by intersection of two quadrics . . 190
Intersection of this developable with either quadric . . . 191
Coordinates of a tangent to the common curve expressed by a parameter . 194
Coordinates of point on curve and of osculating plane how expressed . 195
A system of confocals is touched by a common developable . . . 195
Locus of points whence three rectangular hnes or planes can be drawn to a
quadric . . . . . . .196
Two quadrics having a common curve can be described to touch a line, and
three to touch a plane ...... 197
Properties of confocals deduced from those of a system inscribed m a common
developable . . . . . .197
Method of finding equations of focal conies of quadric given by general equation 198
Quadriplanar equation of system of confocals .... 199
Theorems respecting perpendiculars, how generalized . . . 200
Tangential equation of sphere inscribed in a tetrahedron . . . 201
Equation of circumscribing sphere . . . . .201
Conditions that quadriplanar equation should represent a sphere . . 202
Condition that a plane section should be a parabola or equilateral hyperbola . 202
Coordinates of foci of a plane section ..... 203
Locus of foci of central section of a given quadric .... 203
Jacobian of a system of four quadrics ..... 205
Locus of vertices of cones which pass through six given points . . 205
Jacobian of four spheres . . . . . . 206
Reduction of two quadrics to their canonical form .... 206
Invariants of a system of three quadrics .... 208
Condition for three quadrics to be each the sum of five squares same as for each
to be polar quadric of a point with regard to a cubic . . 210
Tact-invariants . . . . . ■ .211
Discriminants of discriminants ..... 212
Jacobian curve of three quadrics ..... 213
Complex of lines cut in involution by three quadrics . . ,214
CHAPTER X.
CONKS AND SPHER0-C0NIC3.
A cone may have sheets of two kinds . . , . .215
Spherical coordinates . . . . . .216
Cyclic arcs of sphero-conics analogous to asymptotes . . . 220
Siun of focal distances constant ..... 222
Focus and directrix property of sphero-conics .... 223
Any generator of hyperboloid makes with planes of circular section angles the
product of wliose sines is constant .... 223
Difference of squares of reciprocals of axes of central section of a quadric,
proportional to product of sines of angles it makes with cyclic planes . 225
Locus of intersection of rectangular tangents to a sphero-conic , . 225
Equation in spherical coordinates of imaginary circle at infinity . . 227
Equation of sphere inscribed in a tetrahedron .... 227
Equation of a right cone ...... 227
Investigation of Hart's extension of Feuerbach's theorem . . 229
C
XIV
CONTENTS.
CHAPTER XI. .
GENERAL THEORY OF SURFACES.
PAGE
Number of terras in general equation . . . . • 233
Section of surface by tangent plane has point of contact for a double point . 234
A surface in general lias triple tangent planes .... 235
Inflexional tangents defined ..... 236
The indicatrix ; elliptic, hyperbolic, and parabolic points . . . 236
Asymptotic curves on a surface ..... 238
Conjugate tangents ....... 239
Tangent plane at a parabolic point is a double tangent plane . . 240
Double or conical points on a surface ..... 240
Reciprocal of a conical point ..... 241
Application of Joachim stbal's method ..... 242
Number of double tangent lines which can be drawn through a point on the surface 244
Formation of equation of tangent cone to a surface .... 245
Number of inflexional tangents which can be drawn through any point . 245
■Number of double tangents through any point .... 246
Characteristics of tangent cone ..... 247
Polar properties of surfaces in general ..... 247
Degree of reciprocal siu:face ..... 248
Discriminant of a surface ...... 249
Polar quadric of a parabolic point is a cone .... 249
Hessian of a surface ....... 250
Number of stationary tangent planes which pass through a point . . 250
Every right line on a surface touches the Hessian . . . .251
Curvature of Suefaces ..... 252
Radius of curvature of normal section ..... 253
Euler's formula ....... 254
Meunier's theorem ....... 256
Two spheres have stationary contact with a surface . . . 257
Values of principal radii at any point ..... 258
Locus of points where radii are equal and opposite . . . 258
Equation determining directions of principal sections . . . 260
Conditions for an umbilic ...... 261
Lines of spherical curvature ...... 262
Number of umbilics on a surface of the ji*** order . . . 263
Stationary contact implies contact at two points .... 264
Determination of normals which meet a consecutive normal . . 264
Bertrand's theory of curvatiu-e ...... 265
Special case of the umbilic considered .... 266
Lines of curvature ....... 266
Of surfaces of revolution ...... 2G7
Their differential equation ...... 268
Lines of curvature of clliiisoid ..... 269
Dupin's theorem ..:.... 269
If two surfaces cut at right angles, their intersection, if a line of curvature on
one, is so on the other ..... 270
Ix>cus of centres along a line of ciu-vature is a cuspidal edge on sui-facc of normals 271
Properties of surface of centres . , . . . . 271
In what cases it can have a double line .... 272
Geodesic line defined ...... 273
CONTENTS. XV
PAOB
Its osculating plane normal to the surface . . . • 273
This property derived from Meunier's theorem .... 274
Curve ou centro-sm-face answering to line of curvature is a geodesic . 275
If a line of curvature be plane, it makes constant angle with tangent plane . 276
Lancret's theorem of variation of angle between tangent plane and osculating
plane of line of curvature ..... 277
A geodesic line of curvature must be plane .... 277
CHAPTER XII.
CURVES AND DEVELOPABLES.
Sec. I. Projective Properties . . • . . 27&
Different modes of representing a curve in space .... 281
Direction-cosines of tangent to a curve .... 283.
Theoiy of developables explained ..... 284
Envelope of a plane whose equation contains one parameter . ' . 286
Tangent planes to developables touch along a line . . . . 289
Characteristics ....... 290
Cuspidal edge of a developable ..,...• 290,
Stationary points and planes » . . . .291
Cayley's equations connecting singularities of a curve in space . . 291
Developable generated by tangents is of same degree as reciprocal developable 294
Special singularities . . . . . .295
Double or nodal curve on developable .... 297
Table of singularities ....... 298
Sec. II. Classification of Curves .... 299.
A twisted cubic can be described through six points . . .301
Projection of a twisted cubic has a double point . . . 302.
Properties of twisted cubics ........ 303,
Their different species ....... SOS
Singularities of curve of intersection of two surfaces . . . 308.
Number of apparent double points of intersection . . ., 309
Case of sm-faces which touch . . , . . .310.
Equations connecting singularities of curves which together make up intersection
of two surfaces . . . . . ,311
Two distinct families of quartics ..... 312,
Pour quartics of second family through eight points . . . 316
Special case of second family . . . . . .316
Twisted Cartesians . . , , , ,317
Classification of quintics . , . . , .318
Planar and multiplanar developables .... 318
Deficiency of a curve in space . . . . , ,319
Common curve ou three surfaces equivalent to how many points of intersection 321
Singularities of a double curve connected with those of its complementary , 322
Sec. III. Non-Projective Properties of Curves . . . 323
Dh-ection-cosines of normal plane ..... 324
Equation of osculating plane ..... 324
TheheUx . . . . . . . .325
Equation of osculating plane of intersection of two surfaces . . 327
Condition that four consecutive points may lie in a plane . . . 329
Radius of absolute and of spherical curvature , . . .331
XVI
CONTENTS.
ExpresBiona for angle of contact ....
Radius of curvature of intersection of two surfaces
Expression for angle of torsion .....
Osculating right cone .....
Eectifying developable .....
Rectifying surface is surface of centres of original developable
Angle between two successive radii of curvature
Cuspidal edge of polar developable is locus of centres of spherical curvature
Every curve has an infinity of evolutes . , . .
These are geodesies on the polar developable
Characteristics of polar developable
Radius of sphere through four consecutive points
Coordinates of its centre ....
History of theory of non-plane curves
Sec. 17. Curves traced on Surfaces
Gauss's method of representing the position of a point on a surface
Equations of lines of curvature, &c., in Gauss's notation
Differential equation satisfied by coordinates, when p = const., q = const., denote
lines of curvature ,
Gauss's measure of curvature of a surface
It varies inversely as product of two principal radii
Measure of curvature unaltered by deformation
Total curvature of geodesic triangle on any surface
Differential equation of a geodesic .
Line joining extremities of indefinitely near and equal
right angles
Radius of geodesic curvature
pD constant for a geodesic on a quadric
Value of the constant the same for all geodesies through
Mr. M. Roberts's deductions from this theorem
Liouville's transformation of equation pD = constant
Chasles's proofs of this theorem . .
„ „ and extensions of it
Elliptic coordinates
Area of surface of ellipsoid
Second integral of equation of geodesic
Length of geodesic
Geodesic polar coordinates .
Dr. Hart's proof of Mr. Roberts's expressions
Umbihcal geodesies do not return on themselves
Lines of level ...
Lines of greatest slope
geodesies cuts them at
an umbilic
PAQH
332
333
334
335
336
337
337
339
339
340
340
341
341
342
343
343
344
349
350
351
355
356
358
358
360
361
364
365
366
368
369
370
371
372
373
373
376
378
380
381
CHAPTER XIII.
families of surfaces.
Sec. I. Partial Differential Equations
Equations involving a single arbitrary function
Cylindrical surfaces
Conical surfaces .
Conoidal surfaces
383
384
386
387
389
CONTENTS.
XVll
PACK
Surfaces of revolution ...... 390
Order of differential equation of a family involving n functions . . 393
Surfaces generated by lines parallel to a fixed plane . . . 396
Or by lines which meet a fixed axis , . . • . 398
Differential equation of ruled surfaces .... 400
Theory of envelopes ...... 401
Determination of arbitrary functions .... 403
Pai-tial differential equation of developables .... 407
Their Pro-Hessian . . ... . .408
Nature of its intersection with the developable .... 408
Tubular surfaces ...... 409
Differential equation of characteristics . . . . ,410
Of cuspidal edge of enveloping developable .... 413
Differential equation of geodesic on a cone . . . .413
Sec. II. Complexes, Congruencies, Ruled Surfaces . . 416
Complexes ....... 418
Congruencies . . • . • .419
Lines of a congmency in general bitangents to a surface . . . 420
Tangent plane at any point on a ruled surface, how constructed , . 422
Normals along a generator generate paraboloid .... 423
Lines of striction . . . . • .424
Nature of contact along any generator ..... 425
Double curves generally exist on ruled surfaces . . . 428
Surfaces generated by a Une resting on three fixed directors . . . 429
Sui-faces generated by a line which meets a curve twice and another curve once 431
„ by a line which meets a curve thi-ee times . . . 432
Order of condition that three surfaces should have a line in common , 433
Sec. III. Orthogonal Surfaces ..... 436
Differential equation of a system of orthogonal surfaces . . . 441,443
Bouquet's special case of the differential equation .... 449
Systems of orthogonal surfaces by Serret, Darboux, and Roberts . . 450
CHAPTER XIV.
surfaces derived from quadrics.
Wave Surface ....
Its sections by principal planes
Apsidal surfaces ....
Polar reciprocal of apsidal same as apsidal of reciprocal .
Degree of reciprocal of wave surface
Geometrical investigation of planes which touch along circles
Equation in elliptic coordinates
Expression for angle between tangent plane and radius vector
Construction for tangent plane at any point
Lines of curvature of wave surface .
Surface of Centres of Quadric
Its sections by principal planes
Its cuspidal curves
Its nodal curves ....
Characteristics of centro-surface in general
Its class . ^ .
453
454
455
456
457
458
460
463
463
464
465
467
469
470
471
472
XVlU CONTENTS.
PAOE
It3 order . . . . . . . 473
Relation between clas3 and order of a congruency, and of its focal surface . 473
Normopolar surface of quadric, synnormals .... 475
Characteristics of parallel surfaces ..... 475
Theory of derived pedal surfaces ..... 478
Properties of inverse surfaces ..... 479
Order and class of inverse surfaces, and of pedals .... 481
Apphcation of inversion to the obtaining of focal properties . . 481
Lines of curvature of surface of elasticity ..... 481
First negative pedal of a quadric ..... 482
Problem of finding negative pedals identical with that of finding parallel surfaces 483
CHAPTER XV.
SURFACES OF THE THIRD DEGREE.
Cubics having double lines ..... 486
Cubics having double points ...... 488
Analysis of various kinds of double points . , , . 488
Torsal and oscular hues ..,,.. 489
Twenty-three possible kinds of cubics .... 490
Cubics whose tangent cones from points on surface break up into two of second
degree ....... 491
Steiner's quartic ...... 491
Characteristics of surface the equation of whose tangent plane is an algebraic
function of two parameters . . . . .491
Sylvester's canonical form for equation of cubic . . . 492
Steinerians ....... 493
Con-esponding points on the Hessian ..... 493
Relation of the five planes to the Hessian .... 494
Polar cubic of a plane ...... 495
These all touch the Hessian , . . . . , 495
Right lines on cubics ...... 496
Number of triple tangent planes ..... 498
Schliifli's scheme for the twenty-seven lines .... 499
Analysis of species of cubics . . . . . .501
Section by tangent plane, how met by polar plane with regard to Hessian . 502
Invariants and Covariants of Cubics .... 503
Method of obtaining contravariants in five letters . . , 504
Five fundamental invariants ...... 508
Equation of surface which determines twenty-seven right lines . . 510
CHAPTER XVI.
SURFACES OF THE FOURTH ORDER.
Quartics with triple lines . , . . . .512
Their reciprocals . . . . , .514
Quartic scrolls with double lines ..... 516
Pinch points . . . . . .519
Quartics with nodal lines ordinarily contain light lines . . . 622
Different kinds of nodal lines ..... 523
Plucker'a " complex " quartic ...... 524
CONTENTS.
XIX
Quartics with nodal conic . . . , .
Sixteen lines on the surface .....
Cyclides ......
Generated as envelope of spheres ....
Their focal curves .....
Their five-fold generation .....
Identical relation between equations of five mutually orthogonal spheres
Confocal cyclides cut each other orthogonally
Sphero-quartica ......
Cartesians ..••.••
Dupin's cyclide ......
Nodal quartics . , , .
The symmetroid .... . .
The 16-nodal quartic ......
The wave siuiace, a case of the above
PAOK
526
527
527
528
529
630
531
533
535
535
535
537
540
541
642
CHAPTER XVIL,
GENERAL THEOET OP SURFACES.
Jacobian of four surfaces ...... 644
Degree of tact-invariant of three surfaces .... 545
Degi-ee of condition that two surfaces may touch .... 546
Order of developable enveloping a surface along a given curve . . 547
Of developable generated by a line meeting two given curves . . 547
On the properties of systems of surfaces .... 547
Principle of correspondence ...... 549
Unicursal surfaces ...... 553
Correspondence between points of surface and of plane . . . 555
Expression for deficiency of a surface .... 557
Contact of Lines avith Surfaces ..... 558
Locus of points of contact of flecnodal tangents . . . 559
Clebsch's calculation of surface S . , . . • 560
Locus of points of contact of triple tangents .... 569
Number of points at which two tangents are biflecnodal . . . 572
Number of points at which lines can be drawn to meet in five consecutive points 574
Contact of Planes with Surfaces .... 576
Locus of points of contact of double tangent planes . . . 579
Theory of Eeciprocal Surfaces .... 580
Number of triple tangent planes to a surface .... 684
Effect of multiple lines on degree of reciprocal . . . 687
Application to developables of theory of reciprocals . . . 688
Singularities of developable generated by a line restmg twice on a given cui-ve 590
Application to ruled surfaces ..... 590
Hessian of ruled surface where meets the surface .... 591
Prof. Catlei's Addition on the Theory op Reciprocal Surfaces 692
ERRATA, &C.
PAGR
7, note, line ^,for "a," read " a,"
8, line?, supply "- 0."
62, lines 12, 13, read " d^Vi, dw^," as last terms of the equations.
90, line 6 from bottom, and 91 line 8 from bottom i-ead " parallelepiped."
122, „ 5, sujjjjhj " = 0."
136, note, line 3, 7'ead "M. Amiot (see Liouville, viii. p. 161, and X. p. 109)."
214, last line but one, read " are," /or " is."
251, to last line. Art. 286, add " see Art. 607."
273, last line, read " normal," second note, end of line 2 add " of."
276, line 9 from bottom,yb?' " radius," read " axis."
297, „ 6, read "+{k- 2) dti"-^ +," Une 6 from bottom, add " see p. 588."
319, Art. 354, line 2, for " (p. 298)" read (p. 297)."
329, first line. Art. 363, read " four consecutive points."
356, end of first line, add " see p. 374."
376, in figure read " d(p',"for » cZ<^,."
382, Ex. 2 the expression /o?- - is -.
407, line 2 from bottom insei-t "Art 285."
444, „ 10 ,, „ read " condition."
476 3 "If"
568, „ 1 and 8 read " Article 588," for "577 ."
Add at end of Chapter IX.
[It ought to have been stated in this Chapter, that Dr. Casey has remarked in the
Annali di 3Iatem.atica, that the investigation given. Conies, p. 358, is capable of
immediate extension to space of three dimensions ; that vre can thus at once write
down an invariant relation between five quadrics whose equations are each of the form
iS — L' = 0, and which touch another quadric also inscribed in S, and that hence the
equation of the quadric touching four others, all being inscribed in S, is
0,
(12),
(13),
(14),
4{S)-L
(12),
0,
(23),
(24),
4{S) - M
(13),
(23),
0,
(34),
4{S) - N
(14),
(24),
(34),
0,
4{S) - P
S)-L,
-iiS)-
M,
4(S) - N,
^(.S)-P,
0
= 0.
These formula include the invariant condition that five spheres should all touch the
same sixth, and the equation of the sphere touching four given spheres.]
ANALYTIC GEOMETRY OF THREE DIMENSIONS,
CHAPTER I.
THE POINT.
1. We have seen already how the position of a point 0
in a plane Is determined, by referring it to two coordinate
axes OX, OY drawn in the plane. To determine the position
of any point P in space, we have only to add to our apparatus
a third axis OZ not in the plane (see figure next page).
Then, If we knew the distance measured parallel to the line OZ
of the point P from the plane XOY, and also knew the x
and y coordinates of the point 0, where PC parallel to OZ
meets the plane, It is obvious that the position of P would
be completely determined.
Thus, if we were given the three equations x = a^ y = 1), z = c,
the first two equations would determine the point O, and then
drawing through that point a parallel to OZ, and taking on it
a length PC = c, we should have the point P.
We have seen already how a change in the sign of a or
b affects the position of the point C. In like manner the sign
of c will determine on which side of the plane XOY the line
PC Is to be measured. If we conceive the plane XOY to be
horizontal. It is customary to consider lines measured upwards
as positive, and lines measured downwards as negative. In this
case, then, the z of every point above that plane is counted as
positive, and of every point below it as negative. It is obvious
that every point oji the plane has its z = 0.
B
THE POINT.
The angles between the axes may be any whatever; but
the axes are said to be rectangular when the lines OX^ OY
are at right angles to each other, and the line OZ perpendicular
to the plane XOY.
2. We have stated the method of representing a point in
space, in the manner which seemed most simple for readers
ah-eady acquainted with Plane Analytic Geometry. We pro-
ceed now to state the same more symmetrically. Our appa-
ratus evidently consists
of three coordinate axes
OX, OY, OZ meeting
in a point 0, which, as
in Plane Geometry, is
called the origin. The
three axes are called the
axes of X, y, z respec-
tively. These three axes
determine also three co-
ordinate planes, namely,
the planes YOZ, ZOX,
XOYj which we shall
call the planes yz, zx,
xy, respectively. Now since it Is plain that PA= CE=a^
PB= CB = h, we may say that the position of any point P
is known if we are given its three coordinates ; viz. PA drawn
parallel to the axis of x to meet the plane yz, PB parallel to
the axis of y to meet the plane zx, and PC parallel to the
axis of z to meet the plane xy.
Again, since 0D = a, 0E=h, (9i^=c, the point given by
the equations x = a, y = h, z = c may be found by the follow-
ing symmetrical construction : measure on the axis of x, the
length 0D= a, and through B draw the plane PBCB parallel
to the plane yzx measure on the axis of ?/, OE=b, and through
£ draw the plane PA CE parallel to zx : measure on the axis
of z, 0F= c, and through F draw the plane PABF parallel
to xy: the intersection of the three planes so drawn is the
point P, whose construction is required.
THE POINT.
3. The points A, B^ (7, are called the projections of the
point P on the three coordinate planes ; and when the axes are
i^ctangular they are its orthogonal projections. In what fol-
lows we shall be almost exclusively concerned with orthogonal
projections, and therefore when we speak simply of projections,
are to be understood to mean orthogonal projections, unless the
contrary is stated. There are some properties of orthogonal
projections which we shall often have occasion to employ, and
which we therefore collect here, though we have given the proof
of some of them already. (See Conies^ Art. 368).
The length of the orthogonal projection of a Jinite right line
on any plane is equal to the line multiplied hy the cosine of the
angle* which it makes ivitli the plane.
LetPO, P'C" be drawn perpendicular to the plane XOF;
and CC is the orthogonal pro-
jection of the line PP' on that
plane. Complete the rectangle
by drawing PQ parallel to GG\
and PQ will also be equal to
CC\ But PQ = PP' cosP'PQ.
4. The projection on any
plane of any area in another
jplane is equal to the original
area multiplied hy the 'cosine\of
the angle between the planes.
* The angle a line makes with a plane is measured by the angle which the line
makes with its orthogonal projection on that plane.
The angle between two planes is measured by the angle between the perpendiculars
drawn in each plane to their line of intersection at any point of it. It may also be
measured by the angle between the pei-pendiculars let fall on the planes from any point.
The angle between two lines which do not intersect, is measured by the angle
between parallels to both drawn through any point.
When we speak of the angle between two lines, it is desirable to express without
ambiguity whether we mean the acute or the obtuse angle which they make with
each other. When therefore we speak of the angle between two lines (for instance
PP', CC in the figure), we shall understand that these lines are measured in the
directions from P to P' and from Cto C", and that PQ parallel to CC is measured in
the same direction. The angle then between the lines is acute. But if wc spoke of the
4 THE POINT.
For If ordlnates of both figures be drawn perpendicular to
the intersection of the two planes, then, by the last article,
every ordinate of the projection is equal to the corresponding
ordinate of the original figure multiplied by the cosine of the
angle between the planes. But it was proved [Conies^ Art. 394),
that when two figures are such that the ordinates corresponding
to equal abscissae have to each other a constant ratio, then the
areas of the figures have to each other the same ratio.
5. The projection of a point on any line is the point where
the line is met by a plane drawn through the point perpen-
dicular to the line. Thus, in figure, p. 2, if the axes be rect-
angular, jD, E^ F are the projections of the point P on the three
axes.
The projection of a finite riglit line upon another right line
is equal to the first line multiplied hy the cosine of the angle
between the lines.
Let PP' be the given line, and DD' its projection on OX.
Through P draw PQ parallel to z
OX to meet the plane P'C'B' ; and
since it is perpendicular to this
plane, the angle PQP' is right, and
PQ = PP' cos P'PQ. But PQ and
PD' are equal, since they are the
intercepts made by two parallel
planes on two parallel right lines.
p
Q.
n
2.' .
c
V
^
6. If there he any three points P, P', P", the projection of
PP" on any line loill he equal to the sum of the projections on
that line of PP' and P'P".
Let the projections of the three points be P, P', P", then
if P' lie between P and P", BD" is evidently the sum of DD'
angle between PP' and C'C, we should draw the parallel PQ' in the opposite direction,
and should wish to express the obtuse angle made by the lines with each other.
When we speak of the angles made by any line OP with the axes, we shall always
mean the angles between OP and Xhcposkive directions of the axes, viz. OX, Oi', OZ.
THE POINT. 5
and D'D". If D" He between D and D', DD" Is the difference
of DD' and D'D"\ but since the direction from D' to D" Is
the opposite of that from D to D'^ DD" Is still the algebraic
sum of DD' and D'D". It may be otherwise seen that the
projection of P'P" is in the latter case to be taken with a
negative sign, from the consideration that in this case the
length of the projection is found by multiplying P'P" by the
cosine of an obtuse angle (see note, Art. 3). In general, if there
be any number of points P, P', P", P'", &c., the projection
of PP'" on any line is equal to the sum of the projections of
PP'^ P'P", P"P"', &c. The theorem may also be expressed in
the form that the sum of the projections on any line of the sides
of a closed polygon = 0.
7. We shall frequently have occasion to make use of the
following particular case of the preceding.
If the coordinates of any point P he projected on any line,
ilie sum of the three projections is eqiiol to the projection of the
radius vector on that line.
For consider the points 0, P, (7, P (see figure, p. 2) and
the projection of OP must be equal to the sum of the pro-
jections of OD[=x), DG{=y), and CP[=z).
8. Having established those principles concerning projec-
tions which we shall constantly have occasion to employ, we
return now to the more Immediate subject of this chapter.
The coordinates of the point P dividing the distance hetween
two points P' [xyz), P" {x'y'z') so that P'P : PP" : : m : Z, are
Ix + mx" ly + my" Iz' + mz"
^= 7 , ... J y
z
l + m ' -^ l + m ' l + m '
The proof Is precisely the same as that given at Conies, Art. 7,
for the corresponding theorem In Plane Analytic Geometry.
The lines PJ/, QN in the figure there given now represent
the ordinates drawn from the two points to any one of the
coordinate planes.
If we consider the ratio I : m as Indeterminate, we have the
coordinates of any point on the line joining the two given points.
6 THE POINT.
9, Any side of a triangle P"'P" is cut in the ratio m : w, and
the line joining this point to the opposite vertex P' is cut in the
ratio m + n : Z, to find the coordinates of the point of section.
Ans.
X-
Ix + mx" 4- nx'" _ ly + my'' + ny'" Iz' + mz" + nz'
I i-m + n ' "^ l-^ m + n ' l + m + n
This is proved as in Plane Analytic Geometry (see Conicsj
Art. 7). If we consider Z, m, n as indeterminate, we have the
coordinates of any point in the plane determined by the
three points.
Ex. The lines joining middle points of opposite edges of a tetrahedron meet in
a point. The x's of two such middle points are J (x' + x"), ^ {x'" + x""), and the x
of the middle point of the line joining them is ^ [x' + x" + x'" +x"''). The other
coordinates are found in like manner, and their symmetry shews that this is also
a point on the line joining the other middle points. Through this same point will
pass the line joining each vertex to the centre of gravity of the opposite triangle.
For the x of one of these centres of gravity is -^ {x' + x" + a;'"), and if the line join-
ing this to the opposite vertex be cut in the ratio of 3 : 1, we get the same value
as before.
10. To find the distance hetween two points P, P', whose
rectangular coordinates are x'yz\ x"y"z' .
Evidently (see figure, p. 3) PP"' = PQ' ^ QP'\ But
QP' = z'- z", and PQ' = CO" is by Plane Analytic Geometry
= {x' - x"f + (/ - y'y. Hence
PF' = [x' - x'J + (/ - y"f + (/ - z'J.
Cor. The distance of any point x'y'z' from the origin is
given by the equation
OP' = x"-\-y"-^z'\
11. The position of a point is sometimes expressed by Its
radius vector and the angles it makes with three rectangular
axes. Let these angles be a, /S, 7. Then since the coordinates
£c, y, z are the projections of the radius vector on the three
axes, we have
aj = /3Cosa, 3/ = p cos/3, ;; = pcos7.
And, since x^ -^ y"^ -{- z^ = p\ the three cosines (which arc
THE POINT. 7
sometimes called the dlrectlon-cosmes of tie radius vector)
are connected by the relation
cos^a + cos''^/3 + cos'7 = 1 .*
Moreover (compare Art. 7), x cos a + 3/ cos/S + z cosy = p.
The position of a point is also sometimes expressed by the
following polar coordinates — the radius vector, the angle 7 which
the radius vector makes with a fixed axis OZ, and the angle
COD {=<!>) which OC the projection of the radius vector on a
plane perpendicular to OZ (see figure, p. 4) makes with a fixed
line OX in that plane. Since then OC=p sin 7, the formulae
for transforming from rectangular to these polar coordinates are
x = p siny coscf), y = p smy sincf)^ z = p cos<y.
12. The square of the area of any plane figure is equal to
the sum of the squares of its projections on three rectangular
planes.
Let the area be -4, and let a perpendicular to Its plane
make angles a, /9, 7 with the three axes; then (Art. 4) the
projections of this area on the planes ?/,2;, zx.^ xy respectively,
are oleosa, ^ cos/3, ^ C0S7. And the sum of the squares
of these three = A\ since cos'^a + cos^/3 -f cos^7 = 1.
13. To express the cosine of the angle 6 hetween two lines
OP, OP' in terms of the direction-cosities of these lines.
We have proved (Art. 10) that
PP"' ^[x- x'f 4- (^ - y'Y + [z- zj.
* I have followed the usual practice in denoting the position of a line by these
angles, but in one point of view there would be an advantage in using instead the
complementary angles, namely, the angles which the line makes with the coordinate
planes. This appears from the coiTesponding formulse for oblique axes which I have
not thought it worth while to give in the text, as we shall not have occasion to use
them afterwards, laeii^y 13, y be the angles which a line makes with the planes
yz, zx, xy, and let A, B, Cbe the angles which the axis of x makes with the plane
of yz, of y with the plane of z.v, and of z with the plane of xy, then the formulae which
correspond to those in the text are
X sin A = p sin a, y sin B = p sin (3, z sin C = p sin y.
These forrculee are proved by the principle of Art. 7. If we project on a line perpen-
dicular to the plane of yz, since the projections of y and of z on this line vanish, the
projection of x must be equal to that of the radius vector, and the angles made by x
and p with this line are the complements of A and a.
yl
8 THE POINT.
But also PP"' = p' + p" - ^pp COS d.
And s'uice p' = x' + f+ z\ p" = x" + tf + z'\
we have pp cos Q = xx + yy + zz\
or cos ^ = cos a cos a' + cos ^ cos /3' + cos 7 cos 7'.
COK. The condition that two lines should be at right angles
to each other is
cos a cosa' + cos^S cos/3'+ cos 7 cos 7'. — 0>
14. The following formula is also sometimes useful :
sin''^^ = (cos/3 cos 7' — cos 7 coS|S')'^+ (cos 7 cos a' — cos a 0037')^
+ (cos a cos/3' — cos/3 cosa'j"*.
This may be derived from the following elementary theorem
for the sum of the squares of three determinants {Lessons on
Higher Algebra^ Art. 26), but which can also be verified at
once by actual expansion,
{he' - ch'f + [ca' - ac'f + {aV - hay
= [ct' + S'^ + c^) [a' + r + O - {aa' + hh' + cc)'\
For when a, b, c; a, h', c are the direction-cosines of two
lines, the right-hand side becomes ] — cos^^.
Ex. To find the perpendicular distance from a point x't/'z' to a line tkrough the
origin whose direction-angles are a, /3, y.
Let P be the point x'lj'z', OQ the given line, PQ the perpendicular, then it is
plain that PQ = OP sin POQ; and using the value just obtained for sin POQ, and
remembering that x' — OP cos a, &c., we have
PQ- — {y' cosy — z' cos/?)- + {z' cos a — x' cosy)^ + (x' cos^ — y' C0Sa)2.
15. To find the direction-cosines of a line perinndicular to
two given lines^ and therefore perpendicular to their jylane.
Let a'^'y'^ a"/3"7" be the direction-angles of the given lines,
and a/37 of the required line, then we have to find a/3y from
the three equations
cosacosa' + cos/3 cos/3' 4-COS7COS7' =0,
cos a COS a" + cosyQ cosyS" + cos 7 cos 7" =^ 0,
cos'^'a + coii'/3 -f cos^7 = 1 .
TRANSFORMATION OF COORDINATES. 9
From the first two equations we can easily derive, by elimi-
nating in turn cos a, cos/5, cosy,
\ cos a = cosyS' cos 7" — cosyS" cos 7',
\ cos/3 = cos7' cos a" — cos 7" cos a',
\ cos 7 =cosa' cosyQ"- cos a" cos/3',
where X is indeterminate ; and substituting in the third equa-
tion, we get (see Art. 14), if 6 be the angle between the two
given lines,
X' = sin''6'.
This result may be also obtained as follows : take any two
points P, Qj or xy'z'^ x"y"z\ one on each of the two given lines.
Now double the area of the projection on the plane of xy
of the triangle POQ^ is (see Conies^ Art. 36) x'y" — y x\ or
p'p" (cosa' cos/3"— cosa" cos/3'). But double the area of the
triangle is p p" sin ^, and therefore the projection on the plane
of xy is p'p" sin Q cos 7. Hence, as before,
sin Q cos 7 = cos a' cos^Q" — cos a" q,q's,^\
and in like manner
sin^ cos a =cosyQ'cos7" — cos/3" cos 7';
sin^ cos/3 = cos 7' cos a" — cos 7" cos a'.
TRANSFORMATION OF COORDINATES.
16. To transform to parallel axes through a new origin^
whose coordinates referred to the old axes^ are x\ y\ z .
The formulae of transformation are (as in Plane Geometry)
£c = X-|-a;', y=Y-\y\ z = Z-\z.
For let a line drawn through the point P parallel to one
of the axes (for instance z) meet the old plane of xy in a point
C, and the new in a point C ; then PC = PC'+ C C.
But PO is the old z^ PC^ is the new z ; and since parallel
planes make equal intercepts on parallel right lines, C G
must be equal to the line drawn through the new origin 0'
parallel to the axis of s;, to meet the old plane of xy.
17. To pass from a rectangular system of axes to another
system of axes having the same origin.
C
10
TRANSFORMATION OF COORDINATES.
Let the angles made by the new axes of x, y, z with the
old axes be a, /3, 7 ; a', /8', 7' ; a", /S", y" respectively. Then
if we project the new coordinates on one of the old axes, the
sum of the three projections will (Art. 7) be equal to the
projection of the radius vector, which is the corresponding old
coordinate. Thus we get the three equations
ic = Xcosa + Fcosa' +^cosa"'
^ = Zcos/3-|- rcos/3'-f^cos/3"[ {A).
z =Xcos7 + ]rcos7' -f Zcosy\
We have, of course, (Art. 11)
cos"''a + cos'yS -1- cos"''7 = 1, cos^a' -f cos^/3' + cosV = Ij
cos'-'a" + cos'yQ" + cos V = 1
.{B).
Let X, /i, V be the angles between the new axes of y and z^
of 2: and a;, of a; and y respectively, then (Art. 13)
cos A, = cos a cos a' + cos /S' cos )S" + cos y cos Y^'
cos yu, = cos a" cos a +cos/8"cosy3 +cos7"cos7 > ...[C).
cosv = cosa cosa' -f cos/3 cos/3' +C0S7 C0S7' .
18. If the new axes be also rectangular, we have therefore
cosa' cosa'' + cos;S' cos/3''-f COS7' cos7" = 0i
cosa"cosa + cos/S'^ cosy3 +cos7"cos7 =0,' ... (Z>).
cosa cosa' + cos/3 cos^S' +C0S7 COS7' =oJ
When the new axes are rectangular, since a, a', a'' are
the angles made by the old axis of x with the new axes, &c.
we must have
cos'^a + cos^a' + cos"'a" = 1, cos'"*^ + cos*/3' + cos^/3'' = 1,
cos"'^7+cosV + cos'V' = l"» (^),
cos/3 cos 7 + cos/S' cos 7' + cos/3" cos 7" = 0
COS7 cosa +COS7' cosa' +C0S7" cosa" =o[ ... [F)j
cosa cos;8 + cosa' cosyS' + cosa" cosyS" = 0,
and the new coordinates expressed in terms of the old are
X=a;cosa +?/ cos/3 +2;cos7 "j
F=a;cosa' -\-i/cosj3' -f ^ C0S7' > (6^).
Z =x cosa" + 1/ COS/3" 4- z cos 7" J
TRANSFORMATION OF COORDINATES.
11
The two corresponding systems of equations A and O may be
briefly expressed by the diagram
X
Y
Z
X
a
a'
a"
y
/3
/3'
/3"
z
7
r
7
7
It is not difficult to derive analytically equations E^ F^ G^
from equations A^ B, Z), but we shall not spend time on what
is geometrically evident.
19. If we square and add equations {A) (Art. 17), attending
to equations (0), we find
x' + / + z'=X'+Y'-\-Z' + 2 YZ cos\ + 2ZXcosfi + 2Xrcos v.
Thus we obtain the radius vector from the origin to any
point expressed in terms of the oblique coordinates of that point.
It is proved in like manner that the square of the distance
between two points, the axes being oblique, is
{x' - xy + [y' - y'J + (.' - z'J + 2[y'- y") [z' - z'') cos A,
+ 2 (^' - z") {x - x') cos/t + 2 [x - x") {y - y") cosi/.*
20. The. degree of any equation hetween the coordinates is
not altered hy transformation of coordinates.
This is proved, as at Conies^ Art. 11, from the consideration
that the expressions given (Arts. 16, 17) for x^ y, z^ only involve
the new coordinates in the first degree.
* As we rarely require in practice the formulae for transforming from one set
of oblique axes to another, we only give them in a note.
Let A, B, C have the same meaning as at note. p. 7, and let a, (3, y; a', /3', y' ;
a", fi", y" be the angles made by the new axes with the old coordinate _pfe«es; then
by projecting on lines perpendicular to the old coordinate planes, as in the note
referred to, we find
a; sin ^ = X sin a + Y sin a + Z sin a".
ysmB = X sin ft + Y sin ft' + Z sin ft",
z sin C = A" sin y + Y sin y' + Z sin y".
( 12 )
CHAPTER 11.
INTERPRETATION OF EQUATIONS,
21. It appears from the construction of Art. 1 that if we
were given merely the two equations cc = a, y = l>^ and if the
z were left indeterminate, the two given equations would de-
termine the point C, and we should know that the point P
lay somewhere on the line PC. These two equations then
are considered as representing that right line, it being the
locus of all points whose x = a^ and whose y = h. We learn
then that any two equations of the form a; = a, y = h represent
a right line parallel to the axis of z. In particular, the equa-
tions X = 0, 2/ = 0 represent the axis of z Itself. Similarly for
the other axes.
Again, if we were given the single equation x = a^ we
could determine nothing but the point D. Proceeding, as at
the end of Art. 2, we should learn that the point P lay some-
where in the plane PBCD^ but its position in that plane would
be indeterminate. This plane then being the locus of all points
whose x = a^ is represented analytically by that equation. We
learn then that any equation of the form x = a represents a
plane parallel to the plane yz. In particular, the equation
a;=0 denotes the plane yz Itself. Similarly, for the other
two coordinate planes.
22. In general, any single equation hetiveen the coordinates
represents a surface of some hind ; any two simultaneous equations
hetween them represent a line of some kind^ either straight or
curved ; and any three equations denote one or more points.
I. If we are given a single equation, we may take for x
and y any arbitrary values ; and then the given equation
solved for z will determine one or more corresponding values
of z. In other words, if we take arbitrarily any point C in
the plane of xy^ we can always find on the line PG one or
INTERPRETATION OF EQUATIONS. 13
more points whose coordinates will satisfy the given equation.
The assemblage then of points so found on the lines PC will
form a surface which will be the geometrical representation
of the given equation (see Conies^ Art. 16).
II. When we are given tiDo equations, we can, by elimi-
nating z and y alternately between them, throw them into
the form i/ = (p{x), z = '\}r[x). If then we take for x any ar-
bitrary value, the given equations will determine corresponding
values for y and z. In other words, we can no longer take
the point C anyiohere on the plane of xy^ but this point is
limited to a certain locus represented by the equation y = ^ {x).
Taking the point C anywhere on this locus, we determine
as before on the line PC a number of points P, the assemblage
of which is the locus represented by the two equations. And
since the points (7, which are the projections of these latter
points, lie on a certain line, straight or curved, it Is plain that
the points P must also He on a line of some kind, though of
course they do not necessarily lie all in any one plane.
Otherwise thus : when two equations are given, we have
seen in the first part of this article that the locus of points
whose coordinates satisfy either equation separately is a surface.
Consequently, the locus of points whose coordinates satisfy
hoth equations is the assemblage of points common to the
two surfaces which are represented by the two equations con-
sidered separately : that is to say, the locus Is the line of in-
tersection of these surfaces.
III. When three equations are given, it is plain that they
are sufficient to determine absolutely the values of the three
unknown quantities a:, ?/, z^ and therefore that the given
equations represent one or more j)oints. Since each equation
taken separately represents a surface, it follows hence that
any three surfaces have one or more common points of inter-
section, real or imaginary.
23. Sui-faces, like plane curves, are classed according to
the degrees of the equations which represent them. Since
every point in the plane of xy has its s == 0, if in any equation
14 INTERPRETATION OF EQUATIONS.
we make 2; = 0, we get the relation between the x and y
coordinates of the points In which the plane xy meets the
surface represented by the equation : that is to say, we get
the equation of the plane curve of section, and it is obvious
that the equation of this curve will be in general of the same
degree as the equation of the surface. It is evident, in fact,
that the degree of the equation of the section cannot be greater
than that of the surface, but it appears at first as if it might
be less. For instance, the equation
zx^ + ay'^ + y^x = c'
is of the third degree ; but when we make ^ = 0, we get an
equation of the second degree. But since the original equation
would have been unmeaning if it were not homogeneous, every
term must be of the third dimension in some linear unit (see
Conies^ Art. 69), and therefore when we make z = 0^ the re-
maining terms must still be regarded as of three dimensions.
They will form an equation of the second degree multiplied
by a constant, and denote (see Conies^ Art. 67) a conic and
a line at infinity. If then we take into account lines at infinity,
we may say that the section of a surface of the order n
by the plane of xy will be always of the order n ; and
since any plane may be made the plane of xy^ and since
transformation of coordinates does not alter the degree of an
equation, we learn that every plane section of a surface of the
order n is a curve of the order n.
In like manner it is proved that every right line meets a
surface of the order n in n points. The right line may be
made the axis of 2;, and the points where it meets the surface
are found by making x = 0, y — 0 in the equation of the surface,
when in general we get an equation of the degree n to de-
termine z. If the degree of the equation happened to be less
than n, it would only indicate that some of the n points where
the line meets the surface are at infinity [Conies^ Art. 135).
24. Curves in space are classified according to the number
of points in which they are met by any plane. Tivo equations
of the degrees m and n respectively represent a curve of the
order inn. For the surfaces represented by the equations
INTEEPRETATION OF EQUATIONS. 15
are cut by any plane In curves of the orders m and n
respectively, and these curves intersect in 7nn points.
Conversely, if the degree of a curve be decomposed in any
manner into the factors m^ n^ then the curve may be the inter-
section of two surfaces of the degrees w, n respectively ; and it
is in this case said to be a complete intersection. But not every
curve is a complete intersection : in particular we have curves,
the degree of which is a prime number, which are not plane
curves.
■ Three equations of the degrees w, w, a7id p respectively,
denote mnp points.
This follows from the theory of elimination, since if we
eliminate y and z between the equations, we obtain an equation
of the degree mnp to determine x (see Lessons on Higher
Algebra^ Arts. 73, 78). This proves also that three surfaces of
the orders wi, n^ p respectively intersect in mnp points.
25. If an equation only contain two of the variables
^ (a?, y,) = 0, the learner might at first suppose that it represents
a curve in the plane of xy^ and so that it forms an exception
to the rule that it requires two equations to represent a curve.
But it must be remembered that the equation </> [x., y) = 0 will
be satisfied not only for any point of this curve in the plane
of xy, but also for any other point having the same x and y
though a different z ; that is to say, for any point of the
surface generated by a right line moving along this curve,
but remaining parallel to the axis of z.* The curve in the
plane of xy can only be represented by two equations, namely,
z = 0, 4> (jr, y) = 0.
If an equation contain only one of the variables, a?, we
know by the theory of equations that it may be resolved
into n factors of the form cc — a = 0, and therefore (Art. 21)
that it represents n planes parallel to one of the coordinate
planes.
* A surface generated by a right line moving parallel to itself is called a cylindrical
feui-face.
( 16 )
CHAPTER III.
THE PLANE AND THE EIGHT LINE.
26. In the discussion of equations we commence of course
with equations of the first degree, and the first step is to
prove that every equation of the first degree represents a plane^
and conversely, that the equation of a plane is always of the
first degree. We commence with the latter proposition, which
may be established in two or three different ways.
In the first place we have seen (Art. 21) that the plane
of xy is represented by an equation of the first degree, viz.
^ = 0 ; and transformation to any other axes cannot alter the
degree of this equation (Art. 20).
We might arrive at the same result by forming the equation
of the plane determined by three given points, which we can
do by eliminating Z, m^ n from the three equations given
Art. 9, when we should arrive at an equation of the first
degree. The following method, however, of expressing the
equation of a plane leads to one of the forms most useful in
practice.
27. To find the equation of a plane^ the perpendicular on
which from the origin =2^) "'^^ makes angles a, /5, 7 loith the
axes.
The length of the projection on the perpendicular of the
radius vector to any point of the plane is of course =p, and
(Art. 7) this is equal to the sum of the projections on that
line of the three coordinates. Hence we obtain for the equa-
tion of the plane
a; cos a + ?/ cos/3 + 2! cos 7 = p.*
* In wliat follows we suppose the axes rectangular, but this equatioia is true
•whatever be the axes.
THE PLANE. 17
28. Now, conversely, any equation of the first degree
Ax-^By-^ Cz + D = 0^
can be reduced to the form just given, by dividing it by a
factor R. We are to have A = Ii cosa, B = B coS|8, C=B cos 7,
whence, by Art. 11, B is determined to be = ^/ [A^ -{■ B'^ -\- G^).
Hence any equation Ax -i By + Cz-\- D = 0 may be identified
with the equation of a plane, the perpendicular on which from
the origin = -rr-n 7^ — 77^ •, and makes angles with the
axes whose cosines are A, B^ C, respectively divided by the
same square root. We may give to the square root the
sign which will make the perpendicular positive, and then the
signs of the cosines will determine whether the angles which
the perpendicular makes with the positive directions of the
axes are acute or obtuse.
29. To find the angle hetween two planes.
Ax+Bij+Cz + ]J = 0, A'x + B'y+C'z+U=0.
The angle between the planes is the same as the angle
between the perpendiculars on them from the origin. By the
last article we have the angles these perpendiculars make with
the axes, and thence. Arts. 13, 14, we have
AA' ^BR+ CC
cos^ =
s/[{A' + B'+ C) [A'' + B" + G"')] '
sin'''
{BG'-B'Gy + {GA'- C'Af^[AB'-A'BY
[A' + B' + G') [A'-' + B" + 6'"j
Hence the condition that the planes should cut at right angles
is AA' + BB' + CC = 0. "
They will be parallel if we have the conditions
BC' = B'C, GA'=C'A, AB' = A'B;
in other words, if the coefficients A, B, C he proportional to
A'^ B\ C, in which case it is manifest from the last article that
the directions of the perpendiculars on both will be the same.
30. To express the equation of a plane in terms of the in-
tercepts a, &, c, which it makes on the axes.
D
18 THE PLANE.
The intercept made on the axis of x by the plane
is found by making y and z both = 0, when we have Aa + D = 0.
And similarly, Bb + I) = 0, Cc + D = 0. Substituting in the
general equation the values just found for A, B, 0, it becomes
X y z ^
-4- f +- = 1.
a 0 c
If in the general equation any term be wanting, for instance
if ^ = 0, the point where the plane meets the axis of x is at
infinity, or the plane is parallel to the axis of x. If we have
both -4=0, B=0, then the axes of x and y meet at infinity the
given plane which is therefore parallel to the plane of xy (see
also Art. 21). If we have^ = 0, ^=0, C=0, all three axes
meet the plane at infinity, and we see, as at Conies, Art. 67,
that an equation 0.x + 0.y + 0.z + D = 0 must betaken tore-
present a plane at infinity.
.31. To find the equation of the plane determined hy three
points.
Let the equation be Ax-\-By-^ Cz-\-D = ^] and since this
is to be satisfied by the coordinates of each of the given points,
-4, Bj C, D must satisfy the equations
Ax'^By^- Cz-tl) = 0, Ax'^By"^ Cz' + D = Q,
Ax"' + By"'+Cz"'^D = 0.
Eliminating A, B, C, D between the four equations, the
result is the determinant
X, y, 2, 1
x\ y, z\ 1
'ff- fff f'f -t
x , y , z , I
= 0.
Expanding this by the common rule, the equation is
Xjy{z"-Z"') ^y"{z"-z') ^y'"[z'-z")]
■\-y\z [x —X ) -\-z [x -X) +z {x—x)]
+iK(2/"-/") ^^"{f'-y) ^^'"[y'-y")]
= X {y"z" - y"z") + X [y"z - y'z'") -f x'" [y'z" - y"z'].
THE PLANE 19
If we consider x, y^ z as the coordinates of any fourth
point, we have the condition that four points should lie in
one plane.
32. The coefficients of x^Jlx ^ in the preceding equation
are evidently double the areas of the projections on the co-
ordinate planes of the triangle formed by the three points.
If now we take the equation (Art. 27)
X cos ct.-\-y cos yS + 2; cos 7 =p,
and multiply it by twice A {^A being the area of the triangle
formed by the three points), the equation will become identical
with that of the last article, since yi cosa, A cos^S, ^ C0S7J
are the projections of the triangle on the coordinate planes
(Art. 4). The absolute term then must be the same in both
cases. Hence the quantity
X {y z -y z)-^x [y z -yz ) + x {y z -y z)
represents double the area of the triangle formed by the three
points multiplied by the perpendicular on its plane from the
origin ; or, in other words, six times the volume of the triangular
l^yramid^ lohose hase is that triangle^ and whose vertex is the
origin.'^
* If in the preceding values we substitute for x', y', z' ; p cos a', p' cos /?', p cos y',
(fcc, we find that sls times the vokime of this pyramid = p'p'p" multiplied by the
determinant
cos a', cos /3', cos y'
cos a", cos j3", cosy"
cos a'", cos/3'", cosy"'
Kow let us suppose the three radii vectores cut by a sphere whose radius is unity,
having the origin for its centre, and meeting it in a spherical triangle R'R"R"'. Then
if a denote the side R'R", and }} the perpendicular on it from R'", six times the volume
of the pyramid will be p'p'p'" sin a sin /; ; for p'p" sin a is double the area of one face
of the pyramid, and p'" sin/) is the perpendicular ou it from the opposite vertex. It
follows then that the determinant above written is equal to double the function
4 [sin s sin (s — a) sin (s — b) sin (s — c)}
of the sides of the above-mentioned spherical triangle. The same thing may be
proved by forming the square of the same determinant according to the ordinary
rule ; when if we write
cos a" cos a'" + cos/3" cos/3'" + cosy" cos y'" = cos«, ifcc,
20
THE PLANE.
We can at once express A itself in terms of the coordinates
of the three points by Art. 12, and must have 4:A^ equal to
the sum of the squares of the coefficients of cc, ?/, and z, in
the equation of the last article.
33. To find the length of the perpendicular from a given point
xy'z' on a given ^lane^ x cos a + 3/ cos/3 4- z cosy = p.
If we draw through x'g'z' a plane parallel to the given
plane, and let fall on the two planes a common perpendicular
from the origin, then the intercept on this line will be equal
to the length of the perpendicular required, since parallel planes
make equal intercepts on parallel lines. But the length of
the perpendicular on the plane through x'g'z' is, by definition
(Art. 5), the projection on that perpendicular of the radius
vector to xyz\ and therefore (Art. 27) is equal to
ic' cosa + 3/' cos/3 + ^' COS7.
The length required is therefore
a;' cosa + 3/' cos/3 + / COS7 — ^.
N.B. This supposes the perpendicular on the plane through
x'y'z' to be greater than p ; or, in other words, that x'yz and
the origin are on opposite sides of the plane. If they were
on the same side, the length of the perpendicular would be
p — (a;' cosa + ?/'cos/3 + 2;'cos7). If the equation of the plane
bad been given in the form Ax + By -}■ Cz + D = 0, it is re-
1,
cose,
cosi
cose,
1,
cos«
cosb,
cos a,
1
we get
■which expanded is 1 + 2 cos a cos b cos c — cos^a — cos-b — cos-e, which is known to
have the value in question.
It is useful to remark that when the three lines are at right angles to each other
the determinant
cos a', cos/3', cosy'
cos a", cos [3", cos y"
cos a", cos p'", cos y'"
has unity for its value. In fact we see, as above, that its square is
1, 0, 0
0, 1. 0
0, 0, 1
THE PLANE. 21
duced, as In Art. 28, to the form here considered, and the length
of the perpendicular is found to be
Ax'+By' ^ Cz^D
It is plain that all points for which Ax + By -f Cz' -\- D
has the same sign as D^ will be on the same side of the plane
as the origin ; and vice versa when the sign is different.
34. To find the coordinates of the intersection of three planes.
This is only to solve three equations of the first degree
for three unknown quantities (see Lessons on Higher Algebra^
Art. 29). The values of the coordinates will become infinite
if the determinant [AB' C") vanishes, or
A {B'G" - B"G') + A {B"C- BG") + A" [BG' -B'G) = 0.
This then is the condition that the three planes should be
parallel to the same line. For in such a case the line of in-
tersection of any two would be also parallel to this line, and
could not meet the third plane at any finite distance. .
35. To find the condition that four planes should meet in a
point.
This is evidently obtained, by eliminating a;, y, z between
the equations of the four planes, and is therefore the determinant
{ARG"D"'), or
A, B, G, D
A\ B\ G\ B'
A\ B'\ G'\ B"
A'\ B"\ G'", U"
= 0.
36. To find the volume of tJie tetrahedron whose vertices are
any four given points.
If we multiply the area of the triangle formed by three
points, by the perpendicular on their plane from the fourth,
we obtain three times the volume. The length of the per-
pendicular on the plane, whose equation is given (Art. 31), is
found by substituting in that equation the coordinates of the
fourth point, and dividing by the square root of the sura of
the squares of the coefficients of cc, y, z. But (Art. 32) that
*
22 THE PLANE.
square root is double the area of the triangle formed by the
three points. Hence six times the volume of the tetrahedron
in question is equal to the determinant
X, y, 2!', 1
// rr rf ,
X , ?/ , S , 1
/// /// /// 1
X ^ y , z , I
X , y , z ,1
H7. It is evident, as in Plane Geometry (see Conies^ Art. 40),
that if /S^, >S', S" represent any three surfaces, then a8+hS\
where a and b are any constants, represents a surface passing
through the line of intersection of S and /S"; and that
aS+bS' -\- cS" represents a surface passing through the points
of intersection of S, S\ and S". Thus then, if L, M, N denote
any three planes, aL + b3I denotes a plane passing through
the line of intersection of the first two, and aL-{-b3I+cN
denotes a plane passing through the point common to all three.f
As a particular case of the preceding aL + b denotes a plane
parallel to L, and aL 4 bM-^ c denotes a plane parallel to the
intersection of L and M (see Art. 30).
So again, four planes Z, il/, iV, P will pass through the
same point if their equations are connected by an identical
relation
aL + bM+cN-tdP=0,
for then any coordinates which satisfy the first three must
satisfy the fourth. Conversely, given any four planes inter-
secting in a common point, it is easy to obtain such an identical
relation. For multiply the first equation by the determinant
* The volume of the tetrahedron formed by four planes, whose equations are given,
can be found by forming the coordinates of its angular points, and then substituting
in the formula given above. The result is (see Lessotis on Higher Algebra, Art. 30),
that six times the volume is equal to
m_
{AB'C) {A'B"C"') {A"B"'C) {A"'BV')
where R is the determinant {AB'C'D'") Art. 35, and the factors in the denominator
express the conditions (Art. 34) that any three of the planes should be parallel to
the same hne.
\ German writers distinguish the system of planes having a line common by the
name Biischel from the system having only one point common, which they call Biindel.
THE PLANE. 23
{A'B^C"'), the second by - {A'' B'^'C), the third by [A'"BG'),
and the fourth by — [AB' C")^ and add, then [Lessons on Higher
Algebra^ Art. 7) the coefficients of a;, ?/, z vanish identically ;
and the remaining term is the determinant which vanishes
(A.rt. 35), because the planes meet in a point. Their equations
are therefore connected by the identical relation
L {A'B" C") - M {A"B"' C) + N {A"'B C) -P[AB' C") = 0.
38. Given any four planes i, If, N^ P not meeting in a
point, it is easy to see (as at Conies^ Art. 60) that the equation
of any other plane can be thrown into the form
aL + hM -}- cN+ dP= 0.
And in general the equation of any surface of the degree n
can be expressed by a homogeneous equation of the degree n
between Z-, if, iV, P (see Conies^ Art. 289). For the number
of terms in the comjjlete equation of the degree n between three
variables is the same as the number of terms in the homogeneous
equation of the degree n between /owr variables.
Accordingly, in what follows, we shall use these quadri-
planar coordinates, whenever by so doing our equations can
be materially simplified ; that Is, we shall represent the equation
of a surface by a homogeneous equation between four coordinates
a', ?/, 0, w] where these may be considered as denoting the
perpendicular distances, or quantities proportional to the per-
pendicular distances (or to given multiples of the perpendicular
distances) of the point from four given planes a; = 0, ^ = 0,
s = 0, w = <d.
It is at once apparent that, as in Conies^ Art. 70, there is also
a second system of interpretation of our equations, in which an
equation of the first degree represents a point, and the variables
are the coordinates of a plane. In fact, if L'M'N'P' denote the
coordinates of a fixed point, the above plane passes through it if
aU + hM' + cN' + dP' = 0, and the coordinates of any plane
through this point are subject only to this relation. The
quantities, a, J, c, d may be considered as denoting the perpen-
dicular distances, or quantities proportional to the perpendicular
distances (or to given multiples of the perpendicular distances)
of the plane from four given points a = 0, 6 = 0, c = 0, <:/ = 0.
24 ' THE RIGHT LINE.
Ex. 1. To find the equation of the plane passing through x'y'z', and through the
intersection of the planes
Ax -x- By + Cz ■¥ J), A'x + B'y + C'z + B' (see Conies, Art. 40, Ex. 3).
Ans. {A'x'+ B'y'+ Cz'+ D') {Ax + By+Cz + D)^{Ax'+By'+Cz+I>) {A'x+B'y+C'z+ D').
Ex. 2. Find the equation of the plane passing through the points ABC, figure, p. 2,
The equations of the line BC are evidently - = 1, | + - = 1. Hence obviously the
equation of the required plane is — ^" T + ~" =2, since this passes through each of the
three lines joining the three given points.
Ex. 3. Find the equation of the plane PEF in the same figure.
The equations of the line EF are x — 0,-+ - = 1; and forming as above the equa-
tion of the plane joining this line to the point abc, we get j-\ =1.
39. If four planes which intersect in a right line he met hy
any plane^ the anharmonic ratio of the pencil so formed will be
constant. For we could by transformation of coordinates make
the transverse plane the plane of ocy^ and we should then obtain
the equations of the intersections of the four planes with this
plane by making z = Oin the equations. The resulting equations
will be of the form aL + ilf, bL + M,cL + ilf, dL + if, whose
anharmonic ratio (see Conies^ Art. 59) depends solely on the
constants a, J, c, c?; and does not alter when by transformation
of coordinates L and 31 come to represent different lines.
THE RIGHT LINE.
40. The equations of any two planes taken together will
represent their line of intersection, which will include all the
points whose coordinates satisfy both the equations. By elimi-
nating X and ?/ alternately between the equations we reduce
them to a form commonly used, viz.
X = mz -\- a^ y = nz + b.
The first represents the projection of the line on the plane of
xz and the second that on the plane of yz. The reader will ob-
serve that the equations of a right line include four independent
constants.
Wc might form independently the equations of the line
joining two points; for taking the values given (Art. 8) of the
THE RIGHT LINE. 25
coordinates of cany point on that line, solving for the ratio
m : I from each of the three equations there given, and equa-
ting results, we get
x — x _ y — y _ z — z
'^~^' ~ Y^' ~ ^^^' '
for the required equations of the line. It thus appears that
the equations of the projections of the line are the same as the
equations of the lines joining the projections of two points on
the line, as is otherwise evident.
41. Two right lines in space will in general not intersect.
If the first line be represented by any two equations Z = 0,
Jf = 0, and the second by any other two N= 0, P= 0, then if
the two lines meet in a point, each of these four planes must
pass through that point, and the condition that the lines should
intersect is the same as that already given (Art. 35).
Two intersecting lines determine a plane whose equation
can easily be found. For we have seen (Art. 37) that when
the four planes intersect, their equations satisfy an identical
relation
aL + Jif + cN^- dP= 0.
The equations therefore aL + hM=0^ and cN+dP=0 must
be identical and must represent the same plane. But the form
of the first equation shows that this plane passes through the
line i, -M, and that of the second equation shows that it passes
through the line iV, P.
Ex. Wlieu the given lines are represented by equations of the form
X = mz + a, y = 7iz + b; x = m'z + a', y — n'z + V,
the condition that they should intersect is easily found. For solving for z from the
first and third equations, and equating it to the value found by solving from the
second and fourth, we get
a -a' __ h-h'
m — m' n — n' '
Again, if this condition is satisfied, the four equations are connected by the identical
relation
{n — n') {{x — mz — a) — {x — m'z — a')} — {m — vi') [{y — nz — V) — [y — n'z — h')],
and therefore (« — 7i') {x — mz — a) = {m — m') (y — nz — b)
is the equation of the plane containing both lines.
£
26 THE RIGHT LINE.
42. To find the equations of a line passing through the point
xyz\ and maMng angles a, /S, 7 voith the axes.
The projections on the axes, of the distance of x'y'z from
any variable point xyz on the line, are respectively x — x\
y - y\ z — z\ and since these are each equal to that distance
multiplied by the cosine of the angle between the line and the
axis in question, we have
x — x _y —y' _z — z' ^
cos a cos/3 cos 7 '
a form of writing the equations of the line which, although
it includes two superfluous constants, yet on account of its
symmetry between a?, y, z is often used in preference to the
first form in Art. 40.
Reciprocally, if we desire to find the angles made with the
axes by any line, we have only to throw its equation into the
f r r
iy> rv* qi __ /^i V cj
form — ~. — = ,. = — 77— when the direction-cosines of the
ABC
line will be respectively A, B, (7, each divided by the square
root of the sum of the squares of these three quantities.
Ex. 1. To find the direction-cosines ot x = mz + a, y — nz + h. Writing the equa-
X — a _y — b _z
m n 1
tions in the form — ::: — • = — : — = 7 , the direction-cosines are
7>l
4(1 + m- + n2) ' 4(1 +m^ + «-) 4(1 + m"- + n")
Ex. 2, To find the direction-cosines of 7 = — , e = 0. Ans. rrpr, — r , ttttt — s; > 0,
Ex. 3. To find the direction-cosines of
Ax + By+ Cz + D, A'x 4- B';j + C'z + I)'.
Eliminating y and z alternately we reduce to the preceding form, and the
,. . . BC'-B'O CA'-C'A AB'-A'B , ^ . ,
direction-cosmes are p , p , p , where Ji^ is the smn of
the squares of the three numerators.
Ex. 4, To find the equation of the plane throiigh the two intersecting hues
X — x' _y — y' _z — z' X — x' _y — y' _z — z'
cos a cos^ cosy ' cos«' ~ cos/3' "cosy'"
The required plane passes through x'y'z' and its perpendicular is pei-pendicular to two
lines whose direction-cosines are given ; therefore (Art. 15), the required equation is
(jj - xJ) (cos /3 cos y' - cos y cos /3') + (y - y') (cos y cos a — cos y' cos a)
+ {z- z') (cos a cos/3' - cos a' cos/3) = 0.
THE RIGHT LINE. 27
Ex, 5. To find the equation of the plane passing tlirough the two parallel lines
X — x' _y — y' _z — z' x — x" _y — y" _z — z"
cos a cos /3 COS y cos a cos /i cos y
The required plane contains the line joining the given points, whose direction-
cosines are proportional to x' — x", y' — y", z' — z" ; the direction-cosines of the
pei-pendicular to the jjlane are therefore proportional to
(^' — y") cos y — (s' - 2") cos /3, {z' - z") cos a— {x' - x") cos y,
{x' - x") cos ji -{y' ~ y") cos a.
These may therefore be taken as the coefficients of x, y, z, in the required equation,
while the absolute term determined by substituting x'y'z' for xyz in the equation is
iy'z" - y"z') cos a + {z'x" - z"x') cos/? + {x'y" - x"y') cosy.
43. To find the equations of the perpendicular from x'y'z'
on the plane Ax + By + Cz + D. The direction-cosines of the
perpendicular on the plane (Art. 28) are proportional to Aj B^ (7;
hence the equations required are
x — x'_y—y'_z — z'
~A~ " '^^ " ~~G~ '
44. To find the direction-cosines of the bisector of the angle
between two given lines.
As we are only concerned with directions it is of course
sufficient to consider lines through the origin. If we take
points x'yz'^ x"y"z" one on each line, equidistant from the
origin, then the middle point of the line joining these points
is evidently a point on the bisector, whose equation therefore is
xyz
X -\rx y -^ y z ■\- z '
and whose direction-cosines are therefore proportional to
f , // r , II 1,11
X +x , y -\- y , z + z ;
but since cc', 3/', z' ; a;", ?/", z" are evidently proportional to the
direction-cosines of the given lines, the direction-cosines of the
bisector are
cos a' + cos a", cos yS' + cos /3", cos 7' -l- cos 7",
each divided by the square root of the sum of the squares of
these three quantities.
The bisector of the supplemental angle between the lines
is got by substituting for the point x'y'z" a point equidistant
28 THE RIGHT LINE.
from the origin measured in the opposite direction, whose
coordinates are - a*", — y'\ — z" ; and therefore the direction-
cosines of this bisector are
cos OL — cos a", cos /3' — cos /3", cos 7' — cos 7'',
each divided by the square root of the sum of the squares of
these three quantities. The square roots in question are ob-
viously V(2±2 cosS) ; that is, 2 cos^S and 2 sin|S, If S is the
angle between the two lines.
N.B. The equation of the jylane bisecting the angle between
two given jplanes is found precisely as at Conies^ Art. 35, and is
(iCCOsa + ^ C0S/3+ 20037— p) = ± (a7Cosa'+^ cos^S'+z cos7'-j>').
45. To find the angle made with each other hy two li7ies
x—a y — i> z — c^ x — a y — h _z— c
m n I in n
Evidently (Arts. 13, 42),
. W + onra -+- nn
^^^ " slil' ^ m' + n') V(/''+ w^'V n") '
Cor. The lines are at right angles to each other if
ir + mm + nn = 0.
Ex, To find the angle between the lines 5 = -fot = -p. '■> rr^ = y,z = 0. Am. 30°.
46. To find the angle between the plane Ax + By + Cz-\- Dj
,, ,. x—a y — b z— c
ana the line — 7— = = ■ .
I 771 n
The angle between the line and the plane is the complement
of the angle between the line and the perpendicular on the
plane, and we have therefore
Al+Bm + Cn
sin 0 =
V(^' + m' + 7f) V(^' + B'+ C) '
Cor. When Al + Bm-i 0/1 = 0, the line is parallel to the
plane, for it is then perpendicular to a perpendicular on the
plane.
THE RIGHT LINE. 29
47. To find the conditions that a line x = mz + a, i/ = nz + b
should he altogether in a plane Ax + By + Cz-\- D. Substitute
for X and y in the equation of the plane, and solve for ^, when
we have
Aa+Bh-\-D
z =
Am -j- Bn+ G^
and if both numerator and denominator vanish, the value of z
is indeterminate and the line is altogether in the plane. We
have just seen that the vanishing of the denominator expresses
the condition that the line should be parallel to the plane ; while
the vanishing of the numerator expresses that one of the points
of the line is iji the plane, viz. the point ab where the line meets
the plane of xt/.
In like manner in order to find the conditions that a right
line should lie altogether in any surface, we should substitute
for X and y in the equation of the surface, and then equate to
zero the coefficient of every power of z in the resulting equation.
It is plain that the number of conditions thus resulting is one
more than the degree of the surface.*
48. To find the equation of the plane drawn through a given
line perpendicular to a given plane.
Let the line be given by the equations
Ax-\-By+Cz + I) = 0, A'x + B'y+az + J)' = Oj
and let the plane be
A''x-\-B''y+C''z + B'' = 0.
Then any plane through the line will be of the form
\ {Axr\- By+Cz + D) + fi {A'x -^ By + C'z + D') = 0,
and, in order that it should be perpendicular to the plane, we
must have
{\A + fiA') A" + [XB -\ fiB') B" + [\C + fx, C) G" = 0.
* Since the equations of a right line contain four constants, a right line can be
determined which shall satisfy any four conditions. Hence any surface of the second
degree must contain an infinity of right lines, since we have only three conditions to
satisfy ar.d have four constants at oiu- disposal. Every surface of the third degree
must contain a finite number of right hues, since the number of conditions to be
satisfied is equal to the number of disposable constants. A surface of higher degree
will not necessarily contain any i-ight lino lying altogether in the surface.
30 THE EIGHT LINE.
This equation determines X, : /a, and the equation of the required
plane is
{A' A'' + B'B" -}- C C') [Ax 4 By + Cz + JD)
= {A A'' + BB'' 4 CC') [A'x + B'y + C'z + U).
When the equations of the given plane and line are given
in the form
/^ __ ^ 01 qj
a; COS a + 2/ cos/3 + 2! cos 7= J?; r=- — |;
X— X V — V z — z ^
cos a' cos/3' cos 7'
we can otherwise easily determine the equation of the required
plane. For it is to contain the given line whose direction-angles
are a', /S', 7' ; and it is also to contain a perpendicular to the
given plane whose direction-angles are a, /3, 7. Hence (Art. 15)
the direction-cosines of a perpendicular to the required plane are
proportional to
cos/3' COS7— cos/3 COS7', COS7' cosa— COS7 cosa', cosa' cos/3— cosa cos^',
and since the required plane is also to pass through xy'z\ its
equation is
[x—x] (cos/3 cos 7' — cos/S' cos 7) -f {y—y) (cos 7 cosa'- cos 7' cosa)
+ (s - z) (cosa cos^' — cosa' cos/8) = 0.
49. Given two lines to find the equation of a 'plane drawn
through either parallel to the other.
First, let the given lines be the intersections of the planes
X, M] N^ P, whose equations are given in the most general
form. Then proceeding exactly as in Art. 37, we obtain the
identical relation
L{A'B"a"')-M{A"B"'C)+N{A"'BC')-P{AB'C'')={A'B"C"D),
the right-hand side of the equation being the determinant, whose
vanishing expresses that the four planes meet in a point. It is
evident then that the equations
L [A'B" 0'") - M[A"B"' C) = 0, N{A"BG') - P [AR G") = 0
represent parallel planes, since they only differ by a constant
quantity ; but these planes pass each through one of the given
lines.
THE RIGHT LINE. 31
Secondly, let the lines be given by equations of the form
x — x y — y z — z X— X y - y z — z
cos a cos/:^ cos 7 ' cos a' cos/3' 0037'
Then since a perpendicular to the sought plane is perpendicular
to the direction of each of the given lines, its direction-cosines
(Art. 15) are the same as those given in the last example, and
the equations of the sought parallel planes are
{x— x') (cos/3 C0S7' — cos /3' cos 7) +(2/ — ?/'') (cos 7 cos a'— cos 7' cos a)
+ (s — s'Xcosa cos /3' — cos a' cos/3) = 0,
{x — a;")(cos/3 cos 7'- cos/3' cos 7)+ (?/-?/") (cos 7 cos a'— cos 7' cos a)
■\-{z — z") (cos a cos /3' — cos 0! cos (3) = 0.
The perpendicular distance between two parallel planes is equal
to the difference between the perpendiculars let fall on them
from the origin, and is therefore equal to the difference between
their absolute terms, divided bj the square root of the sum of
the squares of the common coefficients of a;, ?/, z. Thus the per-
pendicular distance between the planes last found is
(x^— x") (cos ^ cos 7'- cos ^' C0S7) -f {y—y") (0037 cosa'— C0S7' cosa)
+ (/ — 5;")(cosa cos/3'— cosa' cos /9) divided by sin^,
where Q (see Art. 14) is the angle between the directions of the
given lines. It Is evident that the perpendicular distance here
found is shorter than any other line which can be drawn from
any point of the one plane to any point of the other.
I 50. To find the equations and the magnitude of the shortest
distance between two non-intersecting lines.
The shortest distance between two lines Is a line per-
pendicular to both, which can he found as follows: Draw
through each of the lines, by Art. 48, a plane perpendicular
to either of the parallel planes determined by Art. 49 ; then the
intersection of the two planes so drawn will be perpendicular
to the parallel planes, and therefore to the given lines which
lie In these planes. From the construction it is evident that
the line so determined meets both the given lines. Its mag-
nitude is plainly that determined in the last article. Calculating
1
32 THE RIGHT LINE.
by Art. 48 the equation of a plane passing through a line whose
dh'ectlon-angles are a, /3, 7, and perpendicular to a plane whose
direction-cosines ai'e proportional to
cos/3' COS7-COS/3COS7', cos7'cosa— cos7COsa', cosa'cos/S— cosacos/3',
we find that the line sought is the intersection of the two planes
(x — x') (cos a — cos 6 cos a) + (?/ — ?/') (cos /3' — cos 0 cos/3)
+ (z — z) (cos 7' — cos ^ cos 7) = 0,
{x — x'^) (cos a — cos ^ cos a) + (y- y") (cos ;S — cos Q cos /S')
■\-{z — z") (cos 7 - cos 6 cos 7') = 0.
The direction-cosines of the shortest distance must plainly be
proportional to
cos^'cos7— cos^cos7', cos7'cosa— cos7Cosa', cosa'cos/3—cosa cos/3'.
Ex. To find the shortest distance h between the right line
X cos a -Vy cos /3 + 2 cos y —-p,
X cos a + y cos (3' + z cos y' = />',
and that joining the points P' {x', y', z') and P" {x"y"z").
Denoting by Z-, M the peipendiculars from any point x]jz on the two given planes
and by L'M\ L" M." those from the points P',P"; i + XJ/ = 0 is the equation of
Ix' + rax"
any plane passing through the first right line, and —r— &c. are the coordinates of
any point on the second. Hence, if the point in which this second right line meets
L + XM—dhe taken infinitely remote, or having l+m — Q,\ can be found so as to
determine the plane through the first line parallel to the second. This gives
U + \M' = L" + \M".
Hence LM" — L"M = LM' — L'M is the plane through L, M required.
Again, LM" - L"M = LM" - L'M + L'M" - L"M'
differs from the former only by a constant, therefore is parallel to it, but also this
equation is satisfied by the coordinates of the points P' and P", therefore it passes
through the second line.
Thus by dividing L'M" — L"M' by the square root of the sum of squares of
coefficients of x, y and z in either of these equations, we find the required shortest
distance.
The result of reducing this expression can also be arrived at thus : L'M' are the
lengths of perpendiculars from P' on the two given planes. They are both contained
in a plane through P ' at right angles to the right line LM. In like manner L"M" are
contained in a parallel plane through P" . Now considering projections on either of
these planes, if <^ be the angle between the planes L and M, double the area of the
triangle subtended by the projection of P'P" at the intersection of L, M multiphed
by sin f/> = L'M" — L"M'. But that double area is evidently the product of the
required shortest distance 2 between the two given lines by the projection of P'P".
Hence, calling 0 the angle between the two line?, we tsee that
L'M" - L'M' - {P'P") , a . sin e sin ^p.
THE EIGHT LINE. 33
51. When the equations of a right line are written in the
form — -, — ■ = "^ — — = to any system of coordinate axes
i in n
they appear to involve five Independent quantities, viz. x'yz^
and the ratios I : m : n. But It Is easily seen that xt/z occur In
groups which are not Independent, and the total number of
independent constants Is only four, as we saw In Art. 40. In
fact, if we denote respectively by a, 5, c the quantities mz' — ny\
nx — lz\ ly — mx\ we have at once the relation la + mh + nc = 0,
and subject to this the equations of the right line are any two of
the four equations
ny — mz + a = 0,
'- nx + ?s + i = 0,
mx —ly + c = Oj
ax + hy + CZ = 0,
for by the above relation the remaining two can In all cases be
deduced.
We have now six quantities a, 5, c, I, m, n which serve to
determine the position of a right line provided the relation
la + mh -\-nc = 0 hold, and these we shall call the six coordinates
of the right line. If we examine the conditions, as In Art. 47,
that this right line may be wholly contained in the plane
Ax-^By-^ Cz-^D = Q^
we find they are any two of the four equations
Be - Cb ^Dl = 0,
-Ac H- Ca + Dm = 0,
Ah-Ba -I- Dn = 0,
Al -f Bm + On =0,
from which also by the universal relation al + hm + en = 0, the
remaining two can in all cases be deduced. It Is important to
observe that the quantities a, h, c which are the functions
mz — ny^ nx— Iz, ly — mx of the coordinates x, y, z of any point
on the right line have the same values for each point on it.
We are thus enabled to express in x, y, z coordinates the
relation equivalent to any given relation in «, J, c. Again, if
F
34
THE EIGHT LINE.
we suppose the a?, y, z axes rectangular, so that ?=cosa,
??i = cos/S, w = co37, it is easily seen, by Art. 15, that a, 5, c
are the coordinates of a point on the perpendicular through,
the origin to the plane passing through the origin and the
given line, and at a distance from the origin equal to that of
the given line.
Ex. To express by the coordinates of two right lines the shortest distance between
them.
The expression found at the close of Art. 49 for the product of the shortest
distance 6 between two right lines by the sine of the angle 6 at which they are
inclined may be written
x' — x", cos a, cos a'
y'-y", cos^, cos/?'
z' — z", cos y, cos y'
if we replace cos a, &c , by V, &c., cos a', &c., by I", &c. this may be written
y', m', m"
z', n', n"
X
V, I"
y , ^, »»
z", n', n"
in which we see that the coordinates of the points x', &c. occur only in the groups
mentioned above.
Hence in the notation of this article, also omitting reference to sign,
5 sin e = I'a" + m'b" + n'c" + l"a' + m"b' + n"c'.
This quantity has been called by Prof. Cayley {Trans. Cambridge Phil, Soe.,
vol. XI. part ii. 1868) the moment of the two lines.
52. Before proceeding to further considerations on the co-
ordinates of a right line we introduce some properties of tetra-
hedra obtained by various methods, which will be useful in
the sequel.
To find the relation hetween tJie six lines joining any four
points in a plane.
Let a, 5, c be the sides of the triangle formed by any three
of them ABC, and let c?, e, / be the lines joining the fourth
point D to these three. Let the angles subtended at D by
a, 5, c be a, /3, 7 ; then we have cos a = cos(;8± 7), whence
cos^a + cos^/3 + cos^7 — 2 cos a cos/3 cos 7 = 1.
This relation will be true whatever be the position of i),
either within or without the triangle ABC. But
cosa =
e'+f-a'
2ef
cos/3 =
cos 7:
2de
THE RIGHT LINE. 35
Substituting these values and reducing, we find for the required
relation
a"- [d' - e') {d' -f) + h' [e' -f) [d' - d) + 6' [f - d'] [f - e')
+ d'd [d- -V- &) + W [h' - 6' - d) 4 dy (c' - d- F) +dbV=0,
a relation otherwise deduced ConicSj p. 134.
53. To express the volume of a tetrahedron in terms of its
six edges.
Let the sides of a triangle formed by any face ABGha
a, J, c; the perpendicular on that face from the remaining
vertex be p^ and the distances of the foot of that perpendicular
from A^ B, C be d, e',/'. Then a, 6, c, d\ e\f are connected
by the relation given in the last article. But if c?, f,/be the
remaining edges d = d^ + p^, e^ = e'' -^p^-^ f^ =f'^ •Vp'' ; whence
d — ^ = d'^ — e\ &c., and putting in these values, we get
- F=f {2bV + 26'd + 2dh' -d-b'- c"),
where F is the quantity on the left-hand side of the equation
in the last article. Now the quantity multiplying y"* is 16 times
the square of the area of the triangle ABG, and since p
multiplied by this area is three times the volume of the
pyramid, we have F=— 144 F^
5-4. To find the relation between the six arcs joining four
points on the surface of a sphere.
We proceed precisely as in Art. 52, only substituting for
the formulse there used the corresponding formulas for spherical
triangles, and if a, j3^ 7, S, s, (f> represent the cosines of the six
arcs in question, we get
- 2al3y - 2ai(j> - 2^8(f) - 2^8^ = 1.
This relation may be otherwise proved as follows : Let the
direction-cosines of the radii to the four points be
cos a, cos/3, cos 7,
cos a', cos/3', cos 7',
cos a", cos/3", cos 7",
cosa , cos/3 , C0S7 .
36 THE RIGHT LINE.
Now from this matrix we can form (by the method of Lessons
on Higher Algebra^ Art. 25) a determinant which shall vanish
identically, and which (substituting cos^a + cos'^/3 + cos''^7 = 1,
cos a cos a' + cos/3 cosyS' + cos 7 cos 7' = cosaJ, &c.) is
1, cosa&, cosac, cos at?
cosJa, 1, cos 5c, cosbd
cos ca, cosc5, 1, cosc^
cosda, COS db, eoadc, 1 =0,
which expanded has the value written above.
This relation might have been otherwise derived from the
properties of tetrahedra as follows :
Calliug the areas of the four faces of a tetrahedron
A, Bj C, D', and denoting by AB the internal angle between
the planes A and B^ &c. we have evidently any face equal
to the sum of the projections on it of the other three faces.
Hence we can write down
— A + B cosAB + C cos AC +D cos AD = 0,
AcosBA -B -\- CcosBC + DcosBD = 0,
A cos CA + B cos CB - C + i) cos CD = 0,
'J
A cosBA -f B cosDB + C cosZ> C -D = 0
from which we can eliminate the areas A^ B, C, i), and get
a determinant relation between the six angles of intersection
of the four planes.
Now as these are any four planes, the perpendiculars let
fall on them from any point will meet a sphere described
with that point as centre in four quite arbitrary points, say
a, 5, c, dj and each angle as ab is the supplement of the cor-
responding angle AB between the planes, hence the former
condition.
N.B. The vanishing of a determinant (see Higher Algebra,
Art 33, Ex, 1) shows that the first minors of any one row are
respectively proportional to the corresponding first minors of
any other. We see by this article that the minors of the
second determinant are proportional to the areas of the faces
of the tetrahedron.
The reader will not find it difficult to show that for any
THE RIGHT LINE. 37
four points on the sphere, each first minor of the corresponding
determinant is that function of one of the four spherical triangles
formed by the points which we mentioned in the note to Art. 32
and which has been called by v. Staudt, Crelle^ 24, p. 252,
1842, the sine of the solid angle that triangle subtends at the
centre of the sphere.
55. To find the radius of the sphere circumscribing a tetra-
hedron.
Since any side a of the tetrahedron is the chord of the arc
a^ . . .
whose cosine is a, we have a = 1 — — -.^ , with similar expressions
for /3, 7, &c. ; and making these substitutions, the first formula
of the last paragraph becomes
F 2cWh'd' + We'dT + "icTa'd^ - a'd* - bV - cT
4? "^ 16r« ~ ^'
whence if ad+be + cf= 25,
behave ^^S(S-adHS-ie)i8-cf) ^
which has been otherwise deduced, see Higher Algebra^ Art. 26.
The reader may exercise himself in proving that the shortest
distance between two opposite edges of the tetrahedron is equal
to six times the volume divided by the product of those edges
multiplied by the sine of their angle of inclination to each other,
which may be expressed in terms of the edges by the help of
the relation 2ad cos d = b'' + e^ - c^ —f\
56* We can establish the general formulae for transforma-
tion of quadriplanar coordinates by proceeding one step farther
in finding the centre of mean position than we did in Art. 9.
We see that if in the tetrahedron whose vertices are P^, P^^
P3, P^, the line joining P^ to P^ be cut in P', in the ratio n : m^
then the line joining P' to P^ in P" in the ratio I : m-\- n, and
lastly that joining P" to Pj in P in the ratio k : 1+ 77i + n, the
The student may omit the rest of this chapter on first reading..
38 THE RIGHT LINE.
perpendicular x from Pon. any plane on which the perpendiculars
from P,, P,, P3, P, are x,, a;,, ajg, a?,, is
kx, -f Za::, + w?a;, + nx.
k -\- I ■{■ m -{■ 71
Now it is evident that k '. k-\-l->tm + n as the pyramid on
P^P^P^ whose vertex is at P is to the pyramid on the same base
whose vertex is at P^, or, as the perpendiculars from those points
on the plane P^P^P^. We have similar values for the coefficients
01 fl?^, ajg, x^.
Now suppose we call ^ the perpendicular from P on the
plane P,^P^P^^ t] that from P on the plane P^P^P^^ ^ that on
the plane P^P^P^^ and &> that on P^P^P^. Also if the perpen-
dicular from P, on P^P^P^ be ^q, from P^ on PgPjP,, t/^, from P3
on P^P^P^t ^0) ^^^ ^^^"^ -^4 on P.P^Pa, cOq, we may write our
equation
Evidently similar equations give the perpendiculars fr-om P
on the other planes of reference ; for instance,
So % ?» <"-
0
Thus, writing down these four equations, we have the full
system requisite for a transformation of coordinates from the old
planes of a;, y^ z^ w to the planes ^, ?;, ^, &>.
It will sometimes be convenient to use a single letter for
f : fg &c., whereby our expressions will gain in compactness,
but at the expense of apparent homogeneity.
It is evident that the transformation of coordinates is quite
similar for the coordinates of planes.
57. If we denote by ai^, y^, z^^ 10^ the perpendiculars from
the vertices on the opposite sides of the original tetrahedron,
we have obviously, if A^ P, C, D be the areas of those faces,
Ax^ = By^=Cz^ = Dw^=:^V,
where V denotes the volume of that tetrahedron.
THE EIGHT LINE.
39
By this we may -write down the solutions of the equations
in last article in the form
x^
2/0
w„
where f,, ^^, ^^, ^^ are the perpendiculars on the plane ^ from
the vertices of the original tetrahedron. -
Also the relation which can at once be written down by
equating the volume of the tetrahedron of reference to the sum
of the four tetrahedra which its faces subtend at any point, viz.
Ax ■\-By+ Cz + Dw = SV may be written
X y z w ^
- + - +- + — = 1,
and in like manner we have
as the relations connecting in each system the homogeneous
coordinates with an absolute numerical quantity (cf. ConicSj
Art. 63).
Ex. To express the volume of a tetrahedron by the homogeneous coordinates
of its vertices.
If we multiply the determinant expression, found Art. 36, for six times the
volume TF by ^
cos a , cosjS , cosy , 0 (,
cos a', cos/3', cosy', 0
cos a", cos/j", cosy", 0
0 , 0 , 0,1
which is the same as the determinant in note Art. 32, and as in the transformation
(G) Art. 18, we find
A"
X"
X"
Y' , Z' , 1
Y" , Z", 1
Y'", Z'", 1
X'-, Y'", Z'*-, :
as the product of six times the volume W by the quantity which we may call the
sine of the solid angle [XYZ) Art. 54.
Now these coordinates are measured along the axes, and we want to refer to
perpendiculars on the coordinate planes. Hence we may write the new coordinates
x — X sin;5, y =Y amq, z~ Z sin r, where p, q, r are the angles the axes of
X, Y, Z make with the planes YZ, r^ &c. ; therefore
x' , y' , z' , I
x" , y" ,z",l
x'", y'", z'", 1
x'", y'-, z'", 1
= 6TF s,m.p sin q sin r sin {XYZ),
40
THE RIGHT LINE.
or by the relations
X y z w ^ .
- + - + - + — =1, &c.,
X,
X , y , z , w
«." »." *" ««"
X J y J z f to
x"\ 'f, z"', w"
X'
y
ff
w '
Vi,
= 6 TFWfl sin^ sin g- sin r sin {XYZ).
We may give this another form by remarking that the determinant reduces for the
tetrahedron of reference to the continued product, which is its leading term, hence
3^02/0^0^0 = 6 FWfl sin^ sin q sin r sin {XYZ),
whence, dividing the former equation by this,
{xYz"'w"') _ W
57a. If we had employed quadrlplanar coordinates in
Art. 40, we should have used for the coordinates of any point P
on the line joining Pj, P^,
from which, by eliminating I and ?Wj we find each determinant
of the matrix
w
X, y, z,
X.
1)
X.
'25
s„ w^
'21
w^
= 0.
These four determinants contain the coordinates of P,, P^ only
in the groups
(2/1^2)) (^1^2)5 {^xV^^
(a'.W'a)) (yi^'^Ji (^1^2))
which are connected by the identity
(^,^2) (^,^2) + (^.«'2) (3/1*^2) + (^12/2) (^i^^a) = 0-
Thus these six quantities so connected amount to four
independent ratios determining the equations, and are homo-
geneous coordinates of the right line ; we shall frequently denote
them, for brevity, by the letters
with or without two suffixes to indicate, as may sometimes be
required, the two points determining the right line; in all
cases these quantities are subject to the relation
ps + 2^ + '^'u = 0.
THE RIGHT LINE. 41
The geometrical value of these coordinates was obtained
Ex. Art. 50, where we saw that each of them, as, for instance,
(3/1^2) is the product of the distance P^P^-, by the sine of the angle
between the planes, which are named in it, multiplied into the
shortest distance of P^P^ from the edge in which those planes
intersect and into the sine of the angle between that edge
and P^P^.
Thus the equations connecting the coordinates of any point
with the coordinates of any right line passing through it are
any two of the four
yu — zt + wp = 0,
— xu + zs + wq = 0,
xt — ys + wr =■ 0,
xp + 1/q + zr = 0,
from which always hy ps + qt + ru = 0 the remaining two can be
deduced. These are the equations of a line as locus or ray.
57b. In like manner. Art. 38, if aj)^c^d^^ <^2^2^A ^^ t^®
coordinates of two planes IT,, n^, the coordinates of any plane
through their line of intersection are
hence for a line regarded as envelope or axis, we have the system
of equations
a , J , c , u
o„ Z*,, c„ d^
«2» Kl ^27 ^2
= 0,
which, adopting a notation in analogy with what precedes,
(^.Cj = '^Ul (C.«J = '^125 K^2) = P.2J
may be written, omitting suffixes,
5u - CT + c?7r = 0,
— av +ccr + dK = 0,
ar — h(T ■+ dp = 0,
air + hK-\- cp =0,
G
42 THE RIGHT LINE.
subject to
TTIT + KT + pv = 0.
If this line contain the point P,, since then
ax^ + hy^ + C0J + dw^ = 0,
we may substitute for a and h in terms of c and d and make
the coefficients of c and d vanish ; and similarly for the others,
hence in this case
y,p - ^i« + w,o- = 0,
— £c,p 4- 2!j7r + 10 J = 0,
x^a + T/j + z^v = 0.
In like manner, if in the last article we had sought for
the conditions that the ray should be contained in the plane
a, bj c, dj we should have found
br —cq -\- ds = 0,
- ar + cp + dt = 0,
aq—bp +du = 0,
as + bt + cu =0.
Further, if we have the point P^ also on the axis, we find
p : q : 0^ : s : t : 2i = (T : T : V : TT : k : p^
or in full, if the line joining P^ to P^ be identical with the line in
which n,, n^ intersect, each determinant vanishes in the matrix,
(3/1^.2) > (^i^-J, (rK^yJ, [x^w;), (3/,m;J, (^^wJ
Thus we see, that equations in the homogeneous coordinates
of a right line are capable of being expressed in either system,
the passage from one to the other being effected by an inter-
change of the coordinates p and 5, q and f, r and u.
N.B. These results are merely another way of presenting
the four simultaneous relations
a^x^ -f ^?/i + c,^, + d^io^ = 0,
«2«. + K'Ji + c-/. + <^^\ = 0,
THE RIGHT LINE.
43
57c. The determinant of the homogeneous coordinates of
four points
^ij y.j ^
J "^D
W.
a^2» y^i \^ ^2
^^-^ y^i ^31 ^8
^45 ^4? ^41 ^^4
whose geometric value we deduced in Ex. Art. 57, may be
written out in full, as in Higher Algebra^ Art. 7 ; and it is
easily seen that the terms occur only in the groups of second
minors, which are the homogeneous coordinates of the lines
arrived at in 57a.
Now when the line joining points 1 and 2 intersects the line
joining 3 and 4, the four points are coplanar and the determinant
vanishes.
Hence it appears that the condition that two right lines
\s, t, uj ' \ s, t , u J
should intersect is
jps + sp + gt' + tg + ru + ur = 0.
bid. By what precedes we can see how to determine the lines
which meet four given right lines. For if the coordinates of the
required Hue be ^ ' and of the given lines ^" " '' &c.j
S, tj Uj Sj, f^J Wj,
we have
ps^ + gt^ + ru^ + sp^ + ^2-4 + ur^ = 0,
which determine p^ g^ r, s linearly in terms of t and u^ and when
these values are substituted in the universal relation
ps + gt-{- ru = 0,
a quadratic is found m t : u^ which determines the lines, two in
number, which are required.
44
THE RIGHT LINE.
57e. In the coordinates of a line we have in transformation
to consider the transformed coordinates of two points or planes.
Ex. gr. considering
X = x^X-\ x^ r+ x^Z\ x^ W, x' = x^X + x^ Y' 4 x^Z' + x^ W\
y=y,X+y^Y+y^Z+y^W, y=y,X'^yJ' + y^Z'^y^W\
&c., we have
y^ ^
y\^'
=
y,) y..? 2^3? y^
^\1 ^11 ^3) ^4
or
X , Y , Z , W
X\ Y\ Z\ W
P =i>.3^ + ^3. Q "rp.^R + p,,8 4-i?,, r+ p3, U,
^=^23^+ ^3.^ + ^..^ + '•.4'^+ ^24^+ ^34 ^J
*=«.3^+ «3.<?+ «i2^+ *U^+ ^24^+ ^34 ^»
t=h.P+ ^S^Q+ ^,.^+ ^U'^+ ^.4^+ ^34^,
the coefficients of the transformation evidently being the coor-
dinates of the edges of the new tetrahedron referred to the old.
If we multiply these equations in order by s,^, i,^, t/,^, j^,^,
?i4? ^14 ^^^ ^^^) ^^ evidently solve for P in terms of the old
coordinates, and (Art. 57c) the factor on P is the modulus of
transformation ; it is easy to complete the solution.
( 45 )
CHAPTER IV.
*PEOPERTIES COMMON TO ALL SURFACES OF THE
SECOND DEGREE.
58. We shall write the general equation of the second
degree
(a, &, c, dj, g, h, Z, w, n) [x, y, z, If =
or ax^ + by" + cz' + J + Ifyz -f 2gzx + 2Ax^ + 2lx + 2my + 2ns = 0.
This equation contains ten terms, and since its signification is
not altered, if by division we make one of the coefficients unity,
it appears that nine conditions are sufficient to determine a
surface of the second degree, or, as we shall call it for shortness,
a quadric\ surface. Thus, if we are given nine points on the
surface, by substituting successively the coordinates of each in
the general equation, we obtain nine equations which are
he
sufficient to determine the nine unknown quantities - , - , &c.
^ a a
And, in like manner, the number of conditions necessary to
determine a surface of the 'nP^ degree is one less than the number
of terms in the general equation.
The equation of a quadric may also (see Art. 38) be ex-
pressed as a homogeneous function of the equations of four
given planes x, y^ s, w.
(a, J, c, d^f, ,7, 7«, ?, w, n) (a;, ?/, s, lof ==
or ax^-{ hy'^-^cz'^+dw'-\-2fyz-\- 2gzx-\-'2hxy-\-'ilxw + ^myw + 2nzw = 0.
For the nine independent constants in the equation last written
may be so determined that the surface shall pass through nine
* The reader will compare the corresponding discussion of the equation of the
second degree {Conies, Chap. X.) and observe the identity of the methods now pursued
and the similarity of many of the results obtained.
t In the Treatise on Solid Geometry by Messrs. Frost and Wolstenliolme, surfaces
of the second degree are called conicoids.
46 PROPERTIES COMMON TO ALL SURFACES
given points, and therefore may coincide with any given quadric.
In like manner (see Conies^ Art. 69) any ordinary tc, ?/, z equa-
tions may be made homogeneous by the introduction of the
linear unit (which we shall call lo) ; and we shall frequently
employ equations written in this form for the sake of greater
symmetry in the results. We shall however, for simplicity, com-
mence with cc, 3/, z coordinates.
59. The coordinates are transformed to any parallel axes
drawn through a point xy'z\ by writing a + a;^, y -Yy •, z + z
for £c, 3/, z respectively (Art. 16). The result of this substitu-
tion will be that the coefficients of the highest powers of the
variables (a, ^, c, /, g, h) will remain unaltered, that the new
absolute term will be U' (where U' is the result of substituting
a;', 2/', z' for x, y, z in the given equation), that the new coeffi-
cient of X will be 2 [ax' 4 hy + gz' + 1) or -j-f- , and, in like
manner, that the new coefficients of y and z will be —j~r and
rJTT
-y-^ . We shall find it convenient to use the abbreviations
dz
U U U fori— liE 1^
60. We can transform the general equation to polar co-
ordinates by writing x = Xp, y = jmp, z = vp (where, if the axes
be rectangular, \, /i, v are equal to cos a, cos/3, cos 7 respec-
tively, and if they are oblique (see note, p. 7) X, /t, v are still
quantities depending only on the angles the line makes with
the axes) when the equation becomes
p' (aV + hfi' + cv^ + 2fixv + 2gv\ -f 2AV)
+ 2p [IX + w/i + nv) +d = Q.
This being a quadratic gives two values for the length of the
radius vector corresponding to any given direction ; in ac-
cordance with what was proved (Art. 23), viz. that every right
line meets a quadric in two 2)0ints.
61. Let us consider first the case where the origin is on the
surface (and therefore J = 0), in which case one of the roots of
OF THE SECOND DEGREE. 47
the above quadratic is p = 0 ; and let us seek the condition that
the radius vector should touch the surface at the origin. In
this case obviously the second root of the quadratic will also
vanish, and the required condition is therefore lX-\- miJ,-\- nv = 0.
If we multiply by p and replace \p^ fip^ vp by a;, y, s, this
becomes
Ix + my + ns = 0,
and evidently expresses that the radius vector lies in a certain
fixed plane. And since X, /t, v are subject to no restriction but
that already written, every radius vector through the origin
drawn in this plane touches the surface.
Hence we learn that at a given point on a quadric an In-
finity of tangent lines can be drawn, that these lie all in one
plane which is called the tangent plane at that point ; and that
if the equation of the surface be written in the form u^-\-u^ = 0,
then u^ = 0 is the equation of the tangent plane at the origin.
62. We can find by transformation of coordinates the equa-
tion of the tangent plane at any point x'y'z' in the surface.
For when we transform to this point as origin, the absolute term
vanishes, and the equation of the tangent plane is (Art. 59)
xu;+yu:^zu:=^%
or, transforming back to the old axes,
\x - x') U; + [y - y') U: + {z- z'] U; = 0.
This may be written in a more symmetrical form by the Intro-
duction of the linear unit ly, when, since U Is now a homo-
geneous function, and the point x'y'z'uf is to satisfy the equation
of the surface, we have
x' u; + y' c7; + 2' c/; + to' u; = u'= o.
Adding this to the equation last found, we have the equation
of the tangent plane in the form
xu;-^yv; + zu;-viou:=Q;
or, writing at full length,
x {ax' + hy' + gz' + Iw') + y {Jix + hy' \-fz + mw^
+ z {qx' -Vfy' + cz + mo) + w [Ix' -f my' + nz' -h dw') = 0.
48 PROPERTIES COMMON TO ALL SURFACES
This equation, it will be observed, is symmetrical between xyzio
and xyziv\ and may likewise be written
x'U^+y'U^ + z'U^ + w'U^^O.
63. To find the point of contact of a tangent line or plane
drawn through a given point x'yz'w' not on the surface.
The equation last found expresses a relation between xyzw^
the coordinates of any point on the tangent plane, and x'y'z'w'
Its point of contact ; and since now we wish to indicate that the
former coordinates are given and the latter sought, we have
only to remove the accents from the latter and accentuate the
former coordinates, when we find that the point of contact must
lie in the plane
xv:^yv:^zv:^xov:=%
which is called the polar plane of the given point. Since the
point of contact need satisfy no other condition, the tangent
plane at any of the points where the polar plane meets the
surface will pass through the given point ; and the line joining
that point of contact to the given point will be a tangent line
to the surface. If all the points of intersection of the polar
plane and the surface be joined to the given point, we shall
have all the lines which can be drawn through that point to
touch the surface, and the assemblage of these lines forms what
is called the tangent cone through the given point.
N.B. In general a surface generated by right lines which
all pass through the same point is called a co?ie, and the point
through which the lines pass is called its vertex. A cylinder
(see p. 15) Is the limiting case of a cone when the vertex is
infinitely distant.
64. The polar plane may be also defined as the locus of
harmonic means of radii passing through the pole. In fact, let
us examine the locus of points of harmonic section of radii
passing through the origin ; then if p\ p" be the roots of the
quadratic of Art. 60, and p the radius vector of the locus, we
are to have
2 _ 1 1 _ 2 (XZ + yu,m + i/n)
p p p d '
of THE SECOND DEGREE. 49
or, returning to x, ?/, z coordinates,
Ix + my + ??3 + ^ = 0 ;
but this is the polar plane of the origin, as may be seen by
making x\ y\ z all = 0 in the equation written in full (Art. 62).
From this definition of the polar plane, it is evident that if
a section of a surface be made by a plane passing through any
point, the polar of that point with regard to the section will
be the intersection of the plane of section with the polar plane
of the given point. For the locus of harmonic means of all
radii passing through the point must include the locus of
harmonic means of the radii which lie in the plane of section.
65. If the polar plane of any point A pass through B^ then
the polar plane of B will pass through A.
For since the equation of the polar plane is symmetrical
with respect to xyz^ a^'j/V, we get the same result whether we
substitute the coordinates of the second point in the equation
of the polar plane of the first, or vice versa.
The intersection of the polar planes of A and of B will be
a line which we shall call the polar line, with respect to the
surface, of the line AB. It is easy to see that the polar line
of the line AB is the locus of the poles of all planes which
can be drawn through the line AB.
'o'
66. If in the original equation we had not only tZ=0, but
also Z, m^ n each = 0, then the equation of the tangent plane
at the origin, found (Art. 61), becomes illusory since every term
vanishes ; and no single plane can be called the tangent plane
at the origin. In fact, the coefficient of p (Art. 60) vanishes
whatever be the direction of p, and therefore every line drawn
through the origin meets the surface in two consecutive points,
and the origin is said to be a double point on the surface.
In the present case, the equation denotes a cone whose
vertex is the origin, as in fact does every homogeneous equation
in x^ ?/, z. For if such an equation be satisfied by any co-
ordinates x\ y\ s', it will be satisfied by the coordinates
hx .^ ky\ hz' (where h is any constant), that is to say, by the co-
ordinates of every point on the line joining x'yz' to the origin.
H
50 PEOPERTIES COMMON TO ALL SURFACES
This line then lies wholly in the surface, which must therefore
consist of a series of right lines drawn through the origin.
The equation of the tangent plane at any point of the cone
now under consideration may be written in either of the forms
The former (wanting an absolute term) shews that the tangent
plane at every point on the cone passes through the origin ;
the latter form shews that the tangent plane at any point
xy'z' touches the surface at every point of the line joining
x'y'z' to the vertex ; for the equation will represent the same
plane if we substitute kx\ ky\ kz' for x\ y\ z'.
When the point x^yz' is not on the surface, the equation we
have been last discussing represents the polar of that point, and
it appears in like manner that the polar plane of every point
passes through the vertex of the cone, and also that all points
which lie on the same line passing through the vertex of a cone
have the same polar plane.
To find the polar plane of any point with regard to a cone
we need only take any section through that point, and take
the polar line of the point with regard to that section ; then
the plane joining this polar line to the vertex will be the polar
plane required. For it was proved (Art. 64) that the polar
plane must contain the polar line, and it is now proved that the
polar plane must contain the vertex.
67. We can easily find the condition that the general equa-
tion of the second degree should represent a cone. For if it
does it will be possible by transformation of coordinates to
make the new ?, m, w, d vanish. The coordinates of the new
vertex must therefore (Art. 59) satisfy the conditions
which last combined with the others is equivalent to U^ = 0.
And if we eliminate x\ y\ z from the four equations
ax + hy' -\-gz' ^ 1=0,
hx + hy+/z-\-m = 0,
gx -f fy' + cz' + ?i = 0,
Ix + my + nz -f f? = 0,
OP THE SECOND DEGREE.
51
we obtain the required condition in the form of the determinant
a.
h^ b, /, 771
9, /)
L
c, n
w, Wj d
= 0,
which, written at full length, is
abcd+2afmn + 2hgnl + 2chhn + 2dfgh — hcV - cam'^ — ahri^ — adf'
- Mg' - cdJt" -\-fr -H gW + K'n^ - 2ghnn - 2hfnl - 2fghn = 0.
We shall often write this equation A = 0, and (as in Conies^
p. 153) shall call A the discriminant of the given quadric.
It will be found convenient hereafter to use tbe abbreviations
A^ B, (7, B, 2F, 2G. 2i7, 2X, 2M^ 2iV, to. denote the differential
coefficients of A taken with respect to a, J, c, &c. Thus
A = hcd + 2fmn — bn^ — cvf — df\
B = cda + 2gnl — c^ — ati^ — dg^^
C=dab + 2hlm - ani' - bP - dK\
D = abc + 2fgh - af - bf - ch\
F= amn + dgh - adf +fl^ — hnl - ghn^
G = bnl + dhf — bdg -f gm^ — /^''i — hmn^
11= elm + d/g ^cdh +hn^ —gmn—fnl,
L = bgn -\- dim —bcl ■\-lf* —hfn —gfm^
M=chl ■\- afn — cam + mg^ - fgl — ghn^
N = afm + bgl — abn + nh^ —gJim — hfl.
68. Let us return now to the quadratic of Art. 60, in which
d is not supposed to vanish, and let us examine the condition
that the radius vector should be bisected at the origin. It is
obviously necessary and sufficient that the coefficient of p in
that quadratic should vanish, since we should then get for p
values equal with opposite signs. The condition required
then is
l\ + mji -4- nv =■ 0,
which multiplied by p shews that the radius vector must lie in
the plane Ix -f my + W2 = 0. Hence (Art. 64) every right line
draion tlirough the origin in a plane imrallel to its polar plane
is bisected at the origin.
52 PKOPERTIES COMMON TO ALL SUEFACES
69. If, however, we had 1 = 0, 7/i = 0, n = 0, then everi/ line
drawn through the origin would he bisected and the origin
would be called the centre of the surface. Every quadric has
in general one and hut one centre. For if we seek by trans-
formation of coordinates to make the new Z, m, n = 0, we obtain
three equations, viz.
Z7/ = 0, or ax' + hy' ■]■ gz' + I =0,
Z7/ = 0, or hx' + ly +fz' + ?n = 0,
Zj; = 0, or gx ■\-fy' + cz' -\- n =0,
which are sufficient to determine the three unknowns x\ y , z' .
L , M , N
The resulting values are x = j^, y' =—, z' = -j^, where L, J/,
N, D have the same meaning as in Art. 67.
If, however, J9 = 0, the coordinates of the centre become
infinite and the surface has no finite centre. If we write the
original equation w^ + m, + m^ = 0, it is evident that D is the
discriminant of m^.*
70. To find the locus of the middle points of chords parallel
^. X y z
to a given Line - = - = - .
^ \ fJ, V
If we transform the equation to any point on the locus as
origin, the new Z, m, w must fulfil the condition (Art. 68)
lX + m/j.-\- nv = 0, and therefore (Art. 59) the equation of the
locus is
This denotes a plane through the intersection of the planes
Z7^, ?7j,, C^, that is to say, through the centre of the surface.
* It is possible that the numerators of these fractions might vanish at the same
time with the denominator, in which case the coordinates of the centre would become
indeterminate, and the surface would have an infinity of centres. Thus if the three
planes Z7„ ZT'j, Z/j all pass through the same line, any point on this liae wOl be a
centre. The conditions that this should be the case may be written
a, h, g, I
h, b, f, m
g, f, c, n = 0,
the notation indicating that all the four determinants must = 0, which are got by
erasing any of the vertical lines, We shall reserve the fuller discussion of these
cases for the next chapter.
OP THE SECOND DEGREE. 53
It is called the diametral plane conjugate to the given direction
of the chords.
If xy'z be any point on the radius vector drawn through
the origin parallel to the given direction, the equation of the
diametral plane may be written
If now we take the equation of the polar plane of hx^ ky'^ Jcz'j
hx'U^ + ky'U^+'kz'U^+U^ = Q^
divide it by ^, and then make h infinite, we see that the
diametral plane is the polar of the point at infinity on a line
drawn in the given direction, as we might also have inferred
from geometrical considerations (see Conies^ Art. 324). In like
manner, the centre is the pole of the plane at infinity, for if
the origin be the centre, its polar plane (Art. 64) is c? = 0,
which (Art. 30) represents a plane situated at an infinite
distance.
In the case where the given surface is a cone, it is evident
that the plane which bisects chords parallel to any line drawn
through the vertex is the same as the polar plane of any
point in that line. In fact it was proved that all points on
the line have the same polar plane, therefore the polar of the
point at infinity on that line is the same as the polar plane
of any other point in it.
71. The plane which bisects chords parallel to the axis
of X is found, by making /a = 0, v = 0 in the equation of Art. 70,
to be
f7j = 0, or ax + hy+gz + l=Q^^
and this will be parallel to the axis of y^ \{ h = 0. But this
is also the condition that the plane conjugate to the axis of y
should be parallel to the axis of x. Hence if the plane con-
jugate to a given direction he parallel to a second given line^
the plane conjugate to the latter ivill he parallel to the former.
* It follows that the plane a; = 0 wiU bisect chords parallel to the axis of x, if
A = 0, ^r = 0, Z = 0 ; or, in other words, if the original equation do not contain any
odd power of x. But it is otherwise evident that this must be the case in order that
for any assigned values of y and z we may obtain equal and opposite values of x.
54 PEOPEETIES COMMON TO ALL SURFACES
When A = 0, the axes of x and y are evidently parallel to
a pair of conjugate diameters of the section by the plane of xy ;
and it is otherwise evident that the plane conjugate to one
of two conjugate diameters of a section passes through the other.
For the locus of middle points of all chords of the surface
parallel to a given line must include the locus of the middle
points of all such chords which are contained in a given plane.
Three diametral planes are said to be conjugate when each
is conjugate to the intersection of the other two, and three
diameters are said to be conjugate when each is conjugate to
the plane of the other two. Thus we should obtain a system
of three conjugate diameters by taking two conjugate diameters
of any central section together with the diameter conjugate
to the plane of that section. If we had in the equation /= 0,
g = 0, A = 0, It appears from the commencement of this article
that the coordinate planes are parallel to three conjugate
diametral planes.
When the surface is a cone, It is evident from what was
said (Arts. 66, 70) that a system of three conjugate diameters
meets any plane section in points such that each is the pole
with respect to the section of the line joining the other two.
72. A diametral plane is said to be principal if it be per-
pendicular to the chords to which it is conjugate.
The axes being rectangular, and X, yti, v the direction-
cosines of a chord, we have seen (Art. 70) that the corresponding
diametral plane is
\[ax-^'hy-\-gz + l)+/ju{hx + by -ifz + m)+v (gx+fy -\-cz + n)=0,
and this will be perpendicular to the chord, if (Art. 43) the
coefficients of a;, y, z be respectively proportional to X, /*, v.
This gives us the three equations
\a-\- jJi-^ vg = Jc\ \h + fxb + vf= 7c fx,, \g + fif+ vc = hv.
From these equations, which are linear in X, /*, y, we can
eliminate X, /a, v, when we obtain the determinant
a - h, Ji, g
h, h- k, f
g, /, c-k =0,
OF THE SECOND DEGREE. 55
which expanded gives a cubic for the determination of A-, viz.
k' - k' {a + h + c] + k {bc + ca + ah -f - g' - K')
~ [abc + 2fgh - af - hf - ch') = 0.
And the three values hence found for k being successively
substituted in the preceding equations enables us to determine
the corresponding values of \, //., v. Hence, a quadric has
in general three principal diametral planes, the three diameters
perpendicular to which are called the axes of the surface. We
shall discuss this equation more fully in the next chapter.
Ex. To find the piincipal planes of
7x- + 6_y2 + bz- — 4xy — 4j/z = 6.
The cubic for k is
yL-s _ i8/t2 + 99/.- - 162 = 0,
whose roots are 3, 6, 9. Now our three equations are
7X-2/jL = k\, - 2X + Gfi - 2v - k/x, - 2ju + oi/ = kv.
If in these we substitute k — 3, we find 2\ = /x = i^. Multiplying by p, and sub-
stituting X for \p, &c., we get for the equations of one of the axes 2x — y — z. And
the plane di-awn thi-ough the origin (which is the centre), pei-pendicular to this line,
is a; + 2y + 2z = 0. In like manner the other two principal planes are 2x — 2y + z — Q,
2x + y-2z = a*
73. The sections of a quadric by parallel planes are similar
to each other.
Since any plane may be taken for the plane of xij^ it is
sufficient to consider the section made by it, which is found
by putting z = 0 \n the equation of the surface. But the section
by any parallel plane is found by transforming the equation
to parallel axes through any new origin, and then making z = 0.
If we retain the planes yz and zx, and transfer the plane
xy parallel to itself, the section by this plane Is got at once
by writing ^ = c in the equation of the surface, since it is evident
that it is the same thing whether we write z + c for s, and
then make 2; = 0, or whether we write at once z = c.
* li U denote the terms of highest degree in the equation, and S denote
{be -p) x-2 + (c« - g") y"- + {ab - r-) z' +2{gh- af) yz + 2 {hf- by) zx + 2{fg- ck) xy,
then the equation of the three principal planes, the centre being origin, is denoted
by the determinant
X, y, z
S^, ,S,, S, [ = 0.
56 PROPERTIES COMMON TO ALL SURFACES
And since the coefficients of x\ xy^ and 3/' are unaltered by
this transformation, the curves are similar.
It is easy to prove algebraically, that the locus of centres
of parallel sections is the diameter conjugate to their plane,
as is geometrically evident.
74. If />', p" be the roots of the quadratic of Art. 60,
their product p p" is = J divided by the coefficient of p\ But
if we transform to parallel axes, and consider a radius vector
drawn parallel to the first direction, the coefficient of p^ remains
unchanged, and the product is proportional to the new d.
Hence, if through two given points A^ B^ any parallel chords be
drawn meeting the surface in points i?, i^'; S^ S\ then the
products RA.AB\ SB.JBS' are to each other in a constant
ratio, namely, U' : W where U\ U" are the results of sub-
stituting the coordinates of A and of B in the given equation.
75. We shall conclude this chapter by shewing how the
theorems already deduced from the discussion of lines passing
through the origin might have been derived by a more general
process, such as that employed [Conies^ Art. 91). For sym-
metry we use homogeneous equations with four variables.
To find the ^points where a given quadric is met hy the line
joining two given points xy'z'io\ x"y"z"w" .
Let us take as our unknown quantity the ratio /u, : \, in which
the joining line is cut at the point where it meets the quadric,
then (Art. 8) the coordinates of that point are proportional to
\x' + P'x" 1 "^y + y^y" 1 ^2' + a^^"? ^^^' + /^^^ ' 5
and if we substitute these values in the equation of the surface,
we get for the determination of X : yu., a quadratic
The coefficients of X^ and /x^ are easily seen to be the results
of substituting in the equation of the surface the coordinates
of each of the points, while the coefficient of 2Xyu may be seen
(by Taylor's theoi-em, or otherwise) to be capable of being
written in either of the forms
x'u:'+y'u:'+z'u:'^w'u:\
or x'' u; + y'' u,; + z" u; + w" u;.
OF THE SECOND DEGREE. 57
Having found from this quadratic the values of \ : /i, sub-
stituting each of them in the expressions Xx' + /xar", &c., we
find the coordinates of the points where the quadric is met by
the given line.
76. If x'y'z'w be on the surface, then Z7' = 0, and one of
the roots of the last quadratic is /* = 0, which corresponds to
the point x'yzw'^ as evidently ought to be the case. In order
that the second root should also be /^ = 0, we must have P=0.
If then the line joining xyz'io to x"y"z"w" touch the surface
at the former point, the coordinates of the latter must satisfy
the equation
xV;-^yTJ^-^zV;^wTJl = %
and since x"y"z"'w" may be any point on any tangent line
through x'y'z'w ^ it follows that every such tangent lies in the
plane whose equation has been just written.
77. If x'y'z'w be not on the surface, and yet the relation
P=0 be satisfied, the quadratic of Art. 75 takes the form
X^V -^ fi^U" = 0^ which gives values of X. : /i, equal with op-
posite signs. Hence the line joining the given points is cut by
the surface externally and internally in the same ratio ; that is
to say, is cut harmonically. It follows then that the locus of
points of harmonic section of radii drawn through x'y'z'w' is
the polar plane
xu;-\-yu;\zu;-\-wu; = o.
78. In general if the line joining the two points touch
the surface, the quadratic of Art. 75 must have equal roots,
and the coordinates of the two points must be connected by
the relation TJ'JJ" —F"*. If the point x'y'z'w be fixed, this
relation ought to be fulfilled if the other point lie on any of
the tangent lines which can be drawn through it. Hence the
cone generated by all these tangent lines will have for its
equation W = F\ where
p=xu;+yu.; + zu; + wu;.
Ex. To find the equation of the tangent cone from the point x'y'z' to the surface
x^ y^ z^ ^ , fx"^ y'~ 2:'2 ,v«^ y'' 2* iN f^x' yy' ««' ,\^
I
58
PROPERTIES COMMON TO ALL SURFACES
79. To find the condition that the plane (xx + ^y -\- 'yz ■{■ Zw
should touch the surface given hy the general equation.
First, if a;, y, z, w be the coordinates of the pole of this
plane, and k an indeterminate multiplier, we have (Art. 63)
in general
Jca. = ax + hy -{ gz + Iw^ k^ = hx-\-hy ■\-fz + mw^
ky =gx +fy + cz + nw, kS =lx + my + nz + dw,
to determine the pole of the given plane. Solving for x, y^ 2, w
from these equations, we find
Ax = k [Aa + Ilfi +Gy + iS),
Ay =k{Ha+ B/3 + Fy +M8),
Az =^k{Goi+F^+Cy+M),
Aio = k[La-^ JSW + Ny + DS),
where A, A^ Bj C, &c. have the same meaning as in Art. 67.
Now if these values satisfy the equation ax + fiy-{-yz-{Sw=-Oj
we get by eliminating them
Aa' -i B^' -{ Cy' + M'
+ 2F/3y + 2 Gya + 2Ha^ + 2Xa8 + 231/38 -f 2Ny8 = 0,
which is the required relation that this plane should touch the
surface.
The result of eliminating k, x, y, 2;, w from the four equa-
tions first written, and ax -i- /3y + yz + Bw = 0 may evidently be
written in the determinant form
a, /?, 7, 8
a, a, /?, g, I
^, A, 5, /, m = 0.
7, 9, /, c, n
8, ly w?, 7?, d
Each of these is a form in which we may write the condition
which must be satisfied by the coordinates of a plane if the plane
touch the surface (see Art. 38) ; that is to say, the tangential
OF THE SECOND DEGREE.
5%.
equation of the surface, or the equation of the surface as an
envelope of planes.
80. To find the condition that the surface should he touched
by any line
ax + ^y + yz + Sw = 0, ax + ^'y + '^'z + h'w — 0.
If the line touches, the equation of the tangent plane at the
point of contact will be of the form
(a + \a') a: + (/3 + \^') ?/ + &c. = 0.
If then we write in the first four equations of the last article
a + Xa' for a, &c., and then between these equations and the
two equations of the line, eliminate h^ kX, a?, y^ 2, w^ we have
the result in the determinant form
a,
A 7,
S
a'5
/3', 7',
8'
«J
«',
a.
h 9^
^
^,
^',
h,
^ /,
?n
7,
7,
9->
/, c,
n
S,
5',
?,
w, n,
d
0.
This Is plainly of the second degree in the coefficients of the
quadric, and is also a quadratic function of the determinants
a/3' — /3a', &c., that Is, of the six coordinates of the line.
If in the condition of Art. 79 we write a+Xa'for a, &c.,
and then form the condition that the equation in X should have
equal roots, the result will be the condition as just written
multiplied by the discriminant (Ex. 2, Art. 33, Higher Algebra).
For the two planes which can be drawn through a given line
to touch a quadric, will coincide either if the line touches the
quadric, or if the surface has a double point.
80a.* Given the six coordinates of any right line (p, q^ r,
5, f, u) to determine the coordinates of its polar line (Art. 65).
* The rest of this chapter may be omitted on first reading.
60
PROPERTIES COMMON TO ALL SURFACES
Since the polar line is the intersection of the polar planes
of the two points determining the ray (Art. 57a),
U;x + U:y + Viz 4- C7>,
Ul'x^Vl\j^Ul'z-vVl'w,
its coordinates as an axis (Art. 575) are
'^'=(c^;c4'^ >^'={u:u:\ p'^[v:v:\
<s'={v:v:\ r'={u:u:), v={u:u:),
Now if we expand
{u:u:^)=
X , 1/ , Z , IV
x'\ /', z\ w"
A, 5, /, m
9-, /) Cj ^
as in Art. 57e, and the others likewise, we get, by a trans-
formation of line coordinates, from the ray coordinates of one
line the axial coordinates of its polar line, since all the coefficients
are the second minors of a determinant of the fourth order — in
this case a symmetrical one, viz. the discriminant of the quadric.
As it is sometimes convenient to have abbreviations to denote
these second minors of the discriminant in the determinant form
of Art. 67, we shall adopt a double suffix notation, thus writing
the axial coordinates or their corresponding ray coordinates in
the form
«' = ^^^V + ^22? + %^ + «24« + ^J + «26^ = ^')
P = «3>P + «32? + «3S»* H- «34« + «35^ + «36^ = ^'j
0-' = a^^p + a^^q + a^^r + a^^s + a J + a^^u =p,
^' = %xP + «62? + «63^ + «64« + «65^ + ^66^ = ^'^
Now, if we multiply these equations in order by p, q^ r,
* The following are the values of the coefficients a,„ ffjj, &c. as they stand in the
above equations :
be —f^i fg —ck, hf — bg, hn — gm, bn —fm, fn —cm,
fg — ch , ca — g''' , gh — af, gl —an, fl — hn , cl — gn,
hf— bg , gh — af, ab — h^, am — hi , hm — bl , gm —fl,
hn — gm. gl — an, am — hi, ad —P , hd —ml, gd — nl,
hn—fm, fl —hn, hm—bl, hd—ml, bd — m"^ , fd—nm,
fn — cm, cl — gn, gm —fl, gd — nl , fd — nm, cd — n*.
OF THE SECOND DEGREE.
61
s, f, u and add, the quantity on the right side vanishes if the
line intersect its polar line (57c) ; but this happens only when
the given line is a tangent to one of the plane sections through
itself, that is, when it touches the surface. In this case, there-
fore, each of the lines touches the surface in their common
point.
Thus the condition that the right line should touch is
^uP^ + &c. + a^^u^ -f 'ia^^pq -f . . .+ 'iaju = 0, or briefly ^ = 0.
This can also be derived from the condition in Art. 78, which
may be written
u;', u:\ u;\ u:
X ,
y J
^ J
w
X-,
y'%
^'\
w
= 0,
and reduced by the process of this article, the quantity on the
left is found to be ^.
80J. The same problem may be treated as follows if the
right line be given as the intersection of two planes
ax + ^2/ + <yz -t Sio, a'x -\- ^'y -h 7'^ + h'w.
Forming the coordinates of the right line joining their poles
(Art. 79) we have, for instance, omitting a common multiplier,
V
a, /3, 7, 8
a', /3', 7', S'
H, B, F, M
a, F, C, N
which we may write
q =&.c. =t', &c.,
where BC — F^ = a^^, &c. But, Higher Algebra^ Art. 33, this
= A (ac?- Z^) = Aa^^, and so for each of the others. We thus
see how to solve the six equations in the last article. To find
p, for instance, we must multiply in order by a^^, a^^ a^^ a^^,
^24J ^34? ^^^ ^^^ 5 ^^'^^ S®^^
ay + a^/ + a^ + a^ + a^/ + a^ = Ap
= ^uP + a^d + %/ + ^4/ + <^J -1- «^64W'-
As before, this right line (axis) meets the polar right line
62
PEOPERTIES COMMON TO ALL SUEFACES
(axis) when each touches the surface ; thus the relation that this
may happen may be written in any of the forms
a^y+...+ ay + 2a,,7r/<: +...+ 2a, ju = 0,
or
'2 , f'i 1
.+ 2a,„o-V+...+ ... =0.
80c. To determine th.e points of contact of tangent planes
through the line (j9, q^ r, s, t^ u) to the quadric.
The coordinates of the plane determined by three points
scyzw^ x^y^z^^^ x^^z^w,^ are found by solving between the
equations
aa; 4- % -f C2 + dw =0,
ax^ + hy^ + cz^ + d^w = 0,
ax^ + hy^ -(- cz^ + d.^w = 0,
and with 6 an undetermined multiplier we may write them,
introducing the coordinates p^ j, r, s, i, u of the line 1, 2
yu— zt -^ wp— 6a^
— xu -^^ zs-f wq= 6bj
xt — ys + wr = 6cj
xp + yq -^ zr = — 6d.
These may be regarded as equations determining the coor-
dinates of any plane passing through the right line by means of
the coordinates of any definite point not upon the right line,
through which the plane is to pass.
Now if in the equations just written we assume that a: h:c: d
are the values oi U^ : U^ : U^ : U^ for the point ; this amounts to
enquiring what is the point whose polar plane passes through
the point itself and through the given right line. In other
words, the point of contact of a tangent plane through the given
line.
Thus, by eliminating x, ?/, z^ w we get, to determine 6^ the
biquadratic
da , 6h - w, 6g + t, 61 -p
6h + u, 6b , 6f-s, 6m -q
% - t, 6f + s, 6c , 6n -r
61 + ^, 6m -^ 2, 6n + r, 6d
which evidently reduces to a pure quadratic, and this is found
= 0,
OF THE SECOND DEGREE. 63
to be 6'^ A + ^ = 0. Substituting 6 from tbis equation, we de-
termine the coordinates x, ?/, 2, w of the point of contact by
solving between any three of the four following equations
6a.x-\- {dh-u)y + {0g + t)z+ [6l-p)w = Q^
[6h + u) X -\- &c. =0, &c.
The two points of contact arise from the double sign
Now if we solve the quadratic of Art. 75 we find under the
radical the quantity, - 4^, as noticed in Art. 80a. Hence we
may draw the following inferences as to the reality of the
intersections of a right line with a quadric, and of the tangent
planes which may be drawn through it, viz. we have taking
A positive, 4^ positive; intersections imaginary, contacts imaginary ;
for A positive, ^ negative ; intersections real, contacts real ;
for A negative, 4* positive ; intersections imaginary, contacts real;
for A negative, ^negative; intersections real, contacts imaginary.
As the contacts coincide if ^ = 0 this establishes once more the
relation that the line may touch.
80(^. We have thus found that whether considered as a
ray or as an axis the coordinates of any line touching a surface
of the second degree satisfy a relation of the second order.
We saw already (Art. 57c) that in like manner the coordinates
of any line which meets a given line satisfy a relation of the
first order. But in neither case is the relation the most general
one of its order which can subsist between those six coordinates.
In fact, we saw that instead of the coordinates of the fixed
right line being perfectly arbitrary, the universal relation of
line coordinates must subsist between them. And again, the
relation of the second degree just found instead of containing
the full number (21) of independent constants, has that number
of coefficients indeed, but all of them are functions of the
10 coefficients in the equation of the quadric surface touched.
Pliicker has applied the term complex of lines to the entire
system of lines which satisfy a single relation. In the case
of the complex of lines which satisfy a homogeneous relation of
the first degree between the six ray coordinates of a line, by
64 PROPERTIES COMMON TO ALL QUADRICS.
supposing fixed one of the points determining any ray, we
evidently get the equation of a plane through that point. If
we replace the ray coordinates by the axial coordinates, on
supposing one of the planes determining the line fixed, we
have the equation of a point in that plane. In like manner,
for a relation of the second degree, the ray coordinates give,
for a fixed point, a cone of the second degree with the fixed
point as vertex, and, the axial coordinates, taking a fixed plane
through the axis, give a conic section In that plane. In
particular if the relation be that establishing contact between
the right line and a quadric surface, the cone becomes the
tangent cone from the special point, and the conic the conic
of intersection of the special plane.
80e. To find the conditions that a right line he wholly con-
tained in the surface.
It should be observed that whereas in plane quadrics we
cannot have in the quadratic of Art. 75 each of the coefficients
zero without a certain relation holding between the coefficients
of the conic, in quadric surfaces the vanishing of those co-
efficients implies no such relation. In fact, if we write down
Z7' = 0, P=0, C/" = 0 in full, as
U;x' + U^y' + U^z' + u;w =0,
u; x" + u; y" + c/ ' z" + zj; w" = o,
TJl'x! + V^'y' + V^'z' + Z7>' = 0,
fZ/V + V^'y" 4 V^'z" -f Vl'w" = 0,
we see (as in Art. 57/6) that they imply only the identity
of the line joining the two points with its polar line. Thus as
the quadratic in X : /* is now indeterminate the line is wholly
contained in the surface.
We noticed (Art. 80a) regarding the condition for contact
that ^ =IJ' TJ" — P'\ Hence, differentiating 4' in succession with
regard to each of the coefficients of the quadric, as each result is
of the form 6 U'+ cf) Z7"+ %P, we see, that for a line to be wholly
contained in the quadric, its coordinates satisfy each of the ten
d^ d'^
relations 7 - = 0, &c,, rT?~^i &c., and these amount to no more
than three independent relations.
( 65 )
CHAPTER V.
CLASSIFICATION OF QUADRICS.
81. Our object in this chapter is the reduction of any equa-
tion of the second degree in three variables to the simplest form
of which it is susceptible, and the classification of the different
surfaces which it is capable of representing.
Let us commence by supposing the quantity which we called
D (Art. 67) not to be=0. By transforming the equation to
parallel axes through the centre, the coefficients Z, wi, n are
made to vanish, and the equation becomes
ax^ + ly^ + cz" + 2/3/5; + 2gzx + 2}ixy + c?' = 0,
where d' is the result of substituting the coordinates of the
centre in the equation of the surface. Remembering that
u'=x'u;+y'u.;+z'u;+w'u:,
and that the coordinates of the centre make C7/, ?7/j U^
vanish, it is easy to calculate that
,,_IL + mM+ nN+ dl) _ A
where A, Z>, Lj M^ N have the same meaning as in Art. 67.
82. Having by transformation to parallel axes mrt.de the
coefficients of ic, y^ z vanish, we can next make the co-
efficients of yz^ zx^ and xy vanish by changing the direction
of the axes, retaining the new origin ; and so reduce the
equation to the form
a:x^->rh'y'-^c'z'' + d' = Q.
It is easy to show from Art. 17 that we have constants
enough at our disposal to effect this reduction, but the method
we shall follow is the same as that adopted, Conies^ Art. 157,
K
66 CLASSIFICATION OF QUADEICS.
namely, to prove that there are certain functions of the co-
efficients which remain unaltered when we transform from one
rectangular system to another, and by the help of these relations
to obtain the actual values of the new a, 5, c.
Let us suppose that by using the most general transfor-
mation which is of the form
x = Xx i- f^y + vzj y = X'x + fiy + vz, z = X"« -f fi'y + v"zj
the function ax^ + ly^ + cz^ + Ifyz + 2gzx + "iTixy
becomes ax" + Vy' + cz' + 2/^3 4- 2g'zx + 2h'xy^
which we write for shortness Z7= U. And if both systems of
coordinates be rectangular, we must have
x' -^ y' + z' = x^ -^-y' + 1""^
which we write for shortness 8= S. Then if Jc be any constant,
we must have kS— U=kS— U. Now if the first side be
resolvable into factors, so must also the second. The discrimi-
nants of kS- U and of JcS— U must therefore vanish for the
same values of k. But the first discriminant is
F-F(a + 5 + c) + ^(Jc + ca + «&-/=-/- A')
- [abc + 2fgh - af - hg' - ch').
Equating, then, the coefficients of the different powers of k
to the corresponding coefficients in the second, we learn that
if the equation be transformed from one set of rectangular
axes to another, we must have
a + b + c = a' + V + c\
hc + ca + ah ~r - f - li^ = h'c + c V -f db' -f - f - Ji'%
ahc + 2fyk - af - Ig' - ch' = a'b'c' + y'g'h' - dp - b'g"' - c'h'\*
83. The above three equations at once enable us to trans-
form the equation so that the new y, g^ h shall vanish, since
* There is no difficulty in forming the corresponding equations for oblique co-
ordinates. We should then substitute for S (see Art. 19),
a;2 J. y2 4- ^2 + 2yz COS X 4- 222; cos /u + 2xy COS v,
and proceeding exactly as in the text, we should form a cubic in k, the coefficients of
which would bear to each other ratios unaltered by transformation.
CLASSIFICATION OF QUADRICS. 67
tbey determine the coefficients of the cubic equation whose
roots are the new a, i, c. This cubic is then
'a
-{a^-'b + G)a"'+ {he ^ca + ab -f - / - A*) a'
- {abc + y'gh - af - Ig" - ch*) = 0,
which may also be written
(a'- a)[a'- h)[a- c) -f (a'- a) -f {a'- b) - h' {a- c) - 2fgh = 0.
We give here Cauchy's proof that the roots of this equation
are all real. The proof of a more general theorem, in which
this is included J will be found in Lessons on Higher Algebra^
Lesson VI.
Let the cubic be written In the form
[a - a) {{a' - b) {a - c) -f] - / [a - b) - W {a - c) - 2fgh = 0.
Let a, /She the values of a' which make [a —b) [a —c)—f^=0^
and it is easy to see that the greater of these roots a is greater
than either b or c, and that the less root /3 is less than either.f
Then if we substitute in the given cubic a^ = a, it reduces to
-{{a-b)g'+2fgh+{a-c]Ji:^],
and since the quantity within the brackets is a perfect square
in virtue of the relation (a — b)(a — c) =/*, the result of sub-
stitution is essentially negative. But if we substitute a' = ^,
the result is
{b-l3)f-2fgh+{c-^)h%
which is also a perfect square, and positive. Since then, if
we substitute a' = go , a =a, a' = /3, a' = — go , the results are
alternately positive and negative, the equation has three real
roots lying within the limits just assigned. The three roots are
the coefficients of x\ y'^, z^ in the transformed equation, but it is
of course arbitrary which shall be the coefficient of x^ or of ?/*,
since we may call whichever axis we please the axis of x.
84. Quadrlcs are classified according to the signs of the
roots of the preceding cubic.
* This 13 the same cubic as that found, Art. 72, as the reader will easily see ought
to be the case.
t We may see this either by actually solving the equation, or by substituting suc-
cessively a' = 00 , a' = b, a' = c, a' = — cc, when we get results +, — , -, +, shewing
that one root is greater than b, and the other less than c.
68 CLASSIFICATION OF QUADRICS.
I. First, let all the roots be positive, and the equation can
be transformed to
The surface makes real intercepts on each of the three axes,
and if the intercepts be a, 6, c, it is easy to see that the equation
of the surface may be written in the form
x' f z'
a b c
As it is arbitrary which axis we take for the axis of a;, we
suppose the axes so taken that a the intercept on the axis
of X may be the longest, and c the intercept on the axis of z
may be the shortest.
The equation transformed to polar coordinates is
1 _ cos'* a cos'^/S 008^*7
p a b c '
which (remembering that cos"''a+ cos'*y8 + cos'*7 = 1) may be
written in either of the forms
? = I + (^ - ?) ""^^ + (? - ?) "^'■^
from which it is easy to see that a is the maximum and c
the minimum value of the radius vector. The surface is con-
sequently limited in every direction, and is called an eUijJsoid.
Every section of it is therefore necessarily also an ellipse.
Thus the section by any plane z=-'k\s,—:^-\-j^ = \ — ;j, and we
shall obviously cease to have any real section when h is greater
than c. The surface therefore lies altogether between the planes
e, = ±c. Similarly for the other axes.
If two of the coefficients be equal (for instance, o = 5), then
t I suppose in what follows that d' (= ^ , Art. 81 j is negative. If it were positive
we should only have to change all the signs in the equation. If it were = 0 the
Burface would represent a cone (Art. 67).
CLASSIFICATION OF QUADRICS. 69
all sections by planes parallel to the plane of xy are circles,
and the surface is one of revolution^ generated by the revolution
of an ellipse round its axis major or axis minor, according as
it is the two less or the two greater coefficients which are
equal. These surfaces are also sometimes called the jprolai£
and the oblate spheroid.
If all three coefficients be equal, the surface is a sphere.
85. II. Secondly, let one root of the cubic be negative.
We may then write the equation in the form
a 2 2
V — = 1
2 T^ 7 2 2 -^ )
a 0 c
where a is supposed greater than 5, and where the axis of z
evidently does not meet the surface in real points. Using
the polar equation
1 _ cos^a cos"''/3 008*^7
p a b c
it is evident that the radius vector meets the surface or not
according as the right-hand side of the equation is positive
or negative ; and that putting it = 0, (which corresponds to
p = CO ) we obtain a system of radii which separate the diameters
meeting the surface from those that do not. We obtain thus
the equation of the asymptotic cone
x^ y' z^ ^
d' ^ h' V' ^'
Sections of the surface parallel to the plane of xy are ellipses ;
those parallel to either of the other two principal planes are
hyperbolas. The equation of the elliptic section by the plane
of y^ k^
z = k being -» + 75 = 1 + -, , we see that a real section is found
^ a b c
whatever be the value of k, and therefore that the surface
is continuous. It is called the Hyijerboloid of one sheet.
If a = 5, it is a surface of revolution.
86. III. Thirdly, let two of the roots be negative, and
the equation may be written
222
^ _ ^ _ 1 - 1
d' V 6'~
70 CLASSIFICATION OP QUADEICS.
The sections parallel to two principal planes are hyperbolas,
while that parallel to the plane oi yz is an ellipse
2/' ^' y-" .
- — I — = 1
0 C a
It Is evident that this will not be real so long as k is within
the limits ±a, but that any plane x = k will meet the surface
in a real section provided k is outside these limits. No
portion of the surface will then lie between the planes x = ±a^
but the surface will consist of two separate portions outside
these boundary planes. This surface is called the Hyperboloid
of two sheets. It is of revolution if & = c.
By considering the surfaces of revolution, the reader can
easily form an idea of the distinction between the two kinds
of hyperbololds. Thus, If a common hyperbola revolve round
its transverse axis, the surface generated will evidently consist
of two separate portions ; but if it revolve round the conjugate
axis It will consist but of one portion, and will be a case of
the hyperbolold of one sheet.
IV. If the three roots of the cubic be negative, the equation
'2 2 2
X y s
f-— H — = — 1
a b c
can evidently be satisfied by no real values of the coordinates.
Y. When the absolute term vanishes, we have the cone as
a limiting case of the above. Forms I. and IV. then become
'2 2 2
X y 2 ^
which can be satisfied by no real values of the coordinates, while
forms II. and III. give the equation of the cone in the form
2 !i 2
X y z ^
2 ~ 72 2 — "•
a 0 c
The forms already enumerated exhaust all the varieties of
central surfaces.
Ex. I. 7x'^ + 6/ + bz^ - Ayz - ^.xy = 6.
The discriminating cubic 13 a''-18a'2 + 99a'- 102 = 0,
and the transformed equation a;* + 2^^ + 3^^ = 2, an ellipsoid.
CLASSIFICATION OF QUADRICS. 71
Ex.2. Ux^+10f + 6z'^-12xij-8y3 + 4zx=].2.
Discriminating cubic n'' - 27«'- + 180rt' - 32-i = 0.
Transformed equation a-- + 22/- + 62- = 4, an ellipsoid.
Ex. 3. 7x'^-13y^ + 62-^-i-24xy+12yz-l2zx = ±Si.
Discriminating cubic a'^ — 343a' — 2058 = 0.
Transformed equation x- + 2y- — 3z'^ = + 12,
a hyperboloid of one or of two sheets, according to the sign of the last term.
Ex.4. 2x'' + 3y- + 4z''+exy + 4yz + 8zx = 8.
Discriminating cubic is a'' — 9a'- — 3a' + 20 = 0. ' ^
By Des Cartes's rule of signs this equation has two positive and one negative root,
and therefore represents a hyperboloid of one sheet.
87. Let us proceed now to the case where we have D = 0.
In this case we have seen (Art. 69) that it is generally im-
possible by any change of origin to make the terras of the
first degree in the equation to vanish. But it is in general
quite Indifferent whether we commence, as in Art. 69, by
transforming to a new origin, and so remove the coefficients
of a;, y, z, or whether we first, as in this chapter, transform
to new axes retaining the same origin, and so reduce the terms
of highest degree to the form ax^ + Vy^ + cV. When i) = 0,
the first transformation being Impossible, we must commence
with the latter. And since the absolute term of the cubic of
Art. 83 is Z), one of its roots, that is to say, one of the three
quantities a', h\ c must In this case = 0. The terms of the
second degree are therefore reducible to the form ax^ ± Vy^.
This is otherwise evident from the consideration that D = 0
is the condition that the terms of highest degree should be
resolvable into two real or imaginary factors, In which case
they may obviously be also expressed as the difference or sum
of two squares. In this way the equation Is reduced to the form
a V + b'f + 2rx + 2m y + 2nz -\-d=Q.
We can then, by transforming to a new origin, make the co-
efficients of X and y to vanish, but not that of 2, and the equation
takes the form
«V ± jy + 2nz -^d' = Q.
I. If n = 0. The equation then does not contain 2;, and
therefore (Art. 25) represents a cylinder which Is elliptic or
hyperbolic, according as a and h' have the same or different
signs. Since the terms of the first degree are absent from
72 CLASSIFICATION OF QUADKICS.
the equation the origin is a centre, but so is also equally
every other point on the axis of z, which is called the axis
of the cylinder. The possibility of the surface having a line of
centres is indicated by both numerator and denominator vanishing
in the coordinates of the centre, Art. 69, note.
If it happened that not only vf but also (^' = 0, the surface
would reduce to two intersecting planes.
II. If n' be not =0, we can by a change of origin make
the absolute term vanish, and reduce the equation to the form
aV + b'f + 2nz = 0.
Let us first suppose the sign of h' to be positive. In this
case, while the sections by planes parallel to the planes of xz
or yz are parabolas, those parallel to the plane of xy are ellipses,
and the surface is called the Elliptic Paraboloid. It evidently
extends only in one direction, since the section by any plane
z = h is ax^ 4 Vy"^ = — ^hn^ and will not be real unless the
right-hand side of the equation is positive. When therefore
n' is positive, the surface lies altogether on the negative side
of the plane of xy^ and when n is negative, on the positive side.
III. If the sign of h' be negative, the sections by planes
parallel to that of xy are hyperbolas, and the surface is called
a Hyperbolic Paraboloid. This surface extends indefinitely in
both directions. The section by the plane of xy is a pair of
right lines ; the parallel sections above and below this plane
are hyperbolas having their transverse axes at right angles to
each other, and their asymptotes parallel to the pair of lines
In question, the section by the plane of xy forming the transition
between the two series of hyperbolas : the form of the surface
resembles a saddle or mountain pass.
IV. If V = 0, that is. If two roots of the discriminating cubic
vanish, the equation takes the form
aV + 2m'y + 2nz -f ^ = 0,
but by changing the axes of y and z in their own plane, and
taking for new coordinate planes the plane m^y + nz and a
plane perpendicular to it through the axis of x, the equation
CLASSIFICATION OF QUADRICS. 73
is brought to the form
aV + 2m y + <? = 0,
which (Art. 25) represents a cylinder whose base is a parabola.
V. If we have also 9n' = 0, w' = 0, the equation ax^ -\-d=^0
being resolvable into factors would evidently denote a pair of
parallel planes.
88. The actual work of reducing the equation of a paraboloid
to the form aoc^ -f Vy^ + 2nz = 0 is shortened by observing that
the discriminant is an invariant ; that is to say, a function of
the coefficients which is not altered by transformation of co-
ordinates [Higher Algebra, Art. 120, also noticing that since
we are transforming from one set of rectangular axes
to another the modulus of transformation is unity, as
seen above Note to Art. 32). Now the discriminant of
ax'' + Vy'' -f In'z is simply — a''h'ri'\ which is therefore equal to
the discriminant of the given equation. And as a and h' are
known, being the two roots of the discriminating cubic which
do not vanish, ri is also known. The calculation of the dis-
criminant is facilitated by observing that it is in this case a
perfect square {Higher Algebra, Art. 37). Thus let us take the
example
5x^-y^ + s" + ^zx + 4.xy + 2a; + 4y 4 62; = 8.
Then the discriminating cubic is X^ — bX'^ — 14\ = 0 whose roots
are 0, 7, and —2. We have therefore a' = 7, 5' = — 2. The
discriminant in this case is (Z + 2?n — 3^)'"*, or putting in the
actual values Z=l, 7n=2, n = 3 is 16. Hence we have 14n'*=]6,
A. ft?*
n = —. — -r , and the reduced equation Is 7x^ - 2y' =
Vil4) ' ^ ^ ^ _ Vil4)
If we had not availed ourselves of the discriminant we
should have proceeded, as In Art. 72, to find the principal planes
answering to the roots 0, 7, — 2 of the discriminating cubic, and
should have found
x + 2y-Sz = 0, 4kX + y -i- 2z = Oj x-2y-z = 0.
L
74 CLASSIFICATION OF QUADRICS.
Since the new coordinates are the perpendiculars on these
planes, we are to take
^x + y + 2z = X'^[2\)^x-'2y-z= rV(6), ir + 2^-3^ = Zv/(14),
from which we can express a:, 3/, z In terms of the new co-
ordinates, and the transformed equation becomes
which, finally transformed to parallel axes through a new origin,
gives the same reduced equation as before.
If in the preceding example the coefficients ?, wj, n had been
so taken as to fulfil the relation 1+ 2m — 3?i = 0, the discriminant
would then vanish, but the reduction could be effected with even
greater facility, as the terms in a;, ?/, z could then be expressed
in the form
(4a; + 3/ + 2z) +X[x-2y - z).
Thus the equation
bx'-y'* + 2' + ^zx + ^xy + 2a; + 2^/ + 22 = 8
may be written in the form
[ix-^y + 2zf -{x- 2y-zY + 'i [^xA- y -^2z)-2[x-2y -z) = 2i:j
which, transformed as before, becomes
21a;'- 6/ + 2a;\/(21)- 23/V(6)=24,
and the remainder of the reduction presents no difficulty.
( 75 )
CHAPTER VI.
PROPERTIES OF QUADRICS DEDUCED FROM SPECIAL
FORMS OF THEIR EQUATIONS.
CENTRAL SURFACES.
89. We proceed now to give some properties of central
x" y* z^
quadrlcs derived from the equation — + r^ + -» = 1. This will
Include properties of the hyperbololds as well as of the ellipsoid
if we suppose the signs of i* and of c' to be Indeterminate.
The equation of the polar plane of the point xyz' (or of the
tangent plane, If that point be on the surface) is (Art. 63)
xx' yvf zz'
-J- ^^^ A =1
a 0 c
The length of the perpendicular from the origin on the tangent
plane Is therefore (Art. 33) given by the equation
1'2 ''i 'a
X y z
f a 0 c
And the angles a, y8, 7 which the perpendicular makes with the
axes are given by the equations
vx r. py' pz'
cosa = — T7 , cosp = ,ir , COS7 = -£-7 ,
a 0 c
as is evident by multiplying the equation of the tangent plane
by^, and comparing It with the form
X cosa + y cos/S + z C0S7 =p.
From the preceding equations we can also Immediately get
an expression for the perpendicular In terms of the angles It
makes with the axes, viz.
■p^ = a^ cos'^a + l)^ cos 73 + c^ cos'' 7.
90. To find the condition that the plane ax -\- /3y + <yz + B = 0
should touch the surface.
76 CENTRAL SURFACES.
r r t
Comparing this •with the equation -^ + ^ + -2" = 1 ? we
have at once
a;' aa y _ h^ z' _ cy
and the required condition is
In the same way, the condition that the plane ax + ^y + yz
x^ y^ z^
should touch the cone -s + t^ 3=0 is
a' ¥ c^
aV-f &"';S^-cV = 0.
These might also be deduced as particular cases of Art. 79.
91. The normal is a perpendicular to the tangent plane
erected at the point of contact. Its equations are obviously
a
2
y , ,. e
-[x-x')=-,{y-y')^-,[z-z').
Let the common value of these be i2, then we have
, Ex' , Ru' , Rz'
^-^=-^' y-y=-h'^ "-"=-?-•
Squaring, and adding, we find that the length of the normal
between x'yz\ and any point on it xyz is + — . But if xyz be
taken as the point where the normal meets the plane of xy^ we
have s = 0, and the last of the three preceding equations gives
i? = — c^. Hence the length of the Intercept on the normal
between the point of contact and the plane of xy\% - ,
92. The sum of the squares of the reciprocals of any three
rectangular diameters is constant. This follows Immediately
from adding the equations
1 _ cos'' a cos"/? cos"''7
p a 0 c '
1 cos'"' a' cos"''/3' cos'''7''
Y' ^~^ '^'~~W "^ ^?~'
1 cos' a" cos'/8" cos"' 7"
■J'* ^'*
d' ' b' "*■ &
CONJUGATE DIAMETERS. 77
whence, since cos*^ a + cos^ a' + cos'^ a" = 1 , &c., we have
i J_ _1 - ^ 1 j_
p p p a 0 c
93. In like manner the sum of the squares of three perpen-
diculars on tangent planes, mutually at right angles, is constant,
as appears from adding the equations
p"^ = a' cos' a + ¥ cos'/S + c^ cos"' 7,
p'"-' =a"^cosV +¥cos^/3' +c^cos'y,
rt'i n 2 // , 7 2 2 a" 1 2 'Iff
p " = « COS a +0 cos p +c cos 7 .
Hence the locus of the intersection of three tangent planes
which cut at right angles is a sphere ; since the square of its
distance from the centre of the surface is equal to the sum
of the squares of the three perpendiculars, and therefore to
d' + b' + c'.
CONJUGATE DIAMETERS.
94. The equation of the diametral plane conjugate to the
diameter drawn to the point oj'^/V on the surface is
-^ + ^^- + -^==0, (Art. 70).
It is therefore parallel to the tangent plane at that point.
Since any diameter in the diametral plane is conjugate to that
drawn to the point x'yz\ it is manifest that when two diameters
are conjugate to each other, their direction-cosines are connected
by the relation
cosacosa' cosScosB' COS7COS7'
a 0 c
Since the equation of condition here given is not altered If
we write kd\ kb'\ kc' for a', h\ c\ It is evident that two lines
'2 *1 2
which are conjugate diameters for any surface — + fa + ~2 = Ij
are also conjugate diameters for any similar surface
222
x y 2 _ 7
a b c
And by making h = Q we see In particular that any surface and
its asymptotic cone have common systems of conjugate diameters.
78
CENTRAL SURFACES.
Following the analogy of methods employed In the case of
conies, we may denote the coordinates of any point on the
ellipsoid by a cosX, h cos^, c cosv, where X, /x, v are the
direction-angles of some line; that is to say, are such that
cos'''X + cos*/i- + cos'''i/= 1. In this method the two lines answer-
ing to two conjugate diameters are at right angles to each
other; for writing /> cos a = a cos X, p cos a' = a cos V, (fee, the
relation above written becomes
cosX cosX' + cos/x cosyu,' + cos V cos v' = 0.
95. The sum of the squares of a system of three coyijugate
semi-diameters is constant.
For the square of the length of any semi-diameter x'^-^y'^-^-z'^
is, when expressed in terms of X, /*, v,
d^ cos^ X + J'^ cos'^'yu. + c^ cos'"* f,
which, when added to the sum of
d^ cos'X' + h^ cosV' + c"^ cos'V ,
d' cos'V' + &' cos'^ /i" + c' cos' v",
the whole is equal to a^ + h^ + c''^ ; since X, /i, v, &c. are the
direction angles of three lines mutually at right angles.
96. The parallelepiped whose edges are three conjugate semi-
diameters has a constant volume.
For if xy'z\ x"y"z\ &c. be the extremities of the diameters,
the volume is (Art. 32)
or
ahc
^\y\ ^'
^'\y\^"
x'", y"\ z'"
>
cosX , cos /A , cosv
cosX', COS /a', cos/
COf
sX", COS/i", COS
V
but the value of the last determinant is unity (see note Art. 32) ;
hence the volume of the parallelepiped is abc.
If the axes of any central plane section be a', J', and p the
perpendicular on the parallel tangent plane, then ab'p=-dbc.
CONJUGATE DIAMETEES. 79
For if c be the semi-diameter to the point of contact, and Q the
angle it makes with j9, the volume of the parallelepiped under
the conjugate diameters a', h\ c is ah'c cos^, but c cosd =p.
97. The theorems just given may also with ease be deduced
from the corresponding theorems for conies.
For consider any three conjugate diameters a', b\ c', and let
the plane of a'b' meet the plane of xt/ in a diameter A, and let
C be the diameter conjugate to A in the section a'b\ then we
have A' +C' = a' + b'' ; therefore a'' -f b'' + c' = A^+ C + c\
Again, since A is in the plane a-?/, then if B is the diameter con-
jugate to A in the section by that plane, the plane conjugate to
A will be the plane containing B and containing the axis c, and
C, c are therefore conjugate diameters of the same section as
B, c. Hence we have A^ -\- C + c"' = A' + B'^ -\- 6' ; and since,
finally, A^-vB'^=a^-\^})''^ the theorem is proved. Precisely similar
reasoning proves the theorem about the parallelepipeds.
We might further prove these theorems by obtaining, as in
the note. Art. 82, the relations which exist when the quantity
a-'
/2 + TT^i + — 2 ill oblique coordinates is transformed to ^ + f? H —
a b c a b d*'
in rectangular coordinates. These relations are found to be
V'c^\ed'Wh'=b'\"' sin'-'X + c'V'' sinV + a'V sinV,
a"''6V =a''''&'V^(l- cos'^X -cos'^yti — cos'V-f 2 cosX cos/a cosv).
The first and last equations give the properties already ob-
tained. The second expresses that the sum of the squares of
the parallelograms formed by three conjugate diameters, taken
two by two, is constant, or that the sum of squares of reciprocals
of perpendiculars on tangent planes through three conjugate
vertices is constant.
98. The sum of the squares of the projections of three con-
jugate diameters on any fixed right line is constant.
Let the line make angles a, /3, 7 with the axes, then the
projection on it of the semi-diameter terminating in the point
xyz is a:'cosa + j/'cos/S + 2'cos7, or, by Art. 94, is
a cosX cosa -I- b cosfj, cos/3 -f c cosv COS7.
80 CENTRAL SURFACES.
Similarly, the others are
acosX' cosa + 5 cosyti' cos/3 + c cos/ cos 7,
a cos V cos a + h cosf/^ cos/3 + c cosi/" cos 7 ;
and squaring and adding, we get the sum of the squares
a^ cos^a + h'' cos'^/S + c" cos" 7.
99. The sum of the squares of the projections of three con-
jugate diameters on any fixed plane is constant.
If d^ d\ d" be the three diameters, ^, Q\ 6" the angles made
by them with the perpendicular on the plane, the sum of the
squares of the three projections Is d"^ sln'^^ + o?'" sin"''^' + c?''^ sln^^",
which Is constant, since d''' Q.Qi'iQ-\-d''^ Q,oi''d' ^-d""^ o.O'iQ" is con-
stant by the last article ; and tZ'" + d''^ + d"'^ by Art. 95.
100. To find the locus of the intersection of three tangent planes
at the extremities of three conjugate diameters.
The equations of the three tangent planes are
- cos X -f ^ cos w. + - cos V =\.
a he '
- cos V -1- f- cos At' + - cos / = 1,
a be '
- cos X" + f cos u," + - cos v" =\.
a o c
Squaring and adding, we get for the equation of the locus
a 2 2
X y ^ _ o
a 0 c
101. To find the lengths of the axes of the sectio7i made hy
any plane passing through the centre.
We can readily form the quadratic, whose roots are the
reciprocals of the squares of the axes, since we are given the
sum and the product of these quantities. Let a, y8, 7 be the
angles which a perpendicular to the given plane makes with
the axes, B the intercept by the surface on this perpendicular ;
then we have (Art. 92)
1 1 Jl _ 1 1 i
7''^ V'^ K' ~ a' "*" h' "^ c^ '
CONJUGATE DIAMETERS. 81
- 1 1/111 cos'^'a cos"''/3 cosV
whence —, -^ -tf, = [ -^ + jr, + -r, 5 p -^
a 0 \a oca 0 c
1 ., / » ^^\ 1 7^^ cos^a cos'^^/S cosV
while (Art. 96) -^ = ^, _, = ^^ + -^ + -^ .
The quadratic required Is therefore
1 1 /sin^a sIn'^/3 sln'^7\ cos*a cos'^'/S cos'''7_^
r r \ a b c J be c a a b
This quadratic may also be written In the form
d^ cos^a ¥ cos^/3 c* cos'''7
a — r b —r c —r
This equation may be otherwise obtained from the principles
explained In the next article.
102. Through a given radius OR of a central quadric we can
in general draw one section of which OR shall be an axis.
Describe a sphere with OR as radius, and let a cone be
drawn having the centre as vertex and passing through the
intersection of the surface and the sphere, and let a tangent
plane to the cone be drawn through the radius ORj then OR
will be an axis of the section by that plane. For in it OR is
equal to the next consecutive radius (both being radii of the
same sphere) and is therefore a maximum or minimum ; or,
again, the tangent line at R to the section is perpendicular to
OR, since it is also in the tangent plane to the sphere. OR is
therefore an axis of the section.
The equation of the cone can at once be formed by sub-
tracting one from the other, the equations
when we get
2 2 'i 2 2 'i
X y z , X y z
a be r r r '
2/1 n , 2/1 1
If then any plane x cos a + ?/ cos/3 -f z cos 7' have an axis in
length =r, it must touch this cone, and the condition that it
should touch it, is (Art. 90)
a^ cos'^a F cos^8 c^ cos%
f- • — -I = 0
2 2 "^ 72 2 T^ 2 2 — )
a — r b — r c —r '
which is the equation found in the last article.
M
82
CENTEAL SUEFACES.
In like manner we can find the axes of any section of a
quadric given bj an equation of the form
ax' + hy'^ 4- cz" -f 2fijz -f 2gzx + 2hxy = 1.
The cone of intersection of this quadric with any sphere
is (a -\)x''+{b- X) if ■\[c-\) z' + %fyz + Igzx + ihxy = 0,
and we see, as before, that if A. be the recipi-ocal of the square
of an axis of the section by the plane ic cos a + ?/ cos 18 + 0 cos 7,
this plane must touch the cone whose equation has just been
given. The condition that the plane should touch this cone
(Art. 79) may be written
a — A,, ^, ^, cosa
^, 5 — X, /, cos/3
ff, /, c-X, cos 7
cosa, cos/3, cos 7, =0,
which expanded is
\^~\{{h + c) cos''a + (c + «) cos'/3 4 (« + 5) cos^
— 2/cos/3 C0S7 — 2g C0S7 cosa — 2h cosa cosyS]
+ (be -/') cos'a 4 [ca -g^) cos'yS 4 {ab - A") cos*7
4 2 [9^ - af] cos/3 cos 7 4- 2 (^/— J^) cos 7 cosa
4 2 (^^ — cA) cosa cos/3 = 0.
CIECULAE SECTIONS.
103. We proceed to investigate whether it is possible to
draw a plane which shall cut a given ellipsoid in a circle. As
it has been already proved (Art. 73) that all parallel sections
are similar curves, it is sufficient to consider sections made by
planes through the centre. Imagine that any central section
is a circle with radius r, and conceive a concentric sphere
described with the same radius. Then we have just seen
that
^^(^7)^/(p-^)-1?-^)-
CIRCULAR SECTIONS. 83
represents a cone having the centre for its vertex and passing
through the intersection of the quadric and the sphere. But
if the surfaces have a plane section common, this equation must
necessarily represent two planes, which cannot take place unless
the coefficient of either x\ ij\ or z^ vanish. The plane section
must therefore pass through one or other of the three axes.
Suppose for example we take r = 3, the coefficient of y vanishes,
and there remains
which represents two planes of circular section passing through
the axis of y.
The two planes are easily constructed by drawing in the
plane (^i. xz a semi-diameter equal to h. Then the plane con-
taining the axis of?/, and either of the semi-diameters which
can be so drawn, is a plane of circular section.
In like manner, two planes can be drawn through each of
the other axes, but in the case of the ellipsoid these planes will
be imaginary ; since we evidently cannot draw in the plane of
xy a semi-diameter = c, the least semi-diameter in that section
being = h ; nor, again, in the plane of yz a semi-diameter = a,
the greatest in that section being = h.
In the case of the hyperboloid of one sheet, c" is negative,
and the sections through a are those which are real. In the
hyperboloid of two sheets, where both If and c* are negative,
if we take r' = — 6'' ijf being less than c"^), we get the two real
sections,
"'(^ + ?)+^1o^ -?) = »•
These two real planes through the centre do not meet the
surface, but parallel planes do meet it in circles. In all cases
it will be observed that we have only two real central planes
of circular section, the series of planes parallel to each of which
aflford two different systems of circular sections.
104. Any two surfaces whose coefficients of x\ ?/^, z\ differ
only by a constant, have the same planes of circular section. Thus
Ax' + By-'^-Cz'^l, and iA. + II) x' + [B +11) y' + {C + II) z'=\
84 CENTEAL SUEFACES.
have the same planes of circular section, as easily appears
from the formula in the last article.
The same thing appears by throwing the two equations into
the form
-i = A cos^'a 4- B coa'/3 + G cos'^y,
r
-, = A cos'a -f B cos'yS + Ccos^'y + ^,
from which it appears that the difference of the squares of the
reciprocals of the corresponding radii vectores of the two sur-
faces is constant. If then in any section the radius vector of
the one surface be constant, so must also the radius vector of
the other. The same consideration shews that any plane cuts
both in sections having the same axes, since the maximum or
minimum value of the radius vector will in each correspond
to the same values of a, /3, 7.
Circular sections of a cone are the same as those of a hyper-
boloid to which it is asymptotic.
105. Any two circular sections of opposite systems lie on the
same sphere.
The two planes of section are parallel each to one of the
planes represented by
Now since the equation of two planes agrees with the
equation of two parallel planes as far as terms of the second
degree are concerned, the equation of the two planes must
be of the form
X
(?-r^)+^'G"-?)+:^" (?-.') +".=«>
where u^ represents some plane. If then we subtract this from
the equation of the surface, which every point on the section
must also satisfy, we get
1
r'
which represents a sphere.
RECTILINEAR GENERATORS. 85
106. All parallel sections are, as we have seen, similar. If
now we draw a series of planes parallel to circular sections, the
extreme one will be the parallel tangent plane which must
meet the surface in an infinitely small circle. Its point of
contact is called an umhilic. Some properties of these points
will be mentioned afterwards. The coordinates of the real
umbilics are easily found. We are to draw in the section,
whose axes are a and c, a serai-diameter = 5, and to find the
coordinates of the extremity of its conjugate. Now the- for-
mula for conies h'^ =-a^ — e^x\ applied to this case, gives us
x'
0 — o, ., • »c ,
a
= ^,_^,; similarly ^, = ^.,_^,
whence
There are accordingly in the case of the ellipsoid four real
umbilics in the plane of xz, and four imaginary in each of the
other principal planes.
RECTILINEAR GENERATORS.
107. We have seen that when the central section is an
ellipse all parallel sections are similar ellipses, and the section
by a tangent plane is an infinitely small similar ellipse. In
like manner when the central section Is a hyperbola, the section
by any parallel plane is a similar hyperbola, and that by the
tangent plane reduces Itself to a pair of right lines parallel to
the asymptotes of the central hyperbola. Thus If the equation
referred to any conjugate diameters be
x' f £'
a 0 c
and we consider the section made by any plane parallel to the
plane of xz [y = /3), its equation is
And It is evident that the value /3 = h' reduces the section to
86 CENTRAL SURFACES.
a pair of right lines. Such right lines can only exist on the
hyperboloid of one sheet,* since if we had the equation
2 2 2
X y z
— = 1 -I
the right-hand side of the equation could not vanish for any real
value of z. It is also geometrically evident that a right line
cannot exist either on an ellipsoid, which is a closed surface,
or on a hyperboloid of two sheets, no part of which, as we
saw, lies in the space included between several systems of two
parallel planes, while any right line will of course in general
intersect them all.
108. Throwing the equation of the hyperboloid of one sheet
into the form
it is evident that the intersection of the two planes
a c \ bj ^ \a cj \ b
lies on the surface ; and by giving different values to \ we get
a system of right lines lying in the surface ; while, again, we
get another system by considering the intersection of the planes
What has been just said may be stated more generally as
follows : If a, /S, 7, B represent four planes, then the equation
ary = (5S represents a hyperboloid of one sheet, which may be
generated as the locus of the system of right lines a = X^, X7=S,
or of the system a = XS, A.7 = /3.
Considering four lines in either system as a = Xy3, X7 = S, we
have two pencils of planes which we see by Art. 39 are equi-
anharmonic ; hence the hyperboloid of one sheet may be
regarded as the locus of lines of intersection of two homographic
pencils of planes.
* It -will be understood that tlie remarks in the text apply only to real right,
lines : every quadric surface has upon it an infinity of right lines, real or imaginary,
and (not being a cone) it is a skew surface. See footnote, Art. 112.
RECTILINEAR GENERATORS. 87
In the case of the equation
x" y' z" ^
the lines may be also expressed by the equations
- = - cos ^ + sin ^, 7 = - sin 6 ± cos 9.
a c be
109. Any two lines belonging to opposite systems lie in the
same plaiie.
Consider the two lines
a — X/3, X7 - S,
a — A-'S, X'7 — /3.
Then it is evident that the plane a - X/3 + W7 — X'h contains
both, since it can be written in either of the forms
a-X/S4 V(X7- a), a- VS + \(\7-/3).
It Is evident in like manner that no two lines belonging to
the same system lie in the same plane. In fact, no plane of
the form (a - A./3) + h (ky - 8) can ever be identical with
(a — X'yS) + k' (V7 - 8) if A, and A,' are different. In the same
way we see that both the lines
- = - cos ^ — sin 6, ^ = - sin ^ + cos 0,
a c ^ 0 c
O* 2J 7/21 '^^
- = - cos ^ -f sin 0, y = - sin ^ - cos ^,
(Z C DC
which belong to different systems, lie in the plane
- cos 1 (^ + <^) + I sin 1 (^ -f </.) = - cos i (^ - (^) - sin ^ [6 - cf>),
CI 0 c
Now this plane is parallel to the second line of the first
system
- = - cos d) - sin (b, 'y =■ — sin c£) + cos d>.
a c DC
but it does not pass through it, for the equation of a parallel
plane through this line will be found to be
- cosi [6 +<f>)+-l sini [d + 0) = - cos|(^ - (^) + sln^ [6 - (^),
Cli o c
88 CENTRAL SURFACES.
which differs in the absolute term from the equation of the
plane through the first line.
110. We have seen that any tangent plane to the hyper-
boloid meets the surface in two right lines intersecting in the
point of contact, and of course touches the surface in no other
point. If through one of these right lines we draw any other
plane, we have just seen that it will meet the surface in a new
right line, and this new plane will touch the surface in the
point where these two lines intersect. Conversely, the tangent
plane to the surface at any point on a given right line in the
surface will contain the right line, but the tangent plane will
in general be different for every point of the right line. Thus,
take the surface X(^ = yy\r^ where the line xy lies on the surface,
and j) and -^ represent planes (though the demonstration would
equally hold if they were functions of any higher degree).
Then using the equation of the tangent plane
[x-x')U: + {y-y')U: + [z-z')U: = 0,
and seeking the tangent at the point x = 0, 3/ = 0, z = z' ^ we find
x^' = y^' 1 where ^' and ■>^' are what 0 and i/r become on sub-
stituting these coordinates. And this plane will vary as z varies.
It is easy also to deduce from this that the anharmonic ratio
of four tangent planes passing through a right line in the surface
is equal to that of their four points of contact along the line.
All this is different in the case of the cone. Here every
tangent plane meets the surface in two coincident right lines.
The tangent plane then at every point of this right line is the
same, and the plane touches the surface along the whole length
of the line.
And generally, if the equation of a surface be of the form
x^ + y''-<if = 0,
It is seen precisely, as above, that the tangent plane at every
point of the line xy\'&x = 0.
111. It was proved (Art, 107) that the two lines in which
the tangent plane cuts a hyperboloid are parallel to the asymp-
totes of the parallel central section ; but these asymptotes are
evidently _edges of the asymptotic cone to the surface. Hence
RECTILINEAR GENERATORS. 89
every right line wliich can lie on a hyperboloid is parallel to
some one of the edges of the asymptotic cone. It follows also
that three of these lines (unless two of them are parallel) cannot
all be parallel to the same plane ; since, if they were, a parallel
plane would cut the asymptotic cone In three edges, which
is impossible, the cone being only of the second degree.
112. We have seen that any line of the first system meets
all the lines of the second system. Conversely, the surface
may be conceived as generated by the motion of a right line
which always meets a certain number of fixed right lines.*
Let us remark, in the first place, that when we are seeking
the surface generated by the motion of a right line, it is
necessary that the motion of the right line should be regulated
by three conditions. In fact, since the equations of a right
line include four constants, four conditions would absolutely
determine the position of a right line. When we are given
one condition less, the position of the line Is not determined,
but It is so far limited that the line will always lie on a certain
surface-locus, whose equation can be found as follows : Write
down the general equations of a right line x — mz+p^ 7/=7iz + q'j
then the conditions of the problem establish three relations
between the constants ???, n, p, q. And combining these three
relations with the two equations of the right line, we have
five equations from which we can eliminate the four quantities
wi, ??, 2h 9. ; a^nd the resulting equation In a;, ?/, z will be the
equation of the locus required. Or, again, we may write the
equations of the line In the form
X — x' _y — y' _z- z
cos a cos/3 cos 7 '
then the three conditions give three relations between the con-
stants x\ ?/', /, a, yS, 7, and if between these we eliminate
a, yS, 7, the resulting equation in x\ y\ z' Is the equation of the
required locus, since x'y'z' may be any point on the line.
* A surface generated by the motion of a right line is called a ruled surface. If
every generating line is intersected by the next consecutive one, the sui-face is called
a devehpahh or torse. If not, it is called a skeio surface or scroll. The Iiyiicrboloid
of one sheet, and indeed every quadric surface (not being a cone or cylinder) belongs
to the latter class ; the cone and cylinder to the former.
N
90 CENTRAL SURFACES.
We see then, that It is a determinate problem to find the
surface generated by a right line which moves so as always
to meet three fixed right lines.* For, expressing, by Art. 41,
the condition that the moveable right line shall meet each of
the "fixed lines, we obtain the three necessary relations between
m^n^p^q. Geometrically also we can see that the motion of
the line is completely regulated by the given conditions. For
a line would be completely determined if It were constrained
to pass through a given point and to meet two fixed lines,
since we need only draw planes through the given point and
each of the fixed lines, when the intersection of these planes
would determine the line required. If, then, the point through
which the line is to pass, Itself moves along a third fixed line,
we have a determinate series of right lines, the assemblage of
which forms a surface-locus.
113. Let us then solve the problem suggested by the last
article, viz. to find the surface generated by a right line which
always meets three fixed right lines, no two of which are In
the same plane. In order that the work may be shortened
as much as possible, let us first examine what choice of
axes we must make in order to give the equations of the
fixed right lines the simplest form.
And it occurs at once that we ought to take the axes, one
parallel to each of the three given right lines^f The only
question then is, where the origin can most symmetrically be
placed. Suppose now, that through each of the three right
lines we draw planes parallel to the other two, we get thus
three pairs of parallel planes forming a paralleloplped, of which
the given lines will be edges. And if through the centre of
this paralleloplped we draw lines parallel to these edges, we
shall have the most symmetrical axes. Let then the equations
of the three pairs of planes be
x = ±a^ y = ±h^ ''^ = ± c,
* Or three fixed curves of any kind.
t We could not do tliis indeed if tbe three given right lines happened to be all
parallel to the same plane. This case will be considered in the next section. It will
not occur when the locus is a hyperboloid of one sheet, see Art. 111.
HECTILINEAE aENERATOKS. 9l
then the equations of the three fixed right lines will be
2/ = &, z = — c', z = c, x = — a', x = aj y =- — l).
The equations of any line meeting the first two fixed lines are
z-\ c = \{y — l))\ z- c = fM{xi-a)j
which will intersect the third i( c + fxa + \h = 0', or replacing
for A. and /j, their values,
c{x-\- a){y-h)+ a [z - c) [y - h) ■\- 1 [z + c) (a; -f a),
which reduced is
ayz + hzx -f cxy + ahc = 0.
On applying the criterion of Art. 86, this Is found to represent
a hyperboloid of one sheet, as is otherwise evident, since it
represents a central quadric, and is known to be a ruled
surface. The problem might otherwise be solved thus :
Assuming for the equations of the moveable line
x — x'_y — y'_z—z'
cos a cos/3 cos 7
}
the following three conditions are obtained by expressing that
this intersects each of the fixed lines,
y —h _z' + c z' — c x' ■\-a x^ —a _y' -\-h
cosyS C0S7 ' C0S7 cosa ' cosa cosyS *
We can eliminate a, /3, 7 by multiplying the equations
together, and get for the equation of the locus,
[x -a){y~ h) [z- c) = [x + a) [y -\-h)[z-^ c),
which reduces to ayz -f Izx + cxy + ahc — 0 the same equation as
before.
The last written form of the equation expresses that this
hyperboloid Is the locus of a point, the product of whose dis-
tances from three concurrent faces of a parallelopiped is equal
to the product of its distances from the three opposite faces.
The following is another general solution of the same pro-
blem : Let the first two lines be the intersections of the planes
a, ^ ; 7j ^ ; then the equations of the third can be expressed in
the form a = -47 + 5S, j3 = Cy + BS. The moveable line, since
it meets the first two lines, can be expressed by two equations
of the form a = \/3, 'y = fiB. Substituting these values in the
92 NON-CENTEAL SUEFACES.
equations of the third Hue, we find the condition that it and
the moveable line should intersect, viz.
Afj. + B = \{Cfi + D).
And eliminating \ and /u, between this and the equations of the
moveable line, Ave get for the equation of the locus,
l3{Ay + B8) = a{Cy + DS).
A third general solution is as follows: taking (^^j, q^^ r^,
*i5 *i) '^i)) (i^2? •••)) (Psi •••) ^^ ^^^® ^''^ coordinates of the given
lines respectively, and writing for shortness (^jr) to denote the
determinant 2^1 (s'^^'s ~" S's^'a) + ^"^"j ^^^ ^^ ^^ other cases, then
it can be shewn that the equation of the hyperboloid passing
through the three given lines is
(jJtu) x^ + (qus) if + (rs^) ?J^ + {'pqr) w^
+ \.{]?9f) ~ {w^)] ^^ + [(s'^O + (^^^^ )] y^
+ {{qru) — {pqs)\ yw + [{rtu) + {pst)~\ zx
+ [{rps) — [qrt )j zw + [{pus) + (2^^*)] a^i/ = 0.
114. Four right lines helonging to one system cut all lines
helonging to the other system in a constant anharmonic ratio.
For through the four lines and through any line which
meets them all we can draw four planes; and therefore any
other line which meets the four lines will be divided in a
constant anharmonic ratio (Art. 39).
Conversely, if two non-intersecting lines are divided homo-
graphically in a series of points, that is to say, so that the
anharmonic ratio of any four points on one line is equal to
that of the corresponding points on the other, then the lines
joining corresponding points will be generators of a hyper-
boloid of one sheet.
Let the two given lines be a, ^ ; 7, S. Let any fixed line
which meets them both be a = \'/3, 7 = (juh 5 then, in order that
any other line a = X/3, 'y—jxh should divide them homographically,
we must have ( Conies^ Art. 57) -7 = — ^ , and if we eliminate X
between the equations a = Xy8, X'7 = /li'XS, the result is \^/3y = /u,^aS.
NON-CENTEAL SUEFACES.
115. The reader is recommended to work out for himself
the properties of paraboloids which are analogous to the results
NON-CENTRAL SURFACES. 93
of the preceding articles of this chapter. In particular he may-
show* that : —
The sum or difference of the principal parameter's of any
two conjugate diametral sections of a paraboloid is constant
according as it is elliptic or hyperbolic.
The sum or difference of the parameters of any two conju-
gate diametral sections at a given point of a paraboloid is
constant, according as it is elliptic or hyperbolic.
If from the extremity of any diameter of a paraboloid a Hue
of constant length be measured and a conjugate plane drawn
cutting the paraboloid, the volume under any two conjugate
diameters of the section and this line is constant.
We proceed to determine the circular sections of the para-
boloid given by the equation
— / _ 2£
a 0 c
Consider a circular section through the origin, and describe a
sphere through it having, at the origin, the same tangent plane
[z] as the paraboloid; then (Art. 61) the equation of the sphere
must be of the form
x" + y' + s^ = 2nz.
And the cone of intersection of this sphere with the paraboloid Is
x'[l
cn\ 2 / _ cn\ „ ,^
This will represent two planes if one of the terms vanishes.
It will represent two real planes in the case of the elliptic
paraboloid, if we take -^=1, for the equation then becomes
Z'V = (a^ - h^) y\ But in the case of the hyperbolic paraboloid
there is no real circular section, since the same substitution
would make the equation of the two planes take the imaginary
form5V+(a'-' + i''')/ = 0.
Indeed, it can be proved in general that no section of the
hyperbolic paraboloid can be a closed curve, for if we take its
intersection with any plane s = aa; + /3_y + 7, the projection on
* See Professor AUman, On some Properties of the Paraboloids, Quarterly Journal
of Pur a and Applkd 3Iathematics, 1871.
94 NON-CENTRAL SUEFACES.
the plane of a??/ is -^ - y^ = — ^^ -^ — ^ which is necessarily
a hyperbola.
116. From the general theory explained in Art. 108, it is
plain that the hyperbolic paraboloid may also have right lines
lying altogether In the surface. For the equation — — p = -
(Art. 87) is included in the general form a<y = ^Bj and the
surface contains the two systems of right lines
- ± f = X,X -+ 7 =- .
a 0 ^ \a bj c
The first equation shews that every line on the surface must
be parallel to one or other of the two fixed planes - + r = (^ j
and in this respect is the fundamental difference between right
lines on the paraboloid and on the hyperboloid (see Art. 111).
It is proved, as in Art. 109, that any line of one system
meets every line of the other system, while no two lines of
the same system can intersect.
We give now the investigation of the converse problem, viz.
to find the surface generated by a right line which always meets
three fixed lines which are all parallel to the same plane. Let
the plane to which all are parallel be taken for the plane of a;y,
any line which meets all three for the axis of z^ and let the
axes of X and y be taken parallel to two of the fixed lines.
Then their equations are
33 = 0, z = a) 2/ = 0, z — h\ x = my^ z = c.
The equations of any line meeting the first two fixed lines are
x = \[z- a)^ y = IJb{z — h)^
which will intersect the third if
^{c- a) = mfjb (c — b)^
and the equation of the locus is
{a — c)x {z — b) = m [h- c) y{z- a),
which represents a hyperbolic paraboloid, since the terms of
highest degree break up into two real factors.
^
m
NON-CENTRAL SUEFA.CES. 95
In like manner we might investigate the surface generated
by a riglit line which meets two fixed lines and is always parallel
to a fixed plane. Let it meet the lines
x=Oj 2; = a; 2/ = 0, z = — aj
and be parallel to the plane
X cosa +?/ cosyS + z COS7 =p.
Then the equations of the line are
x = \{z — a), y = fi[z + rt),
which will be parallel to the given plane if
C0S7 + X cosa + /I. cos/3 = 0.
The equation of the required locus is therefore
COS7 [z' - d^) + a; cosa (s + a) + ?/ cos/3 [z- a)= 0,
which is a hyperbolic paraboloid, since the terms of the second
degree break up Into two real factors.
A hyperbolic paraboloid is the limit of the hyperbolold of
one sheet, when the generator in one of Its positions may He
altogether at Infinity.
We have seen (Art. 107) that a plane Is a tangent to a f.
surface of the second degree when It meets It In two real or
imaginary lines; and (Art. 87) that a paraboloid Is met by ^"^ '
the plane at infinity In two real or Imaginary lines. Hence
a paraboloid is always touched by the plane at infinity.
117. In the case of the hyperbolic paraboloid any three
right lines of one system cut all the right lines of the other
in a constant ratio. For since the generators are all parallel
to the same plane, we can draw, through any three generators,
parallels to that plane, and all right lines which meet three
parallel planes are cut by them In a constant ratio.
Conversely, if two finite non-intersecting lines be divided,
each Into the same number of equal parts, the lines joining
corresponding points will be generators of a hyperbolic para-
boloid. By doing this with threads, tlie form of this surface
can be readily exhibited to the eye.
To prove this directly, let the line which joins two corre-
sponding extremities of the given lines be the axis of z ; let
J
96 SURFACES OP REVOLUTION.
the axes of x and y be taken parallel to the given lines, and
let the plane of X7j be half-way between them. Let the lengths
of the given lines be a and bj then the coordinates of two
corresponding points are
s = c, x = fia^ y = 0,
s = -c, aj = 0, y =■ fibj
and the equations of the line joining these points are
X 7/
— f- !| = ju,j 2cx — fxaz — fxac,
whence, eliminating /^, the equation of the locus is
2cx = a (« + c) f- + y
which represents a hyperbolic paraboloid.
SURFACES OF REVOLUTION.
118. Let it be required to find the conditions that the
general equation should represent a surface of revolution. In
this case the equation can be reduced (see Art. 84), if the surface
2 2 a
be central, to the form -j + ^, + —, — + l, and if the surface
2£
a" a" c
when the highest terms are transformed so as to become the
sum of squares of three rectangular coordinates, the coefficients
of two of those squares are equal. It would appear then that
the required condition could be at once obtained by forming
the condition that the discriminating cubic should have equal
roots. Since, however, the roots of the discriminating cubic are
always real, its discriminant can be expressed as the sum
of squares (see Higher Algebra^ Art. 44), and will not vanish (the
coefficients of the given equation being supposed to be real)
unless tv;o conditions are fulfilled, which can be obtained more
easily by the following process. We want to find whether
it is possible so as to transform the equation as to have
ax' + by' + cz' + ^fyz + 2gzx + 2hxy = A [X'' + Y^) + CZ\
x' ii' 2z
be non-central to the form — + "^ = — . In either case then
SURFACES OF REVOLUTION. 97
but we have (Art. 19)
It 13 manifest then that by takuig A- = ^, we should have the
following quantity a perfect square :
[ax' + hf + cz" + 2fyz + 'igzx -f 2hxy) - \ (a;' + ?/" + a'),
and it is required to find the conditions that this should be
possible.
Now it is easy to see that when
Ax' + By"" + Cz" + 2Fyz + 2 Ozx + iHxy
is a perfect square, the six following conditions are fulfilled :*
BG = F% CA=0% AB=H%
AF=GH, BG = EF, CH=FG)
the three former of which are included in the three latter. In
the present case then these latter three equations are
{a-\)f=gh, {b-\)y = hf, {c-\)h=fg.
Solving for \ from each of these equations we see that the
reduction is impossible unless the coefficients of the given equa-
tion be connected by the two relations
a
ah hf fg
f 9 h
If these relations be fulfilled, and if we substitute any of these
common values for X in the function
{a - \) x' -f {b-\) f -f (c - X) z' + Ifijz + 'Igzx + "ihxy^
it becomes, as it ought, a perfect square, viz.
and since the plane ^=0 represents a plane perpendicular to the
axis of revolution of the surface, it follows that - + — + - = 0
represents a plane perpendicular to that axis.
In the special case where the common values vanish which
have been just found for X., the highest terms in the given
* That is to say, the reciprocal equation vanishes identically.
0
98 SURFACES OF REVOLUTION.
equation form a perfect square, and the equation represents
either a parabolic cylinder or two parallel planes (see IV.
and v., Art. 87). These are limiting cases of surfaces of re-
volution, the axis of revolution in the latter case being any
line perpendicular to both planes. The parabolic cyUnder is
the limit of the surface generated by the revolution of an ellipse
round its minor axis, when that axis passes to infinity.
119. If one of the quantities /, g^ h vanish, the surface
cannot be of revolution unless a second also vanish. Suppose
that we have y and g both =0, the preceding conditions become
a — h'- — h — h'- = c,
/ 9
. f
from which, eliminating the indeterminate- , we get
{a-c)[b-c)=h\
This condition might also have been obtained at once by
expressing that
(a - X) x' +{b-X)f+{c- \) z' + Ihxy
should be a perfect square, and it is plain that we must have
X, = c ; {a — c)[h — c)= Ji\
120. The preceding theory might also be obtained from the
consideration that in a surface of revolution the problem of
finding the principal planes becomes indeterminate. For since
every section perpendicular to the axis of revolution is a circle,
any system of parallel chords of one of these circles is bisected
by the plane passing through the axis of revolution and through
the diameter of the circle perpendicular to the chords, a plane
which is perpendicular to the chords. It follows that evert/
plane through the axis of revolution is a principal plane. Now
the chords which are perpendicular to these diametral planes are
given (Art. 72) by the equations
{a-\]x + ky-{-gz=Q, hx+{h — \)i/-+/z = 0, gx+fi/+ {c-\]z=0,
which, when X is one of the roots of the discriminating cubic,
represent three planes meeting in one of the right lines required.
The problem then will not become indeterminate unless these
LOCI.
99
equations call represent the same plane, for which we have the
conditions
a — X h _$'.«— ^_^_ 9
which, expanded, are the same as the conditions found already.
LOCI.
121. We shall conclude this chapter by a few examples of
the application of Algebraic Geometry to the investigation of
Loci.
Ex. 1. To find the locus of a point whose shortest distances from two given non-
iutersecting right lines are equal.
If the equations of the lines are written in their general form, the solution of this 13
obtained immediately by the formula of Art. 15. We may get the result in a simple
form by taking for the axis of z the shortest distance between the two lines, and,
choosing for the other axes the lines bisecting the angle between parallels to the
given lines through the point of bisection of this shortest distance ; then their equa-
tions are of the form
z-c, y — mx ; z — — c, y — — rnx,
and the conditions of the problem give
or cz (1 + w»*) + mxy — 0.
The locus is therefore a hyperbolic paraboloid.
If the shortest distances had been to each other in a given ratio, the locua would
have been
{(1 + A) 2 + (1 - \) c) {(1 -\)z+{i + \)c)
+ T-7-^ Kl + X) 3/ + (1 - X) mx] {(1 - X) y + (1 + \) mx) = 0,
which represents a hyperboloid of one sheet.
Ex. 2. To find the locus of the middle points of all lines parallel to a fixed plane
and terminated by two non-intersecting lines.
Take the plane x = 0 parallel to the fixed plane, and the plane a = 0, as in the last
example, parallel to the two lines and equidistant from them ; then the equations of
the lines are
2 = c, y — mx + n; z = — c, y = vi'x + n'.
The locua is then evidently the right line which is the intersection of the planes
z-0, 2y = {m + m') x + {n + n'). C ^^
Ex. 3. To find the surface of revolution generated by a right line turning round a
fixed axis which it does not intersect.
Let the fixed line be the axis of z, and let any position of the other be x = ot0 + n,
y = m'z + n'. Then since any point of the revolving line describes a circle in a plana
parallel to that of xy, it follows that the value of x" + y'^ is the same for every point in'
such a plane section, and it is plain that the constant value expressed in terms of z is
{mz + n)- + {m'z + n'Y. Hence the equation of the required surface is
a;2 + ^2 - („j, 4. n)- + {m'z + 11')",
which represents a hyperboloid of revolution of one sheet.
100 LOCI.
Ex. 4. Two lines passing through the origin move each in a fixed plane, remaining
perpendicular to each other, to find the surface (necessarily a cone) generated by a
right line, also passing through the origin perpendicular to the other two.
Let the direction-angles of the perpendiculars to the fixed planes be a, b, c; a', V, c',
and let those of the variable line be a,(i,y; then the direction-cosines of the intersec-
tions with the fixed planes, of a plane perpendicular to the variable line, will (Art. 15)
be proportional to
cos /? cose —cosy cos 6, cosy cos a —cos a cose, cos a cos 6 — "cos/Jcosa,
cos /3 cos c' — cos y cos 6', cos y cos a'— cosacosc', cosacosi' — cos/3 cos o',
and the condition that these should be perpendicular to each other is
(cos/3 cose — cosy cos h) (cos/3 cose' — cosy cos 6')
-1- (cos y cos a — cos a cos e) (cos y cos a' — cos a cos c')
-I- (cos a cos b — cos j8 cos a) (cos a cos b' — cos/3 cos a') = 0
which represents a cone of the second degree.
Ex. 5. Two planes mutually perpendicular pass each through a fixed line; to find
the surface generated by their line of intersection.
Take the axes as in Ex. 1. Then the equations of the planes are
\{s-c)+y-mx = Q; \' [z + c) -^^ y + mx - 0,
which will be at right angles if X\' + 1 - wi^ = 0 ; and putting in for X, X' their
values from the pair of equations, we get
^2 _ OT'a;2 -f (1 - m2) [z^ - c^) = 0,
which represents a hyperboloid of one sheet.
Both the hyperboloid of this Example and of Ex. 1 are such that two pairs of
generators are perpendicular to the planes of circular sections. Such hyperboloida
of one sheet have been called orthogonal hyperboloids (Schroter, Crelh's Jour. Vol. 85).
In either case, if the lines intersect, making c = 0, the locus reduces to a cone.
;^ + |5-3-.= li^'>itbogonalif--^-f3j=0.
Ex. 6. To find the locus of a point, whence three tangent lines, mutually at right
iC 11 z
angles, can be drawn to the quadric -5 + t; + -^ = 1.
If the equation were transformed so that these lines should become the axes of co-
ordinates, the equation of the tangent cone would take the form Ayz + Bzx + Cxy - 0,
since these three lines are edges of the cone. But the untransformed equation of the
tangent cone is, see Art. 78,
(^^ + i? + ^ - 1 j 1,^ + 6? + ^. - V ~ U'^ + 6-^ ^ c^ V •
And we have seen (Art. 82) that if this equation be transformed to any rectangular
system of axes, the sum of the coefficients of x"^, y^, and s^ will be constant. We have
only then to express the condition that this sum should vanish, when we obtain as
equation of the required locus,
a;2/l 1\ 3/V1 1\ , zVlo-i^-i 4-i-4-i
a2 [^2 + 35; + j-2 [^2 + a^J + c2 [a^ + b^j " 02 "^ 62 + ^^ •
Ex. 7. To find the equation of the cone whose vertex is x'y'z' and which stands on
the conic in the plane of a;^, — + 7^ = !•
The equations of the line joining any point a/3 of the base to the vertex are
a («' -z)- z'x - x'z, /3 {z' -z) = z'y - y'z.
LOCI. 101
Substituting these values in the equation of the base, we get for the required cone
{z'x - x'zY {z'y - y'zf _
a' + 62 - ^« «J •
The following method may be used in general to find the equation of the cone
whose vertex is x'y'z'w', and base the intersection of any two surfaces U, V. Substitute
in each equation for x, x + Xx' ; for y, y + \y', &c., and let the results be
cr+\^^7+-^^2^7 + (ic. = 0, v + \sv+ ^dw + &c. = o,
then the result of eliminating X between these equations will be the equation of the
required cone. For the points where the hue joining x'y'z'w' to xyzw meets the surface
U are got from the first of these two equations ; those where the same line meets the
surface V are got from the second ; and when the eliminant of the two equations
vanishes they have a common root, or the point xyzw lies on a line passing through
x'y'z'w' and meeting the intersection of the surfaces.
Ex. 8. To find the equation of the cone whose vertex is the centre of an ellipsoid
and base the section made by the polar of any point x'y'z'.
«2 ^2 02
Ex. 9. To find the locus of points on the quadric i + rj + -^ = Ij the normals at
which intersect the normal at the point x'y'z'.
Ans. The points required are the intersection of the surface with the cone.
a^ (y'z - z'y) {x - x') + IT- [z'x - x'z) {y - y') + c- {x'y - y'x) {z-z')-0.
Ex. 10. To find the locus of the poles of the tangent planes of one quadric with
respect to another.
We have only to express the condition that the polar of x'y'z'w', with regard to
the second quadric, should touch the first, and have therefore only to substitute
U^, Cj, U^, L\, for a, p, y, 8 in the condition given Art. 79. The locus is therefore
a quadric.
Ex. 11. To find the cone generated by perpendiculars erected at the vertex of a
given cone to its several tangent planes.
Let the cone be Lx- + My- + Nz^ = 0, and any tangent plane is Lx'x + My'y + Nz'z = 0
the perpendicular to which through the origin is -j—, — -^, = t— 7 . If then the com-
mon value of these fractions be called p, we have a;' = -=^ , «' = -^ , z' = -r^ , substitu-
Jup Mp i\p
ting these values in Lx'^ + My"^ + Nz"^ = 0, we get ^ + |j + "I? = 0. The form
of the equation shews that the relation between the cones is reciprocal, and that
the edges of the first are perpendicular to the tangent planes to the second. It
can easily be seen that this is a particular case of the last example.
If the equation of the cone be given in the form
ax2 + by- + cz"^ + 2fyz + 2gzx + 2hxy = 0,
the equation of the reciprocal cone will be the same as that of the reciprocal curve in
plane geometry, viz.
{be -P) x^ + (ca - g'^) y"^ + {ab - h"-) z'^ + 2{gh- af) yz + 2 {hf- bg) zx + 2(Jg- ch) xy = 0.
102 LOCI.
Ex. 12. A line moves about so that three fixed points on it move on fixed planes ;
to find the locus of any other point on it.
Let the coordinates of the locus point P he a, (i, y ; and let the three fixed planes
be taken for coordinate planes meeting the line in points A, B, 0. Then it is easy
to see that the coordinates of A are 0, -^-^ (3, -p-^ y, where the ratios AB : PB,
AC : PC are known. Expressing then, by Art. 10, that the distance PA is constant,
the locus is at once found to be an ellipsoid.
Ex. 13. A and 0 are two fixed points, the latter being on the surface of a sphere.
Let the line joining any other point D on the sphere to A meet the sphere again in D'.
Then if on OD a portion OP be taken = AD', find the locus of P. [Sir W. R.
Hamilton] .
We have AD^ -A0'^+ Olfi -2A0.0I) cos A OB. But AB varies inversely as the
radius vector of the locus, and OB is given, by the equation of the sphere, in terms of
the angles it makes with fixed axes. Thus the locus is easQy seen to be a quadric of
which 0 is the centre.
Ex. 14. A plane passes through a fixed line, and the Unes in which it meets two
fixed planes are joined by planes each to a fixed point ; find the surface generated by
the line of intersection of the latter two planes.
Ex. 15. The four faces of a tetrahedron pass each through a fixed point. Find
the locus of the vertex if the three edges which do not pass through it move each in a
fixed plane.
The locus is in general a surface of the third degree having the intersection of the
three planes for a double point. It reduces to a cone of the second degree when the
four fixed points lie in one plane.
Ex. 16. Find the locus of the vertex of a tetrahedron, if the three edges which pass
through that vertex each pass through a fixed point, if the opposite face also pass
through a fixed point and the three other vertices move in fixed planes.
Ex. 17. A plane passes through a fixed point, and the points where it meets three
fixed lines are joined by planes, each to one of three other fixed lines ; find the locus of
the intersection of the joining planes.
Ex. 18. The sides of a polygon in space pass through fixed points, and all the
vertices but one move in fixed planes ; find the curve locus of the remaining vertex.
Ex. 19. All the sides of a polygon but one pass through fixed points, the
extremities of the free side move on fixed lines, and all the other vertices on fixed
planes, find the surface generated by the free side.
/Ex. 20. The plane through the extremities of conjugate diameters of an elUpsoid
3.2 y1 g2
envelopes the ellipsoid —„ + tx+-x=^ and touches it in the centre of the section.
'^ a^ 0'' c- ■'■•^
/*." jy2 njS
Ex. 21. The condition that a system of generators of the hyperboloid -5 + tj — 5=1
may admit of three such generators mutually at right angles is found to be
i+i-1.0.
Such hyperboioids have been called equilateral hyperholoids. (Schroter, Ober-
ftdchen zweiter Ordnung, p. 197, 1880),
( 103 )
CHAPTER Vir.
METHODS OF ABRIDGED NOTATION.
THE PRINCIPLE OF DUALITY AND RECIPROCAL POLARS.
122. We shall in this chapter give examples of the appli-
cation to quadrics of methods of abridged notation. It is
convenient, however, first to shew that every figm-e we
employ admits of a two-fold description, and that every theorem
we obtain is accompanied by another reciprocal theorem.
In fact, the reader can see without difficulty that the whole
theory of Reciprocal Polars explained [Conies^ Chap. XV.) is
applicable to space of three dimensions. Being given a fixed
quadric S, and any surface 8^ we can generate a new surface s
by taking the pole with regard to 2 of every tangent plane
to S. If we have thus a point on s corresponding to a tangent
plane of 8^ reciprocally the tangent plane to s at that point
will correspond to the point of contact of the tangent plane
to 8. For the tangent plane to s contains all the points on s
consecutive to the assumed point ; and to it must correspond
the point through which pass all the tangent planes of 8 con-
secutive to the assumed tangent plane ; that is to say, the point
of contact of that plane. Thus to every point connected with
one surface corresponds a plane connected with the other, and
vice versa ; and to a line (joining two points) corresponds a line
(the intersection of two planes). For example the degree of. 9,
being measured by the number of points in which an arbitrary
line meets it, is equal to the number of tangent planes which
can be drawn to 8 through an arbitrary right line. Thus the
reciprocal of a quadric is a quadric, since two tangent planes
can be drawn to a quadric through any arbitrary right line
(Art. 80).
■
123. In order to shew what corresponds to a curve in space
we shall anticipate a little of the theory of curves of double
104 METHODS OP ABRIDGED NOTATION.
curvature to be explained hereafter. A curve in space may be
considered as a series of points in space 1, 2, 3, &c., arranged
according to a certain la^. If each point be joined to its next
consecutive point, we shall have a series of lines 12, 23, 34, &c.,
each line being a tangent to the given curve. The assemblage of
these lines forms a surface, and a developable surface (see note,
Art. 112), since any line 12 intersects the consecutive line 23.
Again, if we consider the planes 123, 234, 345, &c., containing
every three consecutive points, we shall have a series of planes
which are called the osculating planes of the given curve, and
which are tangent planes to the developable generated by its
tangents. Now when we reciprocate, it is plain that to the
series of points, lines, and planes will correspond a series of
planes, lines, and points ; and thus, that the reciprocal of a
series of points forming a curve in space will be a series of
planes touching a developable. If the curve in space lies all
in one plane, the reciprocal planes will all pass through one
point, and will be tangent planes to a cone.
Thus the series of points common to two surfaces forjms a
curve. Reciprocally the series of tangent planes common to two
surfaces touches a developable which envelopes both surfaces.
To the series of tangent planes (enveloping a cone) which can be
drawn to the one surface through any point, corresponds the
series of points on the other which lie in the corresponding plane :
that is to say, to a 'plane section of one surface corresponds a
tangent cone of the reciprocal. It easily follows hence, that to a
point and its polar plane with respect to a quadric, correspond
a plane and its pole with respect to the reciprocal quadric.
124. The reciprocals are frequently taken with regard to a
sphere whose centre is called the origin of reciprocation^ and
as at Conies (Art. 307) mention of the sphere may be omitted,
and the reciprocals spoken of as taken with regard to this origin.
To the origin will evidently correspond the plane at infinity ;
and to the section of one surface by the plane at infinity will
correspond the tangent cone which can be drawn to the other
through the origin. Thus, then, when the origin is without a
quadric, that is to say, is such that real tangent planes can be
METHODS OP ABRIDGED NOTATION. 105
drawn from It to the surface, the reciprocal surface will have
real points at infinity, that is to say, will be a hyperboloid ;
when the origin is inside, the reciprocal is an ellipsoid ; when
the origin is on the surface, the reciprocal will be touchc.l by
the plane at infinity, or what is the same thing (as we shall pre-
sently sec) the reciprocal is a paraboloid.
The reciprocal of a ruled surface (that is to say, of a surface
generated by the motion of a right line) is a ruled surface.
For to a right line corresponds a right line, and to the surface
generated by the motion of one right line will correspond the
surface generated by the motion of the reciprocal linc^ Hence
to a hyperboloid of one sheet always corresponds a hyperboloid
of one sheet unless the origin be on the- surface when the reci-
procal is a hyperbolic paraboloid.
125. When reciprocals are taken with regard to a sphere,
any plane is evidently perpendicular to the line joining the
corresponding point to the origin. Thus to any cone corre-
sponds a plane curve, and the cone whose base is that curve
and vertex the origin has an edge perpendicular to every
tangent plane of the first cone, and vice vei'sd. In general two
cones (which may or may not have a common vertex) arc said
to be reciprocal when every edge of one is perpendicular to a
tangent plane of the other (see Ex. 11, Art. 121). For example,
it appears from the last article, that the tangent cone from the
origin to any surface is in this sense reciprocal to the asymp-
totic cone of the reciprocal surface.
The sections hy any plane of tivo reciprocal concs^ having a
common vertex^ are 'polar recip'ocals with regard to the foot of
the perpendicular on that plane from the common vertex. For,
let the plane meet an edge of one cone in a point P, and the
* Prof. Cayley has remarked, that the degree of any ru'/ed surface is equal to the
degree of its reciprocal. The degree of the reciprocal is equal to the number of
tangent planes which can be drawn through an arbitrary right line. Now it vdW be
formally proved hereafter, but is sufficiently evident in itself, that the tangent plane
at any point on a ruled surface contains the generating line which passes through that
point. The degree of the recipirocal is therefore equal to the number of generating
lines which meet an arbitrary right line. But this is exactly the number of iioints in
which the arbitrary lino meets the siurface, since every point on a generating line is a
point on the surface,
106 METHODS OF ABRIDaED NOTATION.
perpendicular tangent plane to the other in the line QR ; le
be the foot of the perpendicular on the plane from the vertej
I then it is easy to see that the line PM is perpendicular to (
'. and if it meet it in 8^ then since the triangle POS is ri
\ angled, the rectangle PM.M8 is equal to the constant C
The curve therefore which is the locus of the point P is
' same as that got by letting fall from M perpendiculars on
I tangents QB^ and taking on each perpendicular a portion
^ versely as its length.
' The following illustrates the application of the principle
established : Througli the vertex of any cone of the second dc
can he drawn two lines^ called focal lines , such that the sectic
the cone hy a 'plane perpendicular to either line is a conic^ ha
for a focus the i^oiyit where the plane meets the focal line.
form a reciprocal cone by drawing through the vertex ]
perpendicular to the tangent planes of the given cone ;
this cone has two planes of circular section (Art. 104) ;
by the present article, the section of the given cone by a p
parallel to either is a conic having for a focus the foot of
perpendicular on that plane from the vertex. What has been
proved may be stated, the focal lines of a cone are perpc
cular to tlie planes of circular section of the reciprocal cone,
126. The reciprocal of a sphere loith regard to any i
is a surface generated hy the revolution of a conic rounc
transverse axis. This may be proved as at Conies, Art.
It is easily proved that if we have any two points A an<
the distances of these two points from the origin are in the s
ratio as the perpendiculars from each on the plane correspon
to the other [Conies, Art. 101). Now the distance of the cc
of a fixed sphere from the origin, and the perpendicular ]
that centre on any tangent plane to the sphere are
constant. Hence, any point on the reciprocal surface is
that its distance from the origin is in a constant ratio to
perpendicular let fall from it on a fixed plane; namely,
plane corresponding to the centre of the sphere. And
locus is manifestly a surface of revolution, of which the oj
is a locus J an.i the plauc in (question a directrix plane.
Methods op ABnmaED notation.
107
By reciprocating properties of the spliere we thus get pro-
perties of surfaces of rcvokition round the transverse axis. The
left-hand column contains properties of the sphere, the right-
hand those of the surftices of revolution.
Ex. 1. Any tangent plane to a
sphere is perpendicular to the line
joining its point of contact to the
centre.
Ex. 2. Every tangent cone to a
sphere is a right cone, the tangent
planes all making equal angles with
the plane of contact.
The line joining focus to any
point on the surface is perpendi-
cular to the plane through the focus
and the intersection with the direc-
trix plane of the tangent plane at
the point.
The cone whose vertex is the
focus and base any plane section is
a right cone whose axis is the line
joining the focus to the pole of the
plane of section.
A particular case of Ex. 2 is " Every plane section of a
paraboloid of revolution is projected into a circle on the tangent
plane at the vertex."
Ex. 3. Any plane is perpendi-
cular to the line joining the centre to
its pole.
Ex. 4. Any plane through the
centre is perpendicular to the con-
jugate diameter.
Ex. 5. The cone whose base is
any plane section of a sphere has
circular sections parallel to the plane
of section.
Ex. 6. Every cylinder envelop-
ing a sphere is right.
Ex. 7. Any two conjugate* right
lines are mutually perpendicular.
Ex. 8. Any quadric enveloping a
sphere is a surface of revolution;
and its asymptotic cone therefore is
a right cone.
The Hne joining any point to tlie
focus is perpendicular to the plane
joining the focus to the intersection
■with the directrix plane of the polar
plane of the point.
Any plane through the focus is
perpendicular to the line joining the
focus to its pole.
Any tangent cone has for its
focal lines the lines joining the ver-
tex of the cone to the two foci.
Every section passing through
the focus has this focus for a focus.
Any two conjugate lines are such
that the planes joining them to the
focus are at right angles.
If a quadric envelope a surface of
revolution, the cone enveloping the
former, whose vertex is a focus of
the latter, is a cone of revolution.
* The polar planes with respect to a quadric of all the points of a line pass
through a right line, which wo call the conjugate line, or polar line (Art. Go).
108 METHODS OF ABRIDGED NOTATION.
127. The equation of the reciprocal of a central surface
with regard to any point is found as at Conies^ Art. 319. For
the length of the perpendicular from any point on the tangent
plane is (see Art. 89)
yr.2
p=~= sj[d' cos''a+5'cos'yS+c'' cos"' 7) - [x' co^a+y' cos/S-l-s' C0S7),
and the reciprocal is therefore
[xx' + yy' + zz' + ly = aV + VSf + (?z\
Thus the reciprocal with regard to the centre is
a quadric whose axes are the reciprocals of the axes of the
given one.
We have given (Ex. 10, Art. 121) the method in general of
finding the equation of the reciprocal of one quadric with
regard to another. Thus the reciprocal with regard to the
sphere ^ -{-y^ -^ z^ = lc\ is found by substituting x^ y^ z^ — U'' for
a, /3, 7, S in the tangential equation, Art. 79 ; or, more symme-
trically, the tangential equation itself may be considered as the
equation of the reciprocal with regard to v^ + ^ + s^ + w'"* = 0 ;
a, /3, 7, 8 being the coordinates.
The reciprocal of the reciprocal of a quadric is evidently the
quadric itself. If we actually form the equation of the re-
ciprocal of the reciprocal A(f!' + B^^ + &c., the new coefficient of
x'hBCD + 2FMN-BN''-mP-DF\ which, when we sub-
stitute for i?, (7, &c., their values will be found to be aA^ And
A^ will in like manner be a factor in every term, so that the
reciprocal of the reciprocal is the given equation multiplied by
the square of the discriminant (see Lessons on Higher Algebra^
Art. 33).
128. The principle of duality may be established indepen-
dently of the method of reciprocal polars, by shewing in ex-
tension of the remarks made above. Art. 38, (see Conies^
Art. 299) that all the equations we employ admit of a two-
fold interpretation ; and that when interpreted as equations in
tangential coordinates, they yield theorems reciprocal to those
which they give according to the mode of interpretation hitherto
METHODS OF ABRIDGED NOTATION. 109
adopted. Wc may call a, /3, 7, 8 the tangential coordinates
of the plane ax + ^1/ -\- ^^z + 8w. Now the condition that this
plane may pass through a given point, being
ax' + jS/ + ryz' + 810' = 0,
conversely, any equation of the first degree in a, /?, 7, S,
Aa + B^+Cy + D8=0
Is the condition that this plane may pass through a point whoso
coordinates are proportional to -4, B^ G, J); and the equation
just written may be regarded as the tangential equation of that
point. If the tangential coordinates of two planes are a, /6, 7, 8 ;
a', /3', 7', 8' it follows, from Art. 37, that a + ka', ^ + Ji^\ &c.
are the coordinates of a plane passing through the line of Inter-
section of the two given planes. And again. It follows from
Art. 8, that If i^ = 0, M= 0 be the tangential equations of two
points, jL + A.M = 0 denotes a point on the line joining the two
given ones; and similarly (Art. 9), that L + JiM-{- //iV denotes a
point in the plane determined by the three points Z, M, N.
Again, any equation In a, ^S, 7, 8 may be considered as
the tangential equation of a surface touclied by every plane
ax + /3j/ + 7« 4 8w whose coordinates satisfy the given equa-
tion. If the equation be of the n^^ order, the surface will be
of the n^^ class, or such that n tangent planes (fulfilling the
given relation) can be drawn through any line. For If we
substitute in the given equation a' + ka'\ /6' + A-/8'^, &c. for a, yS,
&c., we get an equation of the nth. degree In A-, determining
n planes satisfying the given relation, which can be drawn
through the intersection of the planes a'^'y8\ d'^"'^"8" .
129. The general tangential equation of the second degree
A(^ + B^^ -f Gi' + DZ' + 2i^/57 + 2 6^7a + "lEa^
+ 2ZaS + 2il//3a + 2XY78 = 0
can bo discussed by precisely the same methods as are used above
(Arts. 75-80). If we substitute ^ + M\ &c. for a, &c., we get
a quadratic in /r, which may be written 8' + 2AP+ H^S" = 0. If
the plane a'/3'7'S' touch the surface in question, B' = 0, and one
of the roots of the quadratic Is k = 0. The second root will
be also A; = 0, provided that P=0. In other words, the co-
llO METHODS OF ABEIDGED NOTATION.
oi'dluatcs of any tangent plane consecutive to a/3'<y'h' must
satisfy the condition
dS' ^dS' dS' ^dS' ^
But this equation being of the first degree represents a point,
viz. the point of contact of a'/3V^'} through which every con-
secutive tangent plane must pass.
We may regard the relation just obtained as one connecting
the coordinates of a tangent plane with those of any plane
passing through its point of contact, and from the symmetry
of this relation, we infer (as in Art. 63) that if a\ /3', 7', S' be the
coordinates of any plane, those of the tangent plane at every
point of the surface which lies in that plane, must fulfil the
condition
dS' r,dS' dS' ^d8' ^
But this equation represents a point through which all the
tangent planes in question must pass; in other words, it re-
presents the pole of the given plane. *
We can, by following the process pursued in Art. 79, deduce
from the general tangential equation of the second degree the
corresponding equation to be satisfied by its points. If the
tangential equation of any point on the surface be
£c'a + ?/'/3 + ^7 + 10 h = 0,
and a/37S the coordinates of the corresponding tangent plane,
we infer from the equations already obtained, that if X be an
indeterminate multiplier, we must have
■Kx' = Aa + H/3 + Gy + LS; Xi/ = Ha -\- B/3 -^ Fy + IIS,
Xz'=Ga+F/3+Cry + NS', \w' = La -^ iW + Ny + M.
Solving these equations for a/378, we get the coordinates of the
polar plane cf any assumed point ; and expressing that these
coordinates satisfy the given tangential equation, we get the
relation to be satisfied by the x, ?/, 2;, lo of any point on the
surface, a relation only diff'ering by the substitution of capital
for small letters from that found in Art. 79.
It seems unnecessary to give further examples how all the
preceding discussions may be adapted to the corresponding
METHODS OF ABRIDaED NOTATION. Ill
equations in tangential coordinates. In what follows, we have
only to suppose that the abbreviations denote equations in tan-
gential coordinates, when we get direct proofs of the reciprocals
of the theorems actually obtained.
130. If Z7 and F represent any two quadrics, then U+W
represents a quadric passing through every point common to
U and F, and if \ be indeterminate it represents a series
of quadrics having a common curve of intersection. Since
nine points determine a quadric (Art. 58), U+W is the most
general equation of the quadric passing through eight given
points (see Higher Plane Curves, Art. 29). For if U and F be
two quadrics, each passing through the eight points, U+W
represents a quadric also passing through the eight points, and
the constant \ can be so determined that the surface shall pass
through any ninth point, and can in this way be made to co-
incide with any given quadric through the eight points. It
follows then that all quadrics which pass through eight points
have besides a whole series of common points, forming a com^
mon curve of intersection ; and reciprocally, that all quadrics
which touch eight given planes have a whole series of common
tangent planes determining a fixed developable Avhich envelopes
the whole series of surfaces touching the eight fixed planes.
It is evident also that the problem to describe a quadric
through nine points may become indeterminate. For if the
ninth point lie anywhere on the curve which, as we have just
seen, is determined by the eight fixed points, then eyerj/ quadric
passing through the eight fixed points will pass through the
ninth point, and it is necessary that we should be given a ninth
point, not on this curve, in order to be able to determine the
surface. Thus if U and F be two quadrics through the eight
points, we determine the surface by substituting the coordinates
of the ninth point in U+W=0; but if these coordinates
make U= 0, F= 0, this substitution docs not enable us to de-
termine \.
I'iil. Given seven points [or tangent planes] common to a
series of quadrics, then an eighth point [or tangent plane]
common to the whole system is determined.
112 METHODS OP ABRIDaED NOTATION.
For let Z7, F, W be three quadr'ics, each of which passes
through the seven pomts, then U+XV+fJ.W may represent
auTj quadric which passes through them ; for the constants X, fj,
may be so determined that the surface shall pass through
any two other points, and may in this way be made to co-
incide with any given quadric through the seven points. But
Z7+ X, V+ fi W represents a surface passing through all points
common to ?7, F, TF, and since these intersect in eight points,
it follows that there is a point, in addition to the seven given,
which is common to the whole system of surfaces.
We see thus, that though it was proved in the last article
that eight points in general determine a curve of double curva-
ture common to a system of quadrics, it is possible that they
may not. For we have just seen that there is a particular case
in which to be given eight points is only equivalent to being
given seven. When we say therefore that a quadric is deter-
mined by nine points, and that the intersection of two quadrics
is determined by eight points, it is assumed that the nine or
eight points are perfectly unrestricted in position.*
132. If a system of quadrics have If a system of quadrics be in-
a common curve of intersection, the scribed in the same developable,
polar plane of any fixed point passes the locus of the pole of a fixed plane
through a fixed right line. is a right line.
For if P and Q be the polar planes of a fixed point with
regard to Z7 and F respectively, then P-fX(> is the polar of
the same point with respect to TJ+W.
In particular, the locus of the centres of all quadrics in-
scribed in the same developable is a right line.
133. If a system of quadrics have a common curve of
Intersection [or be inscribed in a common developable], the
polars of a fixed line generate a hyperboloid of one sheet.
* The reader who has studied Illfjher Plane Curves, Arts. 29—34, will have no
difficiUty in developing the corresponding theory for surfaces of any degree. Thus if
we are given one less than the number of points necessary to determine a surface of the
n'" degree, we are given a series of points forming a curve through which the surface
must pass ; and if we are given two less than the number of points necessary to deter-
mine tlie surface, then we arc given a certain number of otlicr points [namely as many
as will make the entire number up to ii^] through which the smface must also pass,
METHODS OF ABRIDGED NOTATION. 113
Let the polars of two points In the line be P+\Q^ P' -\-\Q\
then it is evident that their intersection lies on the hyper-
holo'xdi PQ' = QP\
134. If a system of qnadrlcs have a common curve, the locus
of the pole of a fixed plane is a curve in space of the third
degree. For, eliminating \ between P+\Q^ P'+XQ'^ P"+\(^\
the polars of any three points, each determinant of the system
P, P', P"
vanishes. Now the intersection of the surfaces represented by
PQ'= QP\ PQ"= QP'\ is a curve of the fourth degree, but
this includes the right line PQ^ which is not part of the inter-
section of Pg'= QP'\ P'Q" = qP". There is therefore only
a curve of the third degree common to all three.
Reciprocally, if a system be Inscribed in the same develop-
able, the polar of a fixed point envelopes the developable which
is the reciprocal of a curve of the third degree, being (as will
afterwards be shewn) a developable of the fourth order.
135. Given seven points on a Given seven tangent planes to
quadric, the polar plane of a fixed a quadric, the pole of a fixed plane
point passes through a fixed point. moves in a fixed plane.
For evidently the polar of a fixed point with regard to
Z7+ X F+ /i IF will be of the form P+\Q + jxR^ and will there-
fore pass through a fixed point.*
136. Since the discriminant contains the coefficients in the
fourth degree, it follows that we have a biquadratic equation
to solve to determine \, in order that C/'+XFmay represent
a cone, and therefore that through the intersection of tivo quadrics
four cones may he described. The vertex of each of these cones
is the common Intersection of the four planes.
* Dr. Hesse has derived from this theorem a construction for the quadric passing
through nine given points. Crelle, Yol. XXiv. p. 36. Cambridge and Dublin Mathe-
matical Journal, Vol. iv. p. 44. See also some further developments of the same
problem by Mr. Townsend, ib. Vol. iv, p. 241.
Q
= 0.
114 METHODS OF ABRIDGED NOTATION.
•when X satisfies the biquadratic just referred to, and the four
vertices are got by substituting its four roots in succession in
any three of these equations ; they are therefore the four points
common to the surfaces found by making each of the determinants
u,, u,, u.. u.
V V V V
There are four points whose polars are the same with respect
to all quadrics passing through a common curve of intersection,
namely the vertices of the four cones just referred to. For to
express the conditions that
should represent the same plane, we find the very same set of
determinants. In like manner there are four planes whose poles
are the same with respect to a set of quadrics inscribed in the
same developable.
137. If the surface V break up into two planes, the form
XJ+\V= 0, becomes U-\ \LM= 0, a case deserving of separate
examination.* In general, the intersection of two quadrics is
a curve of double curvature of the fourth degree, which may in
some cases (Art. 134) break up into a right line and a cubic, but
the intersection with C/of any of the surfaces C/+XLJ/, evidently
reduces to the two conies in which U is cut by the planes X and il/.
Any point on the line LM has the same polar plane with regard to all
surfaces of the system Z7+ \LM.-\ For if P be the polar of any
point with regard to f/, its polar with regard to U+\LM v^iW be
P+X{LM' + ML') which reduces to P, wheni.'=0, il/'=0. Thus,
* The case where U also breaks up into two planes has been discussed, Art. 108.
t There are two other points whose polar planes are the same with regard to all the
quadrics, and which therefore (Art. 136) will be vertices of cones containing both the
curves of section. It is only necessary that P, the polar plane of one of these points
with regard to U, should be the same plane as L'3f+ LM' the polar with regard
to LM. Since then the polar plane of the point with regard to U passes through
LM, the point itself must lie on the polar line of LM with regard to U, that is to say,
on the intersection of the tangent planes where LM meets U. Let this polar line
meet Z7in A A', and LM in BB', then the points required will be FF', the foci of the
involution deLermined by AA', BB'. For since FF' form a harmonic system either
with A A' or with BB', the polar plane of /'"either with regard to U or i J/ passes
through /■", and vice versa.
METHODS OF ABRIDGED NOTATION. 115
In particular, at each of the two points where the line LM meets
Uj all the surfaces have the same tangent plane. The form,
then, U-y \LM^ may be regarded as denoting a system of quadrics
having double contact with each other. Conversely, if two
quadrics have double contact, their curve of intersection breaks
up into simpler curves. For if we draw any plane through the
two points of contact and through any point of their intersec-
tion, this plane will meet the quadrics in sections having three
points common, and having common also the two tangents
at the points of contact ; these sections must therefore be
identical, and the curve of intersection breaks up into two plane
curves unless the line joining the points of contact be a
generator of each surface in which case the rest of the curve
of intersection is a curve of the third degree.
In like manner all surfaces of the system are enveloped by
two cones of the second degree. For take the point where
the Intersection of the two given common tangent planes is cut
by any other common tangent plane ; then the cones having
this point for vertex, and enveloping each surface, have common
three tangent planes and two lines of contact, and are therefore
identical. The reciprocals of a pair of quadrics having double
contact will manifestly be a pair of quadrics having double con-
tact, and the two planes of intersection of the one pair will corre-
spond to the vertices of common tangent cones to the other pair.
138. If there he a plane curve common to three quadrics^ each
pair must have also another common plane curve^ and the three
planes of these last common curves pass through the same line.
Let the quadrics be Z7, U-{ LM, U+ LNj then the last two
have evidently for their mutual intersection two plane sections
made by Z, M — N.
139. Similar quadrics belong to the class now under dis-
cussion. Two quadrics are similar and similarly placed when
the terms of the second degree are the same in both (see
Conies^ Art. 234). Their equations then are of the form U= 0,
U+cL — 0. We see then that two such quadrics intersect
in general in one plane curve, the other plane of intersec-
tion being at infinity. If there be three quadrics, similar and
116 METHODS OF ABRIDGED NOTATION.
similarly placed, their three finite planes of intersection pass
through the same right line.
Spheres are all similar quadrics, and therefore are to be
considered as having a common section at infinity, which section
will of course be an imaginary circle.
A plane section of a quadric will be a circle if it passes
through the two points in which its plane meets this imaginary
circle at infinity. We may see thus immediately of how many
solutions the problem of finding the circular sections of a quadric
is susceptible. For the section of the quadric by the plane at
infinity meets the section of a sphere by the same plane in four
points, which can be joined by six right lines, the planes passing
through any one of which meet the quadric in a circle. The
six right lines may be divided into three pairs, each pair inter-
secting in one of the three points whose polars are the same
with respect to the section of the quadric and of the sphere.
And it is easy to see that these three points determine the
directions of the axes of the quadric.
An umbilic (Art. 106) is the point of contact of a tangent
plane which can be drawn through one of these six right lines.
There are in all therefore twelve umbilics, though only four
are real. If a tangent plane be drawn to a quadric through
any line, the generators in that tangent plane evidently pass,
one through each of the points where the line meets the surface.
Thus, then, the umbilics must lie each on some one of the eight
generators, which can be drawn through the four points at
infinity common to the quadric and any sphere. Or, as Sir
W. Hamilton has remarked, the twelve umhilics lie three by three
on eight imaginary right lines.
A surface of revolution is one which has double contact at in-
finity with a sphere. For an equation of the form x^+y^-{-az^=h
can be written in the form
{x' + f + z'-r') + {(a - 1) z'-{h-r')]=Q>,
and the latter part represents two planes. It is easy to see
then why in this case there is but one direction of real circular
sections, determined by the line joining the points of contact
of the sections at infinity of a sphere and of the quadric.
METHODS OF ABRIDGED NOTATION. 117
140. If the two planes i, M coincide, the form Z7-f- \LM
becomes U-\- \U^ which denotes a system of surfaces touching
U at every point of the section of U by the plane L. Two
quadrics cannot touch in three points without their touching all
along a plane curve. For the plane of the three points meets
the quadrics in sections having common those three points and
the tangents at them. The sections are therefore identical.
The equation of the tangent cone to a quadric given Art. 78, is a
particular case of the form V=U. Also two concentric and
similar quadrics (Z7, U—c^) are to be regarded as having plane
contact with each other, the plane of contact being at infinity.
Any plane obviously cuts the surfaces U and U- U in two
conies having double contact with each other, and if the section
of one reduce to a point-circle, that point must plainly be the
focus of the other. Hence when one quadric has plane coyitact
with another^ the tangent plane at the umhilic of one cuts the
other in a conic of which the umhilic is the focus / and if one
surface be a sphere, every tangent plane to the sphere meets
the other surface in a section of which the point of contact
is the focus.
Or these things may be seen by taking the origin at the
umhilic and the tangent plane for the plane of xy^ when on
making 2 = 0, the quantity U— U reduces to x^ + y'^ — F^ and
denotes a conic of which the origin is the focus, and I the
directrix.
Two quadrics having plane contact with the same third quadric
intersect each other in plane curves. Obviously U— L\ U— 31'^.
have the planes L — M^ L-^ M for their planes of intersection.
141. The equation aU + h^P^ cN' + dP% where L, il/, N, P
represent planes, denotes a quadric such that any one of these
four planes is the polar of the intersection of the other three.
For aU + hJSP + cN"^ denotes a cone having the point LMN
for its vertex ; and the equation of the quadric shews that this
cone touches the quadric, P being the plane of contact. The
four planes form what I shall call a self-conjugate tetrahedron
with regard to the surface. It has been proved (Art. 136)
that given two quadrics there are always four planes whose
118 METHODS OF ABRIDGED NOTATION.
poles with regard to both are the same. If these be taken
for the planes Z, J/, N, P, the equations of both can be
transformed to the forms
aL' + b3P + cN' + dP' = 0, a'L' + b'M'' + c'N' + d'F' = 0.
It may also be seen, a priori^ that this is a form to which
It must be possible to bring the system of equations of two
quadrlcs. For X, i/, iV, P involve implicitly three constants
each ; and the equations written above involve explicitly three
independent constants each. The system therefore includes
eighteen constants, and is therefore sufficiently general to ex-
press the equations of any two quadrlcs.
We are misled, however, if we conclude in like manner that
the equations of any three quadrlcs may be written in the form
aU -\hl\P +cN'' +dP +eQ'' =0,
a'L' + VM' -H cN^ + d'P^ -f e Q' = 0,
a'T + b"]\P + c''N' + d'P' 4 e" Q' = 0,
where i, J/, N^ P, Q are five planes whose equations are con-
nected by the relation
p + ii/+^+p+ <g = o.
For though, since X, 3f, N^ P, Q involve implicitly three
constants each, and the equations written above involve explicitly
four independent constants each, the system thus appears to
include twenty-seven constants, it has not really so many. For,
as we shall show in a subsequent chapter, a relation must subsist
among them, and the system is consequently not general enough
to express the equations of any three quadrlcs,
142. The lines joining the vertices of any tetrahedron to the
corresponding vertices of its polar tetrahedron with regard to a
quadric belong to the same system of generators of a hyperboloid
of one sheetj and the intersections of corresponding faces of the
tioo tetrahedra possess tlie same property.
Taking the fundamental tetrahedron and Its polar, the
vertices of the polar tetrahedron (Art. 79) are proportional
METHODS OF ABRIDGED NOTATION. 119
to the horizontal rows In
A, H, O, L,
H, B, F, M,
G, F, C, N,
L, M, N, n,
Thus the equations of the four lines we are considering are
H
z
a''
w
z
F~
w
M
X
w
N~
X
G'
- y
~ F'
X
y _
M
z
Now the condition that any line
ax + ^7/ + yz + Sio = 0, ax + ^'i/ + y'z + S'lo = 0,
should intersect the first of the four, is, by eliminating x between
the last two equations, found to be
ir(ay3' - ;Sa') + G (a/ - 7a') + L (aS' - Sa') = 0,
and the conditions that it should intersect each of the other
three, are in like manner found to be
Zr(ySa' - yQ'a) + F{/3Y - ^'y) + M{/3S' - fi'S) = 0,
G [ya' - 7'a) + F{y/3' - y'0) + N{yS' - y'8) = 0,
L {Ba' - 8'a) + M (S/3' - S'/3) + N[8y' - h'y) = 0.
But these four conditions added together vanish identically.
Any right line therefore which intersects the first three will
intersect the fourth, which is, in other words, the thing to be
proved.*
AVe find the equation of the hyperbolold by any of the
methods in Art. 113, for example, by expressing that the line
wx — w'x toy — w'y loz — iv'z , ^ 1 r ^
= -^ — = meets the first three ot these
s t u
lines. For then
IIw-Ly _ Gw-Lz Fw-Mz _ Hw-Mx Gw-Nx _ Fio-Nij
t u ^ u s ' s t ^
* This theorem is due to M. Chusles. The proof here giveu is by Mr. Ferrers,
Quarterly Journal of Mathematics, (Vol. i., p. 241).
120 METHODS OF ABRIDGED NOTATION.
from which by multiplication, s, f, u are eliminated in the form
{Fw-Mz][ Giv-Nx)[Hw-Ly) = [Fw-Ny] ( Ow -Lz) [Hw -Mx)j
or {HN- GM) [Fwx + Lyz) + [FL - HN) ( Owy + Mzx)
+ ( GM- FL) (Hwz + Nxy) = 0.
142a. This hyperbololdal relation between the four joining
lines has been established by Mr. M'Cay by the following con-
siderations.
First, considering any solid angle formed by three planes;
their poles in regard to any quadric determine a plane, and in
it these three poles form a triangle which is conjugate, in regard
to the curve of section, to the triangle which the solid angle
cuts out In the same plane.
Now conjugate triangles are In perspective, hence the three
planes, — each through an edge of the solid angle, and the pole
of Its opposite face, — all pass through a right line.
If then we have two tetrahedra, polars with regard to a
quadric, having the vertices abcd^ ah'cd\ we see that at any
one (a) of their eight vertices a right line may be found In the
manner described; and since this line is common to the three
planes abb\ acc\ add' it meets the connecting lines hh\ cc', dd' ;
also, since it passes through [a] it meets aa. In this way,
taking each of the eight vertices, we have eight lines each of
which meets aa\ bb\ cc, dd'. The relation is thus demonstrated.
N.B. It appears from what has been stated that, when three
planes are given and two points assumed which are to be poles
to two of them in regard to any quadric, the pole of the third
is limited to a certain plane locus.
Ex. 1. Given three planes and their poles in regard to a quadric, the locus of the
centre is a right line (Mr. M'Cay).
Ex. 2. The four perpendiculars from the vertices on the opposite faces in any
tetrahedron are generators of one system, and the four perpendiculars to the faces at
their orthocentres are generators of the other system of an equilateral hyperboloid.*
In the tetrahedron, whose vertices are a, b, c, d, let the opposite faces be
A, B, C, J>, and the perpendicular from a on A, Xo, from b on B, t/^, &c.
Also let the feet of these perpendiculars be a, (3, y, S. Then since in a spherical
* The equilateral hyperboloid is defined as one which admits of three generators
mutually at right angles, see Ex. 21 Art. 121. Schrbter, as there referred to p. 205^
gives these theorems. The first part of the theorem was given by Steiner, Crelle 2,
p. 98. The second part of the theorem and the determination of the centre Ex. 3
are referred by Baltzer to Joachimsthal, Grunert Archiv, 32, p. 109. Ex. 4 is referred
to Monge, Corresp. sur V Ecole Polytech. II. p. 2G6.
METHODS OF ABRIDGED NOTATION. 121
triangle the perpendiculars intersect, the planes through each edge of the solid angle
(a) perpendicular to the opposite face intersect in a right line. This right line'
therefore, meets the perpendiculars ?/„, z^, «,'(,, and as it passes through («), also a:,,'
In like manner at each other vertex we have a right Hne meeting those four right
lines. They, therefore, belong to the same system of generators of a hyperboloid.
Again, taking through »/(, a parallel plane e to Xq, this plane is orthogonal both to
B and also to A, and, therefore, to their edge of intersection cd. Therefore this
plane passes through a perpendicular of the triangle A.
Repeating this we see that the plane s' through z^ parallel to x^ passes through
the perpendicular from c on bcl in the same triangle A, Thus the intersection tt',
which is parallel to x^, is the perpendicular to A at its orthocentre. This line ee' is
manifestly a generator of the second system of the above hyperboloid, which contains
the four perpendiculars of the tetrahedron.
Further, the plane A intersects this hyperboloid in a conic, which passes through
bed and the orthocentre of A, which is, therefore, an equilateral hyperbola; the
generators parallel to the asymptotes of this hyperbola and the generator x^ are an
orthogonal system, therefore the hyperboloid is equilateral.
The reader will easily perceive that this example is included in the general
theorem.
Ex. 3. If in a tetrahedron a plane be taken through the middle of each edge
normal to the opposite edge, these six planes intersect in a point, the centre of the
above equilateral hyperboloid.
Ex. 4. In a tetrahedron the line joining the centre of the circumsciibed sphere and
the centre of the above equilateral hyperboloid is bisected by the centre of gravity of
the tetrahedron.
143. The second part of the theorem stated In Article 142
is only the polar reciprocal of the first, but, as an exercise, we
give a separate proof of it.
Taking the fundamental tetrahedron and its polar as before,
the equations of the four lines are
x = 0, hi/ + gz + ho = 0,
?/ = 0, hx-\-fz +??2?^ = 0,
2 = 0, gx +fy + mo = 0,
i^j = 0, Ix + my + nz =0.
Now the conditions that any line
ax + /3y + >yz + 8io = 0, ax + /S'y + yz + h'lo = 0,
should intersect each of these are found to be (Art. blh)
hv — gr -f Zvr = 0, —hv -^fa- + 7nK = 0,
.9"^ ~/o" + ''P = 0} —Itt — mK — np = 0,
and, as before, the theorem is proved by the fact that these
122 METHODS OF ABRIDGED NOTATION.
conditions when added vanish identically. The equation of the
hjperboloid Is found to be
x^ghl + y^hfm + zygn + w^hnn
-f {fyz + Ixw) {gm + hn) + {gzx + myiv) [hn +fl)
+ {hx2/ + nziv) [fl + gm).
As a particular case of these theorems the lines joining each
vertex of a circumscribing tetrahedron to the point of contact
of the opposite face are generators of the same hyperboloid.
144. Pascal's theorem for conies may be stated as follows :
" The sides of any triangle intersect a conic in six points lying
in pairs on three lines which Intersect each the opposite side of
the triangle In three points lying in one right line." M. Chasles
has stated the following as an analogous theorem for space
of three dimensions : " The edges of a tetrahedron intersect a
quadric In twelve points, through which can be drawn four
planes, each containing three points lying on edges passing
through the same angle of the tetrahedron ; then the lines
of Intersection of each such plane with the opposite face of
the tetrahedron are generators of the same system of a certain
hyperboloid."
Let the faces of the tetrahedron be a;, ?/, z^ w, and the quadric
x'-\-f + z' + w'- (f+jj yz- [g + -^ zx - {h + ^ xy
then the four planes may be written
x = hy i- gz -\ Iw^ y = hx-{fz -r miv,
z =gx -\-fy + niv^ w = Ix + my + nz^
whose intersections with the planes ic, ?/, z^ w, respectively, are
a system of lines proved in the last article to be generators of
the same hyperboloid.
144a. The conception of a Brianchon's hexagon may be
extended to space, and we may denote by this name any
hexagon whose diagonals meet in a point. Now it is evident
METHODS OF ABRIDGED NOTATION. 123
that if this be the case, each pair of opposite sides of the
hexagon intersect ; and, conversely, if in any skew hexagon
each pair of opposite sides intersect, the diagonals are concurrent.
Thus three alternate sides of such a hexagon are met each
by the other three, hence the odd sides belong to one set
of generators of a hyperboloid of one sheet and the even
to the other. Conversely, any hexagon whose sides lie in a
hyperboloid is a Brianchon's hexagon.*
It is further not difficult to see that if any hexagon U in
space and a point (a) are given, and through (a) three right
lines are drawn cutting the opposite sides of the hexagon in
pairs, their intersections on consecutive sides of C/'are consecutive
vertices of a Brianchon's hexagon F, having [a) as its Brianchon
point. This hexagon F inscribed in U determines uniquely a
hyperboloid on which it lies. But again this hyperboloid is cut
by the sides of the given hexagon U in six other points, which
\u the same order are the vertices of a second Brianchon's
hexagon inscribed in the given one and lying on the same
hyperboloid, but having a different Brianchon point.
144i. Considering further this conception of a Brianchon's
hexagon, there is at each vertex a tangent plane, and this
contains the two sides which meet in that vertex. Now, taking
an opposite pair of these six planes, viz. the plane containing
the lines 1, 2 and the plane containing the lines 4, 5 ; since
1 meets 4 and 2 meets 5, the line of intersection of these two
tangent planes is the same as the line joining the point 1, 4 to
2, 5. In like manner, the axis of 2, 3 with 5, 6 is the same as
the ray from 2, 5 to 3, 6 ; and the axis of 3, 4 with 6, 1 is the
same as the ray from 3, 6 to 1, 4. Hence, the three axes of
intersection of opposite tangent planes at six points are coplanar.
Their plane may be considered a Pascal ^??a?ie to the same
hexagon. Thus, in three dimensions both properties meet in
the same figure. In fact —
* See a posthumous paper of O. Hesse in the 85th vol. of the Journal founded by
Crelle; where, after giving the algebraical treatment of the above geometrically
evident statements, Hesse also treats algebraically the question of the two inscribed
Brianchon's hexagons derived by aid of an arbitrary point from any skew hexagon.
124
METHODS OF ABRIDGED NOTATION.
If the surface and the tangeut planes be cut hy an arhitrary
plane [A)^ since each tangent plane contains two generators,
it will meet {A) In the chord joining two points on the conic
of section, and what we have called the Pascal plane will meet
[A] in the Pascal line of the inscribed hexagon.
But if the whole figure he looked at from any voint (a) to
which the contour of the surface affords a real tangent cone,
each generator of the surface determines a tangent plane to
this cone, and the planes through opposite edges of this cir-
cumscribed hexagon have a common line of intersection, the
raj to the Brianchon point.
Ex. Analytically we may consider the quadric yz — wx, and take the odd Bides of
the form (1) x = Xiy, z = XjW, and the even (2) x — XjZ, y — \ro. These two lines
meet in the point whose coordinates are proiDortional to W, Xj, Xj, 1, and the
equation of the tangent plane at it is t^2 = x — \y — \„z + Ww = 0. The Brianchon
point will then evidently be the intersection of the planes
X — X^y — \^z + X1X4M) = 0,
X — \^ — X^z + X5X2W = 0,
X — X3?/ — XgS + XjXgW = 0,
[, h, c, d
its equation therefore is
-X.
X4, X1X4
X2,
Xr,
and the equation of what we call the Pascal plane, may be written
X,
X1X4,
^5X2)
if we multiply this by
X3X6,
X,
^67 ^3>
W
1
1
1
= 0,
i.e. by X,
'^5, the result is
1,
1,
0,
0,
0,
0,
1,
0,
0,
1,
0.
X1X2
0
1
0,
0,
K
I
(X3 - Xi) (X6 - ^2)
(X3-X,)(\a-\)
Xa
hence, the value of the determinant is (compare Conies, p. 383)
with similar forms in t^^, t^^ and in t^^, t^i : showing that the plane contains the hues
<,2, <45, and these other two lines.
METHODS OF ABRIDGED NOTATION.
125
Also since for any undetermined quantities x, y, z, w
^,
y,
2,
w
1,
-^1,
-K
KK
\ Xj,
Xj,
K
1
1,
-^5,
-K
x,\.
\K,
K
K
1
1,
-^3,
-K,
W
= t
u
*^\
= 0,
'•Hi '■i2^ ■'36
0, 0, {\ - X3) (X, - X,)
0, 0, (X, - X;) (X5 - Xj)
every point xyzw is coplanar with the three points 1, 2 ; 4, 5 ; and that whose coordi-
nates are the determinants in the second matrix. Therefore these last three points
must be collinear ; which is a verification that the diagonals in our hexagon intersect.
( 126 )
CHAPTER VIII.
FOCI AND CONFOCAL SURFACES.*
145. When U represents a sphere, the equation of a
quadric having double contact with it, U= LM expresses, as
at Conies^ Art. 260, that the square of the tangent from any
point on the quadric to the sphere is in a constant ratio to the
rectangle under the distances of the same point from two fixed
planes. The planes L and M are evidently parallel to the
planes of circular section of the quadric, since they are planes
of its intersection with a sphere ; and their intersection is there-
fore parallel to an axis of the quadric (Arts. 103, 139). We
have seen [Conies^ Art. 261) that the focus of a conic may be
considered as an infinitely small circle having double contact
with the conic, the chord of contact being the directrix. In
like manner we may define a focus of a quadric as an infinitely
small sphere having double contact with the quadric, the chord
of contact being then the corresponding directrix. That is to
say, the point a/?7 is a focus if the equation of the quadric can
be expressed in the form
where ^ is the product of the equations of two planes. We
must discuss separately, however, the two cases, where these
planes are real and where they are imaginary. In the one
case the equation is of the form U=LM^ in the other U—U+]\P.
In the first case, the directrix (the line LM) is parallel to that
axis of the surface through which real planes of circular section
can be drawn ; for example, to the mean axis if the surface
be an ellipsoid. In the second case the lino LM is parallel to
one of the other axes.
* The properties treated of in this chapter were first studied in detail by
M. Chasles and by Professor Mac Cullagh, who about the same time independently
arrived at the principal of them. M. Chasles' results will be found in the notes to
his Aperqu Misioriijue, published in 1837.
FOCI AND CONFOCAL SURFACES. 127
We can shew directly that the line LM is parallel to an
axis of the surface. For If the coordinate planes x and y be
any two planes mutually at right angles passing through LM]
then since L and M are both of the form \x + [xy^ the quantities
LM and L^ + M^ will be both of the form ax' -f 2hxy + J/.
Andj as in plane geometry, it is proved that by turning round
the coordinate planes x and ;/, this quantity can be made
to take the form px^±qy\ The equations then, U=-LM^
U= L^ -f IP, written in full, are of the form
{x - 0.Y +iy- ^Y + (^ - 7)' =i^-»' ± qf,
and since the terms yz^ zx, xy do not enter into the equation,
the axes of coordinates are parallel to the axes of the surface.
146. A focus of a plane curve has been defined {Higlier
Plane Curves^ Art. 138j as the point of intersection of two
tangents, passing each through one of the circular points at
infinity. The definition just given of a focus of a quadric may
be stated in an analogous form. When the origin is a focus we
have just seen that the equation of the quadric may be written
in the form U=LM, where U, or {x- aY + {y- fif + {z-y)%
denotes a cone whose vertex is the focus, and which passes
through the imaginary circle at infinity. The form of the
equation shews (Art. 137) that this cone has double contact
with the quadric in the points where the line LM meets it.
The tangent plane to the surface at either point of contact
will then be a tangent plane to the cone, and will therefore
pass through a tangent line of the circle at infinity. We may
thus define a focus as a point through which can be drawn
two lines (7, each touching the surface and meeting the imaginary
circle at infinity, and such that the tangent plane to the surface
through either also touches the circle at infinity. This definition
is not restricted to the case of a quadric, but applies to a surface
of any order.
Starting from this definition, if we desire to find the foci of
any surface, we should consider the tangent planes to the surface
drawn through the tangent lines of the circle at infinity : these
form a singly infinite series of planes, and will envelope a
developable surface. The intersection of two consecutive such
128 FOCI AND CONFOCAL SUEFACES.
planes, will be a line cr, and will be a generator of the developable.
A focus, being a point through which pass two lines o-, that is to
say, two generators of the developable, must be a double point
on the developable. Now we shall see hereafter that a develop-
able has in general a series of double points forming a nodal
curve or curves; we infer, therefore, that the foci of a surface
in general are not detached points, but a series of points forming
a curve or curves. We shall shew directly, in the next article,
that this is so in the case of a quadric. It is evident from this
definition that two surfaces will have the same series of foci,
if the developable, just spoken of, passing through the tangent
lines of the circle at infinity and enveloping the surface, be
common to both.
147. Let us then directly examine whether a given central
quadric necessarily has a focus, and whether it has more than
one. For greater generality instead of taking the directrix for
the axis of 2, we take any parallel line, and the equation of
the last article becomes
and we are about to enquire whether any values can be assigned
to a, /3, 7, a', /3', p? 2', which will make this identical with a
given equation
x^ y' z' ,
A^ B^ G
Now first, in order that the origin may be the centre, we have
ry = 0, a =^a', /8 = q^' ; by the help of which equations, elimi-
nating a', /S', the form written above becomes
(1 -p) x^+[\-q) f + .^ =^ a-^ + ^-^ ^\
C A-G , G B-G
whence ^ -P = -J^ P = ^^'1 1-? = ^?^ = — g— 5
l^Pa^+l^^^=G,
®^ A-G^ B-G~
* When p and q have opposite signs tlie planes of contact of the focus with
the quadric are real, while they are imaginary when 7; and 2 have the same sign.
FOCI AND CONFOCAL SURFACES. 129
Thus It appears that the surface being given, the constants 'p
and 2' are determined, but that the focus may lie anywhere
on the conic
"^ Vf n — ^\
A-G^B-G '
which accordingly is called a. focal conic of the surface.
Since we have purposely said nothing as to either the signs
or the relative magnitudes of the quantities A^ B, C, it follows
that there is a focal conic in each of the three principal planes,
and also that this conic is confocal with the corresponding
principal section of the surface; the conies
A^ B ' A-G^ B-G '
being plainly confocal. Any point a/S on a focal conic being
taken for focus, the corresponding directrix is a perpendicular
to the plane of the conic drawn through the point
These values may be interpreted geometrically by saying that
the foot of the directrix is the pole, with respect to the principal
section of the surface, of the tangent to the focal conic at the
point a/3. For this tangent is
037
-4 1^ =1 or I- ■ = 1
A-G^ B-G ' A ' B
^' , f
which is manifestly the polar of a'/S' with regard to -^ + ^ = 1.
Hence, from the theory of plane confocal conies, the line
joining any focus to the foot of the corresponding directrix is
normal to the focal conic. The feet of the directrices must
evidently lie on that conic which is the locus of the poles of
the tangents of the focal conic with regard to the corresponding
principal section of the quadric. The equation of this conic is
,A-G . ,B-G ,
for if we eliminate a, yS from the equation of the focal conic
and the equations connecting a^, a'/3', we obtain this relation
s
130 FOCI AND CONFOCAL SURFACES.
to be satisfied by the latter pair of coordinates. The directrices
themselves form a cylinder of which the conic just written is
the base.
148. Let us now examine in detail the different classes of
central surfaces, in order to investigate the nature of their focal
conies and to find to which of the two different kinds of foci the
points on each belong. It is plain that the equation
A-G^ B-G
will represent an ellipse when G is algebraically the least of
the three quantities A, B, C] a hyperbola when G is the
middle, and will become imaginary when G is the greatest.
Of the three focal conies therefore of a central quadric, one
is always an ellipse, one a hyperbola, and one imaginary. In
the case of the ellipsoid, for example, the equations of the focal
ellipse and focal hyperbola are respectively
The corresponding equations for the hyperboloid of one sheet
are found by changing the sign of c"'', and those for the hyper-
boloid of two sheets by changing the sign both of h^ and c^
Further, we have seen that foci belong to the class whose planes
of contact are imaginary, or are real, according as p and q have
the same or opposite signs, and that p = {A — G): A^ q= {B — G) : B.
Now if G be the least of the three, in these fractions both nume-
rators are positive, and the denominators are also positive in
the case of the ellipsoid and hyperboloid of one sheet, but in
the case of the hyperboloid of two sheets one of the denomi-
nators is negative. Hence, the points on the focal ellipse are
foci of the class whose planes of contact are imaginary in the
cases of the ellipsoid and of the hyperboloid of one sheet, but
of the opposite class in the case of the hyperboloid of two sheets.
Next, let G be the middle of the three quantities ; then the two
numerators have opposite signs, and the denominators have the
same sign in the case of the ellipsoid, but opposite signs in the
POCT AND CONFOCAL SURFACES. 131
case of either hyperboloid. Hence the points of the focal
hyperbola belong to the class whose planes of contact are real
in the case of the ellipsoid, and to the opposite class in the case
of either hyperboloid. It will be observed then that all the real
foci of the hyperboloid of one sheet belong to the class whose
planes of contact are imaginary ; but that the focal conies of
the other two surfaces contain foci of opposite kinds, the ellipse
of the ellipsoid and the hyperbola of the hyperboloid being
those whose planes of contact are imaginary. This is equi-
valent to what appeared (Art. 145) that foci having real planes of
contact can only He in planes perpendicular to that axis of a
quadric through which real planes of circular section can be drawn.
149. Focal conies with real planes of contact intersect the
surface in real points, while those of the other kind do not.
In fact, if the equation of a surface can be thrown into the
form U= U + M'\ and if the coordinates of any point on the
surface make Z7=0, they must also make ^ = 0, J/=0; that is
to say, the focus must lie on the directrix. But in this case
the surface could only be a cone. For taking the origin at
the focus, the equation ^' +/ + 2' = Z' + ii', where L and M
each pass through the origin, would contain no terms except
those of the highest degree in the variables, and would there-
fore represent a cone (Art. 66).
The focal conic on the other hand, which consists of foci of
the first kind, passes through the umbilics. For if the equa-
tion of the surface can be thrown into the form Z7=Zif, and
the coordinates of a point on the surface make f =0, they
must also make either L or J/=0. But since the surface passes
through the intersection of Z7, i ; if the point U lies on i, the
plane L intersects the surface in an infinitely small circle ; that
is to say, is a tangent at an umbilic.
From the fact that focal conies which consist of foci having
real planes of contact pass through the umbilics. Professor
Mac Cullagh gave them the name umbilicar focal conies.
150. The section of the quadric by a plane passing through
a focus and the corresponding directrix is a conic having the
same point and line for focus and directrix. For, taking the
132 FOCI AND CONFOCAL SURFACES.
origin at the focus, the equation is either x^ + y^ + z^ = LM, or
x^ + y^ -If z^= U + M^. And if we make z = Q^ the equation of
the section is x'' + y^ = Im or = f* + wz^, where Z, m are the sections
of Z, M by the plane z. But if this plane pass through LM^
these sections coincide, and the equation reduces to x^ + y^ = f,
which represents a conic having the origin for the focus and I
for the directrix. Since the plane joining the focus and directrix
is normal to the focal conic (Art. 147) ; we may state the
theorem just proved, as follows : Every plane section normal to
a focal conic has for a focus the point where it is normal to the
focal conic.
x^ y^ ^
151. If the given quadric were a cone "i" + % + 7=r = ^j
the reduction of the equation to the form U= U ± M^ proceeds
exactly as before, and it is proved that the coordinates of the
focus must fulfil the condition —. — j^ + -^ — -p^ = 0, which re-
^ — O x> — O
presents either two right lines or an infinitely small ellipse,
according as ^ — C and B— G have opposite or the same signs.
In other words, in this case the focal hyperbola becomes two
right lines, while the focal ellipse • contracts to the vertex of the
x^ y^ z^
cone. For the cone — + ^ — 2 = ^j the equation of the focal
Imes IS —^ — 72 — ,1 5 = 0.
The focal lines of the cone, asymptotic to any hyperboloid,
are plainly the asymptotes to the focal hyperbola of the surface.
The foci on the focal lines are all of the class whose planes
of contact are imaginary ; but the vertex itself, besides being
in two ways a focus of this kind, may also be a focus of the
other kind, for the equation of the cone just written takes any
of the three forms
2 1 2 7,2 1 2
2 , 2 , 2 « + c „ 6 4 c .^
X ^f^rZ =-^» +— j^2/)
a^-}i'- y^j,c' , P-a' , a' + c' ,
= -^^^+"^^"' or=-^3/+-^^..
The directrix, which corresponds to the vertex considered as
a focus, passes through it.
FOCI AND CONFOCAL SURFACES. 133
The line joining any point on a focal line to the foot of
the corresponding directrix is perpendicular to that focal line.
This follows as a particular case of what has been already proved
for the focal conies in general, but may also be proved directly.
The coordinates of the foot of the directrix have been proved
to be ci.'=-T—ni ^' = p /»> the equation of the line joining
this point to a/3 is
/S a ^( 1 _1 \
and the condition that this should be perpendicular to the focal
line ^x = ay Is -3 ^ + WZTr "^ ^' which we have already
seen is satisfied.
In like manner, as a particular case of Art. 150, the section
of a cone by a plane perpendicular to either of its focal lines
is a conic of which the point in the focal line is a focus. The
focal lines of this article are therefore identical with those de-
fined (Art. 125).
152. The focal lines of a cone are perpendicular to the cir-
cular sections of the reciprocal cone (see Art. 125).
For the circular sections of the cone Ja;* + ^/+ (7s"'' = 0,
are (see Art. 103) parallel to the planes
and the corresponding focal lines of the reciprocal cone
«■* v^ ^^ 1 • aj"* 7"
2 + ^ + -^= 0) are, as we have just seen, ^^37^ + ^ _ ^, = 0,
and the lines represented by the latter equation are evidently
perpendicular to the planes represented by the former.
153. The investigation of the foci of the other species of
quadrlcs proceeds In like manner. Thus for the paraboloids
x^ v^
included in the equation -^ -I- -^ = 2^; ; this equatioji can be
written in either of the forms
(a.-«)^+/ + (.-7r=^^(a;-^ay+(.-7 + ^r,
134 FOCI AND CONFOCAL SURFACES.
a*
where A- B = ^'i' ~ ^'
where -^ : =2y — A.
B— A
It thus appears that a paraboloid has two focal parabolas,
which may easilj be seen to be each confocal with the corre-
sponding principal section. The focus belongs to one or other
of the two kinds already discussed, according to the sign of
the fraction (A — B) : A. In the case of the elliptic paraboloid
therefore, where both A and B are positive, if A be the
greater, then the foci in the plane xz are of the class whose
planes of contact are imaginary, while those in the plane yz
are of the opposite class. But since if we change the sign
either of A or of B, the quantity {A— B) : A remains positive, we
see that all the foci of the hyperbolic paraboloid belong to the
former class, a property we have already seen to be true of the
hyperboloid of one sheet.
It remains true that the line joining any focus to the foot
of the corresponding directrix is normal to the focal curve, and
that the foot of the directrix is the pole with regard to the
principal section of the tangent to the focal conic. The feet
of the directrices lie on a parabola, and the directrices them-
selves generate a parabolic cylinder.
To complete the discussion it remains to notice the foci of
the different kinds of cylinders, but it is found without the
slightest difficulty that when the base of the cylinder is an
eUipse or hyperbola there are two focal lines ; namely, lines
drawn through the foci of the base parallel to the generators
of the cylinder ; while, if the base of the cylinder is a parabola,
there is one focal line passing in like manner through the focus
of the base.
154. The geometrical interpretation of the equation U= LM
has been already given. We learn from it this property of foci
whose planes of contact are real, that tlie square of the distance
of any point on a quadric from such a focus is in a constant
FOCI AND CONFOCAL SURFACES. 135
ratio to the product of the perpendiculars let fall from the point
on the quadric^ on two planes drawn through the corresponding
directrix^ p)arallel to the planes of circular section. The corre-
sponding property of foci of tiie other kind, which is less
obvious, was discovered by Professor Mac CuUagh. It is, that
the distance of any point on the quadric from such a focus is in
a constant ratio to its distance from the corresponding directrix^
the latter distance being measured parallel to either of the planes
of circular section.
Suppose, in fact, we try to express the distance of the point
x'y'z' from a directrix parallel to the axis of z and passing
through the point whose x and y are a', /S', the distance being
measured parallel to a directive plane z == mx. Then a parallel
plane through xyz\ viz. z — z' = m [x — x) meets the directrix
in a point whose x and y of course are a', yS', while its z is
given by the equation z —z' =■ m {a! — x). The square of the
distance required is therefore
{x' - a'Y + [y' - /3')' + m" K - o.y = iy' - /Sy + (1 + m') [x' - aj.
In the equation then of Art. 147,
{X - ay +{y- ISf + z^ = p[x- a.J + q[y- ^')\
where p and q are both positive, and p is supposed greater
than 5, the right-hand side denotes q times the square of the
distance of the point on the quadric from the directrix, the
distance being measured parallel to the plane z = mx where
t^-^ = (p — q)''<l- By putting in the values of p and q^ given
in Art. 147, it may be seen that this is a plane of circular
section, but it is evident geometrically that this must be the
case. For consider the section of the quadric by any plane
parallel to the directive plane, and since evidently the distances
of every point in such a section are measured from the same
point on the directrix, the distance therefore of every point in
the section from this fixed point is in a constant ratio to its
distance from the focus. But when the distances of a variable
point from two fixed points have to each other a constant
ratio, the locus is a sphere. The section therefore is the inter-
section of a plane and a sphere ; that is, a circle.
An exception occurs when the distance from the focus is
136 FOCI AND CONFOCAL SURFACES-
tobe equal to the distance from the directrix. Since the locus
of a point equidistant from two fixed points is a plane, it
appears as before, that in this case the sections parallel to the
directive plane are right lines. By referring to the previous
articles, it will be seen (see Art. 153) that the ratio we are
considering is one of equality [q = 1) only in the case of the
hyperbolic paraboloid, a surface which the directive plane could
not meet in circular sections, seeing that it has not got any.
Professor MacCullagh calls the ratio of the focal distance to
that from the directrix, the modulus of the surface, and the foci
having imaginary planes of contact, he calls modular foci,*
155. It was observed (Art. 137) that all quadrics of the
form C7—jLilf are enveloped by two cones, and when ?7 repre-
sents a sphere, these are cones of revolution as every cone
enveloping a sphere must be. Further, when U reduces to a
point-sphere, these cones coincide in a single one, having that
point for its vertex ; and we may therefore infer that the cone
enveloping a quadric and having any focus for its vertex is one
of revolution.
This theorem being of importance, we give a direct alge-
braical proof of it. First, it will be observed, that any equa-
tion of the form oc'' -\- y^ -\- z^ = [ax + hy + czf represents a right
cone. For if the axes be transformed, remaining rectangular,
but so that the plane denoted by ax + hy -\- cz may become one
of the coordinate planes, the equation of the cone will become
X'^ + Y'^ -\- Z'^ = \X'\ which denotes a cone of revolution, since
the coefficients of Y^ and Z'^ are equal.
* In the year 1836 Professor MacCullagh published this modular method of
generation of quadrics. In 1842 I published the supplementary property possessed
by the non-modular foci. Not long after, M. Amyot independently noticed the same
property, but owing to his not being acquainted with Professor Mac Cullagh's method
of generation, M. Amyot failed to obtain the complete theory of the foci. Professor
MacCullagh has published a detailed account of the focal properties of quadrics,
which will be found in the Proceedings of the Royal Irish Academy, vol. ii., p. 446 :
reprinted at p. 2G0 of his Collected Works, Dublin, 1880. Mr. Townsend also has
published a valuable paper {Cambridge and Dublin Mathematical Journal, vol. ill.,
pp. 1, 97, 148) in which the properties of foci, considered as the limits of spheres
having double contact with a quadiic, are very fully investigated.
FOCI AND CONFOCAL SURFACES. 137
But now if we form, by the rule of Art. 78, the equation
of the cone whose vertex is the origin and circumscribing
x' + if + z^- U - M'\ where
L = ax + h7/ + cz-\- d, M= ax + h'y + cz + c?',
it is found to be ,
{d'' + d'') [x' + f + z'- U - IP) + [dL + d'Mf = 0,
or [d'' + d"') {x^ + 2/' + z') - {d'L - dM)' = 0,
which we have seen represents a right cone.
Cor. Since, in reciprocation, the circumscribing cone whose
vertex is the origin corresponds to the asymptotic cone of the
reciprocal surface, it follows from this article, that the reciprocal
of a quadric with regard to any focus is a surface of revolution.
A few additional properties of foci easily deduced from the
principles laid down are left as an exercise to the reader.
Ex. 1. The polar of any directrix is the tangent to the focal conic at the coiTe-
sponding focus.
Ex. 2. The polar plane of any point on a directrix is perpendicular to the line
joining that point to the corresponding focus,
Ex. 3. If a Hne be drawn through a fixed point 0 cutting any dii'ectrix of a quadric,
and meeting the quadric in the points A, B ; then if F be the corresponding focus,
tan J^i^O. tan ^B/"6> is constant. This is proved as the corresponding theorem for
plane conies. Conies, Art. 226, Ex. 8.
Ex. 4. This remains true if tlie point 0 move on any other quadric having the
same focus, directrix, and planes of circular section.
Ex. 5. If two such quadrics be cut by any line passing through the common direc-
trix, the angles subtended at the focus by the intercepts are equal.
Ex. 6. If a line tlu'ough a directrix touch one of the quadrics, the chord intercepted
on the other subtends a constant angle at the focus.
156. The product of the perpendiculars from the two foci
of a surface of revolution round the transverse axis, on any
tangent plane, is evidently constant. Now if we reciprocate
this property with regard to any point by the method used in
Art. 126, we learn that the square of the distance from the
origin of any point on the reciprocal surface is in a constant
ratio to the product of the distances of the point from two
fixed planes.
T
138 FOCI AND CONFOCAL SUEFACES.
It appears from Art. 126, Ex. 5, that the two planes are
planes of circular section of the asymptotic cone to the new
surface, and therefore of the new surface itself. The intersection
of the two planes is the reciprocal of the line joining the two
foci; that is to say, of the axis of the surface of revolution.
The property just proved,* belongs as we know (Art. 154) to
every point on the urabilicar focal conic ; hence the reciprocal of
any quadric with regard to an umbilicar focus, is a surface
of revolution round the tranverse axis ; but with regard to a
modular focus is a surface of revolution round the conjugate
axis.
By reciprocating properties of surfaces of revolution, we
obtain propei'ties of any quadric with regard to focus and
corresponding directrix. It is to be noted, that the axis of the
figure of revolution of either kind is the reciprocal of the
directrix corresponding to the given focus ; and is parallel to
the tangent to the focal conic at the given focus (see Art. 147).
The left-hand column contains properties of surfaces of re-
volution, the right-hand of quadrics in general.
Ex. 1. The tangent cone whose The cone whose vertex is a focus
vertex is any point on the axis is and base any section whose plane
a right cone whose tangent planes passes thi'ough the corresponding
make a constant angle with the directrix, is a right cone, whose axis
plane of contact, which plane is is the line joining the focus to the
perpendicular to the axis. pole of the plane of section, and this
right line is perpendicular to the
plane through focus and directrix.
Ex. 2. Any tangent plane is at The line joining a focus to any
right angles with the plane through point on the surface is at right
the point of contact and the axis. angles to the line joining the focus
to the point where the corresponding
tangent plane meets the directrix.
Ex. 3, The polar plane of any The line joining a focus to any
point is at riglit angles to the plane point is at right angles to the
containing that point and the axis. line joining the focus to the point
where the polar plane meets the
directrix.
* It was in this way I was first led to this property, and to observe the distinction
between the two kinds of foci.
FOCAL CONICS AND CONFOCAL SURFACES. 139
Ex. 4. Any two conjugate lines Any two conjugate lines pierce
are such that the planes joining a plane through a directrix parallel
them to the focus are at right to circular sections, in two points
angles. (Ex. 7, Art. 126). which subtend a right angle at the
corresponding focus.
Ex. 5. If a cone circumscribe a The cone whose base is any plane
surface of revolution, one principal section of a quadric and vertex any
plane is the plane of vertex and focus has for one axis the line join-
axis. - ing focus to the point where the
plane meets the directrix.
Ex. 6. The cone whose vertex The cone is a right cone whose
o*
is a focus and base any plane sec- vertex is a focus and base the sec-
tion is a right cone. (Ex. 2, tion made by any tangent cone on
Art. 126). a plane through the corresponding
directrix parallel to those of the
circular sections.
FOCAL CONICS AND CONFOCAL SURFACES.
157. In the preceding section an account has been given
of the relations which each focus of a quadric considered
separately bears to the surface. We shall in this section give
an account of the properties of the conies which are the as-
semblage of foci, and of the properties of confocal surfaces.
And we commence by pointing out a method by which we
should be led to the consideration of the focal conies of a quadric
independently of the method followed in the last section.
Two concentric and coaxal conies are said to be confocal
when the difference of the squares of the axes is the same for
. x^ ?/
both. Thus given an ellipse — ^ + ^= 1, any conic is confocal
with it whose equation is of the form
2 2
^ y
I 72 . ^2 ■■■•
If we give the positive sign to X\ the confocal conic will be
an ellipse; it will also be an ellipse when X^ is negative as
long as it is less than l)\ When X'' is between If and aj' the
confocal curve is a hyperbola, and when X' is greater than a*
the curve is imaginary. If V = y\ the equation reducing itself
to y'' = 0, the axis of x itself is the limit which separates con-
140 FOCAL CONICS AND CONFOCAL SURFACES.
focal ellipses from hyperbolas. But the two foci belong to
this limit in a special sense. In fact, through a given point
x'y can in general be drawn two conies confocal to a given
one, since we have a quadratic to determine X"'^, viz.
or \* - X"- {a' + }f- x" - if) + a^h' - Vx'' - dY = 0-
When y = 0 this quadratic becomes (V- V) {X'-d' + a;") = 0,
and one of its roots is V = h^ ; but if we have also x'^ = a'"* — b^
the second root is also X^ = h'\ and therefore the two foci are
in a special sense points corresponding to the value X^= J'"*. If
a a 'i
X 2/ V
in the equation ^, _ ^, + p^. = 1, we make X' = h% ^, ^ ^, = 0,
x^
we get the equation of the two foci — — tt^ = 1.
158. Now in like manner two quadrlcs are said to be
confocal if the differences of the squares of the axes be the
a u 2
same for both. Thus given the ellipsoid ^ 4- yj- + — = 1, any
surface is confocal whose equation is of the form
x' y^ z^ _
^^7x' ^ h'±x' "^ ?Tv ~
If we give X^ the positive sign, or if we take it negative and
less than c^, the surface Is an ellipsoid. A sphere of Infinite radius
is the limit of all ellipsoids of the system, being what the equa-
tion represents when X^ = cc . When X^ Is negative and between
c^ and ¥ the surface is a hyperboloid of one sheet. When
it is between b'^ and a^ it is a hyperboloid of two sheets. When
V = c* the surface reduces itself to the plane z = 0, but if we
make in the equation X^ = c^, r-g ^ = 0, the points on the conic
x^ V^
thus found, viz. -^ ^ + r^ — 5^ = 1) belong In a special sense
to the limit separating ellipsoids and hyperbololds. In fact,
in general through any point x'y'z' can be drawn three surfaces
confocal to a given one; for regarding V as the unknown
FOCAL CONICS AND CUNFOCAL SURFACES. 141
quantity, we have evidently a cubic for the determination of
it ; namely,
or x" {b' - X') [6' - X') + /' {c' - X') {a' - V) + z" {a'- V) {b'- X')
= {a:'-X'){b'-X'){d'-X').
If z' = 0, one of the roots of this cubic is X^ = c^, the other two
being given by the equation
x" {¥ - V) + y" {a' - X') = {a' - V) {b' - X')^
and a root of this equation will also be V = c^, if
-J^ + JL- =1
a —c b — c
The points on the focal ellipse therefore belong in a special
sense to the value X^ = c\ In like manner the plane ^ = 0
separates hyperboloids of one sheet from those of two, and to
this limit belongs in a special sense the hyperbola in that
x'' z^
plane —. — tt, + -r — 77, = I. The focal conic in the third principal
^ a — b c — b
plane is imaginary.
159. The three quadrics which can be drawn through a given
point confocal to a given one are respectively an ellipsoid^ a
hyperboloid of one sheet^ and one of two. For if we substitute
in the cubic of the last article successively
X' = a^^ X' = V\ X' = c\ \' = -co,
we get results successively 4- — I — , which prove that the equa-
tion has always three real roots, one of which is less than c^,
the second between 6^ and 6^, and the third between h^ and a'' ;
and It was shown in the last article that the surfaces corre-
sponding to these values of X^ are respectively an ellipsoid, a
hyperboloid of one sheet, and one of two.
160. Another convenient way of solving the problem to
describe through a given point quadrics confocal to a given
one, is to take for the unknown quantity the primary axis
of the sought confocal surface. Then since we are given
142 FOCAL CONICS AND CONFOCAL SUEFACES.
a'* — Z*'* and a^ - c'' which we shall call 1^ and U\ we have the
equation
or a'« - a'* (A'' + U' + a;''' + y'' + s'^)
+ «"= [WW + ^''^ (/i''' + le) + ^'''A;^ + ^'^A^} - :x^''h^h^ = 0.
From this equation we can at once express the coordinates
of the Intersection of three confocal surfaces In terms of their
axes. Thus if a'^, a''\ ci"'^ be the roots of the above equation,
the last term of it gives us at once x"^Wh^ = a'V'V'^, or
a; =
And bj parity of reasoning, since we might have taken W or &
for our unknown, we have
N.B. In the above we suppose V\ U''\ &c., to Involve their
signs implicitly. Thus c'"^ belonging to a hyperboloid of one
sheet is essentially negative, as are also h"'' and c
rm
161. The preceding cubic also enables us to express the
radius vector to the point of intersection in terms of the axes.
For the second term of It gives us
ic'^ + if + z"' + {d' - V') + (a'-' - c^j - a' + a'"' + a'"',
or x'-'-^y'^^z-' = o:''^-h"^^d"\
This expression might also have been worked out directly from
the values given for x'^, ?/'^, z''' In the last article, by a process
which may be employed in reducing other symmetrical functions
of these coordinates. For on substituting the preceding values
and reducing to a common denominator, x"'' -Vy'"^ + ^'^ becomes
c^'ol'-'a:"' [h' - c') + b%'''b"'' jc' - gp + cW'"' (g'^ - b')
[h' - c') {a' - 6') [d' - b'')
* These expressions enable us easily to remember the coordinates of the umbilics.
The umbilics are the points (Art. 149) where e.g. an ellipsoid is met by its focal
hyperbola. But for the focal hyperbola a""- a'""- «-— b". The coordinates are therefore
a^ — (p-^ •> ' a?— c^
FOCAL CONICS AND CONFOCAL SURFACES. 143
But the numerator obviously vanishes if we suppose either
h^ = c\ (? = d\ a' = V. It is therefore divisible by the de-
nominator. The division then is performed as follows: Any
term, for example a'VV'V, when divided by a' -V' (or by
its equal a'* - V'^) gives a quotient a''^a"'^c\ and a remainder
&' VV'V. This remainder divided by a'"^ - V^ gives a quotient
Va"'^G^ and a remainder J'^'yVV, which divided in like manner
by a'"' - 1""' gives a quotient 5"'5"V and a remainder V%"W\\
which is destroyed by another term in the dividend. Proceeding
step by step in this manner we get the result already obtained.
162. Tioo confocal surfaces cut each other everywhere at
right angles.
Let cc'?/V be any point common to the two surfaces, p' and^"
the lengths of the perpendiculars from the centre on the tangent
plane to each at that point, then (Art. 89) the direction-cosines
of these two perpendiculars are
j)x py 'p z p X p y p z
And the condition that the two should be at right angles,
is, (Art. 13)
^ ^' \a"a"' "^ h"'h'" "^ c'V'^J ~
But since the coordinates x'y'z' satisfy the equations of both
surfaces we have
f'i /2 /2 /H '2 ''i
X y ^ _ 1 ^ y ^_i
a o c 'a o c
And if we subtract one of these equations from the other,
and remember that a"'^ — a'^ = h"'^ — l"^ = c"^ — c'\ the remainder is
{'2 '2 /2 \
7V^^ + J'2^. + ^7^2| = y,
which was to be proved.
At the point therefore where three confocals intersect, each
tangent plane cuts the other two perpendicularly, and the
tangent plane to any one contains the normals to the other two.
163. If a plane he drawn through the centre parallel to any
tangent plane to a quadriCj the axes of tlie section made hy that
144
FOCAL CONICS AND CONFOCAL SURFACES.
plane are parallel to the normals to the two confocals through
the point of contact.
It has been proved that the parallels to the normals are at
right angles to each other, and it only remains to be proved
that they are conjugate diameters in their section. But (Art. 94)
the condition that two lines should be conjugate diameters is
cos a cos a' cos /3 cos /3' cos 7 cos 7'
a"' ^ h"' ^ e
The direction-cosines then of the normals being
p X p y p z p X p y p z
we have to prove that
p p
X
[a a a
+
y
'2
n-itrtrtrr-i
h%"'b
-f
c"'c'"'c"'''\
= 0.
But the truth of this equation appears at once on subtracting
one from the other the equations which have been proved in
the last article,
-r'* ■?/''■' z'^ x''^ iP z''^
'•I //a
a a
c c
164. To find the lengths of the axes of the central section of a
quadric hy a plane parallel to the tangent plane at the point xyz.
From the equation of the surface the length of a central
radius vector whose direction-angles are a, /3, 7 is given by
the equation
cos^a
cos^/3 cos%
_) — -I -'
p' a " h'
Put for a, /3, 7 the values given in the last article, and we find
for the length of one of these axes,
= V
x
[a a
fl /2
11 z
4 ~ 1-
b'%'
c c
Now we have the equations,
x"' _y^ z''
a'\r "^ h"b'"' "*" c'c"-' ~ '
j'i
j'i
„'2
£c' y ^' _ 1
FOCAL CONICS AND CONFOCAL SURFACES. 145
Subtracting we have
/2 '2 'Z ■«
X y z 1
4 ~ — 4- —
And substituting this value in the expression already found
for p'' we get p'^ = a'' — a"^. In like manner the square of the
other axis is a"^ — a'"^.
Hence, if two confocal quadrics intersect, and a radius of
one be drawn parallel to the normal to the other at any point
of their curve of intersection, this radius is of constant length.
165. Since the product of the axes of a central section by
the perpendicular on a parallel tangent plane is equal to ahc
(Art. 96), we get immediately expressions for the lengths
Pi Pi P" • ^^ have
P I ''!■ ^,"'i\ I y,'^ n"''i\ 1 P
[a — a )[a —a ) ^ [a ~ a ] [a — a )
V//2
a''"'b"'V'
P / •'//2 /2\ / /'/2 //2\ •
■^ [a - a ) [a —a ]
These values might have been also obtained by substituting
in the equation
« -t fZ 'a f'i
I _x y z
pa 0 C '
the values already found for cc'^, ?/"^, ^'* and reducing the re-
sulting value forp''^ by the method of Art. 161.
The reader will observe the symmetry which exists between
these values for p'^, p''^, p''^'\ and the values already found for
x'^, y'\ z"^. If the three tangent planes had been taken as
coordinate planes, p\ p\ p'" would be the coordinates of the
centre of the surface. The analogy then between the values for
P' 1 v" 1 v"i ^"^^ those for x\ y\ /, may be stated as follows : With
the point x'y'z' as centre three confocal s may be described
having the three tangent planes for principal planes and inter-
secting in the centre of the original system of surfaces. The
axes of the new system of confocals are a', a", a'"; J', V\ &"';
c', c", c"\ The three tangent planes to the new system are the
three principal planes of the original system.
U
146 FOCAL CONICS AND CONFOCAL SURFACES.
If a central section tbrough x'y'z be parallel to one of these
principal planes (the plane of yz for instance) in the surface to
which this latter is a tangent plane, it appears from Art. 164
that the squares of its axes are d^ — V^ d^ — c\ It follows then
that the directions and magnitudes of the axes of the section are
the same, no matter where the point x'yz' be situated. The
squares of the axes are equal, with signs changed, to the squares
of the axes of the corresponding focal conic.
166. If D be the diameter of a quadric parallel to the
tangent line at any point of its intersection with a confocal,
and p the perpendicular on the tangent plane at that point,
then pD is constant for every point on that curve of intersec-
tion. For the tangent line at any point of the curve of inter-
section of two surfaces is the intersection of their tangent planes
at that point, which in this case (Art. 162) is normal to the third
confocal through the point. Hence (Art. 164) D'' = a^ — d"'\
and therefore (Art. 165) p^D'^ = —^, »„ which is constant if
a — a ^
a', a' be given.
167. To find the locus of the pole of a given plane with regard
to a system of confocal surfaces.
Let the given plane be Ax-]- By + Cz=\^ and its pole l^?^;
then we mast identify the given equation with
d'-X' U'-X' c'^-V
Whence ^,-_ ^,=A, ^, =^.73^^= C'.
Eliminating V between these equations we find, for the equa-
tions of the locus,
■^ 2 y -I'i. ^ vj
The locus is therefore a right line perpendicular to the given
plane.
The theorem just proved implicitly contains the solution of
the problem, " to describe a surface confocal to a given one to
FOCAL CONICS AND CONFOCAL SURFACES. 147
touch a given plane." For, since the pole of a tangent plane
to a surface is its point of contact, it is evident that but one
surface can be described to touch the given plane, its point of
contact being the point where the locus line just determined
meets the plane. The theorem of this article may also be
stated — " The locus of the pole of a tangent plane to any
quadric, with regard to any coufocal, is the normal to the first
surface."
168. To find an expression for the distance heticeen the 'point
of contact of any tangent p)lane^ and its pole with regard to any
confocal surface.
Let xy'z' be the point of contact of a tangent plane to the
surface whose axes are a, J, c ; ^rjt, the pole of the same
plane with regard to the surface whose axes are a', h\ c.
Then, as in the last article, we have
?!=-L yL-1^ ^L-^
a" a" ' b' h" ' e c^ '
whence g - a; = ^, a; , 'n-y= — -^ y ^ ^-z= 5— g ,
squaring and adding
whence D = , where p is the perpendicular from the centre
on the plane.
169. The axes of any tangent cone to a quadric are the
normals to the three confocals which can he drawn through the
vertex of the cone.
Consider the tangent plane to one of these three surfaces
which pass through the vertex xy'z'^ then the pole of that
plane with regard to the original surface lies (Art. 65) on the
polar plane of xyz\ and (Art. 167) on the normal to the ex-
terior surface. It is therefore the point where that normal
meets the polar plane of x'yz\ that is to say, the plane of
contact of the cone.
It follows, then (Art. 64), that the three normals meet
148 FOCAL CONICS AND CONFOCAL SURFACES.
this plane of contact in three points, such that each is the
pole of the line joining the other two with respect to the
section of the surface by that plane. But since this is also
a section of the cone, it follows (Art. 71) that the three normals
are a system of conjugate diameters of the cone, and since they
are mutually at right angles they are its axes.
170. If at any point on a quadric a line be drawn touching
the surface and through that line two tangent planes to any
confocal, these two planes will make equal angles with the
tangent plane at the given point on the first quadric. For, by
the last article, that tangent plane is a principal plane of the
cone touching the confocal surface and having the given point
for its vertex, and the two tangent planes will be tangent
planes of that cone. But two tangent planes to any cone
drawn through a line in a principal plane make equal angles
with that plane.
The focal cones (that is to say, the cones whose vertices are
any points and which stand on the focal conies) are limiting
cases of cones enveloping confocal surfaces, and it is still true
that the two tangent planes to a focal cone drawn through any
tangent line on a surface make equal angles with the tangent
plane in which that tangent line lies. If the surface be a cone
its focal conic reduces to two right lines, and the theorem just
stated in this case becomes, that any tangent plane to a cone
makes equal angles with the planes containing its edge of
contact and each of the focal lines. This theorem, however,
will be proved independently in Chap. x.
171. It follows, from Art. 169, that if the three normals be
made the axes of coordinates, the equation of the cone must
take the form Ax' + Bif + Cz' = 0. To verify this by actual
transformation will give us an independent proof of the theorem
of Art. 169, and a knowledge of the actual values of A^ B^ G
will be useful to us afterwards.
The equation of the tangent cone given, Art. 78, is
fx'^ y'^ z'^ \(x^ f z^ \ (XX yy' zz' \»
FOCAL CONICS AND CONFOCAL SURFACES. 149
If the axes be transformed to parallel axes passing through the
vertex of the cone, this equation becomes, as is easily seen,
Now to transform to the three normals as axes, we have to
substitute the direction-cosines of these lines in the formulae
of Art. 17, and we see that we have to substitute
„ p X p X p X
for X^^X + ^ y + ^-TTTT ^)
(J, (J/ (Ai
for y, ^ « -f ^ 2/ + ^ ^.
o o o
172. In order more easily to see the result of this substitu-
tion the following preliminary formulse will be useful :
f'i /n /a
^' ^ + 1^ + 7-1 = ^.*
then smce ^"^F^"^^"'
we have -^-7-, + -^, +
In hke manner ^y^. + ^. + ^ = ^tt^Z^ ,
1 1 ^" y" «" S
x''^ y'^ s'^ 1
Lastly, since — + |^ + _ = _ ,
, cb''' y'' z'' S
and + ^ +
, a;'^ y'-^ ^"^ >Sf
we have + ^ 4.
a'V ' Z^'V ' cV (a^'-'-a^^ p"'' (a'^* - a^) *
* It may be observed that this quantity S is equal to
(ft'- - g^) (a"2 - fl^) («"'2 - g'')
g-6V '
for a2 - a'\ a? ~ a"% a?- - a'""- are the roots of the cubic of Art. 158, whoso absolute
term is a^6V<S.
150 FOCAL CONICS AND CONFOCAL SUEFACES.
173. When now we make the transformation du'ected, in
the left-hand side of the equation of Art. 171, the coefficient
of a;'^ is found to be
X^ ] ^/*^2 ~ //■*A''2 '
[a a 0 0 c c )
and that of xi/ is
The left-hand side therefore of the transformed equation Is
\a —a a —a a —a J \a —a a —a a —a)
But the quantity -^ + -rir + —^ treated in like manner becomes
a 0 c
Its square therefore destroys the first group of terms on the
other side of the equation, and the equation of the cone becomes
x^ if ^
— I ^ 1 = 0
/2 'l ' rfi, 2 ' ■'//2 2 ")
a —a a —a a —a
which is the required transformed equation of the tangent cone.
174. As a particular case of the preceding may be found
the equation of either focal cone (Art. 170); that is to say, the
cone whose vertex is any point cc'?/V and which stands on the
focal ellipse or focal hyperbola. These answer to the values
c^ — c^, c^ — Z>'^ for the square of the primary axis : the equa-
tions therefore are
+
Z^
y; + J./2
+
z'
7///2 ^'
These equations might also have been found, by forming, as In
Ex. 7, Art. 121, the equations of the focal cones, and then
transforming them as in the last articles.
It may be seen without difficulty that any normal and the
corresponding tangent plane meet any of the principal planes
FOCAL CONICS AND CONFOCAL SURFACES. 151
in a point and line which are pole and polar with regard to
the focal conic in that plane. This is a particular case of
Art. 169.
The formulas employed in the articles immediately preced-
ing enable us to transform to the same new axes any other
equations.
Ex, 1. To transform the equation of the quadric itself to the three normals through
any point x'y'z' as axes. The equation transformed to parallel axes becomes
And when the axes are turned round, we get
22
i^ b- c- \a- b- c^ J
U'- -a? a"2 - a2 a'"^ - a^ ) a'^ -a"'^ a"- - a^ a"
The quantity under the brackets on the left-hand side of the equation is evidently the
transformed equation of the polar plane of the point.
Ex. 2. The preceding equation is somewhat modified if the point x'l/'z' is on the
surface. The equation transformed to parallel axes is
g.2 „2 1.2 /^j.' yy> ^Z'X .
-, + f5 + -5 + 2 Hr + <';- + -;r =0.
62 c2 ' V«- 62
then the equation, transformed to the three normals as axes, is
^2 y'^ z^ '2p'xy 'ip"xz 2a; _
^ "^ o^^^o^ "'' ^^^"^"2 ~> (a2 - a'2) ~ p[aP- - a"-') '^ ^ ~
It is to be observed that y is the diameter parallel to the normal at the point x'y'z',
and that we have
and the transformed equation may be otherwise written
{p'x-pyf {p"x~pz)
■ - «'2
+ (X + p)2 = p2.
a^- a'^ a- — a ^^
Ex. 3. To transform the equation of the reciprocal surface with regard to any point
to the three normals through the point. The equation is (Art. 127)
{xx' + yy' + zz! + Ti^f = a-x^ + h^' + cV,
and the transformed equation is found to be
(a'2 - ciF) X- + (a"2 - a^) f- + (a'"* - a^) z^ + 24^ (^j'x + js"?/ +i)"'«) + /t* = 0.
175. To return to the equation of the tangent cone (Art. 173).
Its form proves that all cones having a common vertex and cir-
cumscribing a series of coufocal surfaces are coaxal and confocal.
For the three normals through the common vertex are axes to
every one of the system of cones ; and the form of the equation
shows that the differences of the squares of the axes are iude-
152 FOCAL CONICS AND CONFOCAL SURFACES.
pendent of a^ The equations of the common focal lines of the
cones are (Art. 151)
^/2 ^//a „//2 „///!! J y — "•
a —a a — a
But it was proved (Art. 164) that the central section of the
hjperboloid of one sheet which passes through x''y'z' is
a i2
X Z
/2 "T //2 ,^"''i >
rn . . . . «
a — a a —a
and the section of the hjperboloid by the tangent plane itself is
similar to this, or is also
^ .=0
/•2 //2 "2. 'rr-i
a — a a —a
Hence the/ocaZ lines of the system of cones are the generating
lines of the hijperholoid which passes through the point — a theorem
due to Chasles, Liouville, xi. 121, and also noticed by Jacobi
{Crelle, Yol. Xll. p. 137).
This may also be proved thus : Take any edge of one of the
system of cones, and through it draw a tangent plane to that
cone and also planes containing the generating lines of the
hyperbolold ; these latter planes are tangent planes to the hyper-
boloid, and therefore (Art. 170) make equal angles with the
tangent plane to the cone. The two generators are therefore
such that the planes drawn through them and through any
edge of the cone make equal angles with the tangent plane to
the cone ; but this is a property of the focal lines (Art. 1 70).
Cor. 1. The reciprocals of a system of confocals, with
regard to any point, have the same planes of circular section.
For the reciprocals of the tangent cones from that point have
the same planes of circular section (Art. 152), and these reci-
procals are the asymptotic cones of the reciprocal surfaces.
Cor. 2. If a system of confocals be projected orthogonally
on any plane, the projections are confocal conies. The pro-
jections are the sections by that plane of cylinders perpendicular
to it, and enveloping the quadrics. And these cylinders may
be considered as a system of enveloping cones whose vertex
is the point at infinity on the common direction of their
generators.
FOCAL CONICS AND CONFOCAL SURFACES. 153
176. Two confocal surfaces can he drawn to touch a given line.
Take on the line any point x'yz' ; let the axes of the three
surfaces passing through it be a\ a", a"\ and the angles the
line makes with the three normals a, y3, 7. Then it appears,
from Art. 173, that a is determined by the quadratic
cos'^'a cos^/3 cosV
a — a a — a a —a
If a and a' be the roots of this quadratic, the two cones
'22 a 2 2 2
X y ^ _ A ^ y ^ _ A
a^ — Q.' a '—Si' a:"—dir ' a'-ai' a '—a" a " — a
have the given line as a common edge, and it is proved, pre-
cisely as at Art. 162, that the tangent planes to the cones
through this line are at right angles to each other. And since
the tangent planes to a tangent cone to a surface, by definition
touch that surface, it follows that the tangent planes drawn
through any right line to the two con/ocals which it touches are
at right angles to each other.
The property that the tangent cones from any point to
two intersecting confocals cut each other at right angles is
sometimes expressed as follows : two confocals seen from any
point a'pjpear to intersect everywhere at right angles.
Vll. If through a given line tangent planes he drawn to a
system of confocals^ the corresponding normals generate a hyper-
holic paraboloid.
The normals are evidently parallel to one plane ; namely,
the plane perpendicular to the given line ; and if we consider
any one of the confocals, then, by Art. 167, the normal to any
plane through the line contains the pole of that plane with
regard to the assumed confocal, which pole is a point on the
polar line of the given line with regard to that confocal. Hence,
every normal meets the polar line of the given line with regard
to any confocal. The surface generated by the normals is
therefore a hyperbolic paraboloid (Art 116). It is evident that
the surface generated by the polar lines, just referred to, is
the same paraboloid, of which they form the other system of
generators.
X
154 FOCAL. CONICS AND CONFOCAL SURFACES.
The points In which this paraboloid meets the given lln
are the two points where this line touches confocals.
A special case occurs when the given line is Itself a norma
to a surface U of the system. The normal corresponding t
any plane drawn through that line is found by letting fall ;
perpendicular on that plane from the pole of the same plan
with regard to U (Art. 167), but it is evident that both pol
and perpendicular must lie In the tangent plane to U to whic'
the given line Is normal. Hence, in this case all the normal
lie in the same plane.
From the principle that the anharmonic ratio of four plane
passing through a line is the same as that of their four poles wit]
regard to any quadric, It Is found at once that any four of th
normals divide horaographlcally all the polar lines correspond
ing to the given line with respect to the system of surfaces. Ii
the special case now under consideration, the normals wil
therefore envelope a conic, which conic will be a parabola, sine
the normal in one of Its positions may lie at Infinity 5 namelj
when the surface is an Infinite sphere (Art. 158). The poin
where the given line meets the surface to which it Is norma
lies on the directrix of this parabola.
178. If a, /9, 7 be the direction-angles, referred to the thre
normals through the vertex, of the perpendicular to a tangen
plane of the cone of Arts. 171, &c., since this perpendicular lie
on the reciprocal cone, a, /3, 7 must satisfy the relation
[a"' - d') cos'a + {a'"' - a') cos'yS + [a"" - d') cos^ = 0,
or a'^ cos'ot + 0!'^ cos'jS + a"' cos'^7 = d\
This relation enables us at once to determine the axis of th
surface which touches any plane, for if we take any point 01
the plane, we know a', a", a" for that point, as also the angle
which the three normals through the point make with the plane
and therefore o^ is known.
179. If the relation of the last article were proved Inde
pendently, we should, by reversing the steps of the demon
stratiou, obtain a proof without transformation of coordinate
FOCAL CONICS AND CONFOCAL SURFACES. 155
of the equation of the tangent cone (Art. 173). The following
proof is due to M. Chasles : The quantity
d' cos'a + a' co3"^/3 + a'"' cos^
is the sum of the squares of the projections on a perpen-
dicular to the given plane of the lines a', a", a" . We have
seen [Kx\. 165) that these are the axes of a surface having
xy'z' for its centre and passing through the original centre.
And it was proved in the same article that three other con-
jugate diameters of the same surface are the radius vector
from the centre to xyz\ together with two lines parallel to
two axes of the surface and whose squares are a* — J', a^ — c*.
It was also proved (Art. 98) that the "sum of the squares of
the projections on any line of three conjugate diameters of a
quadric is equal to that of any other three conjugate diameters.
It follows then that the quantity
a" cos'a + a'"" cos'/S + a'"'' cos'V
is equal to the sum of the squares of the projections on the
perpendicular from the centre on the given plane, of the radius
vector, and of two lines whose magnitude aud direction are
known. The projections of the last two lines are constant,
while the projection of the radius vector is the perpendicular
itself which is constant if xy'z belong to the given plane.
It is proved then that the quantity
a" cos'^a + a"' co3'/3 + a"" cos"'7
is constant while the point xy'z moves in a given plane ; and
it is evident that the constant value is the of' of the surface
which touches the given plane, since for it we have
cosa = l, cos;S = 0, cos7 = 0.
180. The locus of the intersection of three planes mutually at
right angles^ each of which touches one of three confocals is a sphere.
This is proved as in Art. 93.
Add together
p^ = a^ cos'^a + y^ cos'^/3 + c^ cos^,
f = a"' cos'^a' + h" cos'/3' + c" cosV,
p"' = a'" cos'-'a" -f b'" cos'^/3" + c'" cosV,
when we get p^ = d' + Z>' + c' -f {ct!' - d') -f [a"'' - d'),
156 FOCAL CONICS AND CONFOOAL SURFACES.
where p is the distance from the centre of the intersection of
the planes.
Again, bj subtracting one from the other, the two equations
^'=a'''cos"'a+S'^cos'''/3 + c'''cosV, p'''=a'^ cos'a+¥''cos'^ + c'^ cos^,
we learn that the difference of the squares of the perpendiculars
on two parallel tangent planes to two confocals is constant and
equal a" - a'^.
It may be remarked that the reciprocal of the theorem of
Art. 93 is that if from any point 0 there be drawn three radii
vectores to a quadric, mutually at right angles, the plane joining
their extremities envelopes a surface of revolution. If 0 be on
the quadric, the plane passes through a fixed point.
181. Two cones having a common vertex envelope two con-
focals /[ to find the length of the intercept made on one of their
common edges hy a plane through the centre parallel to the tangent
plan6 to a confocal through the vertex. The intercepts made
on the four common edges are of course all equal, since the
edges are equally inclined to the plane of section which is
parallel to a common principal plane of both cones.
Let there be any two confocal cones
then for their intersection, we have
• ^a /./'• «*
X y z
and if the common value of these be X'', we have
cc' + / + ^' = V {a' - ^') {IS' - 7") (a^ - 7').
Putting in the values of a", /3', 7' from the equations of the
tangent cones (Art. 176), and determining V by the equation
(see Art. 165) x''= ,—r, ,,., . ,, _ ,,,2. , we get for the square
of the required intercept
a''b"c''
(a'*-a'^)(a'*-a'^)*
FOCAL CONICS AND CONFOCAL SURFACES. 157
If then the confocals be all of different kinds this value shews
that the intercept is equal to the perpendicular from the centre
on the tangent plane at their intersection.
In the particular case where the two cones considered are
the cones standing on the focal ellipse, and on the focal hyper-
bola, we have a* = d^ — c\ s!^ = a' — F, and the intercept reduces
to a. Hence, if through any point on an ellipsoid he drawn
a chord meeting both focal conies, the intercept on this chord hy
a plane through the centre parallel to the tangent plane at the
point will be equal to the serni-axis-major of the surface. This
theorem, due to Prof. MacCullagh, is analagous to the theorem
for plane curves, that a line through the centre parallel to a
tangent to an ellipse cuts off on the focal radii portions equal
to the semi-axis-major.
182. M. Chasles has used the principles just established to
solve the problem to determine the magnitude and direction of
the axes of a central quadric being given a system of three
conjugate diameters.
Consider first the plane of any two of the conjugate dia-
meters, and we can by plane geometry determine in magnitude
and direction the axes of the section by that plane. The
tangent plane at P, the extremity of the remaining diameter,
will be parallel to the same plane. Now the centre of the
given quadric is the point of intersection of three confocals
determined as in Art. 165, having the point P for their
centre. If now we could construct the focal conies of this new
system of confocals, then the two focal cones, whose common
vertex is the centre of the original quadric, determine by their
mutual intersection four right lines. The six planes containing
these four right lines intersect two by two in the directions of
the required axes, while (Art. 181) planes through the point
P parallel to the principal planes, cut off on these four lines
parts equal in length to the axes.
The focal conies required are immediately constructed. We
know the planes in which they lie and the directions of their axes.
The squares of their semi-axes are to be a^—a"'\ a'^-a"'^ 5 a^—d"^^
^1 _ ^n^ jg^j^ ^^^ ^^ squares of the semi-axes of the given
158 FOCAL CONICS AND CONFOCAL SUKFACE8.
section are (i—(i'\ d^ — a""^ (Art. 164), and these latter axes
being known, the axes of the focal conies are immediately found.
183. If through any point P on a quadric a chord be
drawn, as in Art. 181, touching two confocals, we can find
an expression for the length of that chord. Draw a parallel
semi-diameter through the centre, the length of which we shall
call U. Now if through P there be drawn a plane conjugate
to this diameter, and a tangent plane, thej will intercept
(counting from the centre) portions on the diameter whose
product = 1^. But the portion intercepted bj the conjugate
plane is half the chord required, and the portion intercepted
by the tangent plane is the intercept found (Art. 181). Hence
2PV{(a^''-a'0(a^'-a^^)}
at) c
When the chord Is that which meets the two focal conies ;
a'^ = a'' - c'% a'^ = oT' - l"% and C = ^^^'
a
184. To find the locus of the vertices of right cones which
can envelope a given surface.
x^ y^ z^
In order that the equation —p. -., + -~- — 5 + -7775 — —a = 0
^ a — a a — a a —a
may represent a light cone, two of the coefficients must be
equal ; that is to say, ol' = a', or ci' = a" ^ or in other words,
for the point x'y'z' the equation of Art. 158 must have two
equal roots, but from what was proved as to the limits within
which the roots lie, it is evident that we cannot have equal
roots except when X is equal to one of the principal semi-axes,
or when xy'z' is on one of the focal conies. This agrees with
what was proved (Art. 155).
It appears, hence, as has been already remarked, that the
reciprocal of a surface, with regard to a point on a focal conic,
is a surface of revolution ; and that the reciprocal, with regard
to an umbilic, is a paraboloid of revolution. For an umbilic
is a point on a focal conic (Art. 149), and since it is on the
surface the reciprocal with regard to it is a paraboloid.
Another particular case of this theorem is, that two right
cylinders can be circurascriLed to a central quadric, the edges
FOCAL CONICS AND CONFOCAL SURFACES. 159
of the cylinders being parallel to the asymptotes of the focal
hyperbola. For a cone whose vertex is at infinity is a cylinder.
As a particular case of the theorem of this article, the cone
standing on the focal ellipse will be a right cone only when
its vertex is on the focal hyperbola, and vice versa. This
theorem of course may be stated without any reference to the
quadrics of which the two conies are focal conies ; that the
locus of the vertices of right cones which stand on a given conic
is a conic of opposite species in a perpendicular plane. If the
equation of one conic be — + j:^ = 1, that of the other will
2 2
, a; _ i - 1
a^-lf b'~
It was proved (Ex. 8, Art. 126) that if a quadric circumscribe
a surface of revolution, the cone enveloping the former whose
vertex is a focus of the latter is of revolution. From this
article then we see that the focal conies of a quadric are the
locus of the foci of all possible surfaces of revolution which
can circumscribe that quadric.
185. It appears from what has been already said that the
focal ellipse and hyperbola are in planes at right angles to each
other, and such that the vertices of each coincide with the foci
of the other. Two conies so related are each (so to speak) a
locus of foci of the other ; viz. any pair of fixed points F, G on
the one conic may be regarded as foci of the other, the sum or
difference of the distances FP^ GP to a variable point P on the
other, being constant.
Taking the equations of the conies
x^ y' _ x' _ ^ _ ,
and introducing the parameters ^, 0, as at Conies, Arts. 229, 232,
the coordinates of a point on each conic may be expressed,
a cos 6, b sin 6, 0 ; sec (j) \J[d^ — W), 0, h tan <j> j
and the square of the distance between these points is
a'cos'^-2acos^sec^ \/{d'-¥)-\- {d'-h^) sec^0-f ^'•'sin'''(9-f J^tan"'^,
or «■■' sec^0 — 2a cos 6 sec^ \l[a^ — V^) + (a^ - h^) cos^O
= {a i:ec(j> — cos 6 V(«" — h'^)Y.
160 FOCAL CONICS AND CON FOCAL SURFACES.
And, plainly, the sum or difference of two distances
+ {asec0 — cos^ \l[o^— ^^)|) + {a sec (^ — cos ^' V(«^ — ^^)}
is independent of 0 ; and of two distances
+ {asec^ — cos^ V(«''- &^)}} ± (asec^' — cos^ \/(«'^- ^^)}
is independent of 6.
Attending to the signs the theorem is this, that if we take
two fixed points F^ O on the ellipse, the difference FP— GP is
constant, being = + a when P is a point on one branch of the
hyperbola, and — a when P is on the other. In particular, when
F^ O are the vertices of the ellipse we have the ordinary focal
property of the hyperbola. Again, taking F^ G two points on
different branches of the hyperbola, the sum FP-\- GP is con-
stant, and when P, G are the vertices of the hyperbola we have
the ordinary focal property of the ellipse. If F^ G be taken
instead on the same branch of the hyperbola, it is the difference
between FP and GP which is constant ; and if i^ and G coincide
at a vertex, we have merely the Identity FP— FP= 0, and not
a new property of the ellipse in piano.
186. The following examples will serve further to illustrate
the principles which have been laid down :
Ex. 1 . To find the locus of the intersection of generators to a hyperboloid which
cut at right angles.
The section parallel to the tangent plane -which contains the generators must
be an equilateral hyperbola, so that (Art. 164) {a'"^ -a'") + {a"--a""^)-0. But
(Art. 161) the square of the radius vector to the point is
«"2 + 5"2 4. c"2 _ (a"2 _ a'2) _ („"2 _ «"'2)_
We have, therefore, the locus a sphere, the square of whose radius is equal to
a"2 + 6"2+c"2. Othervnse thus: If two generators are at right angles, their plane
together with the plane of each and of the normal at the point, are a system of three
tangent planes to the surface, mutually at right angles, whose intersection lies on the
sphere r' = a"* + 6"2+ c"^ (Art. 93).
Ex. 2. To find the locus of the intersection of three tangent lines to a quadric
mutually at right angles (see Ex. 6, Art. 121).
Let a, (i, y be the angles made by one of these tangents with the normals through
the locus point, and since each of these tangents lies in the tangent cone through
that point, we have the conditions
cos^a cos^fl cos^v
■ 1 — 4- ■ — = 0
„'2 _ „2 ^ a"2 _ a2 ^ a"'2 - a' '
cos^ a' cos^ fi' cos- y' _ f.
008^ a" cos' 13" cos' y"
FOCAL CONICS AND CONPOCAL SURFACES. 161
Adding, we have ~, + —. r, + -n^ i = 0.
But a-~a'\ a^-a'"^, a^-a""^ are the three roots of the cubic of Art. 158 which
arranged in terms of X* is
X« + X< {3? + y2 + z' -a?-Vi- c^) _ \^ {(J^ + c^) x' + (c« + a^) / + {a? + V^) 2*
- SV - c^a? - a2J2} + JVx^ + c^a^ + ft^fiijj _ g_Wc^ = 0.
And the sum of the reciprocals of the roots will vanish when the coefficient of X'* = 0.
This, therefore, gives us the equation of the locus required.
Ex. 3. The section of an ellipsoid by the tangent plane to the asymptotic cone
of a confocal hyperboloid is of constant area.
The area (Art. 96) is inversely proportional to the perpendicular on a parallel
tangent plane, and we have
p2 = a^ cos' a + J2 cos'/3 + c' cos'y.
But since the perpendicular is an edge of the cone reciprocal to the asymptotic cone
of the hyperboloid, we have
0 = a" cos'a + J'2 cos'/? + c'2 coa'y,
whence p^ = a^ — a"^.
Ex. 4. To find the length of the perpendicular from the centre on the polar plane
of x'y't' in terms of the axes of the confocals which pass through that point.
Ant. K a'« - a' = h\ o"' - a' = A-', a'"' - a' = P,
1 AU-'Z' n
b2 a'6V
/i 1 1 i 1 u
W "^ A^ "^ c« "^ A« "^ A« "^ p]'
187. Two points, one on each of two confocal ellipsoids,
are said to correspond if
x_X y _Y z _Z
a~ A' b~ B' c~ G'
It is evident that the intersection of two confocal hyper*
boloids pierces a system of ellipsoids in corresponding points;
for from the value (Art. 160) x'^ = --~i — jii — 5 j: j t^© quantity
^, [a — o j [a — c)
— is constant as long as the hyperboloids, having a"^ ol'' for
Cv
axes, are constant.
It will be observed that, the principal planes being limits
of confocal surfaces, points on the principal planes determined
x'^ X^ y'^ Y^
by equations of the form — j = -^ j, ^^ =„ j, Z=Oj
correspond to any point x'y'z' on a surface, and when x'y'z' is
in the principal plane, the corresponding point is on the focal
conic.
Y
162 FOCAL CONICS AXD CONFOCAL SURFACES.
188. The points on the plane of xy^ which correspond to
the intersection of an ellipsoid with a series of confocal surfaces,
form a series of confocal conies, of which the points corre-
sponding to the umbilics are the common foci.
Eliminating 2' between the equations
cc'' 1? s' , o? 1? z^
— h^-i — = 1 \-- I = 1
Tvetind^ ^V. + b'b'^ -^'
whence the corresponding points are connected by the relation
^2 Y^ ^
This is evidently an ellipse for the intersections with hyper-
boloids of one sheet, and a hyperbola for the intersections with
hyperboloids of two.
The coordinates of the umbilics are
2 2 a —b ,-
the points corresponding to which are
x'=a'-b\ r=o,
which are therefore the foci of the system of confocal conies.
Curves on the ellipsoid are sometimes expressed by what
are called elliptic coordinates ; that is to say, by an equation
of the form cf) (a', a") = 0, expressing a relation between the
axes of the confocal hyperboloids which can be drawn through
the point. Now since it appears from this article that a is half
the sum and a" half the difference of the distances of the
points corresponding to the points of the locus from the points
which correspond to the umbilics, we can from the equation
^ [a J a") = 0 obtain an equation (f){p + p',p — p) = 0, from which
we can form the equation of the curve on the principal plane
which corresponds to the given locus.
189. If the intersection of a sphere and a concentric ellipsoid
be projected on either plane of circular section by lines parallel
to the least (or greatest) axis, the projection will be a circle.
FOCAL CONICS AND CONFOCAL SURFACES. 163
This theorem Is only a particular case of the following :
"if any two quadrlcs have common planes of circular
section, any quadric through their intersection will have the
same;" a theorem which is evident, since if by making z = 0 in
U and in F, the result in each case represents a circle, making
z = 0 in U+kV, must also represent a circle.
It will be useful, however, to Investigate this particular
theorem directly. If we take as axes the axis of y which is
a line in the plane of circular section and a perpendicular to
it In that plane, the y will remain unaltered, and the new
a;" = the old x^ 4- z^. But since by the equation of the plane
or circular section z = — . -^^ , x . the new ic = -, . tz 2 * •
a' V — c^ ' a 0 - c
But for the Intersection of
3 3 3
we have 5— x ^ f^— ir=r- c ,
which, on substituting for a;',
c
, -:; . -r-.X? bcCOmeS — rr^ — (x' -1- if') = r' - c*.
It win be observed that to obtain the projection on the
planes of circular sections we left y unaltered, and substituted
for a*', — 5 . — cc". But to obtain the points corresponding
to any point, as in Art. 187, we substitute for a;*, -^ ^ a;',
and for j/^, r^y"^. Now the squares of the former coordinates
have to those of the latter the constant ratio (5^- c') : 5^ Hence
we may Immediately infer from the last article that the pro-
jection of the Intersection of two confocal quadrlcs on a
plane of circular section of one of them Is a conic whose foci
are the similar projections of the umbilics ; and, again, that
given any curve 0 («', a") on the ellipsoid we can obtain the
algebraic equation of the projection of that curve on the plane
of circular section.
164 FOCAL CONICS AND CONFOCAL SURFACES.
190. The distance between two points^ one on each of two
confocal ellipsoids is equal to the distance between the two corre-
sponding points.
We have
= x' + if + z' ^ X' + Y' -^ Z" -2[xX-V yY-^ zZ).
Now (Art. 161)
a;' + y + a' = a'4 5'=' + c'", Z'+ Y" + Z'^A" -^ R' + C"\
But for the corresponding points
X'' + Y" + Z" = ^' + V' + c''\ x" + y" + z" = a' + B" + C''\
The sum of the squares therefore of the central radii to the
two points is the same as that for the two corresponding points.
But the quantities xX^ yY^ zZ are evidently respectively equal
to x'X\yY\zZ\ since aX'=Ax^ Ax'=aX, &c. The theorem
of this article, due to Sir J. Ivory, is of use in the theory of
attractions.
Ex. Similarly it may be shewn that if P„ Pj be points on a generator of
jjS yi ^2 2-2 J.2 g2
"5 + T^ - Y- '' ^^^ ^i'> ^-i points d a generator of -tj + ttz + V2 = 1» such that
mC sr cr cc
— = '\ , — = -7 , &c., the distance P1P3 is equal to the distance of the correspond-
ing points Pj'Pj' on the second hyperbola.
191. In order to obtain a property of quadrlcs analogous
to the property of conies that the sum of the focal distances
is constant, Jacob! states the latter property as follows : Take
the two points C and C on the ellipse at the extremity of the
axis-major, then the same relation p + p' = 2a which connects
the distances from C and C" of any point on the line joining
these points, connects also the distances from the foci of any
point on the ellipse. Now, in like manner, if we take on the
principal section of an ellipsoid the three points (-4, B^ C) which
correspond In the sense explained (Art. 3 87) to any three points
(a, &, c) on the focal ellipse, the same relation which connects
the distances from the former points of any point [D) In their
plane will also connect the distances from the latter points of any
FOCAL CONICS AND CONFOCAL SURFACE. 165
point [d) on the surface * In fact, by Art. 190, the distances
of the points on the confocal conic from a point on the surface
will be equal to the distances of the point on the principal plane
■which corresponds to the point on the surface, from the three
points in the principal section. t
192. Conversely, let it be required to find the locus of
a point whose distances from three fixed points are connected
by the same relation as that which connects the distances from
the vertices of a triangle, whose sides are a, Z>, c, to any point
in its plaae. Let p, p', p" be the three distances, then (Art. 52)
the relation which connects them is
a' ip' - p'^) [p' - p"^) + h^ fp" - p'j (/'^ - p"') + c^ {/" - p') [p"' - p')
-d'{F-^c'-a')p'-b'{c'-]-a'-b')p'-c'{a' + b'-c')p'''-^a'bV==0.
But p^ - p'^^ &c. being only functions of the coordinates of the
first degree, the locus is manifestly only of the second degree.
That any of the points from which the distances are measured
is a focus, is proved by shewing that this equation is of the form
Z7+ LM=0^ where Z7is the infinitely small sphere whose centre
is this point. In other words, it is required to prove that the
result of making p* = 0 in the preceding equation is the product
of two equations of the first degree. But that result is
d' [p" - 6') ip'"' - ¥) + [by - dY") ip'' - p'" + V' - c') = 0.
* In a note by Joachimsthal, published since his death, Crelle 73, p. 207, it is shown,
with a similar analogy to the ellipse, that the normal to the ellipsoid is constructed
by measuring from d on da, db, dc lengths da', db', dc' which would represent equili-
brating forces if measured from I> along DA, DB, JDC. The resultant of da', db', dc'
is the normal of the ellipsoid.
t Mr Townsend has shewn from geometrical considerations {Cambridge and
Dublin Mathematical Journal, vol. ill. p. 154) that this property only belongs to
points on the modular focal conies, and in fact the points in the plane y which
correspond to any point x'y'z on an ellipsoid are imaginary, as easily appears from
the formula of Art. 189. Mr. Townsend easily derives Jacobi's mode of generation
from Mac CuUagh's modular property. For if through any point on the surface we
draw a plane parallel to a circular section, it will cut the directrices corresponding
to the three fixed foci in a triangle of invariable magnitude and figure, and the
distances of the point on the surface from the three foci will be in a constant ratio
to its distances from the vertices pf this triangle. And a similar triangle can be
formed with its sides increased or diminished in a fixed ratio, the distances from the
vertices of which to the point x'y'z' shall be equal to its distances from the foci.
1G6 FOCAL CONICS AND CONFOCAL SURFACES.
Let now the planes represented by p''* — p'"* - c\ p"* — p" - Z>' be
L and J/, then the result of making p^ = 0 in the equation is
a'LM^ [VL - c'M) [L - M) = 0,
or FU - 2hcLM cos A + c^M' = 0,
where A is the angle opposite a in the triangle ahc. But this
breaks up into two imaginary factors, shewing that the point
we are discussing is a focus of the modular kind.
193. If several parallel tangent planes touch a series] of
confocals^ the locus of their points of contact is an equilateral
hyperbola.
Let a, /S, 7 be the direction-angles of the perpendicular on
the tangent planes. Then the direction-cosines of the radius
a^cosa b' cos 13 c^ cosy
vector to any pomt or contact are , — , J
J f rp rp ' rp
as easily appears by substituting in the formula a^ cosa — px'
(Art. 89), r cos a' for x' and solving for cos a'. Forming then,
by Art. 15, the direction-cosines of the perpendicular to the
plane of the radius vector and the perpendicular on the tangeat
plane, we find them to be
{b^ — c^) cos/3 cos 7 [c' — a^) cos 7 cos a {a^ — W) cos a cos^S
rp sin</) ' rp sin0 ' rp sin</) '
where ^ is the angle between the radius vector and the per-
pendicular. Now the denominator is double the area of the
triangle of which the radius vector and perpendicular are sides.
Double the projections, therefore, of this triangle on the co-
ordinate planes are
[y^ — c') cos^ cos 7, (c^ — d^) cos 7 cos a, {(f — b^) cos a cosyS.
Now these projections being constant for a system of confocal
surfaces, we learn that for such a system, both the plane of
the triangle and its magnitude is constant. If then G3I be
the perpendicular on the series of parallel tangent planes and
PM the perpendicular on that line from any point of contact
P, we have proved that the plane and the magnitude of the
triangle CPM are constant, and therefore the locus of P is an
equilateral hyperbola of which CM is an asymptote.
CURVATURE OF QUADRICS, 167
193a. Writing down the equations of the normals to
A'^ B'^ (7~ '
at two points, we find as the condition that they may intersect
A (a;' - x") {yz" - y"z') ^ B {7/ - y") [z'x'' - z'x')
+ C{z'-z")[xY-xy) = Q,
or, calling a, yS, 7 the direction angles of the line which joins the
points, and a^, /3,, 7, those of the perpendicular to the central
plane containing the two points, the condition becomes
A cos a cosa, +Bcos^ cos/3j + CC0S7 cos7j = 0.
This relation obviously still holds If A, B, C be replaced by
kA + l, IcB+l, kC+l. Hence, we see that if the normals at
the two points of Intersection of any right line with any central
quadric Intersect, the normals at Its two points of intersection
with any confocal, or with a similar and similarly placed con-
centric quadric likewise Intersect.*
As a special case of this, we may consider the three confocals
w, v, 10 which meet in any point P. The normal at P to m
meets u again In Q, therefore meets the normal at Q. Hence,
if normals be drawn to v at the points In which it Is met by PQ
they must Intersect, and, in like manner, the normals at the
points where PQ meets w, Intersect. But the line PQ Is a
tangent line both to v and to w. Hence, normals to either
surface taken at consecutive points along their common curve
intersect. A curve possessing this property Is defined to be a
line of curvature on either surface.
CURVATURE OF QUADRICS.
194. The general theory of the curvature of surfaces will
be explained in Chap, xi., but It will be convenient to state
here some theorems on the curvature of quadrics which are
immediately connected with the subject of this chapter.
If a normal section he made at any point on a quadric^ its
radius of curvature at that point is equal to ^^:p^ where /S is the
* See a paper by Mr.F. Purser, QuarUrhj Journal 0/ J£athematics,lp. ^6, vol. viii.
168 CURVATURE OF QUADRICS.
semi-diameter parallel to the trace of the section on the tangent
plane, and p is the perpendicular from the centre on the tangent
plane.
We repeat the following proof by the method of infini-
tesimals from Conies^ Art. 398, which see.
Let P, Q be any two points on a quadric; let a plane
through Q parallel to the tangent plane at P meet the central
radius CP in P, and the normal at P in ;S^, then the radius
of a circle through the points P, Q having its centre on PS
is PQ^ : 2P8. But if the point Q approach indefinitely near to P,
QP is in the limit equal to QR] and if we denote CP and
the central radius parallel to QR by a and /i?, and if P' be
the other extremity of the diameter GP^ then (Art. 74)
/3^ : a'* :: QR' : PR . RP'{=2a' .PR)',
,, r np. 2/3\PP .,..' c , ^" P^
theretore QR = -. — and the radms ot curvature = — ? . -j^ .
a a jTO
But if from the centre we let fall a perpendicular (73/ on the
tangent plane, the right-angled triangle CMP is similar to
PRS^ and PR : PS:: a' : p. And the radius of curvature ia -
therefore —;. — = — ; which was to be proved.
a p p
If the circle through PQ have its centre not on PS, but on
any line PS\ making an angle 0 with PS, the only change
PQ'
is that the radius of the circle is ^-^7 , S' being still on the
plane drawn through Q parallel to the tangent plane at P.
But PS evidently = P/S" cos ^. The radius of curvature is
PQ''
therefore — rrr~,cos6, or the value for the radius of curvature
2PS ' -^
of an oblique section is the radius of curvature of the normal
section through PQ, multiplied hy cos^.
195. These theorems may also easily be proved analytically.
It is proved {Conies, Art. 241) that if ax' -^-^hxy -{-hy' -{■2gx = 0
be the equation of any conic, the radius of curvature at the
origin is ^ 4- Z>. If then the equation of any quadric, the plane
of xy being a tangent plane, be
ax' + 2hxy + hy'' + ^gzx + 2fyz + cz' + 2nz = 0,
CURVATURE OP QUADRICS. 169
the radii of curvature by the sections ^ = 0, x = 0 are respec-
tively w : a, n : b. But if the equation be transformed to
parallel ax^s through the centre, the terms of highest degree
remain unaltered, and the equation becomes
ax^ + 2hxy + hy^ + 2gzx + 2fyz -1- cz^ = D.
The squares of the intercepts on the axes of ic and y are D:a^D'.h.
This proves that the radii of curvature are proportional to the
squares of the parallel semi-diameters of a central section. And
since, by the theory of conies, the radius of curvature of that
section which contains the perpendicular on the tangent plane
is ^^ :p^ the same is the form of the radius of every other section.
The same may be proved by using the equation of the
quadric transformed to any normal and the normals to two
confocals as axes (see Ex. 2, Art. 174), viz.
a?" y'' z^ ^pxy 2p'xz 2a; _
7 a —a a —a p[a — a ) p[a —a ) p
The radii of curvature of the sections by the planes 2; = 0, y = 0
are respectively , . ihe numerators are the
p p
squares of the semi-axes of the section by a plane parallel to
the tangent plane (Art. 164).
The equation of the section made by a plane making an
angle 6 with the plane of y is found by first turning the
axes of coordinates round through an angle 6^ by substituting
^ cos ^ — 2 slij ^, y^itid + zcosd for y and z^ and then making
1 B^
the new z = 0. Then, if the new coeflScient of V'' is — , , — is the
' ^ 0'^ p
corresponding radius of curvature. But this coefficient is at
once found to be
cos'^ sin'^^
a —a a — a
and this coefficient of y^ is evidently the inverse square of that
semi-diameter of the central section, which makes an angle 0
with the axis y.
196. It follows from the theorem enunciated in Art. 194,
that at any point on a central quadric the radius of curvature
z ^N
170 CUEVATUEE OF QUxiDEICS.
of a normal section has a maximum and minimum value^ the
directions of the sections for these values heing parallel to the
axis-major and axis-minor of the central section hy a plane
parallel to the tangent plane.
These maximum and minimum values are called the prin-
cipal radii of curvature for that point, and the sections to
which they belong are called the principal sections. It appears
(from Art. 163) that the principal sections contain each the
normal to one of the confocals through the point. The Inter-
section of a quadric with a confocal is a curve such that at
every point of it the tangent to the curve Is one of the prin-
cipal directions of curvature. Such a curve is called a line
of curvature on the surface, and this definition agrees with that
of Art. 193a.
In the case of the hyperboloid of one sheet the central
section is a hyperbola, and the sections whose traces on the
tangent plane are parallel to the asymptotes of that hyperbola
will have their radii of curvature infinite ; that is to say, they
will be right lines, as we know already. In passing through
one of those sections the radius of curvature changes sign ; that
is to say, the direction of the convexity of sections on one
side of one of those Hues is opposite to that of those on the
other.
197. The two j^rincipal centres of curvature are tJie two
poles of the tangent plane ivith regard to the two confocal surfaces
which pass through the point of contact. For these poles lie
on the normal to that plane (Art. 167), and at distances from
it = and (Art. 168), but these have been just
proved to be the lengths of the principal radii of curvature.
We can also hence find, by Art. 168, the coordinates of the
centres of the two principal circles of curvature, viz.
ax by c ^ . ^ ^ ^^ y ^ ^
h' ' d' ' a'' ' -^ b'
a' ' " 0' ' c ' a ' ~ 0' ' cf'
198. If at each point of a quadric we take the two principal
centres of curvature, the locus of all these centres Is a surface
of two sheets, which Is called the surface of centres.
CURVATURE OF QUADRICS. 171
We shall shew how to find its equation in the next chapter,
but we can see h priori the nature of its sections bj the
principal planes. In fact, one of the principal radii of cur-
vature at any point on a principal section is the radius of
curvature of the section itself, and the locus of the centres
corresponding is evidently the evolute of that section. The
other radius of curvature corresponding to any point in the
section by the plane of xy is c'"' :^, as appears from the for-
mula of Art. 194, since c is an axis in every section drawn
through the axis of z. From the formulse of Art. 197 the
2 2 7 2 2
coordinates of the corresponding centre are — r^ — x\ — ^^ — y']
that is to say, they are the poles with regard to the focal
conic of the tangent at the point xy' to the principal section.
The locus of the centres will be the reciprocal of the principal
section, taken with regard to the focal conic, viz.
The section then by a principal plane of the surface (which Is
of the twelfth degree) consists of the evolute of a conic, which
is of the sixth degree, and of the conic (it will be found)
three times over, this conic being a cuspidal line on the surface.
The section by the plane at infinity is of a similar nature to
those by the principal planes. It may be added, that the
conic touches the evolute in four points (real for the principal
plane through the greatest and least axes, or umbillcar plane)
and besides cuts it in four points.
199. The recijprocal of the surface of centres is a surface of
the fourth degree.
It will appear from the general theory of the curvature of
surfaces, to be explained in Chap. XI., that the tangent plane
to either of the confocal surfaces through xy'z' Is also a tangent
plane to the surface of centres. The reciprocals of the intercepts
which the tangent plane makes on the axes are given by the
equations
f r r
a
U
172 CURVATURE OF QUADRICS.
The relation
a a 0 0 CQ
gives between ^, 17, ^ the relation
(r+,-+r)=(a'-a")(|: + ^4+^:),
and the relation
x'' y" z" ,
gives (aT + W + cX' - 1) = (a^ - a'^) (f + 97^ + TO-
Eliminating a" — a'^, we have
(I' ^rf^ Ky = (I + ^' + ^) («^r + 2'''?' + «^r - 1).
But it is evident (as at Higher Plane Curves^ Art. 21) that ^, 77, ^
may be understood to be coordinates of the reciprocal surface ;
since, if f, 17, ^ be the coordinates of the pole of the tangent
plane with regard to the sphere x^ -'ry'^ + z^ = \^ the equation
x^ + yr) + z^=l being identical with that of the tangent plane,
^, 77, f will be also the reciprocals of the intercepts made by
the tangent plane on the axes.
♦ This equation was first given, as far as I am aware, by Dr. Booth, Tangential
Coordinates, Dublin, 1840.
*
5^\
( 173 )
CHAPTER IX.
INVAKIANTS AND COVARIANTS OF SYSTEMS OF QUADRICS,
200. It was proved (Art. 136) that there are four values
of \ for which \U+ V represents a cone. The biquadratic
which determines \ Is obtained by equating to nothing the dis-
criminant of XU+ V. We ehall write it
X*A + V0 + X'^ + X0' + A' = 0.
The values of \, for which \ U+ V represents a cone, are
evidently independent of the system of coordinates in which
U and V are expressed. The coefficients A, 0, &c. are there-
fore invariants whose mutual ratios are unaltered by transforma-
tion of coordinates. The following exercises in calculating
these invariants include some of the cases of most frequent
occurrence.
Ex. 1. Let both quadrics be referred to their common self-conjugate tetrahedron
(Art. 141). We may take
Z7 = aa^2 4. i,yi ^ cz^ + ^,„2^ y = x- + y- + z'^ + w^,
(see Art. 141, and Conies, Ex. 1, Art. 371), then
A = abed, Q = bed + cda + dab + abc, ^ = bc + ca-'r ab + ad + bd + cd,
Q'=a + b + c->rd, ^'-\.
Ex. 2. Let V, as before, be x^ + ^" + a^ + w"^, and let U represent the general
equation. The general value of 0 is
a' A + b'B + c'C + d'D + 2f'F +2g'G + 2h'n + 21' L + 'im'M + In'N,
where A, B, &c. have the same meaning as in Art. 67. In the present case therefore
Q = A + B+C-\- D, Q' = a + b + c + d;
we have also <t> = 6c -/^ + ca — g'^ + ab - h^ + ad — l'^ + bd - m^ + cd - n\
Similarly, if U is ax~ + by^ + cz'- + dvfl, and V is the general equation,
e is a'bcd + Veda + c'dab 4- d'abc, 0' is aA' + bB' + cC + dU .
Ex. 3. Let U and V represent two spheres,
a;2 + 2,2 + 22 _ p2^ (^ _ „)2 + (y _ /3)2 + (^ _ y)2 _ ^'2,
and let the distance between the centres be D, {a^ + /32 + y^ _ j}2)^ t^en
A=-f)^ A' = -/2^ 0 = i)2 _ 3p2 _ ^'2^ e' - JT- - p^ - 3p'^, ^ = 2LP - 3p^ - 3p'^,
and the biquadratic which determines \ ia
(X + 1)2 (- p2X2 + (/)2 _ p2 _ p'2) X _ p'2} = 0.
174 INVAEIANTS AND COVARIANTS OF
1/2
Ex. 4. Let U represent ^ + 1; + ^ - 1, while V is the sphere
(x - a)2 + (y - (iy- +{z- y)2 - p^.
o =
a"b~c'^
Since \Z7+ F admits of being written in the form AX^ + BT^ + CZ"^ + BW", it
is easily seen that the biquadratic found by equating to nothing the discriminant
of \U ■\- V may be written
a'^ + X *2 + X, c2 + X ^ A."
Ex. 5. Let U represent the paraboloid ax'^ + by- + 2nz and V the sphere as in
the last example.
Ans. A = - ahn-, A' = - p"^,
e = - 7i2 (rt + J) + 2ahny, Q' = aa^ + S/S^ + 2»y - (a + 5) /j^,
* = «6 {a"' + /32 - p2) 4. 2 (« + J) rty - n2 ;
and the biquadratic may be written by a similar method
Ex. 6. In general the value of $ is
(*c -P) {a'd' - V^) + {ca - g'^) {b'd' - m'"-) + [ah - h"-) [c'd' - n'^)
+ [ad - P) [b'c' -/'2) + {bd - OT=) {c'a' - ff'^) + [cd - m=) {a'b' - h'^)
+ 2{gm- hn) {g'm' - h'n') + 2 {hn -fT) {h'n' -f'l') + 2{fl- gm) {f'V - g'm!)
+ 2 {mh - lb) [I'c' - n'g') + 2 {nf- mc) [m'a' - I'h') + 2 [Ig - no) {n'b' - m'f)
+ 2 (m'A' - I'b') {Ic -ng) + 2 {ii'f - m'c') {ma - Ik) + 2 {I'g' - n'a') {nb - mf)
+ 2{fd- mn) {g'h' - a'f) +2{gd- nl) {h'f - b'g') + 2 {hd - Im) [f'g' - c'h')
+ 2 if'd' - m'n') [gh - «/) + 2 {g'd' - n'l') [lif - bg) + 2 {h'd' - I'm') [fg - ch).
Thus * is a function of the same quantities which occur in the condition (Art. 80a)
that a line should touch a quadric. This condition is a quadratic function of the six
coordinates of the line ; and if we write the coefficients which affect the squares of the
coordinates in that condition «„, aoi-.-aei^, and those which affect the double rectangles
Cut "i3> '^c., writing the coiTesponding quantities for the second quadric c^, c^^, &c.,
then $ is ane^^ + a^fy^ + asjCge + "44^11 + '^65''22 + "ee^ss + ^<^u^ii + '^C. In like
manner, writing the discriminant in any of the three forms,
^ = «n«44 + «12«43 + «13^46 + «-14 + «15«42 + «'lG«43
= «21«54 + «22«55 + «23«56 + «24«51 + «-25 + «26«53
= «3l'''64 + «32«65 + «33"66 + «34"61 + «35«62 + «'36)
if by the substitution of a + Xa' &c. for a Ac, «„ become a^ + Xb^ + X-c^i &c.,
different methods of \vi-iting the invariants are foimd.
201. To examine the geometrical meaning of the condition 0 — 0
and of the condition 4> = 0. It appears, from Art. 200, Ex. 2,
that when U\b referred to a self-conjugate tetrahedron,
0 = hcda' + cdaV + dabc + ahcd\
SYSTEMS OF QUADRICS. 175
which will vanish when a\ h\ c', d' all vanish. Hence 0 will
vanish whenever it is possible to inscribe in V a tetrahedron which
shall he self-conjugate with regard to U. In like manner 0' will
vanish for this form of U whenever A\ B\ C", U vanish. But
^' = 0 is the condition that the plane x shall touch V. Hence 0'
loill vanish whenever it is possible to find a tetrahedron self-conju-
gate with regard to U whose faces touch V. By the first part of this
article 0' = 0 is also the condition that it may be possible to
inscribe m U a tetrahedron self-conjugate with regard to V.
Hence when one of these things is possible, so is the other also.
4) 1=0 will be fulfilled, if the edges of a self-conjugate tetra-
hedron, with respect to either, all touch the other.
Ex. 1. The vertices of two self -conjugate tetrahedra, with, respect to a quadric
form a system of eight points, such that every quadric through seven will pass through
the eighth (Hesse, Crelle, vol. XX., p. 297).
Let any quadric be described through the four vertices of one tetrahedron, and
through three vertices of the second, whose faces we take for x, y, z, to. Then
because the quadric circumscribes the first tetrahedron, G' =: 0, or « + S + c + (f = 0
(Art. 200, Ex. 2) ; and because it passes through three vertices of xyzw, we have
a — Q, 6 = 0, c = 0; therefore d — 0, or the quadric goes through the remaining
vertex. It is proved, in like manner, that any quadric which touches seven of the
faces of the two tetrahedra touches the eighth.
Ex. 2, If a sphere be circumscribed about a self-conjugate tetrahedron, the length
of the tangent to it from the centre of the quadric is constant. For (Art. 200, Ex. 4)
the condition B = 0 gives the square of the tangent a^ + (S^ + y"^ — p'^ = n~ + U^ + c".
This corresponds to M. Faure's theorem {Conies, Art. 375, Ex. 2). It may be other-
wise stated: "The sphere which circumscribes a self -conjugate tetrahedron cuts
orthogonally the sphere which is the intersection of three tangent planes at right
angles" (Art. 93).
x"^ y- z- 111
Ex. 3. If a hyperboloid — 1-^ + — = 1 be such that - + -^ + - = 0, then the
■^ a b c a 0 c
centre of a sphere inscribed in a self -con jugate tetrahedron hes on the surface. This
follows from the condition B' - 0 (Art. 200, Ex. 4).
Ex. 4. The locus of the centre of a sphere circumscribing a tetrahedron, self-
conjugate with regard to a paraboloid, is a plane (Art. 200, Ex. 5).
202. To find the condition that two quadrics U, V should
touch each other. As in the case of conies [Conies^ Art. 372)
the biquadratic of Art. 200 will have two equal roots when
the quadrics touch. This is most easily proved by taking
the origin at the point of contact, and the tangent plane for
the coordinate plane z. Then, for both the quadrics, we
have J = 0, l — O, m = 0; and since, if we substitute these values
176 INVARIANTS AND COVARIANTS OF
in the discriminant (Art. 67), it reduces to 71^ {h^ - ah), tlie bi-
quadratic becomes
(Xn + 7iy {{\h + hy - {\a + a') {Xh + ¥)} = 0,
which has two equal roots. The required condition is there-
fore found by equating to zero the discriminant of the biquadratic
of Art. 200.
Ex. 1. To find tlie condition that two spheres may touch. The biquadratic for
this case (Art. 200, Ex. 3) has always two equal roots. This is because two spheres
having common a plane section at infinity, always have double contact at infinity
(Art, 137). The condition that they should besides have finite contact is got by
expressing the condition that the other two factors of the biquadratic should be
equal and is (D^ — r^ — r'^)^ = irh-"^, whence D — r + ?•'.*
Ex. 2. Find the locus of the centre of a sphere of constant radius touching a
central quadric. The equation got by forming the discriminant with respect to X.
of the biquadratic of Art. 200, Ex. 4, is of the twelfth degree in a, /3, y. "When we
make jo = 0, it reduces to the quadric taken twice, together with the equation of
the eighth degree considered below (Art 221). The problem considered in this
example is the same as that of finding the equation of the surface parallel to the
quadric (see Conies, Ex. 3, Art. 372) ; namely, the surface generated by measuiing
from the surface on each normal a constant length equal to p. The equation is of
the sixth degree in p'^, and gives the lengths of the six normals which can be drawn
from any point xyz to the surface {Conies, Art. 372, Ex. 3). To find the section of the
surface by one of the principal planes, we use the principle that the discriminant with
respect to X of any algebraic expression of the form (\ — a) <^ (\) is the square of
<p (a) multiplied by the discriminant of <p (\). If then we make 2 = 0 in the
equation, the discriminant of
is the conic h ,-^ 1 + — t
a — c 0 — c c
taken twice, this curve being a double line on the surface, together with the dis-
criminant of the function within the brackets ; this latter representing the curve of
the eighth order, parallel to the principal section of the ellipsoid.
Ex. 3. The equation of the surface parallel to a paraboloid is found in like
manner by forming the discriminant of the biquadratic of Ex. 5, Art. 200. The
result represents a surface of the tenth degree, and when /o = 0, reduces to the
paraboloid taken twice, together with the surface of the sixth degi-ee considered
below (Art. 222). The equation is of the fifth degree in p'^, shewing that only five
normals can be drawn from any point to the surface. It is seen, as in the last
example, that the section by either principal plane is a parabola taken twice, together
with the curve parallel to a parabola.
203. It is to be remarked that when two surfaces touch,
the point of contact is a double point on their curve of
* Generally the biquadratic (Art. 200) will have two pairs of equal roots when
the quadrics have a generator common, the conditions for this may be WTitten down
as in Art. 214 Higher Algebra.
SYSTEMS OF QUADEICS. 177
intersection. In general, two surfaces of the m^^ and n^^ degrees
respectively intersect in a curve of the mn^^ order. And at
each point of the curve of intersection there is a single tangent
line, namely the intersection of the tangent planes at that point
to the two surfaces. For any plane drawn through this line
meets the surfaces in two curves which touch : such a plane
therefore passes through two coincident points of the curve of
intersection. But if the surfaces touch, then everi/ plane through
the point of contact meets them in two curves which touch,
and eve7y such plane therefore passes through two coincident
points of the curve of intersection. The point of contact Is
therefore a double point on this curve.
And we can shew that, as in plane curves, there are two
tangents at the double point. For there are two directions
in the common tangent plane to the surfaces, any plane through
either of which meets the surfaces in curves having three points
in common.
Take the tangent plane for the plane of xt/, and let the
equations of the surfaces be
z + ax^ + 2hxy -f hjf + «&c.,
z 4 ax'' 4- 'ih'xy + h'y'' + &c.,
then any plane y^fJix cuts the surfaces in curves which oscu-
late (see Conies^ Art. 239), if
a + 2hiM + V' = «' + 2A> -f VfM.
The two required directions then are given by the equation
{a - a) x'^2[h- h') xy + [b- h') y' = 0.
The same may be otherwise proved thus. It will be shown
hereafter precisely as at Higher Plane Curves^ Arts. 36, 37, that
if the equation of a surface be w, + u.^ + u^ + &c. = 0, the origin
will be on the surface, and u^ will include all the right lines
which meet the surface in two consecutive points at the origin ;
while if Mj is identically 0, the surface has the origin for a
double point, and u^ includes all the right lines which meet the
surface at the origin in three consecutive points. Now in the
case we are considering, by subtracting one equation from the
other, we get a surface through the curve of intersection, viz.
(« - a) x' + 2 [h - h') xy+{b- V) / + &c.,
AA
178 INVARIANTS AND COVAEIANTS OF
in which surface the origin is a double point, and the two
lines just written are two lines which meet the surface in
three consecutive points.
204. When these lines coincide there is a cusp or stationary
point (see Higher Plane Curves^ Art. 38) on the curve of inter-
section. We shall call the contact in this case stationary
contact. The condition that this should be the case, the axes
being assumed as above, is
{a-a'){h-h') = {h-h')\
Now if we compare the biquadratic for X, given Art. 202,
remembering also that in the form we are now working with,
we have n = w', we shall see that when this condition is
fulfilled, three roots of the biquadratic become equal to — 1.
The conditions then for stationary contact are found by forming
the conditions that the biquadratic should have three equal roots^
viz. these conditions are >8'=0, 2'=0, 8 and T being the two
invariants of the biquadratic.
205. Every sphere whose centre is on a normal to a quadrlc,
and which passes through the point where the normal meets
the surface, of course touches the surface. But it will have
stationary contact when the length of the radius of the sphere
is equal to one of the princiinil radii of curvature (Art. 196).
Let us take the tangent plane for plane of ocy^ and the two
directions of maximum and minimum curvature (Art. 196) for
the axes of x and y. Then since these directions are parallel
to the axes of parallel sections, the term xy will not appear in
the equation, which will be of the form z -f ax^ + by'' + &c. = 0.
By the last article, any sphere z -\- \ [x'' + y'^ A- z^) will have
stationary contact with this if (X — a) {\ — h)= 0, for we have
h and h' each = 0. We must therefore have X equal either to
a or h. Now if we make y = 0, the circle z + a{x^ -{■ z') is
evidently that which osculates the section z -f ax'' + &c. ; and,
in like manner, the circle z-^h{7f ■\- z'^) osculates z-^hy^ -{^ &c.
206. To find the equation of the surface of centres of a
quadric. If we form, for the biquadratic of Ex. 4, Art. 200,
the two equations /S' = 0, 2'=0j we have two equations con-
SYSTEMS OP QUADRICS. 179
necting a, /3, 7, the coordinates of the centre of curvature of
any principal section, and p its radius. One of these equations
is a quadratic and the other a cubic in p^ ; and if we eliminate
p'^ between thera, we evidently have the equation of the locus
of the centres of curvature of all principal sections. The
problem may also be stated thus : If U and U' be any two
algebraical equations of the same degree and k a variable
parameter, it is generally possible to determine k so that the
equation U+ kU' = 0 may have two equal roots. But it is
uot possible to determine k, so that the same equation may
have three equal roots, unless a certain invariant relation subsist
between the coefficients of U and U\ Now the present problem
is a particular case of the general problem of finding such an
invariant relation. It is in fact to find the condition that it
may be possible to determine k so that the following biquadratic
in \ may have three equal roots :
x' f z" k
+ x^^r-T+ -r— 7- = H-T.
a' + X h'^X c' + X " ' X
The following are the leading terms in the resulting equation :
the remaining terms can be added from the symmetry of the
letters. We use the abbreviations J/ — d^= a, 6^ — d^ = /3,
o"' — 6'^ = 7 ; and further we write x^ 3/, z Instead of ace, hy^ cz :
a'x'' + 3 [a' + /3'0 a'x'Y + 3 (a* 4- 3a^/3'^ + /3*) a'xY
-h 3 (2a* + 3d'l3' + Say - 7/3V) d'xYz'
+ (a' -;- /3^ + 9a*y3'^ + da'/S') x'f
+ 3 (a' + 6a'l3' + So^y + Sa'jS* + /Sy - 21a^/3V) x^i/z'
-f 9 (a*/3'^ + ^'d' + /Sy + ^y + y'd' + y'a' - Ud^y) xYz*
- 3 (/3^ + 7''') a^'.*'*' - 3 [2/3* 4 3/3V + 3/3V - 7y'd') aV/
-S{(d' + 6^*d' + S^y + S^'a' + aV' - 21a^/3V) oi'xY
+ S[U{a'^'+a'^^ /3V+^V+7V+7V)+20a^/3y}aVyV
+3[47«-77^(a^+/3'Vl987'a'^/Q'-'+68a'''/Qy(a'+/S')+4:2a*/3>y2''
+ 3(^* + 3/SV + y)a''>'C«
4 3 (/3« + 6/3y 4 3/3V' 4 3^V 4 aV - 21a''/3V) aV/
4 9 (c(*/3'^ 4 a^'/3* 4 /Sy + ^y + 7'a' + 7'a' - 1 4a^^V') a'/5 V?/*
180 INVARIANTS AND COVARIANTS OF
- 198a*/3Y + 68a'''/3V {^' + 7"'') 4 42/3^'} aV?/V
- 3 (7' + 67*/3^ -f 37^a^ + 37'''yS* + a'^' - 21a^/3V) a*/3Vy
+ 3 {14 (a*/3^ + a^/S* + ^V + fi'y + ^a^ + 7'^a*)
+ 20a"'^yS'''7^} a'lSyxyz'
+ 3 (/S^ + 3/3^7'^ + 7*) a'^yx'
+ 3 (27* + 37V + 37^^/3'^ - Ta-'/S^) a*/3>Vy
- 3 (/3^ + 7'0 a'/S^V -f a'^'^y' = 0.
If we make in this equation ^ = 0, we obtain
(aV + /Sy - a'/S'-y {(^' 4 2/' - 7'T 4 27a;y7^}, see Art. 198.
The section by the plane at infinity is of a similar kind to that
by the principal planes, the highest terms in the equation being
{x' 4 f 4 zj {(aV 4 iSy + y'z'f - ^Id'^yxY^].
In like manner we find the surface of centres of the paraboloid
ax'' + hf 4 2 no?. If we write
a—h=m^ a+b=2}, ah=q^ hx^+ai/=V^ x^-ry^=^p^^ qz'+pnz + n^=W^
the equation is
8 [q'z V+ qn (JV 4 ay) 4 2m'n W]' + 27 T= 0,
where
T= q'n V - IGmYn TTV/ 4 GmYn'z V^ - 5(jmYn'z Vx'?/
+ 8jnynVy TF4 1277iYifz' V 4 QmYn'p' V- Vo^mYn'xYp''
4 4:8m'2)q'n'xY F4 SmYn*z' F4 247nYw*^p' V+ 2-imYn'p'z^
+UmYn'p*-{ASinY7i'xy'-\-2im^z7iYax'+bf)-\-877i\a:'x'-{-by)n\
The section by either plane x or ?/, is a parabola, counted three
times, and the evolute of a parabola.
207. To find the condition that two quadrics shall he such that
a tetrahedron can be mscribed in one having two pairs of opposite
edges on the surface of the otherJ^ The one quadric then can
* This problem and its reciprocal appear to answer to the plane problem of
finding the condition that a triangle can be inscribed in one conic and circumscribed
about another. Mr. Purser {Quarterly Journal, vol. viii., p. 149) has determined the
envelope of the fourth face of a tetrahedron whose other three faces touch a quadric U
when two pairs of its opposite edges are generators of another quadric V to be a
quadric passing through the curve of intersection of the given quadrics.
SYSTEMS OF QUADRICS. 181
have Its equation thrown Into the form Fyz + Lxw = ()^ while
the coefficients «, Z*, c, d are wanting in the equation of the
other. We have, then,
A = r'L\ Q=2FL{Fl+Lf], <P = {Fl + LfY + 2FL{fl-gm-7m)^
0' = 2 ( /^ - rjm - hn) [Fl + Lf).
And the required condition is
Siiuilarly the condition that It may be possible to find a tetra-
hedron having two pairs of opposite edges on the surface of
one, and whose four faces touch the other, is
4A'0'* = 0" + 8A"'0.
This may be derived from the equation examined In the next
article.
208. To find the general form of the equation of a quadric
which touches the four faces a;, y^ z^ w of the tetrahedron of
reference. The reciprocal quadric will pass through the four
vertices of the tetrahedron, and its equation will be of the
form
^.fyz + 2(jzx + 2hxy + 2lxw + 2myw + 2nzw = 0.
The equation of the reciprocal of this is (Arts. 67, 79)
2fmnd' + 2gnI/3'' 4 2hhnY + 2fgh^''
-f 2 [fl - gm - hn) [I^y +/aS) + 2 [gjn - hn -fl) [mja + g^8)
+ 2 (hn -fl-gm) [na^ ■+ hy8) = 0.
If now we write for a >^J[fmn)^ l3 \J{gnl)^ y \/[hhn)j 8 '^{fgh)j
a;, y, 2, w respectively, this equation becomes
., „ ., 2 fl — qm — hn , .
x' + y^ + z' + vf V—ry--, r- {yz + Xio)
N\ghinn)
am — hn — fl, , hn —fl — am , ,
+ ,n4.-n' [^^ + y^'^) + //,- / \ [xy + zio) = 0.
^/{hfnl) -" '^i/glm) -^ '
Now It is easy to see that these three coefficients are re-
spectively — 2cos^, — 2cos^, - 2cosC, where A^ B^ C are
the angles of a plane triangle whose sides are \/[fl)^ Vli/wi),
f^{hn). Hence, the general form of the equation of a quadric
touching the four planes of reference Is
x^ + y'' -\- z'' + vf - 2p [yz + xw) — 2q {zx + yw) - 2r [xy + zw) = 0,
182 INVARIANTS AND COVARIANTS OF
where p, q^ r are the cosines of the angles of a plane triangle,
or, in other words, are subject to the condition \ — 2pqr=p'-^q^-'cr^.
It may be seen otherwise that the surface whose equation has
been written is actually touched by the four planes ; for the
condition just stated is the condition of the vanishing of the
discriminant of the conic obtained by writing a;, y^ z, or w = 0,
in the equation of the quadric. The section therefore by each
of the four planes being two real or imaginary lines, each of
these planes is a tangent plane.
209. If V represents a cone we have A' = 0, and we proceed
to examine the meaning in this case of 0, *t>, 0'. For simplicity
we may take the origin as the vertex of F, or Z', m\ n\ d' all = 0.
We have then & = d[a:h'c' ^Ifgh! -af -Vg'^ - ch"'), or 0'
vanishes either if the cone break up into two planes, or if the
vertex of the cone be on the surface TJ. Let the cone whose
vertex is the origin and which circumscribes Z/, viz.
d [ax'' + h-if' + cz^ + 2^2; + Igzx + llixy) — {Ix + my + nzf
be written
?ix^ + hy^ + cz' + 2?yz + 2gzx + 2hxy = 0,
then 4> may be written
a [b'c' -D + b (c'a' - g") 4- c {a'V - ¥')
+ 2f [g'K - a'/) + 2g (A/ - Vg') + 2h (// - c'A').
Hence, by the theory of the invariants of plane conies [Conies^
Art 375) <i> = 0 expresses the condition that it shall be possible
to draw three tangent lines to U from the vertex of the cone K,
which shall form a system self-conjugate with regard to V. In
like manner
de = a' (be - f •■') + V (ca - g^) + &c.,
or 0 vanishes whenever three tangent planes to U can be drawn
from the vertex of the cone V which shall form a system self-
conjugate with regard to V. The discriminant of the cubic in
X will vanish when the cone V touches U.
"When V represents two planes, both A' and 0' vanish.
Let the two planes be x and ?/, then V reduces to 2h'xy^ and 4>
SYSTEMS OF QUADEICS. 183
reduces to Ji'^ (w'"* - cd)^ <^ will vanish therefore in this case when
the intersection of the two planes touches U. We have @=2h'Hj
(see Art. 67) and its vanishing expresses the condition that the
two planes should be conjugate with respect to Z7; or, in other
words, that the pole of either, with regard to ?7, should lie on
the other. For (see Art. 79) the coordinates of the pole
of the plane x are proportional to A, 11^ G, L, and the pole
will therefore lie in the plane ?/ when 11=0. The condition
©^ = 4A<I> is satisfied if either of the two planes touches U.
210. The plane at Infinitj cuts any sphere In an Imaginary
circle the cone standing on which, and whose vertex Is the
origin, is x^ -\- y^ + z^ = 0. Further, since this cone is also an
Infinitely small sphere, any diameter is perpendicular to the
conjugate plane. If now we form the Invariants of x^ + y'^ + z'^^
and the quadric given by the general equation, we get 0 = 0,
or A-\-B+ C=0, as the condition that the origin shall be a
point whence three rectangular tangent planes can be drawn
to the surface, and <i> = 0, or
ad — r -\- bd— vt' + cd — ti^ = 0,
as the condition that the origin shall be a point whence three
rectangular tangent lines can be drawn to the surface. In
particular if the origin be the centre and therefore Z, «?, n all = 0,
and (the surface not being a cone) d not = 0, the cubic is
the same as that worked out (Art. 82). The condition ^ = 0
reduces to a+ h + c = 0, as the condition that it shall be possible
to draw systems of three rectangular asymptotic lines to the
surface ; and the condition 0 = 0, gives
bc + ca + ah -f - g^ - h^ = 0,
as the condition that it shall be possible to draw systems
of three rectangular asymptotic planes to the surface. These
two kinds of hyperboloids answer to equilateral hyperbolas in
the theory of plane curves (see Ex. 3, Art. 201) ; the former
were called equilateral hyperboloids, (Ex. 21, p. 102). But
orthogonal hyperboloids (Ex. 5, p. 100) are of a distinct kind,
answering In a similar manner to circles in the theory of plane
184 INVARIANTS AND COVARIANTS OF
curves, and the relation among the coefficients can be found by
investigating when the pole of one of the chords of intersection
at infinity of x'' -¥ y^ + z^ and the general cone with regard to
the former lies on the latter curve.
Ex. Every equilateral hyperbola which passes through three fixed points passes
through a fourth ; what corresponds in the theory of quadrics ? It will be seen
that the truth of the plane theorem depends on the fact that the condition that the
general equation of a conic shall represent an equilateral hyperbola is linear in the
coefficients. Thus, then, every rectangular hyperboloid (viz. hyperboloid fulfilling
such a relation ssa + b + c — Q) which passes through seven points passes through a fixed
curve, and which passes through six fixed points passes through two other fixed points.
Eor the conditions that the surface shall pass through seven points together with the
given relation enable i^s to determine all the coefficients of the quadric except one.
Its equation therefore containing but one indeterminate is of the form U + hV which
passes through a fixed curve. And when six points are given the equation can be
brought to the form U + JcV + IW which passes through eight fixed points.
211. Since any tangent plane to the cone x^ + if + z^ is
xx' + yy' -f zz' = 0, where x''^ + y'^ + z'^ — 0, and since any parallel
plane passes through the same line at infinity, we see that
a'^ + /3^ + 7'' == 0 is the condition that the plane ax + ^y + 72; + 8
shall pass through one of the tangent lines to the imaginary
circle at infinity through which all spheres pass. And therefore
a"*' + fi'^ -f 7'' = 0 may be said to be the tangential equation of
this circle. The invariants formed with a^ -f ^'^ + 7'' and the
tangential equation of the surface are
0 = A' (a + Z^ + c), * = A [he -f + ca - / + ah - li')^
the geometrical meaning of which has been stated in the last
article.
The condition that two planes should be at right angles
viz. aa' + /3/3' + 77' = 0 (Art. 29), being the same as the con-
dition that two planes should be conjugate with regard to
a'^ + yS^ + 7'', we see that two planes at right angles are con-
jugate with regard to the imaginary circle at infinity ; or, what
is the same thing, their intersections with the plane infinity
are conjugate in regard to the circle.
212. In general, the tangential equation of a curve in space
expresses the condition that any plane should pass through one
of the tangents of the curve. For instance, the condition
SYSTEMS OF QUADRICS. 185
(Art. 80) that the intersection of the planes ax + ^y + 'yz + 8wj
ax+^'y + jz + B'iu should touch a quadric, may be considered
as the tangential equation of the conic in which the quadric
is met by the plane ax + /S'^/ + jz + B'lo.
The reciprocal of a plane curve is a cone (Art. 123), and since
an ordinary equation of the second degree whose discriminant
vanishes represents a cone, so a tangential equation of the second
degree whose discriminant vanishes represents a plane conic
From such a tangential equation Aa^ + J5/3* + &c. we can derive
the ordinary equations of the curve, by first forming the reci-
procal of the given tangential equation according to the ordinary
rules, [BCD + &c.)x^ -\- &C.J when we shall obtain a perfect
square, viz. the square of the equation of the plane of the curve.
And the conic is determined, by combining with this the equation
x'[BG-F') +f{CA - G')+z' {AB- E')
+ 2i/z ( GH- AF) + 2zx [HF- BG) -h 2xy [FG - CH) = 0,
which represents the cone joining the conic to the origin.
213. To find the equation of tlie cone wliicli touches a quadric
U along the section made in it hy any plane ax + /3y +<yz + Sio.
The equation of any quadric touching t/ along this plane section
helng kU'+ {ax + ^y + 'yz + Bw)' = 0, it is required to deter-
mine k so that this shall represent a cone. We find in this
case 4>, 0', A' all = 0. And if we denote by a the quantity
Aa^ -{■ B^^ -\- &c. (Art. 79), the equation to determine k has
three roots =0, the fourth root being given by the equation
kA + o- = 0. The equation of the required cone is therefore
o- U= A [ax ■+ ^y + yz + Sivy. When the given plane touches
Z7, we have cr = 0, Art. 79, and the cone reduces to the tangent
plane itself, as evidently ought to be the case. Under the
problem of this article is included that of finding the equation of
the asymptotic cone to a quadric given by the general equation.
214. The condition cr = 0, that ax -{■ ^y + <yz + Sio should
touch f7, is a contravariant (see Conies, Art. 380) of the third
order in the coefficients. If we substitute for each coefficient
a, a + \a', &c., we shall get the condition that ax-\- ^y + <yz + 8w
shall touch the surface U+XV, a condition which will be of
BB
186 INVARIANTS AND COA^ARIANTS OP
the form cr + Xr + X'^r' + Vo-' = 0. The functions o", o-', t, r'
each contahi a, /3, &c. in the second degree, and the coefficients
of U and V in the third degree. In terms of these functions
can be expressed the condition that the plane ax + ^y + 7^ + Sm;
should have any permanent relation to the surfaces f/, F; as
for instance that it should cut them in sections m, ?;, connected
by such permanent relations as can be expressed by relations
between the coefficients of the discriminant of u + A,y. Thus if
we form the discriminant with respect to \ of o- + X.T + \V + X.'cr',
we get the condition that ax + /3i/ + <yz + 8iv should meet the
surfaces in sections which touch ; or, in other words, the con-
dition that this plane should pass through a tangent line of the
curve of intersection of U and V. This condition is of the
eighth order in a, /?, 7, S, and of the sixth order in the coeffi-
cients of each of the surfaces. Thus, again, t = 0 expresses the
condition that the plane should cut the surfaces in two sections
such that a triangle self-conjugate with respect to one can be
inscribed in the other, &c.
The equation cr = 0 may also be regarded as the tangential
equation of the surface Z7; and, in like manner, r = 0, t' = 0
are tangential equations of quadrics having fixed relations to
U and V. Thus, from what we have just seen, t = 0 is the
envelope of a plane cutting the surface In two sections having
to each other the relation just stated. And the discriminant of
cr + Xt + A,V + A,V Is the tangential equation of the curve of
intersection of f/'and V.
Or, again, a- = 0 may be regarded as the equation of the
surface reciprocal to U with regard to x' + y^ + z' + w^ (Art. 127).
And, In like manner, tr + \t -f XV + X^a Is the equation of the
surface reciprocal to U+XV. Since, If \ varies, V + XVde-
notes a series of quadrics passing through a common curve,
the reciprocal system touches a common developable, which is
the reciprocal of the curve UV. And the discriminant of
ct + Xt + XW + X'V may be regarded at pleasure as the tan-
gential equation of the curve C/F, or as the equation of the
reciprocal developable. This equation is, as was remarked
above, of the eighth degree In the new variables, and of the
sixth in the coefficients of each surface.
SYSTEMS OF QITADRICS. • 187
When A=(), a is the square of a linear function of a, ^S, 7, S;
and when the surftice consists of two planes it is easily seen by-
putting in the values of the coefficients, that each first minor
of A vanishes, and therefore in this case a vanishes identically.
215. We can reciprocate the process employed in the last
article. If o- = 0, o-' = 0 be the tangential equations of two
quadrics, we can form the equation in ordinary coordinates
answering to a + A,o-'. This will be of the form
A"'Z7+XAr+VA'r + X'A"^7=0,
and will represent a system of quadrics all touching a common
developable, whose equation is found by forming the discri-
minant of the equation last written. Thus, for example, using
the canonical forms, let
U= ax' + hi/ + cz' + dw\ r= aV + J'/ + cV + d'w'' ;
then o- = Arx^ + ^/3^ + Ci' + Dh\ a = A'o^ + B'^^ + (7 V + D'h\
where A = bcd^ B = cda^ &c., and the reciprocal of o- + A-cr' Is
[BCDx^ + &c.| + \ [{BCD' + CDD' + DBG') x' + &c.}
-f V{{B'C'D + C'D'B + D'B'C) ^'+ &c.} + X'[B'C'D'x'+&c.]=0.
Putting in the values for B, C, D^ &c., we find
BCDx' + &c. = A'U,
while the coefficient of \ is
A {aa' {b^cd + c'd'b + d'b'c) x' + &c.}.
Just as all contravariants of the system o-, o-' can be ex-
pressed in terms of two fixed contravariants t, t together with
cr, o-', so all covariants of the system t/, V can be expressed in
terms of the two fixed covariants 2\ T' together with Z7, V and
the invariants (Art. 200). Reciprocating what was stated in the
last article we can see that the quadric T is the locus of a point
whence cones circumscribing TJ and V are so related that three
edges cf one can be found, which form a self-conjugate system
with regard to the second, and three tangent planes of the second
which form a self-conjugate system with regard to the first.
If we please we may use instead of T and T' the quadric 8^
which is the locus of the poles with respect to V of all the
188 INVARIANTS AND COVARIANTS OF
tangent planes to ZZ, and S' the locus of the poles with respect
to Uof all the tangent planes to F (see Ex. 10, Art. 121). By
the help of the canonical form we can see what relations connect
S and S' with T and T'. Thus we easily find
S = hcda^'x' + cdah'Sf 4 dabe£' + ahcd:\o\
But T = aa' {hcd' + cdV + dhc') cc' + &c.
= {hcda + cdah' + ddbc + ahcd') [ax'' + &c.) — {bcda'^x^ + &c.),
hence T' = QV- S, and in like manner T= @'U-S\ It ap-
pears thus that Z7, /S", T have a common curve of intersection.
Ex. 1. To find the locus of a point whose polar planes with respect to U touch
U + XV. We have then in a- + \t + X-t' + X^<t' to substitute Z/,, U^, U^, U^ for
^t /3i y, 5- The result is expressible in terms of the covariants by means of the
canonical forms U = x^ + y"^ + z- + lo^, F = wa;^ + by'^ + cz^ + dw"^. For the result is
x^ + &,c. + X{{b + c + d) x'- + &c.} + X" {{be + cd + db) x- + &c.} + A.» (Jcc^a;'^ + &c.) = 0,
or AU +X{QV - /\V) ^X^i^U - T') + X^e'U- T) = 0.
In like manner the locus of points, whose polar planes with respect to V touch
U+XV, is
S=eV- T' + X{i>V- T) + X^{Q'V- A'U) +\3A'F=0.
Ex. 2. To find the locus of a point whose polar planes with respect to U and V
are a conjugate pair with regard to U+XV. In the same manner that the con-
dition that two points should be conjugate with respect to Fis ax'x" + by'y" + &c. = 0,
BO the condition that two planes should be conjugate is Aaa' + Bj3jy + &c. = 0.
Applying this to the case where a, (i are Z7,, U^, &c , we get for the canonical form
ax" + &c. + X{{b + c+d)ax' + &c.} + X- [(be + cd + db) ax- + &c.} + X^ahcd (a;^ + &c.)
or AV+XT' + \"-r+X^A'U=0.
Ex, 3. To find the discriminant of T. Ans. A A' {B'2$ _ a' (69' - AA')}.
216. What has been stated in the last article enables us
to write down the equation of the developable circumscribing
two given quadrics Z7, V. We have seen that its equation is
the discriminant of the cubic A'U-^ XATi-X'A'T i-X'A''Vj
where if
U= ax' + hf + cz' + dw\ T= aa [Vcd + c'd'h + dih'c) x' + &c.
Clearing the discriminant of the factor A'^A''^, the result is
27AWZ7'F' + 4A'C7r' + 4AFr= T'T"' -{■ l^AA'TTUV^
an equation of the eighth degree in the variables, and the tenth
in the coefficients of each of the quadrics. By making U= 0,
we see that the developable touches U along the curve UT^
SYSTEMS OF QUADJIICS. 189
and that it meets U again in the curve of intersection of U
with T'^ — 4AFT. We shall presently see that the latter locus
represents eight right lines, real or imaginary generators of the
quadric U.
It is otherwise evident what is the curve of contact of the
developable with U. For the point of contact with U of a.
common tangent plane to UV is the pole with regard to U
of a tangent plane to F, and therefore is a point on the surface
S' ; and we have proved, in the last article, that the curves
US\ TU^re the same.
The section of the developable by one of the principal planes
(iv) is most easily obtained by reverting to the process whence
the equation was formed. The common tangent developable
of x^ + if ■\-z^ •\- w\ ax^ -f hy^ + cz^ + dw'^ is the discriminant of
ax^ hii^ cz^ dw'^
+ r^ + .— +; 7=0.
A, + a X-\-b X + c \-[-d
Hence, as in Art. 202, Ex. 2, if we make lo = 0, the discriminant
will be
ax^ 1)7^ cz^ Y"*
\a — d h — d c — dj ^
multiplied by the discriminant of
ax^ hi/ cz*
\-{- a \ + b X + c
In order to obtain the latter discriminant, differentiate with
regard to X, when we have
ax' b£_ cz" _aV_ hY c^z'
{\-\-af ^ (X. + ^y^ ^ {\^cf~ ' i^-Vaf ^ (A, + 6/ "^ (^Tf^^"^'
, ax'' . In/ cz'
whence = b- c, ,^ '^ , , = c-a, ~ = a-b:
[x + ay ' {X + by ' (x + cy '
and, substituting in the given equation, the result is
X \/[a[b — c)} ±y sj[b (c- a)] ±z V[c(a- J)] =0.
The section therefore is a conic counted twice and four right
lines.
217. To find the condition that a given line should pass
through the curve of intersection of two quadrics U and V.
{Suppose that we have found, by Arts. 80, &c., the condition,
190 INVARIANTS AND COVARIANTS OF
^ = 0, that the line should touch Z7, and that we substitute in it for
each coefficient a, a+A,a', the condition becomes 4'+X4'j+V4''=0;
and it" the line have any arbitrary position, we can by solving
this quadratic for X,, determine two surfaces passing through
the curve of intersection C/F and touching the given line. But
if the line itself pass through Z7F, then it is easy to see that
these two surfaces must coincide, for the line cannot, in general,
be touched by a surface of the system anywhere but in the
point where it meets UV. The condition therefore which we
are seeking is ^^^ = 44'4''. It is of the second order in the
coefficients of each of the surfaces and of the fourth in the
coefficients of each of the planes determining the right line :
these (see Art. 80) enter through the combinations ct/S' — a'/3,
&c., viz. the equation contains, and that in the fourth degree,
the six coordinates of the line of intersection of the two planes.
In the case where the two quadrics are ax^ + hy^ + cz^ + dw^^
a^ + &y + cV + d'w^^ and the right line is aa; + /Sy + 72; + S?^,
aa; + /3'?/ + 7'2; + S'w;, the quantity 4' is (see Art. 80) '2.ab{^l' -^l)^^
by which notation we mean to express the sum of the six terms
of like form, such as cd (oc/S' — c^^)\ &c. When the line is
expressed by its ray coordinates (p. 40) the relation which holds for
contact is hep' + ca(f-\- ahr^+ ads' + hdf-\- cdu = 0, which is satis-
tied by each of the complex of lines which touch the quadric Z7(see
Art.80(?). Then 4', is S(Z)c'+Z>'c)p'^,and its vanishing is the relation
for the complex of all lines which are cut harmonically by the
quadrics f7and F, as it is easily seen that *, = U' W+ U" V'-2PQ
in the notation of Art. 75. Also ^^^ - 44'4'' is
S {bc')Y + 22 [be) {ac)pY + 22 {[ab') {cd'] + {ac') {bd')] pV,
and vanishes for the complex of right lines intersecting the
common curve.
218. To find the equation of the developable generated by the
tangent lines of the curve common to U and V.
If we consider any point on any tangent to this curve, the
polar plane of this point with regard to either U or F passes
evidently through the point of contact of the tangent on which
it lies. The intersection therefore of the two polar planes
meets the curve UV. We find thus the equation of the
SYSTEMS OF QUADRICS. 191
developable required, by substituting in the condition of the
last article, for a, yS, &c., a', /S', &c., the differential coefficients
Z/j, ZJ^, &c., Fj, F^, &c. This developable will be of the eighth
degree in the variables and of the sixth in the coefficients
of each surface. When we use the canonical form of the
quadrics, it then easily appears that the result is
2 [ahj {cd')* zW + 22 [ah') [ac') {cdj [hdy^z^'w* + 2xyzW
x[{aV)[cd')-{ad'){lcyi[{ad')[hcY\hd')[ca')][{^^^^^
When we make in the above equation w = 0 we obtain a perfect
square, hence each of the four planes cr, y, s, w meets the de-
velopable in plane curves of the fourth degree which are double
lines on the surface.* This is, a priori^ evident since it is plain
from the symmetry of the figure, that through any point in
one of these four planes through which one tangent line of
the curve C/F passes, a second tangent can also be drawn.
By the help of the canonical form the previous result can
be expressed in terms of the covariant quadrics when the de-
velopable is found to be
4 (0 UV- r U- A V) (0' UV- TV- A' U') = (* UV- TU- T V]\
The curve W is manifestly a double linef on the locus re-
presented by this equation, as we otherwise know it to be, and
the locus meets TJ again in the line of the eighth order deter-
mined by the intersection of TJ with T'^ — 4ATF. This is the
same line as that found in Art. 216.
* See Cambridge and Dublin Mathematical Journal, vol. ill., p. 171, where, though
only the geometrical proof is given, I had arrived at the result by actual formation
of the equation of the developable. Pee Ibid, vol. ir., p. 68. The equations were
also worked out by Mr. Cayley, Ibid, vol. v., pp. 50, 51.
t It is proved, as at Higher Plane Curves, Art. 51, (see also Art. 110 of this volume)
that when the equation of a surface is U"<p + UV\}/ + V'x = 0, then UV is a double
line on the surface, the two tangent planes at any point of it being given by the equation
u-(f>' + uvxp' + r'x' = 0> where u, v are the tangent planes at that point to U and V,
and <j)' is the result of substituting in <f> the coordinates of this point, &c. Applying
this to the above equation it is immediately found that the two tangent planes are given
by the equation {TU — T'V)- = 0, where in T, T' the coordinates of the point are
supposed to be substituted. Thus the two tangent planes at every point of the double
curve coincide, and the curve is accordingly called a cuspidal curve on the surface.
192 INVARIANTS AND COVARIANTS OF
219. We can shew geometrically (as was stated Art. 216)
that a generator of the quadric U at eacli of the eight pointa
of intersection of the three surfaces ZJ, F, 6", (or C/", F, T) is
also a generator of the developable, and that therefore these
eight lines form the locus of the eighth order, Z/, T'^ — ^^TV.
For the surface S' being the locus of the poles with regard
to U of the tangent planes to F, the tangent plane to F at
one of the eight points in question is also a tangent plane to Z/,
and therefore passes through one of the generators to U at the
same point. This generator is therefore the line of intersection
of the tangent planes to U and F, and therefore is a generator
of the developable in question.
220. The calculation in Art. 218 may also be made as
follows : When we write in the determinant of Art. 80 for a,
a + Xa &c., and for a, /3 &c. Z7j, U^ &c., for a', /S' &c. Fj, F^ &c.,
we can reduce It by subtracting from the first column the sum of
the third multiplied by a;, of the fourth, fifth, and sixth multiplied
respectively by ?/, z^ and w^ and then, removing the terms — A, F, &c.
in the first column by means of F, &c. in the second ; when
we deal similarly with the rows, the determinant becomes
where —S\s the value of the determinant of Art. 79, when
a &c. are replaced by a + \a &c. and a &c. by F^ &c. But
the last result of Ex. 1, Art. 215, determined the value of S.
Putting in that value we find, as it should be, that \ occurs
in no higher power than the second, and the determinant
becomes
[QUV-rU- I^V')+\{<^UV-TU- TV)
+ V {&UV- TV- A'U') = 0.
Thus then we see that QUV= TU+AVis the condition
that the intersection of the two polar planes should touch U]
while <i>UV= TU+ TV is the condition that it should be cut
harmonically by the surfaces Z7, F; and again the equation of
the developable is
4 (0Z7F- T'U-AV) [&UV-TV-L'U')=[^UV-TU- TVf,
SYSTEMS OF QUADRICS. 193
220a. The equation of this developable has been otherwise
derived by Mr. W. R. Roberts as follows : When the line
whose ray coordinates are p, g-, r, s, t, u is a, generator of
ax^ + hf + cz^ + dw' = 0,
we have (Art. 80c)
0= c<f+hr''^ds' ,
0 = cp*^ + ar' 4- df ,
0 = hp'' + a(f + du^^
0 = as' -I- hf + cu"^
which are equivalent to the four equations
OG CCt Q/0
Now a generator of any one of the system of quadrics
through the curve common to U and F is a line which meets
that curve in two points ; hence the line whose coordinates
are related as follows :
,_ ^{a-^\a){d+\d') ,_ ^[h + \'b'){d^\d')
^' ~^ [h^W) (c+ Xc')' ^ "^ {c-\-\c)[a + \a)'
^'="'|twtw) ' (''+^<'>''+(*+ ^^'''■'+(^+^''>=»'
Is a generator of Z7+\Fand a chord of the curve of intersection of
U= ax' + Inf + cz" + did' = 0,
V= ax' + by + c'z' + d'to' = 0.
220J. Again, when a line touches the curve UV^ it touches
both Z7and F, hence, in this case
hep' + ca(f + ahr' + ads' + hdt' + cdu' = 0,
Vc'p^ + c'a'q^ + a'b'r' -f a'd's' + h'd'i' + c'du' = 0,
therefore by the fourth relation In last article
[led' -f Wed) f + [cad' + \cad) (f + {aW + \ah\l) r' = 0,
or, replacing f'^ <f^ r'^ by their values In s^, t'^ ii',
{bed' + We'd] [a + \a7 s' + {cad' + Xe'a'd) {b + Wf f
+ \abd' + Xa'b'd) (c + XcJ u' = 0,
CC
194 INVAKIANTS AND COVARIANTH OF
solving between this and
(a + \a) s' +[b + W) f + (c + \c) u' = 0,
we get s\ i\ u\ and accordingly also jp^ q\ r^.
Omitting a common factor, the results may be written
- / = {ic) (ad') {a + \a) [d + Xd'},
q' = [ca) [hd') [h + \¥) {d + \d'],
r'=[ab') {cd') (c + \c) [d + \d'),
s* = {W) {ad') [b + W) [c + \c),
f = (ca) [bd') [c + Xc) (a -t- Xa),
v' = {a¥) {cd') {a + Xa) {b -f Xb') ,
and evidently admit of]js-i-qt + ru = 0 being identically satisfied.
220c. From these expressions in the parameter X, for the
coordinates of any generator, the equation of the developable
may be found in ordinary coordinates by the usual method.
For any point on the line we must have, for instance,
px + q7/ + rz = 0,
but we have also U-j- XV =0, hence the surface Is
X {{be') {ad') {aV-a'U)]i + y[{ca') {bd') {bV- b'U)]^
+ z{{ab'){cd'){cV-cU)]i=^0,
and the section by the plane 2: = 0 is seen at once to be a
double curve which is a trinodal quartic ; and similarly for the
other planes of reference. Again, this equation of the surface
evidently, on rationalisation, becomes of the form
U'<i>-\-UVyjr-\-V'x,
whence UV is a double line on it; also, making U=0,^/V
becomes a factor, and the eight right lines forming the remain-
ing intersection with f/are at once found.
220^. If the line pqr, &c. be contained in the plane
a.x-\- ^y + 72; + Zw = 0 its coordinates satisfy as + j3t -\- <yu = 0 &c.
(Art. !Jlb). If the consecutive line also lie in this plane
ds „ dt du
SYSTEMS OP QUADKICS. 195
By these, determining a, y3, 7, It is seen that the following
are symmetrical expressions fur the coordinates of the plane
of two consecutive generators of the developable, or of two
consecutive tangents to the common curve UV, omitting a
common factor,
a' [ah') (ac) [ad') = {a + \a)%
/3^ [be) [bd') [ba) = (J + \b')%
ry' icd') [ca] [cb') = (c + \c')%
h'{da')[db') [dc) =[d + \d')%
also the expressions
x^ [ab') (ac) [ad') = a + /j.a\
if [be') {bd') [ba') = & + fib',
z' [cd') [ca') [cb') =c-\- fic\
ic^ [da') [db') [dc) = d+nd',
are easily seen to be those for the coordinates of any point on
the curve UV.
221. The equation ax^+bi/'+ cz^+ \ [x^-{-y^-\- z^) = 1 denotes
(Art. 101) a system of concentric quadrics having common
planes of circular section. And the form of the equation shews
that the system in question has common the imaginary curve
in which the point sphere x^ + f + z"^ meets any quadric of the
system. Again, since the tangential equation of the system
of confocal quadrics
222
X y z
it follows reciprocally that a system of confocal quadrics is
touched by a common imaginary developable (see Art. 146) ;
namely, that enveloped by the tangent planes drawn to any
surface of the svstein, throu2:h the tanojeut lines to the Ima-
ginary circle at infinity. The equation of this developable is
found by forming the discriminant with regard to \ of the
196 INVARIANTS AND COVARIANTS OF
equation of the system of quadrics. If we write b-c=pj
c— a = qj a—b = r^ the equation is
(«' + ^f -f zy (pV + qY + rV - 2^/-yV - 2rp3 V - ^pqx'if)
+ 2y [q - r) x^ + 2^* {r -qy) if + 2/' {p-q)z''
+ 2p (^r - Sj'') a;*^'- 2^ (^z* - 3p^) a;'/- 2p [pq - 3r') x*z*
+ 2r {qr - Sp") a;V + 2q {qp - 3r') y'z^ - 2r {rp - Sq') z\f
■\-2[p— q){q — r) (/• -p) x'y^z' + [p* — Qp'qr) x*
+ {q* — Gq^pr) y* + (r* — Qr'^pq) z* ■+ 2pq [pq — Sr'"*) x^y^
+ 2qr [qr — Zp^) y^z^ + 2rp [rp — 2>q^) sV^ + 2p^qr (r - q) x^
+ 2(frp [p — r)y'^-\- 2r^pq [q —p) ^ -\-p^c[r'^ = 0.
It may be deduced from this equation, or as in Art. 202,
that the focal conies, and the imaginary circle at infinity, are
double lines on the surface.
222. In like manner, if cr = 0 be the tangential equation of a
quadric, and if we form the reciprocal of cr + X (a'^ + /S'^ + 7^),
we get
A* Z7+ X A [{a (5 + c) - (f - K'] x' 4 (J (c + «) - K' -f] f
+ [c [a + h] -r-cf] z'+{cl[a + &+ c) - I'-m'-n']
+ 27JZ [of- gh) + 2zx [hg - hf) + 2xy [cli -fg)
+ 2x {(J + c) Z - km - gn] + 2y [(c + a) m —fn — hi]
■^2z[[a + h)n- gl -fm]']
+ X'{D[x''-]-7/-]-z')-hA + B+C-2Lx-2My-2Nz]+\^=0.
This is the equation of a series of confocal surfaces, and its
discriminant with respect to X will represent the developable
considered in the last article. If we write the coefficients of
X and X^ respectively T and T', then T= 0 denotes the locus
of points whence three rectangular lines can be drawn to touch
the given quadric, and T' = 0 the locus of points whence three
rectangular tangent planes can be drawn to the same quadric.
x^ li^
If the paraboloid h "4- + 2s; be treated in the same way,
^ a b
we obtain, as the equation of a system of confocal surfaces,
(&a;'+ a/+ 2al>z) + X [x^^if^2 [a+b] z-ab]+ X'{2z - (a+b)] - X'= 0,
SYSTEMS OF QUADRICS. 197
and the developable which they all touch Is, if we write a — h = r^
+ 42 [x' + y') [ax' + hi/) + 16rV + 32rV (a;' -f if)
+ 24r ( Ja:'^+ «/) 2'+ (a^' + 5/)'+ «'' (^•^'+ a/) [^'- f) + I'^r'xY
+ 1 6 (a + J) r'z [x' + / + 2"'') - 1 2r'z [ax' + 5/)
+ Urabz [x' - 7/) + Ar'z^ {a' -f Aab + Z>'^) + 4r' (iV + a'/)
+ 2ahr [ax' - by') + 4.i^ah (a -f J) 0 + a'^> V = 0.
The locus of intersection of three rectangular tangent planes
to the paraboloid is the plane 2z = a-\- b^ and of three rect-
angular tangent lines is the paraboloid of revolution
x' + / + 2 (a + &) s = ah.
223. We shall now shew that several properties of confocal
surfaces are particular cases of properties of systems inscribed
in a common developable. It will be rather more convenient
to state first the reciprocal properties of systems having a
common curve.
Since the condition that a quadric should touch a plane
(A.rt. 79) involves the coefficients in the third degree, it follows
that of a system of quadrics passing through a common curve,
three can be drawn to touch a given plane, and reciprocally,
tliat of a system inscribed in the same developable, three can
be described through a given point. It is obvious that in the
former case one can be described through a given point, and
in the latter, one to touch a given plane. In either case, two
can be described to touch a given line ; for the condition that
a quadric should touch a right line (Art. 80) involves the co-
efficients of the quadric in the second degree.
It is also evident geometrically, that only three quadrics
of a system having a common curve can be drawn to touch
a given plane. For this plane meets the common curve in four
points, through which the section by that plane of every surface
of the system must pass. Now, since a tangent plane meets
a quadric in two right lines, real or imaginary, (Art. 107)
these right lines in this case can be only some one of the three
pairs of right lines which can be drawn through the four points.
198 INVARIANTS AND COVARIANTS OF
The points of contact which are the points where the lines of
each pair intersect, are ( Conies^ Art. 146, Ex. 1) each the pole
of the line joining the other two with regard to any conic passing
through the four points. Hence (Art. 71) if the vertices of one
of the four cones of the system be joined to the three points,
the joining lines are conjugate diameters of this cone.
224. Now let there be a system of quadrics of the form
S+'^[x^ + y' -{-z')^ since oc^ ■{- y"^ + z^ is a cone, the origin is
one of the four vertices of cones of the system. And since
x^^-y'^^-z^ is an infinitely small sphere, any three conjugate
diameters are at right angles, and we conclude that three
surfaces of the system can be drawn to touch any plane, and
that the lines joining the three points of contact to the origin
are at right angles to each other. Moreover as a system of
concentric and confocal quadrics is reciprocal to a system of the
form S-\-\{x^ ■\-y'^-'r z')^ we infer that three confocal quadrics
can be drawn through any point and that they cut at right
angles.
Again (Art. 132) the polar planes of any point with regard
to a system of the form S-\-X[x^ ■'t-y'^ + z^) pass through a right
line, the plane joining which to the origin is perpendicular to
the line joining the given point to the origin ; as is evident
from considering the particular surface of the system «;'' + ?/''' + s^
Reciprocally then the locus of the poles of a given plane with
regard to a system of confocals is a line perpendicular to that
plane.
225. We have seen that cr + \ (a'"' ■+ ^'^ + 7') is the tangential
equation of a system of confocals : and when the discriminant
of this equation vanishes it represents one of the focal conies.
We can therefore find the tangential equation of the focal
conies of a given surface by determining X from the equation
Z)X,' + {he + ca + ah -f - / - K') AX' +[a + h-tc) A'X + A' = 0.
Thus, let the surface be
7x' -f 6/ + 5z' - Ayz - 'ixy + \0x -\- iy i- Gz + i = 0,
we have A = -972, and the cubic is
1 62\'' -f 99X'A + 1 8 A'X + A' = 0,
SYSTEMS OF QUADRICS. 199
whose factors are 3A, + A, 6\ + A, 9A. + A, whence X = 108, 162,
or 324.
The tangential equation of the given surface divided by 6 is
a'-8/3'-ll7'''+27S''+2687 + 4G7a+34ai8-54aS-54/8S-547S = 0.
Thus then the tangential equations of the three focal conies are
obtained by altering the first three terms of the equation last
written Into
1 9a' + 1 0/3' 4 77'', 28a' + 19/3' + I67', 55a' + 46/3' + 437",
respectively. Their ordinary equations are found, as in Art.
212, to be the intersections of
2x-2f/ + z + IV, Ux' + 44/ -f 1 10' - S22/Z + 2zx - iOxT/ ;
X + 2i/ -j- 2z -\- 5w, 67a;" + 68/ -f 83s' - 24j/s - 62sa; - 32a;3/ ;
2x-^?/-2z-\- 10, bx' - 3/ + 9^' + 2ijz -l&zx^ 2xy.
226. In order to find in quadriplanar coordinates the tan-
gential equation of a surf;ice confocal to a given one, it is
necessary to find the equivalent in quadriplanar coordinates to
the equation a' + 8'' -f 7' = 0.* It is evident that if x, y, z, w re-
present any four planes, and if their equations referred to any
three rectangular axes be Xcosyl+ F cosi?+^cos(7 = p, &c.,
then the coefficient of X in ax + /3i/ + jz + Sw is
a cos^ + /S cos^' + 7 cos 4" + 8 cos A'",
and the sum of the squares of the coefficients of X, Y, Z is
a' + /S' + 7"' + S' - 2^37 cos [yz) - 27a cos izx) — 2a/3 cos [xy)
- 2aS cos [xw) - 2/3S cos {yio) — 27S cos izw)^
where [yz^ denotes the angle between the planes y, z, &c.
This quantity, equated to nothing is the tangential equation
of the Imaginary circle at infinity. The processes of the last
articles then can be repeated by substituting the quantity just
written for a' + /3''' + 7*. We thus find, without difficulty, the
condition that the general equation in quadriplanar coordinates
should represent a paraboloid, or either class of rectangular
♦ This condition evidently expresses that the length is infinite of the perpendicular
let fall from any point on any of the planes which satisfy the equation.
200 INVARIANTS AND COVARIANTS OF
Lyperboloid ; the equations of the loci of points whence systems
of three tangent planes or tangent lines are at right angles;
the equations of the focal conies, &c.
227. We have seen (Art. 211) that the condition in rect-
angular coordinates aa' + /3/3' + 77' = 0, that the planes ax + &c.,
a'ic + &c. should be at right angles, expresses that the planes
should be conjugate with respect to the imaginary circle at
infinity. It follows that the condition of perpendicularity in
quadriplanar coordinates is
a' {a — /3 cos [xy) — 7 cos [xz] - S cos {xw)\
+ /S' {— a cos [xy) + /S - 7 cos [yz) — S cos (?/w?)} + &c. = 0.
Any theorems concerning perpendiculars may be generalized
projectively by substituting any fixed conic for the imaginary
circle at infinity ; and thus, instead of a perpendicular line and
plane, we get a line and plane which meet the plane of the
fixed conic in a point and line which are pole and polar with
respect to that conic (see Conies^ Art. 356). The theorems may
be extended further (see Conies^ Art. 385) by substituting for the
fixed conic a fixed quadric, when instead of a line perpendicular
to a plane, we should have a line passing through the pole of
the plane with regard to the fixed quadric. These latter ex-
tensions, however, are theorems suggested, not proved.
Ex. Any tangent plane to a sphere Any plane section of a quadric is met
is perpendicular to the corresponding in a conjugate line and point, by any
radius. tangent plane and the line joining its
point of contact to the pole of the plane
of section.
228. The tangential equation of a sphere, in rectangular
coordinates, is written down at once by expressing that the
distance of the centre from any tangenfplane Is constant. The
equation Is therefore
[ax' + /33/' + 70' + hf = r'' [d' + ^'' + 7^).
If then x\ 3/', z\ to' be the coordinates of the centre of a
sphere, the tangential equation of the sphere In quadriplanar
coordinates must be
[ax' + ^y' + r^z' + Zio'f = r' {a*+ /3"''+ 7'=+ 5'-'- 2a^ cos [xy) - &c.}.
SYSTEMS OF QLADRICS. 201
If the sphere touch the four pKanes a;, ?/, s, to, the coefficients
of a'^, ^'\ <y\ B^ must vanish, and the tangential equation of
such a sphere must therefore be
{a±^±y ±8)^ = a' + 13' -i- y' + S' -2oL^ cos (xi/) - &c.
There are therefore eight spheres which touch the faces of a
tetrahedron. Taking all positive signs, we get the tangential
equation of the inscribed sphere
/S7 cos''* 5 (yz) + 7a cos^l [zx] + a/3 cos* | [xy)
+ aS cos""*^ [xiv] + /3S cos'"*^ (?/zo) + yS cos^^ [zw) = 0.
The corresponding quadriplanar equation is obtained from this
as in Art. 208.
229. The equation of the sphere circumscribing a tetra-
hedron may be most simply obtained as follows : Let the
four perpendiculars on each face from the opposite^ vertex be
^0? ^oi ^0? ^^0- Now the equation i'n i^lano of the circle circum-
scribing any triangle ohc may be written in the form
(M'^^ {caYzx {ahyxy ^^
y,% ^o«^o a^o^/o '
where a?, cc^, &c. denote perpendiculars on the sides of a triangle
the lengths of which are (Z)c), &c. But it is evident that for
any point in the face w, the ratio x : x^ is the same whether
x and x^ denote perpendiculars on the plane x or on the line
xw. We are thus led to the equation required, viz.
{bcf yz [ca^ zx [ah)'^ xy [adYxw [hdfyw {cd)'^zw
~yJ7 ^^ ^o2/o a^o^o 2/0^0 ^o^^o
For this is a quadric whose intersection with each of the four
faces Is the circle circumscribing the triangle of which that
face consists. If this equation be reduced to rectangular co-
ordinates it will be found that the coefficients of a:^, y\ z^ are
each =—1. Hence if we substitute the coordinates of any
point, we get - the square of the tangent from that point to
the sphere.
Cor. The square of the distance between the centres of the
inscribed and circumscribing spheres is
1 3/0^0 ^0-^0 a^o3/o ^0^0 ^o^'o V^oJ*
D D
202 INVARIANTS AND COVARIANTS OF
280. The equation of any other sphere can only differ from
the preceding by terms of the first degree, which must be of
f nr II z 77? \
the form [ax + ^ii -f 7.^ -f S?t?) V ^ A h — ) , the second
factor denoting the plane at infinity (Art. 57). If then we add to
the equation of the last article the product of these two factors,
identify with the general equation of the second degree and
eliminate the indeterminate constants, we obtain the conditions
that the general equation of the second degree in quadri planar
coordinates aa;^-f b?/*''+ &c. may represent a sphere, viz.
{hcf [caf [ah)''
[adf [bdf [cdf
231. It was shewn (Art. 214) that by forming the con-
dition that ax -\- ^y + fyz -{■ hw should touch Z7+XF, we get
an equation in X whose coefficients are the Invariants in
piano A, A', 0, 0' of the sections of U and V by the given
plane. It was also shewn [Conies, Art. 382) that if we form
the Invariants of any conic and the pair of circular points at
infinity, 0 = 0 Is the condition that the curve should be a
parabola, 0' = 0 the condition that it should be an equilateral
hyperbola, and 0"* = 40 the condition that the curve should
pass through either circular point at infinity. Applying then
these principles to any quadric in rectangular coordinates and
the tangential equation of the imaginary circle a^+ /S^ + 7*,
we get for the condition, 0 = 0, that any section should be
a parabola,
(be -D a- + [ca - cf) ^' + [ah - ¥) 7^
+ 2{cjli-af)^y + 2[lif-hg)^a + 2[fcj-c]i)a^ = 0',
for the condition 0' = O that It should represent an equilateral
hyperbola
[h + c) a"-' + (c + a) /3' + [a + i) 7" - 2//3y - 2gya - 2ha^ = 0,
while 0"^ = 40 (a^ + ^'^ + 7'"') Is the condition that the plane
should pass through any of the four points at Infinity common
to the quadric and any sphere.
SYSTEMS OF QUADRICS. 203
232. We know from the theory of conies that If cr = 0 be
the tangential equation of a conic, and a = 0 the tangential
equation of the two circular points at infinity In its plane,
<T + \a = 0 Is the tangential equation of any confocal conic.
Now the tangential equation of the pair of points where the
Imaginary cIrclea''' + /S^+7''Is met by the plane ax-\-^'y^'^'z-\- h'w
is evidently {al'' + ^"' + 7") (a' + /3''' + 7') - (aa + y8^' + 77')' = 0.
Thus then the tangential equation of all conies confocal to the
section by a'a; + ^'y + 7'^ + I'xo of a^^ + hy^ + cz^ + dw\ Is
d' [[am" + dhy''^ + hch'') + \ (/3"^ + 7'^)}
+ ^' [ [cda" + day'' + ach") + X [a." + y'')] .
+ 7"'' {{bdd' + da^"-]- aW) + X [a" + /3'0}
+ S^ [hca" + ca/3'^ + a J7"'') - 2 (a J + \) /3 7'/S7
-2{bd+ X) yaya - 2 (ct/ + X) a/3'a/3
- 2hcaB'aS - 2ca/3'S'/3S - 2ahy8'ryS = 0.
If we form the reciprocal of this according to the ordinary
rules, we get the square of ax + ^'y + yz + h'w multiplied by
S' + \20' + X' (a' + /3' + 7') 0 where 2 Is the condition that
a' J7 4- /3'v/ + 7'.3 + S't(? should touch the given quadric, and 0', 0
have the same signification as In the last article. By equating
the second factor to nothing we obtain the values of \ which
give the tangential equations of the foci of the plane section
in question.
Ex. 1. To find the foci of the section of 4x- + ?/- — 42- + 1 by a; + ?/ + z. The
equation for X is found to be SX^ + 2\ = 16, whence X = 2 or = — §. The equation
of the last article, for the values a' = /3' = y' = 1, and the given values of cr, h, c, d, is
a' (- 3 + 2\) + 2X/32 + (5 + 2X) y2 _ 16^2 _ 2 (4 + X) /3y - 2 (1 + X) 7 a + 2 (4 - X) a/J = 0.
Substituting X = 2 it becomes (a + 2/3 — Sy)^ — 165^, whence the coordinates of the
foci are + i, i h + h '^^^ other value of X gives the imaginary foci.
Ex. 2. To find the locus of the foci of all central sections of the quadric
ax^ + by- + cz- + 1. Making 0' = 0, the equation for X is found to be
_fL:_ + _^ + jyi. = 0.
a 4- X 6 + X c + X
By the help of this relation the tangential equation of the foci is reduced to the form
( ««' /3/3' jyylY_ f>ra'- + ear- + ah'- g. ^ p
V« + X l) + \ c + \J {u + Xj (6 + XJ (c + X)
Thus then the coordinates of the foci are
_ _a^ _ _£_ _ j/_ „ _ 6c-a'2 + cafr- + ahy'"-
"^ ~ rt + X ' ^ ~ 6 + X ' ^ ~ c + X ' '''■' ~ (a + X) (* + X) (c + \) •
204 INVARIANTS AND COVARIANTS OF
Solving for a', /3', y' from the first three equations and substituting in the equation
for X, we get
{ax^ + hif + cz^) + \ (.-c2 + / + 32) = 0 5
solving for \ and substituting in the value for itfl, we get the equation of the locus, viz.
= vfl {{a -b)y-^+{a- c) z'] {{b - c) z" + {b - a) x^] {[c - a) x" + [c - b) f],
a surface of the eighth degree having the centre of the given quadric as a multiple
point.
The left'hand side of the equation may be written in the simpler form
{x? + 2/2 + 2.2) (Q,a:2 + by^ + cz^) {a {b - cf y^z^ +l{c- a)"- zV + c (« - by x^f\.
For a discussion of this surface see a paper by M. Painvin, Nouvelles Annates,
Second Series iii. 481,
From the property that if a point be a focus of a plane section of a quadric,
the plane is a cyclic plane of the tangent cone from the point ; Mr. M' Cay
writes down immediately this locus in the coordinate system of Art. 160.
In fact the equation of the tangent cone (173) being
a-2 y2 22
a"- - a'i "^ a2 - a"2 "^ a" - o'"* ~ '
one of its pairs of cyclic planes is
a'2-a"2 „
7.2 - n"i
X^
But, for central sections, since the coordinates of the centre satisfy this equation,
we may replace x by p' and z by p'", Art. 165. Substituting these values, we get
a'2J'2c'2 a"'W'"-c""'
fi - n!'-
a'- — a -
.(1).
It is easily derived from this by the cubic equation of Art. 158, taking a- — a"^ = X^^
ffl - a"'2 = k2, and n^ the third root, that ^i2 = -^ — - , where p^ =^x'^+ y" + z^, and
o 4- 1
2;2 ^2 g2
/S = — +-7-2 + — — 1; and this value of /u2 substituted in the cubic gives an equation
of the eighth degree in a-, y, z as above. It is similarly seen that each side of (1), also
_ a"'^b"V2
~ a2 - o"2 •
Ex. 3. To find the locus of foci of sections parallel to an axis (say a' = 0). The
equation which must break up into factors is in this case
a2 {{c + \) /3'2 + {b + \) y'2 + bcd'^] + /a2 {{a + \) y'2 + acr-} + y^ ((« + X) /3'2 + «J5'2}
+ d-a {c^'"- + Jy'2) _ 2 (a + X) /S'y'/Jy - Ica^'c'ftd - 2aby'd'y6 = 0.
The condition that the resolution into factors shall be possible is
(a + X) (Sy'2 + c/3'2) + abco'^ =r 0.
Subject to this condition the equation becomes
whence /3' = by, y' =: cz, ad' = (« + X) io, substituting which values in the equaticai
of condition we have {a + X) w'^ + acz- + aby^ = 0 ; whence again substituting in
he {a + -\) x"--{c + X) /?'2 + {b + X) y'2 + beo'^,
SYSTEMS OF QUADRICS. 205
we get for the required locu3
{bi/ + cz^) {V^ (a -c) f- + c"' {a -h)-<? - nbcx'} = vT- [IT- [a - c) f + c^ {a - h) z"~].
It is obvious that the methods of this and the preceding article can be applied to
equations in quadriplanar coordinates.
233. Given four quadrics the locus of a point whose polar
planes with respect to all four 7neet in a point is a surface of
the fourth degree^ which we call the Jacobian of the system, of
quadrics (see Conies^ Art. 388). Its equation in fact is evidently
got by equating to nothing the determinant formed with the
four sets of differential coefficients C7,, U^^ ZJ^, U^ ; F,, F^, &c.
It is evident that when the polars of any point with regard
to U^ F", TF, T meet in a point, the polar with respect to
X C/"-!- /A F+ v IF -t- TT jT will pass through the same point. The
Jacobian is also the locus of the vertices of all cones which
can be represented by X.Z7+ /iF4- vW-\- ttT. Thus, then, given
six points the locus of the vertices of all cones of the second
degree which can pass through them is a surface of the fourth
degree. For if 7", t/, F, IF be any quadrics through the six
points, every quadric through them can be represented by
XU+ /jlV+vU+ttT^ since this last form contains the three
independent constants which are necessary to complete the
determination of the surface. It is geometrically obvious that
this quartic surface passes through each of the fifteen lines join-
ing any two of the given points, and also through each of the
ten lines which are the intersections of two planes passing
through the given points.
If in any case \U+ /jbV+ vW+irTcKn represent two planes,
the intersection of those planes lies on the Jacobian.
If the four surfaces have a common self-conjugate tetra-
hedron the Jacobian reduces to four planes. For let the
surfaces be ax^ 4 hf'' + cz^ + dw\ ax^ + Vy'^ + &c., &c., then we
have f/j = ax^ F, = a'cc, &c., and it is easy to see that the
Jacobian is xyzw multiplied by the determinant [ab'c'd'").
If one of the quantities t/" be a perfect square X^, Z is a
factor in C^, C/, &c., and the Jacobian consists of a plane and
a surface of the third order. If the surfaces have common
four points in a plane, it is evident geometrically that this
plane is part of the Jacobian ; and if they have a plane section
206 • INVARIANTS AND COVARIANTS OP
common to all, this plane counts doubly in the Jacobian, which
is only a surface of the second degree besides. Thus the
Jacobian of four spheres is a sphere cutting the others at
right angles.
Cor. If a surface of the system \U+ /jlV+vW touch T,
the point of contact is evidently a point on the locus considered
in this article, and therefore lies somewhere on the curve of
intersection of T with the Jacobian. Again, if a surface of
the system XU+ fj,Viouch the curve of intersection of J', TF;
that is to say, if at one of the points where X U+ ix V meets
T, TF, the tangent plane to the first pass through the inter-
section of the tangent planes to the two others, the point of
contact is evidently a point on the Jacobian of the system.
It follows that sixteen surfaces of the system X Z7+ /* V can be
drawn to touch '1\ W\ for since three surfaces of degrees
m^ w, p meet in mnp points, the Jacobian, which is of the fourth
degree, meets the intersection of the two quadrics T^ W in
sixteen points.
234. To reduce a pair of quadrics U, V to the carionical
form x^ \ if' ■]: z^ ^ w^ ^ ax^ -\- hy'' -^ cz'' ■\- dvo'^ . In the first place
the constants a, 5, c, d are given by the biquadratic
AX* - 0X' + ct>\'' - 0'\ + A' = 0.
Then solving the equations
x' + 2/' + •■2' + ^o' =U^ a [he ^cd + db) x' -|- &c. = T^
a{l)-[ c + d) x' + &c. = T\ ax' + &c. = T,
we find x\ ?/, z'^^ ic\ in terms of the known functions Z7, V
T, T'. Strictly speaking we ought to commence by dividing
V and V by the fourth root of A, in order to reduce them to
a form in which the discriminant of U shall be 1. But it will
come to the same thing if leaving U and V unchanged we
divide by A, T and T' as calculated from tlie coefficients of
the given equation.
Ex. 1. To reduce to the canonical form
5x2 _ 11,^2 _ 11^2 _ 6^,2 j^ 24i/z + 22zx - 20xy + Syio + izw - 0,
2bjr- - 10;/ - 15s" - bnf- + 38y.5 + 4G«x - ZOxy - \Qxw + IQtjio + ISzw - 0,
SYSTEMS OF QUADRICS. 207
The reciproccals of these equations are
550a2+1036/32+85072_32452+2120|3y + 500ya-520a/3-180a5 + 2088/3o + 1980ya = 0,
3950a2+800^2+2750y2- 9720^2 + 11200/3y + 4900ya-4160a/3+25920/3o + 16200yo = 0.
And the biquadratic is
8100 {X* - l0/\3 + 35\2 - 50\ + 24} = 0 ;
whence a, b, c, d are 1, 2, 3, 4. We then calculate T and T' by the formula
T=x^B'{ab - h^) +C'{ac -g"") +D'{ad- 1'') +2F'{af-gh) '+ 2M '{am -kl)+ 'IN '{an -gl)}
+ 2yz [A' {of- gh) + D' {df- mn) + M' {mf - bn) + N' {nf- cm)
+ G' {fg - ch) + K' {fh - bg) + F' (/^ - be) + L' {21/- mg -^h)} + &c.,
and dividing T and T' so calculated by A (= 8100), we write
A'2 + 1-2 + ^2 + TF2
= 5x- - 11?/- - 11.?= - Giv- + 24yz + 22zx - 20xg + Syio + izw,
A2 + 2r2 + 3Z2 + 4TF2
= 25a;2 - 10^2 _ 15^2 _ 5j„2 + 38,^32 + 4(3^3. _ ^q^^ _ jOa-^o + [Ogw + 18sw,
9Z2 + 16r2 + 21^2 + 241F2
= 161a;2 - 100*/2 - 135^2 - 55w- + 30G//Z + M2zx - 250a-?/ - IQxiv + 70gw + \25zw,
26A'2 + 38F2 + 42Z2 + 44PF2
= 280x2 _ 300^2 _ 360^2 - 1702«2 + n2yz + llQzx - &2%xy - lQ%xw + \SOyw + 2b2zic.
Then from 2i.U - V+T' -T,we get
6 A' 2 = - 6 {2x + 3y-2z- 2iv}^.
And, in like manner,
Y"~ = - {x + 2y-3z + 2iof, Z'^={3x-y + z- loY, W^ = {x + y + z + wf.
Ex. 2. It having been shewn that x", y'^, 7?, iifl can be expressed in terms of
Z7, F, T, r', it follows that the square of the Jacobian of these four surfaces can also
be expressed as a function of them. We find thus
J2 = AT* _ QT^T' + ^r2r'2 _ Q'TT'^ - Q'T*
+ Y ((92 _ 2A<t>) T^ + (e<I' - Se'A) T-T' + (09' - 4AA') TT'- - A'BT's)
+ U {(9'2 _ 2A'*) 7"3 + (9'* - 39A') T'-T+ (99' - 4AA') T'T^ ~ AB'T^}
+ AV {(*2 - 299' + 2AA') ^2 - {Q'^ - 39A') TT' + $A'7"2}
+ A't'"2 {((1>2 _ 299' + 2AA') T'"~ - (9'i> - 3A0') TT' + A^T~}
+ T {(9'2 - 2A''f>) F^A" - (9'*2 - 299'2 + 59'A'A - e<l>A') V^UA
+ (92* - 2*2 A - 09'A + 4A'A2) A'Ft^2_ ^^'20^7*}
+ T' ((92 - 2A*) r^A'^ - (9<I>- - 29'92 + 59AA' - 0'<i>A) Z72FA'
+ (0'2$ _ 2$2A' - 09'A' + 4AA'2) aUV^ - A2A'0'F3}
+ A3A'-'F* + A2A'3f74 - C^F3A2 {0'' - 39'<I'A' + 39A'2} _ J73 ^A'2 {9' - 30$ A + 30'A2}
+ AA'r2F2 [4>3 _ 3$AA' + 302A' + 30'2A - 300'*}.
Ex. 3. The formulfe for the coordinates of a point on the curve TJV, given
Art. 220cZ, evidently result from the determination of this Article. We proceed to
treat similarly the tangential equations.
208 INVARIANTS AND COVAEIANTS OF
Writing down the four contravariants (214) in the form
a^. (bc'd' + cd'V + d¥c') + (3^ ) + = t',
a2. {cdb' + dbc' + bed') + ft"- { ) + = t ,
a'^.bcd+ ^'^.cda+ = a ,
these give, when solved for a?, ^, y^, S-,
{ab') {ac') {ad') a^ = a^o-' - a^a'-r' + aa"^T - a'^a, &c.
Hence, for any tangent plane common to TJ and V,
{ab') {ac') {ad') a- = aa' («'t - a-r'), &c.
The coordinates of the line in which this intersects a consecutive common tangent
plane, i. e. the coordinates of a generator of the circumscribed developable are derived
from these by taking the consecutive tangent plane
<7a _ a'd-r - ad-r' d(i b 'd-r - bd-r'
a « T — ax /3 6 V — Ot
whence, by taking the difference of these two and substituting for a, (3, we get the
value for the coordinate
p2 = aa' {ab') {cd') {c't - ct') {d'-r - d-r'),
and for the other coordinates values coiTesponding, omitting a common factor.
Prom these the tangential equation of the circumscribed developable may be found.
235. If we form the discriminant of \U-\- fiV+vW^ the
coefficients of the several powers of X, ytt, v will evidently be
invariants of the system Z7, F, W. There are three invariants
however of this system, (which we shall call A*, /, /) which
* In the former editions it had been supposed that the equations of any three
quadrics could be reduced to the form
V -a x"- + b 7f- + c z^ + d u" + e v^,
V = a' X- + b'f + c' z- + d' u- + e' v^,
W - a"x^ + b"y"- + c"z"- + d"ii' + e'V,
a form containing 12 independent constants expressed and 15 implicitly, or, in all,
the right number 27 (see Art. 141). Doubt was cast on the validity of this argument
when Clebsch observed that a similar argument does not hold good for plane
quartics. The form
ax* + by^ + cz* + du* + ev*,
contains the right number of constants for representing a general quartic ; yet for
this form it is easily shown that an invariant vanishes which in general is not = 0
(see Higher Plane Curves, Art, 294). The same thing is true of the form
abode
- + - +- + - + -,
X y z u V
■which though containing the right number of constants will not represent a quartic
in general, but only one for which a certain invariant relation is fulfilled. Frahm
SYSTEMS OF QUADRICS. 209
deserve special attention as being also invariants of any three
quadrics of the system \U'+ fjuV+vW; or, what is the same
thing, as being also comhinants.
The invariant A vanishes, when each of the three quadrics
Z7, F, W is the polar quadric of a point with regard to a
surface of the third degree. In fact it is easy to see that, taking
two points 1, 2 and a cubic surface, the polar plane of 1 with
respect to the polar quadric of 2 must be the same as the polar
plane of 2 with regard to the polar quadric of 1. Supposing
then U^ F, W to be the polar quadrics of points 1, 2, 3
respectively, and expressing that the polar plane of 1 in
respect of V is identical with that of 2 in respect of ?7, we get
by comparing coefficients of aj, ?/, 2, lo four equations linear
in a-j, ?/j, a;^, &c. Similarly two other sets of four are got by
comparing the surfaces Z7, W\ F, W. Eliminating then linearly
the twelve unknown variables iCj, ?/,, &c., x^, &c., the result
of elimination can be written at once in the determinant form
• ? ~
-«"-
7 // //
-a — g ~
- 1"
^ 1
r
a
h' g' I'
• J ~
r-
r-r-
rr
m ,
K
V / ^n
• ? ~
/'-
f- c"-
ft
9
f c n'
• ) ~
r-
ff ff
m — 71 -
d",
I'
m n a
/'
r
•
m •
) ~
-a -
h-rj - I
/"
m\
•
•
? ~
-h-
b- f-m
C
ft
n ,
■
• •
) "
-.9 -
- f-c - n
n"
d'\
•
• •
) ~
■I -
771 — n — d
9 f
— a — li - g' — Z' , a h g
-]{ -h' -f -m\ h h f
t /}> r t X
-.9' -/ -c - *i , 9 J
— I' — in —n — d\ I 711 71 d ,
' 3
c n ,
= 0,
showed {Math. Annal, vii.) that there is in fact an intimate relation between the
theory of three quadrics and that of a plane quartic. Form the discriminant of
\U + fiV + fir and we get a result which is a ternary quartic in X, [x, v of the most
general kind. Now the discriminant of
ax- + bif- + cz" + du" + ev^,
is easily seen to be
a 0 c a e
EE
210 INVARIANTS AND COVARIANTS OF
but as this is a skew symmetrical determinant of even order,
it is a perfect square, thus the condition in question is of the
second order only, in the coefficients of each of the surfaces.
Reducing this determinant by assuming two of the surfaces in
the forms
/ 2,7' i?, 'a, V 2
•" 2 , 7 // 2 I '/ 2 , J'f 2
ax-\-oy-\-cz-'rdw^
which is always admissible 5 it is found to be in this case
0 , [h'a") h, [ccr) f/, {d'a:') I
[ah")h, 0 , {cb")f, {d'h")m
{a'c")g, [h'c")f, ^ 0 , {d'c")n
[a'd")l^ {h'd")m^ {c'd")n^ 0
which is also skew symmetrical and is the square of
(JV) {ad")fl + [ca!') \h'd") gm + [a'h") {cd") hn.
In this form it is easily seen that A vanishes if Z7, F, W
each admit of being written as sum of five squares. In fact
we can in this case eliminate one variable between each
pair of equations reducing two to the forms just written,
making each of them the sura of four squares ; and the third
becomes, by replacing the fifth variable from the universal
linear relation,
ax' + hf + cz' + chv' -\- c[x-\- y ^ z + ivY = 0,
whence // = ^m = hn = e\ and these values substituted in the
expression just found for A evidently make it vanish.
And, therefore, if U, V, W be three qiiadrics of this form the discriminant of
\U+/xV+v]V is got by writing Xa + fxa' + m" for a, &c., in the above. Andaccording
to what has been just stated this ia only a ternary quartic of a special form. If
then we write down the invariant condition that the discriminant of W + fj.V + vW
considered as a ternary quartic in X, fx, v should be capable of being reduced to the
special form just mentioned, we have at the same time the condition that these
quadrics should be such that their equations may be written as the sum of squares
of the same five linear functions. Toeplitz {Math. AnnaJ, xi.) gave the form of A
definitely as in the text, and also by determining its symbolical expression showed
that it can be expressed in terms of the functions of the coefficients which occur in
the conditions that a right line should touch U, V, ]V respectively. The condition
that a line should touch a surface may be expressed symbolically (see Arts. 80, 217)
as (12a/3)2, The symbolical function (12a/3) (12a'/3') expresses that two hues are
harmonic conjugates with regard to a surface, and is a function of the same coefficients
of the quadric. And, if taking a, P; a', fi' as symbols with respect to two other
surfaces we multiply by {afta'ft') we get the symbol which expresses A.
SYSTEMS OF QUADRICS. 211
236. The invariant which we call / vanishes, whenever
any four of the points of intersection of U, F, W lie in a
plane, (a condition which implies that the other four points
of intersection lie in a plane), or, in other words, whenever it Is
possible to find values of A., /x, v, which will make A,C74-/aF+ vW
represent two planes. Now in this case the tangential equation
vanishes (Art. 214), hence, writing for a, \a-\- ijlci -\- va'\ &c.
in cr, let the result be denoted by o"o„(,V + o-qoi^V + ^oq^'^ + = 0,
the ten coefficients of this quadric in a, /3, 7, 5, therefore vanish,
whence we can write down the required condition as the
determinant of the tenth order got by eliminating X, /z, v;
but each coefficient is of the third order in the original
coefficients, hence this invariant, involving symmetrically each
surface, must be of the tenth degree in the coefficients of each
surface (compare Conies^ 389a). That / is of the tenth degree
in the coefficients of each surface may be otherwise seen
as follows: Let t/, U\ F, W be four quadrics passing
each through the same six points ; then since through these
points twenty planes [ten pairs of planesj can be drawn,
it follows that the problem to determine A-, yti, v so tha,t
U+\U' + fiV+vW may represent two planes, admits of ten
solutions. But \ might also be determined by forming the
invariant / of the system U, F, TF, and then substituting
for each coefficient a of Z7, a + \a. And since there are ten
values of \, the result of substitution must contain X, in the
tenth degree ; and therefore / must contain the coefficients
of U in the same degree.
237. The invariant which we call J vanishes, whenever any
two of the eight points of intersection of the surfaces U, F, IF
coincide.* Thus, if at any point common to the three surfaces,
their three tangent planes pass through a common line, the
consecutive point on this line will also be common to all the
surfaces. Such a point will also be the vertex of a cone of
the system \U-{- fiV+vW. For take the point as origin, and
if the tangent planes be ar, ?/, ax-i-bf/, the equations of the
* This invariant is called by Professoi- Cayley the tact-invariant of a system of
three quadrics, as that considered Art. 202 is the tact-invariant of a system of two.
212 INVARIANTS AND COVARIANTS OF
surfaces are x + w^, y + Vj, ax-^-hy + to^^ where m.^, •y^, w^ de-
note terms of the second degree. And it is evident that
a U+ h V— TF is a cone having the origin for its vertex.
The invariant J is of the sixteenth degree in the coefficients
of each of the surfaces. For if in J we substitute for each coeffi-
cient a of ZZ, a-\- \a where a is the corresponding coefficient of
another surface Z7', it is evident that the degree of the result
in \ is the same as the number of surfaces of the system
U+ X U' which can be drawn to touch the curve of intersection
of F, W\ that is to say, sixteen (Cor., Art. 233).
238. If ax^ -f ly^ + cz^ + du^ + ev^ represent a cone, the co-
ordinates of the vertex satisfy the four equations got by diffe-
rentiating with respect to a?, 3/, 2;, w; that is to say, (remem-
bering that x-\-y + z + u-\-v\'& supposed to =0) ax = ev, by = ev,
&c. The coordinates of the vertex may then be written
- , J-, - , -^ , - , substituting which values in the condition
connecting x, y, z, m, v^ we obtain the discriminant of the
surface, viz.
1111 1 ^
a 0 c a e
Thus, then, when the equations of Z7, F, W admit of being
written in the form here used, the discriminant \U+ fJbV+vWis
+ T-F- — 7/ . 7// + &c. = 0 ;
Xa + fia + vd' \h + yJj + vh'
and when XZ7+/iF+ vTF represents a cone, if we substitute the
coordinates of its vertex in the equation of each of the surfaces
in succession, we get
(\a + iio! ■\ yay (,U + fx¥ + vhy
But these equations are the differentials of the discriminant
with respect to X, /i, v. Hence we derive the theorem that in the
case in question if we form the discriminant of X, U+ /x F+ v TF,
SYSTEMS OF QUADKICS.
213
and then tlie dlscrlrainant of this again with respect to X, ^, v ;
J will be a factor in the result. It may be shewn easily
that / must also be a factor in this result, and the result is
in fact /v.*
238a. Given three quadrics the locus of a point whose polar
planes with respect to all three meet in a line is a curve of
the sixth order, which may be called the Jacobian curve of
the system. For such a point must evidently satisfy all the
equations got by equating to nothing the determinants of the
system of differential coefficients U^ &c., of Z7, F, &c., of F, &c.,
£^., f^., f^3, ^.
V V V V
^„ tf;, >n, K
but equating to zero any two of these determinants as (123) and
(124) we get two surfaces of the third order which have common
the cubic curve (Art. 134) whose equations are got by the
vanishing of
u., v.. w,
u.. n, w,
and this does not belong to the other cubic surfaces. Hence
there is only a sextic curve common.
* An analogous theorem, due to Professor Cayley, is that if U and V be homo-
geneous functions of two variables of the nth degree; and if we form the discri-
minant of U +\V and then the discriminant of this with respect to X, the result
will be AE^C^ where A is the result of elimination between U and V ; B (of the
degree 2 (w — 2) (« — 3) in both sets of coefficients) vanishes whenever \ can be so
determined that f''+ \T" shall have two pairs of equal factors; and C (of the degree
3 [n — 2) vanishes whenever X can be determined so that U -\-XV shall have three
equal factors. In like manner, if U and V be homogeneous functions of three varia-
bles, the discriminant with regard to X of the discriminant of f/'+ XF is still AB-C*
where A (of the degree Zn (« — 1) in each set of coefficients) is the condition that U
and V should touch, B vanishes whenever it is possible to determine X so that
^7+ XF may have two double points; and C, so that it may have a cusp. Lastly,
when U, V, \V are three conies, the discriminant with respect to X, /u, v of the dis-
criminant of W + t^V+vWia AB'^, where ^ = 0 is the condition that the three
curves should intersect and £ = 0 is the condition that \U + ixV + vV/ should ever
be a perfect square.
214
INVARIANTS AND COVAEIANTS.
238b. If we express the relation that the right line joining
the points 1 and 2 may be cut in involution by three quadrlcs
U, F, TF, writing the quadratic of Art. 75 in the form
^u^' + 2 f^i.^/^ + U^,l^' = 0, &c.
that relation is
w w
but this may be written in the form
a jb , c , c? ,y ...
tf:
22
= 0,
0 =
a ,b ,c ,d ,/
X.
Vx
U 5 ^1 J 2?/,2;,...
< ,.^2' 5<' ,< ,2?/,^«2---
and it can be seen without difficulty that each determinant in
the second matrix consists of powers and products of the six
coordinates of the right line I, 2. Hence we have the relation
in question as a complex of the third order the coefficients of
which are linear in the coefficients of each quadric. Employing
a usual method of squaring, we find by multiplying
C/., , Z7„, U
'xi )
12 5
V V V
w w w
22
^22,-2F,., F„
^'^2, -2^F,,, TF„
2*
00 5
4'.„, *„
10?
^1, 2VI'.,,
*.
*,
02?
*,2? 24',,
where 4'^^ is the condition for the line to touch C/, &c. and
St'gj for it to be cut harmonically by U and F, &c, (Art. 217).
Hence it is seen that the squares and products of the coefficients
in M can be expressed by the combinations of the original
coefficients which arise from the second minors of the dis-
criminant Ex. 6, Art. 200. Again, the complex M Is the
same for any three surfaces of the system \U-\- /juV-]- vTF. Also
il/=0 if for such a surface we have XC/,, +/xF,, -f vTF,, = 0,
X L\^ + /x F,^ + V TF;, = 0, X U^, + /x F,, + y W^,^ = 0, hence (Art. 80c)
it contains all the right lines which are contained in surfaces
of the system. This complex M may be also written in axial
coordinates: Toeplitz has noticed that when the products of
corresponding coefficients of both forms is summed, the inva-
riant A is the result.
{ 215 )
CHAPTER X.
CONES AND SPHERO-CONICS.
239. If a cone of any degree be cut by any sphere, wbose
centre is the vertex of the cone, the curve of section will
evidently be such that the angle between two edges of the cone
is measured by the arc joining the two corresponding points
on the sphere. When the cone is of the second degree, the
curve of section is called a sphero-coyiic. By stating many of
the properties of cones of the second degree as properties of
sphero-conics, the analogy between them and corresponding
properties of conies becomes more striking.*
Strictly speaking, the intersection of a sphere with a cone
of the n^"^ degree is a curve of the 2n*^ degree : but when the
cone is concentric with the sphere, the curve of intersection
may be divided, in an infinity of ways, into two symmetrical
and equal portions, either of which may be regarded as analo-
gous to a plane curve of the oi^^ degree. For if we consider
the points of the curve of intersection which lie in any hemi-
sphere, the points diametrically opposite evidently trace out
a perfectly symmetrical curve in the opposite hemisphere.f
* See M. Chasles's Memoir on Sphero-conics (published in the Sixth Volume of the
Transactions of the Roijal Academy at Brussels, and translated by Professor Graves,
now Bishop of Limerick, Dublin, 1837), from which the enunciations of many of
the theorems in this chapter are taken. See also M. Chasles's later papers Comptes
Eendiis, March and June, 1860.
t It has been remarked {Higher Plane Curves, Art. 198) that a cone of any order
may comprise two forms of sheet, viz. (1) a twin-pair sheet which meets a concentric
sphere in a pair of closed curves, such that each point of the one curve is opposite
to a point of the other curve (of this kind are cones of the second order) ; or (2) a
single sheet which meets a concentric sphere in a closed curve, such that each point
of the curve is opposite to another point of the curve; (the plane affords an ex-
ample of such a cone) see Mbbius, Abhandlungen der K. Sachs. Gesellschajl, Vol. i.
216 CONES AND SPHERO-CONICS.
Thus, then, a sphero-conic may be regarded as analogous
either to an ellipse or to a hyperbola. A cone of the second
degree evidently intersects a concentric sphere in two similar
closed curves diametrically opposite to each other. One of
the principal planes of the cone meets neither curve, and if we
look at either of the hemispheres into which this plane divides
the sphere, we see a closed curve analogous to an ellipse.
The other principal planes divide the sphere into hemispheres
containing each hemisphere a half of the two opposite curves,
and in particular the principal plane not passing through the
focal lines of the cone (supra. Art. 151) divides the sphere
into two hemispheres each containing a curve consisting of
two opposite branches like the hyperbola.
The curve of intersection of any quadric with a concentric
sphere is evidently a sphero-conic.
240. The properties of spherical curves have been studied
by means of systems of spherical coordinates formed on the
model of Cartesian coordinates. Choose for axes of coordi-
nates any two great circles OX, OY intersecting at right
angles, and on them let fall perpendiculars FM, PN from any
point P on the sphere. These perpendiculars are not, as in
plane coordinates, equal to the opposite sides of the quad-
rilateral OMPN; and therefore it would seem that there is
a certain latitude admissible in our selection of spherical co-
ordinates, according as we choose for coordinates the per-
pendiculars Pilf, PN, or the intercepts Oil/, ON which they
make on the axes.
M. Gudermann of Cleves has chosen for coordinates the
tangents of the intercepts OM, ON (see Crelle's Journal,
vol. VI., p. 240), and the reader will find an elaborate discussion
of this system of coordinates in the appendix to Graves's
translation of Chasles's Memoir on Sphero-conics. It is easy
to see, however, that if we draw a tangent plane to the sphere
at the point 0, and if the lines joining the centre to the points
J/, iV, P, meet that plane in points m, n, ij ; then Om, On will
be the Cartesian coordinates of the point ^). But Om, On
CONES AND SPHERO-CONICS. 217
are the tangents of the arcs Oil/, ON. Hence the equation
of a spherical curve in Gudermann's system of coordinates
is in reality nothing but the ordinary equation of the plane
curve in which the cone joining the spherical curve to the centre
of the sphere is met by the tangent plane at the point 0.
So, again, if we choose for coordinates the sines of the per-
pendiculars Pil/, PN^ it is easy to see, in like manner, that the
equation of a spherical curve in such coordinates is only the
equation of the orthogonal projection of that curve on a plane
parallel to the tangent plane at the point 0.
It seems, however, to us, that the properties of spherical
curves are obtained more simply and directly from the equa-
tions of the cones which join them to the centre, than from
the equations of any of the plane curves into which they can
be projected.
241. Let the coordinates of any point P on the sphere be
substituted in the equation of any plane passing through the
centre (which we take for origin of coordinates), and meeting
the sphere in a great circle AB^ the result will be the length of
the perpendicular from P on that plane ; which varies as the sine
of the spherical arc let fall perpendicular from P on the great
circle AB. By the help of this principle the equations of
cones are Interpreted so as to yield properties of spherical
curves in a manner precisely corresponding to that used in
interpreting the equations of plane curves.
Thus, let a, /3 be the equations of any two planes through
the centre, which may also be regarded as the equations of the
great circles in which they meet the sphere, then (as at Conies^
Art. 54) a — A/S denotes a great circle, such that the sine of the
perpendicular arc from any point of it on a is In a constant
ratio to the sine of the perpendicular on /3; that is to say,
a great circle dividing the angle between a and ^ into parts
whose sines are In the same ratio.
Thus, again, a — A/S, a — k'/3 denote arcs forming with a
k
and /3 a pencil whose anharmonlc ratio is -7 . And a - A-73,
a + A/3 denotes arcs forming with a, /3 a harmonic pencil.
FF
21 S CONES AND SPHERO-CONICS.
It may be noted here that if A' be the middle point of
an arc AB, then B', the fourth harmonic to A\ A, and B, is
a point distant from^' by 90°. For if we join these points
to the centre C, CA' is the internal bisector of the angle AGB^
and therefore GB' must be the external bisector. Conversely,
if two corresponding points of a harmonic system are distant
from each other by 90°, each is equidistant from the other two
points of the system.
It is convenient also to mention here that if xyz' be the
coordinates of any point on the sphere, then xx -\- yj/ -{■ zz
denotes the great circle having x'y'z for its pole. It is in
fact the equation of the plane perpendicular to the line joining
the centre to the point x'y'z'.
242. We can now iraraedlately apply to spherical triangles
the methods used for plane triangles [Conies^ Chap. IV., &c,).
Thus, if a, /S, 7 denote the three sides, then lot. = m^ = ny
denote three great circles meeting in a point, each of which
passes through one of the vertices : while
m^ -\- ny — /a, riy -{-la — m/3^ la + m^ — ny
are the sides of the triangle formed by connecting the points
where each of these joining lines meets the opposite sides of
the given triangle ; and la + w/S + ny passes through the inter-
sections of corresponding sides of this new triangle and of the
given triangle.
The equations a = /3 = 7 evidently represent the three bi-
sectors of the angles of the triangle. And if A^ B, C be the
angles of the triangle, it is easily proved that, as in plane
triangles, a cos ^ = /8 cos Z?= 7 cos C denote the three perpen-
diculars. It remains true, as at Conies^ Art. 54, that if the
perpendiculars from the vertices of one triangle on the sides
of another meet in a point, so will the perpendiculars from the
vertices of the second on the sides of the lirst.
The three bisectors of sides are a sin ^ =yS sin -B= 7 sin C.
The arc a sin ^1 + /3 sin^+ 7 sin C passes through the three
points where each side is met by the arc joining the middle
l)oints of the other tvvoj oi", again, it passes through the
CONES AND SPHERO-CONICS. 219
point on each side 90° distant from its middle point, fur
a sin A ± /3 sin B meet 7 in two points which are harmonic
conjugates with the points in which a, /3 meet it, and since
one is the middle point the other must be 90° distant from it
(Art. 241). It follows from what has been just said, that the
point where a sin J. -f ^ sin i? 4- 7 sin (7 meets any side is the
pole of the great circle perpendicular to that side at its middle
point, and hence, that the intersection of the three per-
pendiculars of this kind (that is to say, the centre of
the circumscribing circle) is the pole of the great circle
a sin A i- 13 s'\nB+ J sin G. The equations of the lines joining
the vertices of the triangle to the centre of the circumscribing
circle are found to be
a ^ ^ 7
sini(5+ C-A) s\n^{C+A-B) Bm^{A + B- C)'
243, The condition that two great circles ax -i- bi/ + cZj.
axi- b'i/ + cz should be perpendicular is manifestly
aa + hb' + cc = 0.
The condition that aa -i- bi3 + 07, a'a -\- h' (3 -f c'7 should be per-
pendicular is easily found from this by substituting tor a, /i, 7
their expressions in terms of .r, ?/, z. The result is exactly the
same as for the corresponding case in the plane, viz.
aa!-^bb'^cc—[bc-^b'c) cos ^4 — [ca'-\-ca) cos B ~ {ab'-\-ba') cosC=0.
In like manner the sine of the arc perpendicular to an + bfS + 07,
and passing through a given point is found by substituting tlie
coordinates of that point in a% + btS -{■ cy and dividing by the
square root of
a" + b' + c* - 2bc cos A - 2ca cos B — 2ab cos C.
244. Passing now to equations of the second degree, we
may consider the equation ay = ^u3'^ either as denoting a cone
having a and 7 for tangent planes, while /3 passes through
the edges of contact, or as denoting a sphero-conic, having
a and 7 for tangents, and /3 for their arc of contact. The
equation plainly asserts that the product of the sines of per-
pendiculars from any point of a sphero-conic on two of it3
220 CONES AND SPHERO-CONICS.
tangents is in a constant ratio to the square of the sine of the
perpendicular from the same point on the arc of contact.
In like manner the equation ay = k^S asserts that the pro-
duct of the sines of the perpendiculars from any point of a
sphero-conic on two opposite sides of an inscribed quadrilateral
is in a constant ratio to the product of sines of perpendiculars
on the other two sides. And from this property again may be
deduced, precisely as at Conies, Art. 259, that the anharmonic
ratio of the four arcs joining four fixed points on a sphero-
conic to any other point on the curve is constant. In like
manner almost all the proofs of theorems respecting plane
conies (given Conies^ Chap, xiv.) apply equally to sphero-
conics.
245. If a, /S represent the planes of circular section (or
cyclic 'planes) of a cone, the equation of the cone is of the
form ic^ + 3/^ + s''*=Z:a/3 ( Art. 103), which interpreted, as in the
last article, shews that the product of the sines of perpen-
diculars from any point of a sphero-conic on the two cyclic arcs
is constant. Or, again, that, " Given the base of a spherical
triangle and the product of cosines of sides, the locus of vertex
is a sphero-conic, the cyclic arcs of which are the great circles
having for their poles the extremities of the given base." The
form of the equation shews that the cyclic arcs of sphero-conics
are analogous to the asymptotes of plane conies.
Every property of a sphero-conic can be doubled by con-
sidering the sphero-conic formed by the cone reciprocal to
the given one. Thus (Art. 125) it was proved that the cyclic
planes of one cone are perpendicular to the focal lines of the
reciprocal cone. If then the points in which the focal lines
meet the sphere be called the foci of the sphero-conic, the
property established in this article proves that the product
of the sines of the perpendiculars let fall from the two foci
on any tangent to a sphero-conic is constant.
24G. If any greM circle meet a sphero-conic in two 2Joints
P, Q, and the cyclic arcs in points A, B, then AP= BQ.
This is deduced from the property of the last article in
CONES AND SPHEKO-CONICS. 221
the same way as the corresponding property of the plane
hyperbola is proved. The ratio of the sines of the perpen-
diculars from P and () on a is equal to the ratio of tlie sines
of perpendiculars from Q and P on /3, But the sines of
the perpendiculars from P and Q on a are in the ratio
sin AP : sin A Q^ and tlierefore we have
sin AP : s'm A Q :: Hin B Q : sin JBP,
whence it may easily be inferred that AP = BQ.
Reciprocally, the two tangents from any point to a sphero-
conic make equal angles with the arcs joining that point to
the two foci.
247. As a particular case of the theorem of Art. 246 we
learn that the portion of any tangent to a sphero-conic intercepted
between the two cyclic arcs is bisected at the point of contact.
This theorem may also be obtained directly from the equation
of a tangent, viz.
2 {xx + yy + zz) = k (a'/3 + ayQ').
The form of this equation shews that the tangent at any point
is constructed by joining that point to the Intersection of its
polar {xx' -\- yij -V zz\ see Art. 241) with a'/3 + fi'o. which is the
fourth harmonic to the cyclic arcs a, /5, and the line joining
the given point to their Intersection. Since then the given
point Is 90° distant from its harmonic conjugate in respect of
the two points where the tangent at that point meets the
cyclic arcs, it is equidistant from these points (Art. 241).
Reciprocally, the lines joining any point on a sphero-conic
to the two foci make equal angles with the tangent at that
point.
248. From the fact that the intercept by the cyclic arcs
on any tangent Is bisected at the point of contact, it may at
once be inferred by the method of infinitesimals (see Conies^
Art. 396) that every tangent to a sphero-conic forms with the
cyclic arcs a triangle of constant area, or a triangle the sura of
whose base angles Is constant. This may also be Inferred tri-
gonometrically from the fact that the product of sines of per-
222 CONES AND SPHERO-CONICS.
pendiculars on the cyclic arcs is constant. For if we call tliG
intercept on the tangent c, and the angles it makes with the
cyclic arcs A and i?, the sines of the perpendiculars on a
and /S are respectively sin |c sin J, sin^csini?. But consider-
ing the triangle of which c is the base and A and B the base
angles, then, by spherical trigonometry,
single sin^ sin5= — cos /S cos (/S^- C).
But C is given, therefore 8, the half sura of the angles, is given.
Reciprocally, the sum of the arcs joining the two fuci to
any point on a sphero-conic is constant. Or the same may be
deduced by the method of infinitesimals (see Conies^ Art. 39"J)
from the theorem that the focal radii make equal angles with
the tangent at any point.*
249. Conversely, again, we can find the locus of a point
on a sphere, such that the sum of its distances from two
fixed points on the sphere may be constant. The equation
cos(p 4-p') =cosa may be written
cos'^p -I- cos^p' — 2 cosp cosp' cos a = sin'^a.
If then a and yS denote the planes which are the polars of
the two given points, since we have a = coS|0, the equation
of the locus Is
o^ + /3'' - 2a/3 cosa = sin' a [x" + ?/ + z^).
In order to prove that the planes a and j8 are perpendicular
to focal lines of this cone, it is only necessary to shew that
sections parallel to either plane have a focus on the line per-
pendicular to It. Thus let a', a" be two planes perpendicular
* Here, again, wc can see that a sphero-conic may be regarded either as an
ellipse or hyperbola. The focal lines each evidently meet the sphere ii; two dia-
metrically opposite points. If we clioo.se for foci two points within one of tlie
closed curves in w^hich the cone meets the sphere, then the sum of the focal dis-
tances is constant. But if we substitute for one of the focal distances FP, the
focal distance from the diamctric;illy opposite point, then since F'P — 180° — FP,
we have the difference of the focal distances constant.
In like manner we may say that a variable tangent makes witli the cyclic arcs
angles whose difference is constant, if we substitute its supplement for one of the
angles at the beginning of this article.
CONES AND SPHEIIO-CONICS. 223
to each other and to a, and therefore passhig through the
line which we want to prove a focal line. Then since
•J 1 Si 1 2 2 , '2 , "2
the equation of the locus becomes
sin'^a (a'" + a""'^) = (/3 — a cosa)^
If, then, this locus be cut by any plane parallel to a, a'^ + a"'^
is the square of the distance of a point on the section from
the intersection of a'a", and we see that this distance is in a
constant ratio to the distance from the line in which /3 — a cosa
is cut by the same plane. This line is therefore the directrix
of the section, the point aV being the focus.
We see thus also that the general equation of a cone having
the line xy for a focal line is of the form x^-\-y^= [ax -}- by -j- czf ;
whence again it follows that tJie sine of the distance of any 'point
on a sphero-conic from a focus is in a constant ratio to the sine
of the distance of the same point from a certain directrix arc.
250. Any two variable tangents meet the cyclic arcs in four
points which lie on a circle. For if Z, M be two tangents
and R the chord of contact, the equation of the sphero-conic
may be written in the iorm LM= B'] but this must be iden-
tical with al3 — x^ -\- y'^ ■]■ z^ . Hence a^ — LM is identical with
x^ + y'^ + z^—R\ The latter quantity represents a small circle,
having the same pole as R^ and the form of the other shews that
that circle circumscribes the quadrilateral aL^M.
Reciprocally, the focal radii to any two points on a sphero-
conic form a spherical quadrilateral in which a small circle can
be inscribed. From this property, again, may be deduced the
theorem that the sum or ditFerence of the focal i-adii is con-
stant, since the difference or sum of two opposite sides of such
a quadrilateral is equal to the difference or sum of the re-
maining two.
251. From the properties just proved for cones can be
deduced properties of quadrics in general. Thus the product
of the sines of the angles that any generator of a hyperboloid
makes vnth the planes of circular section is constant. For the
generator is parallel to an edge of the asymptotic cone whose
224
CONES AND SPHEEO-CONICS.
circular sections are the same as those of the surface. Again,
since the focal lines of the asymptotic cone are the asymptotes
of the focal hyperbola, it follows from Art. 248 that the sum
or difference is constant of the angles which any generator of
a hyperboloid makes with the asymptotes to the focal hyper-
bola. Again, given one axis of a central section of a quadric^
the sum or difference is given of the angles which its plane
makes with the planes of circular section. For (Art. 102) given
one axis of a central section its plane touches a cone concyclic
with the given quadric, and therefore the present theorem
follows at once from Art. 249.
We get an expression for the sum or difference of the angles,
in terms of the given axis, by considering the principal sec-
tion containing the greatest and least axes of the quadric.
We obtain the cyclic planes by inflecting in that section,
semi-diameters OB^ OB' each =h.
Then the planes containing these
lines and perpendicular to the
plane of the figure are the cyclic
planes. Now if we draw any
semi-diameter a' making an angle
a with 0(7, we have
1
cos^a
a
/2
+
Sin a
But a is obviously an axis of the section which passes
through it and is perpendicular to the plane of the figure,
and (if a be greater than h) a. is evidently half the sum of
the angles BOA\ B'OA' which the plane of the section makes
with the cyclic planes. If a' be less than h^ OA' falls between
OB, OB', and a is half the difference of BOA', B'OA'. But
this sum or difference is the same for all sections having the
same axis. Hence, if a', b' be the axes of any central section,
making angles, 6, 6' with the cyclic planes, we have
1 _ co&me-e') &mme-0')
1
a
n
cos'-'i (6> + ^o ^^\ [e + e')
•2 "I
a
CONES AND SPHERO-CONICS. 225
Subtracting, we have
775 7i = (-,- ~tA sin d sin 6\
or, the difference of the squares of the reciprocals of the axes of
a central section is proportional to the 'product of the sines of
the angles it makes with the cyclic planes.
252. We saw (Art. 246) that, given two sphero-conics
having the same cyclic arcs, the intercept made by the outer
on any tangent to the inner is bisected at the point of contact ;
and hence, by the method of infinitesimals, that tangent cuts
off from the outer a segment of constant area [Conies^ Art. 396).
Again, if two sphero-conics have the same foci, and if
tangents be drawn to the inner from any point on the outer,
these tangents are equally inclined to the tangent to the outer
at that point. Hence, by infinitesimals (see Conies^ Art. 399),
the excess of the sum of the two tangents over the included
arc of the inner conic is constant. This theorem is the reci-
procal of the first theorem of this article, and it is so that
it was obtained by Dr. Graves (see his Translation of Chasles's
Memoir, p. 77).
253. To find the locus of the intersection of two tangents to
a sphero-conic lohich cut at right angles. This is, in other words,
to find the cone generated by the intersection of two rect-
angular tangent planes to a given cone ^ + tj + 77=0* Let
the direction-angles of the perpendiculars to the two tangent
planes be a'/3'7', a"/3'V') then they fulfil the relations
A cosV + B cos';8'+ C cos V=0, A cos V'+ B cos'-'/S'-f C co&Y = 0-
But if a, /3, 7 be the direction-cosines of the line perpendicular
to both, we have cos'^a = 1 — cosV — cosV, &c. Therefore
adding the two preceding equations, we have for the equation
of the locus,
Ax' + By'-v C£' = [A-^B+C){x' + f + z'),
a cone concyclic with the reciprocal of the given cone. Reci-
GG
22G CONES AND SPHEEO-CONICS.
procally, the envelope of a chord 90° In length is a sphero-
conic, confocal with the reciprocal of the given cone.
254. To find tlie locus of the foot of the perpendicular from
the focus of a sphero-conic on the tangent. The work of this
question is precisely the same as that of the corresponding
problem in plane conies, and the only difference is in the inter-
pretation of the result. Let the equation of the sphero-conic
(Art. 249) be x' -{■ y"^ = f where t = ax-\-hy -{^ cz^ then the equa-
tion of the tangent is
OCX -i yy =^i i
and of a perpendicular to it through the origin is
[x' — at') y — [y — lit') a? = 0.
Solving for x\ y\ and t' from these two equations, and sub-
stituting in x''^ + y'' — Z'^, we get for the locus required,
[x' + f) {{a' ^W-\) [x' -f %f) + 2cs [ax -f lij) + cV} = 0.
The quantity within the brackets denotes a cone whose circular
sections are parallel to the plane z.
255. It may be inferred from Art. 242 that the quantity
a sin ^4-/3 sin ^ -I- 7 sin G
has not, as in p)Iano^ a fixed value for the perpendiculars
from any point. It remains then to ask how the three per-
pendiculars from any point on three fixed great circles are
connected. But this question we have implicitly answered
already, for the three perpendiculars are each the complement
of one of the three distances from the three poles of the sides
of the triangle of reference. If then a, J, c be the sides ;
A, B, C the angles of the triangle of reference, then a, /3, 7
the sines of the perpendiculars on the sides from any point
are connected by the following relation, which is only a trans-
formation of that of Art. 54,
-1-2/37 sin 5 sin 0 cos a -f 27a sin C sin ^ cosZ'-l-2a/3 sln^ sin ^ cose
= 1 - cos" -4 - cos'" B — cos'" 0-2 cos A cosB cos C.
CONES AND SPHEUO-CONI(;S. 227
The equation in this form represents a relation between the
sines of the arcs represented by a, /?, 7, If we want to get
a relation between the perpendiculars from any point of the
sphere on the planes represented by a, /S, 7, we have evidently
only to multiply the right-hand side of the preceding equation
by r\ and that equation in a, yS, 7 will be the transformation
of the equation x'^ + y^ f z'' = /•''.
Hence, it appears that if we equate the left-hand side of
the preceding equation to zero, the equation will be the same
as ic'' + j/'^ + 2;^ = 0, and therefore denotes the imaginary circle
which is the intersection of two concentric spheres; that is to
say, the imaginary circle at infinity (see Art. 139).
256. This equation may be used to find the equation of the
sphere inscribed in a given tetrahedron, whose faces are
a, /3, 7, S. If through the centre three planes be drawn
parallel to a, /S, 7, the perpendiculars on them from any point
will be a — r, /3 — r, 7 — r. The equation of the sphere is
therefore
{a.-rf sin^^l + (/3 - r)' s'm'B + &c.
= r^ ( 1 — cos''* A — cos^ B — cos" C —2 cosA cos B cos C).
But if Z, i)/, iV, P denote the areas of the four faces, we have
La + M/3 + Ny + F8 = {L + M+ N+ P) r.
Hence, by eliminating ?•, we arrive at a result reducible to the
form of Art. 228.
257. The equation of a small circle (or right cone) is easily
expressed. The sine of the distance of any point of the circle
from the polar of the centre is constant. Hence, if a be that
polar, the equation of the circle is oif' = cos'^p [x^ 4 u'^ + z^).
All small circles then being given by equations of the form
S=a\ their properties are all cases of those of conies having
double contact with the same conic.
The theory of invariants may be applied to small circles.
Let two circles S, S' ho
x' + f + z' - a' sec'/j, x' -f / -f z' - /3' sec'p',
228 CONES AND SPHERO-CONICS.
and let us form the condition that \S+ 8^ should break up
into factors. This cubic being
\'A + \'0 + X0' + A' = 0,
we have A = — tan^/a, A' = — tan**/?',
0 = sec'''|0 sec'''p' sin^D — 2 tan'^p — tan'^'p',
0' = sec^p sec'"^p' sin'^i) - 2 tan^p' - tan^p,
where D is the distance between the centres.
Now the corresponding values for two circles in a plane are
Hence, if any Invariant relation between two circles In a plane
is expressed as a function of the radii and of the distance
between their centres, the corresponding relation for circles
on a sphere is obtained by substituting for r, /, D] tanp, tanp',
and seep seep' sin Z).
Thus the condition that two circles in a plane should touch
is obtained by forming the discriminant of the cubic equation,
and Is either D = 0 or D = r±r. The corresponding equation
therefore for two circles on a sphere is
tan p + tan p' = seep seep' sin Z), or sinZ) = sin (p + p').
Again, If two circles In a plane be the one inscribed in,
the other circumscribed about, the same triangle, the Invariant
relation is fulfilled 0''' = 4A0', which gives for the distance
between their centres the expression D^ = E^ — 2Rr.
The distance therefore between the centres of the inscribed
and circumscribed circles of a spherical triangle Is given by
the formula
sec^Psec^p ^\n^D = tan^P- 2 tanPtanp.
So, In like manner, we can get the relation between two
circles Inscribed in, and circumscribed about, the same spherical
polygon.
258. The equation of any small circle (or right cone) In
trilinear coordinates must (Art. 255) be of the form
a' sin'^ + yS'^ sln'P + i' sin' G
+2/37 sin P sin C cos a + 27a sin (7 sin ^ cosZ> +2a/3 sin^ sin 5 cose
= [la + m^ + W7)''.
CONES AND SPHERO-CONICS. 229
If now the small circle circumscribe the triangle a/37, the
coeflScients of a", ^'\ and 7^ must vanish, and we must therefore
have la. + ?/i/3 + 727 = a sin ^ + /3 sin i? + 7 sin G. Hence, as was
proved before, this represents the polar of the centre of the
circumscribing circle. Substituting the values sin^, sin5, sin (7
for Zj m, w, the equation of the small circle becomes
^7 tan I a + 7a tan \h + a^S tan |c = 0.
The equation of the inscribed circle turns out to be of
exactly the same form as in the case of plane triangles, viz.
cos| J. \/(«) ± COS 1^5 Vl/S) ± cos^C \/(7) = 0«
The tangential equation of a small circle may either be derived
by forming the reciprocal of that given at the commencement
of this article, or directly from Art. 243, by expressing that the
perpendicular from the centre on \a + /i/3 + vy is constant.
We find thus for the tangential equation of the circle whose
centre is a'yS'7' and radius p
sin"''/) (A,^ +/*''+ v^ — 2fiv cos J. — 2v\co^B — 2\/jL cos C)
a form also shewing (see Art. 257) that every circle has double
contact with the imaginary circle at infinity.
259. As a concluding exercise on the formulas of this
chapter, we investigate Dr. Hart's extension of Feuerbach's
theorem for plane triangles, viz. that the four circles which
touch the sides are all touched by the same circle.
It is easier to work with the tangential equations. The
tangential equations of circles which touch the sides of the
triangle of reference must want the terms X\ f^\ v\ and there-
fore evidently are
V + /x"'' + v^ — 2/iv cos A - 2v\ cos B - 2\fj, cosC={X±/j,± vf ;
or ytivcos'^^-f vXcos'^^^ + X/i cos''^(7=0 (1),
^vcos'^^-v\ m\'^B-\fi sin'''|O=0 (2),
- /iv sin'''^^ + v\ cos'|5-X/i sin'^C=0 (3),
- fiv sin^^-vX sin'l^+XyLt cos'|C=0 (4),
230 CONES AND SPHERO-CONICS.
all which four are touched by the circle (5)
\^ -f fi' + v^ — 2fiv cos A - 2v\ cos 5 - 2\/j, cosC
= {X cos [B-C)-\- /M cos (C- A)-hv cos {A - B}]\
For the centres of similitude of the circles (1) and (5) are given
by the tangential equations
(\ 4- /A + v) ± {\ cos{B- C)+ficos{C-A) + v cos {A - B)] = 0,
one of them therefore is
\ sin^i {B- C)-{fi sin^i (C- A) + v sin^^i {A - B).
And [Conies^ Art. 127) the condition that this point should be
on the circle (1) is
cos\A sin^(i?-C) + cos|5sin^(6'-vl) + cos^Osin|(^-J5) = 0,
which is satisfied. The coordinates of the point of contact are
accordingly
sin'-'i [B- C), sln'^i [C~A], sln^i {A- B).
It is proved, in like manner, that the circle (5) touches the
three other circles.
260. The coordinates of the centre of Dr. Hart's circle
have been proved to be cos(5-C), cos ((7—^4), co's,[A- B).
This point therefore lies on the line joining the point whose
coordinates are cosjScos(7, cosCcos^, cos ^ cos 5 to the point
whose coordinates are sln^slnC, sinCsin>4, sin^lsin^; that
is to say, (Art. 242) on the line joining the intersection of per-
pendiculars to the intersection of bisectors of sides. Since
cos^-cos(i?- C) = 2B\n\[A + B-C)s\Ql [C+A-B];
the centre lies also on the line joining the point cos^, cosi?,
cos (7 to the point
sm{S-B)sm[8-C), sm[S-C) sm{S-A), sm{8-A)sm{S-B).
The first point is the intersection of lines drawn through each
vertex making the same angle with one side that the per-
pendicular makes with the other ; the second point is the in-
tersection of perpendiculars let fall from each vertex on the
line joining the middle points of the adjacent sides. The centre
of Dr. Hart's circle is thus constructed as the intersection of
two known lines.
CONES AND Sl'HERO-CONlCS. 231
261. The problem might also have been investigated by
the direct equation. We write a sIn-<4 =a;, &c., so that the
equation of the imaginary circle at infinity Is U= 0, where
U= x^ -f y^ + z^-\- 2yz cosa + 2zx cosh + 2x7/ cose.
Then the equation of the inscribed circle is
U= {x cos [s — a) -t y cos {s — b) -\- z cos (s — c)}^,
where 2s = a + h + c. For this equation expanded is
x^ sin^(s- a)+y^ sm^{s — 'b) + z'' sm^{s—c)—2yz sin(s — Z') sin (s — c)
— 2zx sin {s — c) sin [s - a) — 2xy sin [s — a) sin [s — h)= 0.
U is not altered if we change the sign of either a, h, or c.
Consequently we get three other circles also touching a?, y^ z
if we change the signs of either a, 5, or c in the equation of
the inscribed circle. All four circles will be touched by
^,_ faj cos^J cos^c 3/cos^ccos|a ^ cos^a cos-|i]'''
COS fa cos^o cos^c J
This last equation not being altered by changing the sign
of a, 5, or c, it is evident that if it touches one it touches all.
Now one of its common chords with the inscribed circle is
f . . cosAJ cos^c) f , 7. cosiccosAa)
a; -^ cos (5 -a) ^— ^ — ~\ +?/-^cos s- J ^ \
\ cos|a J ( cos^i I
f , . cosiffl cosAS)
+ z \cmis -c) ? — -. — ^ - ,
( cos^c j '
which reduced is
X y z
+ ^-7 r^-^, r + -^-7 ^ ^^ — ^^ = 0.
sin(s— J) -sin(s— c) sin(s-c) — sin(s-a) sin(s-«) — sin(s— Z>)
But the condition that the line Ax + By -\- Cz shall touch
^/{ax)-i■W{,by) + ^/{cz) '^^ -j + -^ + -fi ' ^Ppljing this condition,
the line we are considering will touch the inscribed circle if
sin (s - a) {sin [s — h) — sin (s- c)]
4-sin(5— 5) (sinfs-c)- sin(s-a)|4 sin(s-c) [sin(s-a) — sIn(s-5)]=0 ;
a condition which is evidently fulfilled. It will be seen that the
condition is also fulfilled that the common tangent in question
should touch \/(a^) + ■\/{y] + Vl^) 5 that is to say, the sphero-conic
282 CONES AND SPHERO-CONICS.
•which touches at the middle points of the sides ; a fact remarked
by Sir Wm. Hamilton, and which leads at once to a construction
for that tangent as the fourth common tangent to two conic3
which have three known tangents common.
The polar of the centre of Dr. Hart's circle has been thus
proved to be
. . cos i 5 cos Ac „ . T^cosiccosAa . „cosiacosi5
asm^ ? — -, — - +/3sm5 ^— , , ^ +7smC — ^-^- = 0,
cos^a cos^o cos^c
or a tan|a4- /3 tau^J + 7 tan|c = 0,
which may be also written
a cos {S-A)+^ cos {S- B) + y cos {S- C) = 0,
forms which lead to other constructions for the centre of this
circle.
The radius of the circle touching three others whose centres
are known, and whose radii are r, /, /' may be determined by
substituting r + R^ r + R, r" + R for cZ, e, / in the formulae of
Arts. 52, 54, and solving for R. Applying this method to the
three escribed circles I have found that the tangent of the
radius of Dr. Hart's circle is half the tangent of the radius of
the circumscribing circle of the triangle.
( 233 )
CHAPTER XI.
GENERAL THEORY OF SURFACES,
INTRODUCTORY CHAPTER.
262. Reserving for a future chapter a more detailed ex-
amination of the properties of surfaces in general, we shall
in this chapter give an account of such parts of the general
theory as can be obtained with least trouble.
Let the general equation of a surface be written in the form
A
■^ Bx-\Cy+Dz
+ Ex" + Fy' + Gz' + 2Hyz + 2Kzx + ^Lxy
+ &C. = 0,
or, as we shall write it often for shortness,
u^ 4- 2«i + ?*._, + M3 + &c. = 0,
where u,^ means the aggregate of terms of the second degree,
&c. Then it is evident that u^ consists of one term, u^ of three,
u^ of six, &c. The total number of terms in the equation is
therefore the sum of ?i-f 1 terms of the series 1, 3, 6, 10, &c.,
,ha, is ,0 say, '" + ^»'' +f '" + =^' .
The number of conditions necessary to determine a surface
ca till . 1 ,, ^, . n [n^ -\- Qn ■^r \l)
ot the n degree is one less than this, or = — ^^ .
The equation above written can be thrown into the form
of a polar equation by writing p cos a, p cos/3, p cosy for
X, ?/, z, when we obviously obtain an equation of the n^^ degree,
which will determine n values of the radius vector answering
to any assigned values of the direction-angles a, /S, 7.
HH
234 GENERAL THEORY OF SURFACES.
263. If now the origin be on tlie surface, we have tt^ = 0,
and one of the roots of the equation is always p = 0. Bat a
second root of the equation will be p = 0 if a, ;3, 7 be con-
nected hj the relation
B cosa + C COS/3 + D C0S7 = 0.
Now multiplying this equation by p it becomes Bx + Ci/+I)z=0,
and we see that it expresses merely that the radius vector must
lie In the plane m^ = 0. No other condition is necessary in order
that the radius should meet the surface in two coincident
points. Thus we see that in general through an assumed
230int on a surface we can draic an wfinity of radii vectores
which will there meet the surface in two coincident points ; that
ts to say J an infinity of tangetit lines to the surface; and these
lines lie all in one plane^ called the tangent plane^ determined
hy the equation n^ = 0.
264. The section of any surface made hy a tangent plane
is a curve having the point of contact for a doidde pointj^
Every radius vector to the surface, which lies in the tangent
plane, is of course also a radius vector to the section made
by that plane; and since every. such radius vector (Art. 263)
meets the section at the origin in two coincident points, the
origin Is, by definition, a double point (see Higher Plane
Curves^ Art. 37).
We have already had an illustration of this In the case
of hyperboloids of one sheet, which are met by any tangent
plane in a conic having a double point, that Is to say, in
two right lines. And the point of contact of the tangent
plane to a quadric of any other species is equally to be con-
sidered as the intersection of two Imaginary right lines.
From this article it follows conversely, that any plane
* I had supposed that this remark was first made by Cayley : Gregoiy's Solid
Genmetr;!, p. 132. I am informed, however, by Professor Cremona that the point
had been previously noticed by the Italian geometer, Bedetti, in a memoir read before
the Academy of Bologna, 1841. The theorem is a particular case of that of Art. 203.
Observe that the tangents at the double point are the inflexional tangents of Art. 265,
and that these may be considered as identical with the asymptotes of the indicatrix
Art. 2GG. There is thus an anticipation of the theorem by Dupin (1813).
GENERAL THEORY OP SURFACES. 235
Mfleeting a surface In a curve having a double point touches
the surface, the double point being the point of contact. If
the section have two double points, the plane will be a double
tangent plane ; and if it have three double points, the plane
will be a triple tangent plane. Since the equation of a plane
contains three constants, it is possible to determine a plane
which will satisfy any three conditions, and therefore a finite
number of planes can in general be determined which will
meet a given surface In a curve having three double points:
that is to say, a surface lias in general a determinate number
of triple tangent planes. It will also have an infinity of double
tangent planes, the points of contact lying on a certain curve
locus on the surface. The degree of this curve, and the
number of triple tangent planes will be subjects of investi-
gation hereafter.
265. Through an assumed point on a surface it is generally
possible to draw two lines which shall there meet the surface
in three coincident points-.
In order that the radius vector may meet the surface in
three coincident points, we must not only, as in Art. 263,
have the condition fulfilled
B cos a + C cos;8 + D cos 7 = 0,
but also E cos'' a. + F cos^'yS + Cr cos' 7
+ 2Zf cos/3 cos 7 + 2K cos 7 cosa + 2L cos a cos^ = 0.
For if these conditions were fulfilled, A being already supposed
to vanish, the equation of the n"^ degree v/hich determines p,
becomes divisible by p^, and has therefore three roots =0.
The first condition expresses that the radius vector must He
in the tangent plane u^. The second expresses that the radius
vector must lie in the surface m^ = 0, or
Ex' + Eg' + Gz' -t 2Hgz + 2Kzx + 2Lxy = 0.
This surface is a cone of the second degree (Art. 66) and
since every such cone Is met by a plane passing through its
vertex In two right lines, two right lines can be found to
fulfil the required conditions.
Every plane (besides the tangent plane) drawn through
236 GENERAL THEORY OF SURFACES.
either of these lines meets the surface in a section having
the point of contact for a point of inflexion. For a point of
inflexion is a point, the tangent at which meets the curve
in three coincident points {Higher Plane Curves^ Art. 46). On
this account we shall call the two lines which meet the surface
in three coincident points, the inflexional tangents at the point,*
The existence of these two lines may be otherwise perceived
thus. We have proved that the point of contact is a double
point in the section made by the tangent plane. And it has
been proved [Higher Plane Curves^ Art. 37) that at a double
point can always be drawn two lines meeting the section (and
therefore the surface) in three coincident points.
266. A double point may be one of three difi'erent kinds,
according as the tangents at it are real, coincident, or Imaginary.
Accordingly the contact of a plane with a surface may be of
three kinds according as the tangent plane meets it in a section
having a node, a cusp, or a conjugate point; or, in other
words, according as the inflexional tangents are real, coincident,
or imaginary.
If instead of the tangent plane we consider with Dupln, a
parallel plane indefinitely near thereto, the section of the surface
by this plane may be regarded as a curve of the second order,
which (as the theorem is usually but inaccurately stated) may
be an ellipse, hyperbola, or parabola ; this curve of the second
order is called the lndicatr{x.\ Analytically, if taking the
given point of the surface for origin, we take the normal for
the axis of z^ and the axes of a', y In the tangent plane ; then
considering .r, y as infinitesimals of the first order, and conse-
quently z as an infinitesimal of the second order, the equation
of the surface, regarding ^ as a given constant, gives the equa-
tion of the section, and if herein we neglect infinitesimals of
an order superior to the second, this reduces Itself to an equation
* They are called by German writers the " Haupt-tangenten."
f Dupin, see the Biveloppements cle Gikmictrie (1813), p. 49, is quite correct, he
saj's : '• En general, une courbe du second degre, dont le centre P nous est donne, ne
peut etre qu'une ellipse ou une hyperbole. Elle peut cependant etre une parabola :
alors elle se presente sous la forme dc deux lignes droites paralleles c'juidistantes
lie hvr centre."
GENERAL THEORY OF SURFACES. 237
of the form z + ax^ -]- 2hx?/ + hif =■ 0, an equation of the second
order representing the indicatrix ; viz. according as ah — h^ is
positive, negative, or zero, this is an ellipse, hyperbola or pair
of parallel lines.* Geometrically, the section of the surface is
either a closed curve, such as the ellipse ; or, attending only to the
curve in the neighbourhood of the given point, it consists of
two arcs having their convexities turned towards each other,
and which may be considered as portions of the two branches of
a hyperbola ; or the convexity vanishes, and the arcs are
infinitesimal portions of two parallel right lines.
If points on a surface be called elliptic, hyperbolic, or para-
bolic, according to the nature of the Indicatrix, we shall pre-
sently shew that In general the parabolic points form a curve
locus on the surface, this curve separating the elliptic from the
hyperbolic points.
In the case of a surface of the second order, taking the axes
as above, the equation of the surface is
z + ax"- + 2]ixy + hy' + 2(jxz + 2fyz + cz"^ = 0,
which equation, if we regard therein x and ;/ as infinitesimals
of the first order, and therefore z as infinitesimal of the second
order, reduces itself to ^ + ax' + 2hxy + hy'^ = 0, viz. z being
regarded as a constant, this is an equation of the form already
mentioned as that of the indicatrix for a surface of any order
whatever. The original equation, regarding therein 0 as a
given constant, is the equation of the section of the surface
by a plane parallel to the tangent plane, but it is not the proper
equation of the indicatrix. To further explain this, suppose that
the surface were of the third or any higher order, then besides
the terms written down, there would have been in the equation
terms (a;, yf^ &c. ; to obtain the indicatrix as a curve of the
second order, we must of necessity neglect these terms of the
third order, and there is therefore no meaning in taking into
* This is sometimes expressed as follows : When the plane of xij is the tangent
plane, and the equation of the surface is expressed in the form z — <p {x, y), we have
/ d'-z \2
an elliptic, hyperbolic, or parabolic point, according as I - — — J is less, greater than,
or equal to ( y-^l \~j-^ • It wiU be easily seen that this is equivalent to the state-
ment in the text.
238 GENERAL THEORY Ot SURFACES.
account tlie terms 2gxz + 2f7/s also of the third order, or the
term cs' which is of the fourth order.*
In the case where the indicatrix Is a hyperbola, then sup-
posing the parallel plane to coincide with the tangent plane,
this hyperbola becomes a pair of real lines ; viz. these are the
Inflexional tangents of Art. 265. And generally the two in-
flexional tangents may be regarded as the asymptotes (real
or imaginary) of the Indicatrix considered as lying in the
tangent plane ; they have been on this account termed the
asymptotic lines of the point of the surface. If from any point
of the surface we pass along one of these lines to a consecutive
point, and thence along the consecutive line to a second point
on the surface, and so on, we obtain a curve ; and we have
thus on the surface two series of curves, which are the asymp-
totic curves. In the case of a quadric surface, these are the
two series of right lines on the surface.
267. Knowing the equation of the tangent plane when
the origin is on the surface, we can, by transformation of
coordinates, find the equation of the tangent plane at any
point. It is proved, precisely as at Art. 62, that. this equation
may be written in either of the forms
dV dU' dU dTJ'
ax ay az dw
268. Let It be required now to find the tangent plane at
a point, Indefinitely near the origin, on the surface
z + ax' + llixy + J?/ + Icjxz + y'yz + cr} + &c. = 0.
We have to suppose x\ y so small that their squares may be
neglected ; while, since the consecutive point Is on the tangent
plane, we have g' = 0 ; or, more accurately, the equation of
the surface shews that z' is a quantity of the same order as
the squares of a;' and y. Then, either by the formula of the
last article, or else directly by putting x + x' ^ y + y for x
* Sse Messenger of Mathematics, Vol. v. (1870), p, 187.
GENERAL THEOliY OF SURFACES. '. 26y
and 1/^ and taking the linear part of the transformed equation,
the equation of a consecutive tangent pLane is found to be
z + 2 [ax + /</) x + 2 [hx + ly) y = Q.
Now (see Conies^ Art. 141) [ax -\- hy') x -\- [hx -k- hy') y denotes
the diameter of the conic ax^ -\-2kxy + hy^ =1, which is con-
jugate to that to the point x'y'. Hence any tangent plane is
intersected hy a consecutive tangent plane in the diameter of the
indicatrix conjugate to the direction in which the consecutive
point is taken.
This, in fact, is geometrically evident from Dupin's point
of view. For if we admit that the points consecutive to the
given one lie on an infinitely small conic, we see that the tan-
gent plane at any of them will pass through the tangent line to
that conic; and this tangent line ultimately coincides with
the diameter conjugate to that drawn to the point of contact ;
for the tangent line is parallel to this conjugate diameter and
infinitely close to it.
Thus, then, all the tangent lines which can be drawn at
a point on a surface may be distributed into pairs, such that the
tangent plane at a consecutive point on either will pass through
the other. Two tangent lines so related are called conjugate
tangents.
In the case where the two inflexional tangents are real,
the relation between two conjugate tangents may be otherwise
stated. Take the Inflexional tangents for the axes of x and y,
which is equivalent to making a and 5 = 0 in the preceding
equation; then the equation of a consecutive tangent plane is
z^-2h[xy-^yx)-=^. xVnd since the lines a:, y^ xy + y'x^
x'y—y'x form a harmonic pencil, we learn that a pair of
conjugate tangents form^ with the inflexional tangents^ a harmonic
pencil. This is in fact the theorem that a pair of conjugate
diameters of a conic are harmonics in regard to the asymptotes.
2G9. In the case where the origin is a parabolic point,
the equation of the surface can be thrown into the form
z + ay~ + &c. = 0, and the equation of a consecutive tangent
plane will ho z + 2ayy = 0. Hence the tangent plane at every
point consecutive to a parabolic point passes through the ia-
240 GENERAL THEOEY OF SUKFACES.
flexional tangent ; and if the consecutive point be taken in
this direction, so as to have y = 0, then the consecutive tangent
plane coincides with the given one. Hence the tangent plane
at a -parabolic point is to he considered as a double tangent
plane^ since it touches the surface in two consecutive points.*
In this way parabolic points on surfaces may be considered
as analogous to points of inflexion on plane curves: for we
have proved [Higher Plane Curves, Art. 46) that the tangent
line at a point of inflexion is in like manner to be regarded
as a double tangent. A further analogy between parabolic
points and points of inflexion will be afterwards stated.
It is necessary to have a name to distinguish double
tangent planes which touch in two distinct points, from those
now under consideration, where the two points of contact coin-
cide. We shall therefore call the latter stationary tangent
planes, the word expressing that the tangent plane being
supposed to move round as we pass from one point of the
surface to another, in this case it remains for an instant in
the same position. For the same reason we have called the
tangent lines at points of inflexion in plane curves, stationary
tangents.
270. If on transforming the equation to any point on a
surface as origin we have not only ic^ = 0, but also all the terms
in u^ = 0, so that the equation takes the form
ax^ + hf + cz" 4 2/7/2; + 2(jzx + 2hxy + u^ + &c. = 0,
then it is easy to see, in like manner, that every line through
the origin meets the curve in two coincident points ; and the
origin is then called a double or conical point. It is easy
to see also that a line through the origin there meets the
surface in three coincident points, provided that its direction-
cosines satisfy the equation
a cos'''a + b cos"''/3 -l- c cos'''7
+ 2fcos^ C0S7 + 2g COS7 cosa + 2A cosa cos/3 = 0.
* I believe this was first i^ointed out in a pa^oer of mine, Cambridge and Dublin
Mathematiml Journal, vol. III., p. -15.
GENERAL THEORY OF SURFACES. 241
In other words, through a conical jJOint on a surface can be
draicn an infinity of lines which will meet the surface in three
coincident points^ and these will all lie on a cone of the second
degree whose equation is u^ = 0. Further, of these lines six will
meet the surface in four coincident points; namely, the lines
of intersection of the cone u^ with the cone of the third degree
Double points on surfaces might be classified according to
the number of these lines which are real, or according as two
or more of them coincide, but we shall not enter into these
details. The only special case which it is important to mention
is when the cone w^ resolves itself into two planes ; and this
again includes the still more special case when these two
planes coincide; that is to say, when u^ is a perfect square.
271. Every plane drawn through a conical point may, in
one sense, be regarded as a tangent plane to the surface, since
it meets the surface in a section having a double point, but
in a special sense the tangent planes to the cone u^ are to be
regarded as tangent planes to the surface, and the sections
of the surface by these planes will each have the origin as a
cusp. To a conical point, then, on a surface (which is a point
through which can be drawn an infinity of tangent planes),
will in general correspond on the reciprocal surface a plane
touching the surface in an infinity of points, which will in
general lie on a conic. If, however, the cone u^ resolves itself
into two planes, the point is in the strict sense a double point,
and there corresponds to it on the reciprocal surface a double
tangent plane having two points of contact.
272. The results obtained in the preceding articles, by taking
as our origin the point we are discussing, we shall now extend
to the case Avhere the point has any position whatever. Let us
first remind the reader (see p. 29) that since the equations of a
right line contain four constants, a finite number of right lines
can be determined to fulfil four conditions (as, for instance,
to touch a surface four times), while an infinity of lines can
be found to satisfy three conditions (as, for instance, to touch
II
242 GENERAL THEORY OF SURFACES.
a surface three times), these right lines generating a certain
surface, and their points of contact lying on a certain locus.
In a subsequent chapter we shall return to the problem to
determine in general the number of solutions when four con-
ditions are given, and to determine the degree of the surface
generated, and of the locus of points of contact, when three
conditions are given. In this chapter we confine ourselves to
the case when the right line is required to pass through a
given point, whether on the surface or not. This is equivalent
to two conditions ; and an infinity of right lines (forming a
cone) can be drawn to satisfy one other condition, while a
fiinite number of right lines can be drawn to satisfy two other
conditions.
We use Joachimsthal's method employed. Conies^ Art. 290,
Higher Plane Curves^ Art. 59, and Art. 75 of this volume.
If the quadriplanar coordinates of two points be xyzio\
x"y"z"w'\ then the points in which the line joining them is
cut by the surface are found by substituting in the equation
of the surface, for x, Xa;' + ytta;", for y, \y' + fiy'^ &c. The
result will give an equation of the n^ degree in X : yu., whose
roots will be the ratios of the segments in which the line joining
the two given points is cut by the surface at any of the points
where it meets it. And the coordinates of any of the points
ot meetmg are kx + fi x ,a?/ + i^i y ,a.z + /j, z , \w ■+ /xw ,
where X' : /j,' is one of the roots of the equation of the n^ degree.
All this will present no difficulty to any reader who has mastered
the corresponding theory for plane curves. And, as in plane
curves, the result of the substitution in question may be written
X" U' + X'-'fiA U' + iX"-VA''' U' + &c. = 0,
where A represents the operation
d d d d
«^wr' + ^:7:/+^x-'-'-^^
dx' dy dz' dw '
Following the analogy of plane curves we shall call the surface
represented by
* As at Art. 59, f,, U^, U3, U^ denote the difEerential coefficients of U with
regard to ;r, y, z, n\
GENERAL THEORY OF SURFACES.
243
the first polar of the pohit xy'z'io'. We shall call
the second polar, and so on ; the polar plane of the same point
being
Each polar surface is manifestly also a polar of the point x'y'zw'
with regard to all the other polars of higher degree.
If a point be on a surface all its polars touch the tangent
plane at that point ; for the polar plane with regard to the
surface is the tangent plane ; and this must also be the polar
plane with regard to the several polar surfaces. This may
also be seen by taking the polar of the origin with regard to
II ^lo' + m,m;"~' -f u,^w'~' + &c.,
where we have made the equation homogeneous by the In-
troduction of a new variable w. The polar surfaces of the origin
are got by differentiating with regard to this new variable.
Thus the first polar is
nuy^ + (n - 1) uy"" -f ('i - 2) wy + &c.,
and if u^ = 0, the terms of the first degree, both in the surface
and in the polar, will be u^.
273. If now the point x'yz'w be on the surface, V vanishes,
and one of the roots of the equation in X, : /i will be /x = 0.
A second root of that equation will be yu- = 0, and the line
will meet the surface in two coincident points at the point
x'y'zw\ provided that the coefficient of X,""^yu. vanish In the
equation referred to. And in order that this should be the
case, it is manifestly sufficient that x"y"z"io" should satisfy the
equation of the plane
It Is proved, then, that all the tangent lines to a surface which
can be drawn at a given point lie in a plane whose equation
is that just written. By subtracting from this equation, the
identity
xu:^y'u:^zv:^wu:^^,
244 GENERAL THEORY OF SURFACES.
we get the ordinary Cartesian equation of the tangent plane, viz.
Hence, again, by Art. 43, can immediately be deduced the
equations of the normal, viz.
X— X _y —y' z— z'
274. The right line will meet the surface in three con-
secutive points, or the equation we are considering will have
for three of its roots /* = 0, if not only the coefficients of X" and
V~^/* vanish, but also that of X""^yu-"'' ; that is to say, if the line
we are considering not only lies in the tangent plane, but
also in the polar quadric,
d d d d V y.,, ^
(=
dx' dy dz dw )
Now (Art. 272) when a point is on a surface all its polars
touch the surface. The tangent plane therefore, touching the
polar quadric, meets it in two right lines, real or imaginary,
which are the two inflexional tangents to the surface.
(Art. 265).
275. ThrougJi a point on a surface can he draion [n ■\-'2){n — 3)
tangents which will also touch the surface elsewhere.
In order that the line should touch at the point x'y'z'w\
we must, as before, have the coefficients of X" and A,"'^ = 0 ;
in consequence of which the equation we are considering be-
comes one of the (w — 2;"" degree, and if the line touch the
surface a second time, this reduced equation must have equal
roots. The condition that this should be the case involves
the coefficients of that equation in the degree ?i — 3 ; one term,
for instance, being (A'^C/''. Z7)"~^. By considering that term we
see that this discriminant involves the coordinates x'yz'io in
the degree [n — 2) {n — 3), and xyzw in the degree (n + 2) (w — 3).
When therefore xy'zio is iixed, it denotes a surface which
Is met by the tangent plane in {n + 2) (n — 3) right lines.
Thus, then, we have proved that at any point on a surface
an infinity of tangent lines can be drawn : that these in general
GENERAL THEORY OF SURFACES. 245
lie in a plane ; that two of them pass through three consecutive
points, and {n + 2) (w — 3) of them touch the surface again.
276. Let us proceed next to consider the case of tangents
drawn through a point not on the surface. Since we have
in the preceding articles estahlished relations which connect
the coordinates of any point on a tangent with those of the
point of contact, we can, by an interchange of accented and
unaccented letters, express that it is the former point which
is now supposed to be known, and the latter sought.
Thus, for example, making this interchange in the equation
of Art. 273, we see that the points of contact of all tangent
lines (or of all tangent planes) which can be drawn through
xy'zto lie on the first polar, which is of the degree (w— 1) : viz.
And since the points of contact lie also on the given surface,
their locus is the curve of the degree n(w— 1), which is the
intersection of the surface with the polar.
277. The assemblage of the tangent lines which can be
drawn through xyzw form a cone, the tangent planes to which
are also tangent planes to the surface. The equation of this
cone is found by forming the discriminant of the equation of
the n^ degree in \ (Art. 272). For this discriminant expresses
that the line joining the fixed point to xyzw meets the surface
in two coincident points ; and therefore xyzio may be a point
on any tangent line through xyzw. The discriminant is easily
seen to be of the degree n (n — 1), and it is otherwise evident
that this must be the degree of the tangent cone. For its
degree is the same as the number of lines in which it is met
by any plane through the vertex. But such a plane meets the
surface in a curve to which n (n — 1 ) tangents can be drawn
through the fixed point, and these tangents are also the tangent
lines which can be drawn to the surface through the given point.
278. Through a 'point not on the surface can in general he
draivn n[n—l){n — 2] inflexional tangents. We have seen
(Art. 274) that the coordinates of any point on an inflexional
246 GENERAL THEORY OF SURFACES.
tangent are connected with those of its point of contact by
the relations f/' = 0, AU' = 0, C^^U' = 0. If, then, we consider
the xyzw of any point on the tangent as known, its point of
contact is determined as one of the intersections of the given
surface C/, which is of the n^^ degree, with its first polar A U^
which is of the [n— 1)*^, and with the second polar A^^ZJ, which
is of the [n — 2^^. There are therefore n [n — 1) [n — 2) such
intersections. If the point be on the surface, this number is
diminished by six.
279. Through a point not on the surface can in general he
drawn ^n [n — 1) (w — 2) {n — 3) double tangents to it. The points
of contact of such lines are proved by Art. 275 to be the
intersections of the given surface, of the first polar, and of the
surface represented by the discriminant discussed in Art. 275,
and which we there saw contained the coordinates of the point
of contact in the degree (w — 2)(w — 3). There are therefore
n [n—\) [n — 2) {n — 3) points of contact ; and since there are
two points of contact on each double tangent, there are half
this number of double tangents. If the point be on the surface,
the double tangents at the point (Art. 275) count each for two,
and the number of lines through the point which touch the
surface in two other points is
■^«(w-l)(w-2)(?2-3)-2(n-f 2)(n-3)=i(n'+w + 2)(w-3)(w-4).
Thus, then, we have completed the discussion of tangent
lines which pass through a given point. We have shewn that
their points of contact lie on the intersection of the surface
with one of the degree w - 1, that their assemblage forms a
cone of the degree n {n — 1), that n[n — 1) [n — 2) of them are
inflexional, and \n [n — 1) [n — 2) [n — 3) of them are double.
These latter double tangents are also plainly double edges
of the tangent cone, since they belong to the cone in virtue of
each contact. Along such an edge can be drawn two tangent
planes to the cone, namely, the tangent planes to the surface
at the two contacts.
The inflexional tangents, however, arc also to be regarded
as double tangents to the surface : since the line passing through
GENERAL THEORY OF SURFACES. 247
three consecutive points is a double tangent in virtue of joining
tlie first and second, and also of joining the second and third.
The inflexional tangents are therefore double tangents whose
points of contact coincide. They are therefore double edges
of the tangent cone ; but the two tangent planes along any
such edge coincide. They are therefore cuspidal edges of
the cone. We have proved, then, that the tangent cone which
is of the degree n[n — l) has n (n — 1) [n— 2) cuspidal edgeSj
and ^n [n — 1) {n — 2) (n — S) double edges; that is to say, any
plane meets the cone in a section having such a number of
cusps and such a number of double points.
280. It is proved precisely as for plane curves {Higher Plane
Curves^ Art. 132), that if we take on each radius vector a length
whose reciprocal is the n^^ part of the sum of the reciprocals
of the n radii vectores to the surface, then the locus of the
extremity will be the polar plane of the point ; that if the
point be on the surface, the locus of the extremity of the mean
between the reciprocals of the w — 1 radii vectores will be the
polar quadric, &c.
By interchanging accented and unaccented letters in the
equation of the polar plane, it is seen that the locus of the
poles of all planes which pass through a given point is the
first polar of that point. The locus of the pole of a plane
which passes through two fixed points is hence seen to be a
curve of the [n — 1)^ degree, namely, the intersection of the
two first polars of these points. We see also that the first
polar of every point on the line joining these two points must
pass through the same curve. And in like manner the first
polars of any three points on a plane determine by their in-
tersection [n — ly points, any one of which is a pole of the
plane, and through these points the first polar of every other
point on the plane must pass.
281. From the theory of tangent lines drawn through a
point we can in two ways derive the degree of the reciprocal
surface. First, the number of points in which an arbitrary
line meets the reciprocal Is equal to the number of tangent
248 GENERAL THEORY OF SURFACES.
planes which can be drawn to the given surface through a
given line. Consider now any two points A and JB on that
line, and let C be the point of contact of any tangent plane
passing through AB. Then, since the line AC touches the
surface, C lies on the first polar of A ; and for the like reason
it lies on the first polar of B. The points of contact, therefore,
are the intersection of the given surface, which is of the 7i^^
degree, with the two polar surfaces, which are each of the degree
[n — 1). The number of points of contact, and therefore the
degree of the reciprocal^ is n[n- \f.
282, Otherwise thus : let a tangent cone be drawn to the
surface having the point A for its vertex ; then since every
tangent plane to the surface drawn through A touches this
cone, the problem is, to find how many tangent planes to the
cone can be drawn through any line AB] or if we cut the
cone by any plane through B^ the problem is to find how many
tangent lines can be drawn through B to the section of the
cone. But the class of a curve whose degree is n {n — 1), which
has n{n—l]{n — 2) cusps, and \n [n — \) [n — 2) [n - 2>) double
points, is
n[n-\) [n {n-\)-\]-?>n [n- 1) [n - 2)
-n{n- 1) [n - 2) (w - 3) = w (w - 1 )'.
Generally the section of the reciprocal surface by any plane
corresponds to the tangent cone to the original surface through
any point. And it is easy to see that the degree of the tangent
cone to the reciprocal surface (as well as to the original surface)
through any point is w (w — 1).
283. Returning to the condition that a line should touch
a surface
we see that if all four differentials be made to vanish by the
coordinates of any point, then every line through the point
meets the surface in two coincident points, and the point is
therefore a double point. The condition that a given surface
may have a double point is obtained by eliminating the vari-
GENERAL THEORY OF SURFACES. 249
ables between the four equations U^ = 0, &c., and the function
equated to zero is called the discriminant of the given surface
{Lessons on Higher Algehra^ Art. 105). The discriminant being
the result of elimination between four equations, each of the
degree w— 1, contains the coefficients of each in the degree
(n — 1)^, and is therefore of the degree 4 [n- If in the coeffi-
cients of the original equation.
It is obvious from what has been said, that when a surface
has a double point, the first polar of every point passes through
the double point.
The surfaces represented by Z7j, C7) ^^- ™^y happen not
merely to have points in common, but to have a whole curve
common to all four surfaces. This curve- will then be a double
curve on the surface ZJ, and every point of it will be a double
point, such that the tangent cone resolves itself into a pair of
planes. Now we saw (Art. 264) that the surface represented
by the general Cartesian equation of the n^^ degree will, in
general, have an infinity of double tangent planes; the re-
ciprocal surface therefore will, in general, have an infinity of
double points, which will be ranged on a certain curve. The
existence then of these double curves is to be regarded among
the " ordinary singularities" of surfaces.
When the point xy'zw is a double point, V and At/'
vanish identically ; and any line through the double point meets
the surface in three consecutive points if it satisfies the equation
A'''?7' = 0, which represents a cone of the second degree.
284. The polar quadric of a parabolic point on a surface
is a cone.
The polar quadric of the origin with regard to any surface
u
^w + u^w + u,^io + ccc. = 0,
(where, as in Art. 272, we have introduced w so as to make
the equation homogeneous) is found by differentiating n — 2
times with respect to w. Dividing out by (n — 2) (n- 3)...3,
and making w;=l, the polar quadric is
n (n - 1) v^ + 2 (n - 1) u^ + 2w, = 0.
KK
250 GENERAL THEORY OF SURFACES.
Now the origin being a parabolic point, we have seen, Art. 266,
that the equation is of the form
2 + (7/ + 2Dzx + 2Ezy + Fz" + &c.,
or, in other words, u^ = 0, and u^ Is of the form u^v^ + w^.
The polar quadric then is
2 (w - 1 + 'iDx -I- 2% + Fz) + Cy"= 0.
But any equation represents a cone when it is a homogeneous
function of three quantities, each of the first degree. The
equation just written therefore represents a cone whose vertex
is the intersection of the three planes, 0, n — \+ iBx + iFy + Fz^
and y. The two former planes are tangent planes to this cone,
and y the plane of contact.
285. It follows from the last article, that tlie locus of
points ichose polar quadrics are cones meets the given surface
in its parabolic points. This locus is found by writing down
the discriminant of A*Z7' = 0. If a, &, &c., denote the second
72 TJf j-i jjr
differential coefficients ■■ , ,., , ■ , ,., , &c., this discriminant will
ax ay .
be a determinant formed with these coefficients, the developed
result being (Art. 67)
ahcd + 2afmn + 2hgnl + 2chlm + '2dfg]i — hcf — cani^ — ahn' - adf'^
- bdf - cdW + IT + ^«y + '^^'1^' - '^rangli - 2nlhf- 2lmfg = 0.
This denotes a surface of the degree 4 [n — 2), which we shall
call the Hessian of the given surface. In the same manner
then, as the intersection of a plane curve with its Hessian de-
termines the points of inflexion, so the intersection of a surface
with its Hessian determines a curve of the degree An [n — 2),
which is the locus of parabolic points (see Art. 269).
286. It follows from what has been just proved that through
a given point can he drawn An {n — V) [n — 2) stationary tangent
planes (see Art. 269). For since the tangent plane passes
through a fixed point. Its point of contact lies on the polar
surface, whose degree is n — \\ and the intersection of this
surface with the surface Z7, and the surface determined in the
GENERAL THEORY OF SURFACES. 251
last article as the locus of points of contact of stationary tangent
planes, determine 4n (?2 - 1) [n - 2) points.
Otherwise thus : the stationary tangent planes to the surface
through any point are also stationary tangent planes to the
tangent cone through that point, and if the cone be cut by
any plane, these planes meet it in the tangents at the points
of inflexion of the section. But the number of points of in-
flexion on a plane curve is determined by the formula [Higher
Plane Curves^ Art. 82)
But in this case, Art. 282, we have v = n (n - 1)'', /^ = ?i [n - 1) ;
therefore v - /* = ?i (?z - 1) (?? - 2), /c = n [n - 1) (n - 2). Hence,
as before, L = 'in[n— \][n — 2).
The numher of double tangent planes to the cone is de-
termined by the formula
2(T-S) = (i/-yLt)(v + /:t-9),
where (Art. 282)
28 = n[n-\) (n - 2) (r* - 3) ; (v + ^ - 9) = n' - n' - 9.
Hence 2r = n [n - 1) [n - 2) {n^ - n' + n- 12).
It follows then, that through any point can be drawn r double
tangent planes to the surface, where r is the number just de-
termined. It will be proved hereafter, that the points of contact
of double tangent planes lie on the intersection of the surface
with one whose degree is [n - 2) (n* — n^-\-n— 12).
287. If a right line lie altogether in a surface it xoill touch
the Hessian and therefore the parabolic curve [Cambridge and
Dublin Mathematical Journal,, vol. IV., p. 255).
Let the equation of the surface be xcf) + y^ = 0, and let
us seek the result of making x and ?/ = 0 in the equation of
the Hessian, so as thus to find the points where the line meets
•11 d'U d'U d'U ,,
that surface. JNow, evidently, -r-z , ^-^ , -j-^- , all contaui
' •' ' dz dio dzdw
£c or ?/ as a factor, and therefore vanish on this supposition.
And if we make c = 0, tZ = 0, ?? = 0 in the equation of the
Hessian, it becomes a perfect square [fl-gni]\ shewing that
the right line touches the Hessian at every point where it
meets it. If we make cc = 0, ?/ = 0 in fl-gm^ it reduces to
252 CURVATURE OF SUEFACES.
-^ _l_ _, _i^ -^ . It Is evident that when the tangent plane
dz dio aw dz
touches all along any line, straight or curved, this line lies
altogether in the Hessian, and not only so, but in the case of a
straight line, it can be shewn that the surface and the Hessian
touch along this line.* The reader can verify this without
difficulty, with regard to the surface xj> + y'^-^.
CURVATURE OF SURFACES.
288. We proceed next to investigate the curvature at any
point on a surface of the various sections which can be made
by planes passing through that point.
In the first place let it be premised that if the equation of
a curve be w, 4 w.^ + ^3 + &c. = 0, the radius of curvature at the
origin is the same as for the conic u^ + u^. For it will be
remembered that the ordinary expression for the radius of
curvature includes only the coordinates of the point and the
values of the first and second differential coefficients for that
point. But if we differentiate the equation not more than twice,
the terms got from differentiating M3, w^, &c. contain powers
of X and ?/, and will therefore vanish for a; = 0, y = 0. The
values therefore of the differential coefficients for the origin are
the same as if they were obtained from the equation u^A- u^ = 0.
It follows hence that the radius of curvature at the origin
(the axes being rectangular) of 3/ + ax^ + '2.bxy + cy'^ + &c, = 0
is -— (see Conies. Art. 241) : or this value can easily be found
directly from the ordinary expression for the radius of curva-
ture [Higher Plane Curves^ Art. 100).
289. Let now the equation of a surface referred to any
tangent plane as plane of xy^ and the corresponding normal
as axis of 2;, be
z + Ax' + 'iBxy + (7/ + 2Dxz + 2Eyz -f Fz" + &c. = 0,
and let us investigate the curvature of any normal section, that
* Cayley, " On Eeciprocal Surfaces," Phil. Trans., vol. 159, 1869, see p. 208.
CURVATURE OF SURFACES. 253
is, of tlie section by any plane passing through the axis of z.
Thus, to find the radius of ciu'vature of the section by the
plane xz^ we have only to make 3/ = 0 in the equation, and
we get a curve whose radius of curvature is half the reciprocal
of A. In like manner the section by the plane yz has its
radius of curvature = half the reciprocal of C. And in order
to find the radius of curvature of any section whose plane makes
an angle d with the plane xz^ we have only to turn the axes of
X and y through an angle 6 (by substituting x cosd — y &\n9
for cc, and x sin^ + y cos^ for ?/, Conies^ Art. 9) ; and by then
putting ?/ = 0 it appears, as before, that the radius of curvature
is half the reciprocal of the new coefiiclent of x^ ; that is to say,
-^ = ^ cos'^ + 25 cos ^ sin ^ + C sln'^.
290. The reader will not fail to observe that this expression
for the radius of curvature of a normal section is identical in
form with the expression for the square of the diameter of a
central conic in terms of the angles which it makes with the
axes of coordinates. Thus if p be the semi-diameter answering
to an angle 6 of the conic Ax' + 2Bxy + Cy'^ = |^, we have R — p\
It may be seen, otherwise, that the radii of curvature are
connected with their directions in the same manner as the
squares of the diameters of a central conic. For we have
seen that the radii of curvature depend only on the terms in
u^ and u^. The radii of curvature therefore of all the sections
of Mj + Mg -1- Mg + &c. are the same as those of the sections of
the quadric u^-\-u,^] and it was proved (Art. 194) that these are
all proportional to the squares of the diameters of the central
section parallel to the tangent plane.
It is plain that the conic, the squares of whose radii are pro-
portional to the radii of curvature, Is similar to the indicatrix.
291. We can now at once apply to the theory of these
radii of curvature all the results that we have obtained for
the diameters of central conies. Thus we know that the
quantity ^ cos^^ + 2j5cos^ sin^+ Oslu'"'^ admits of a maxi-
mum and minimum value; that the values of 6 which corre-
254 CURVATURE OF SURFACES.
spond to the maximum and minimum are always real, and
belong to directions at right angles to each other; and that
those values of 6 are given by the equation (see Conies^ Art. 155)
B co&'d - [A - C) cos d sin 9-B ^xn'O = 0.
Hence, at any point on a surface there are among the normal
sections, one for which the value of the radius of curvature
is a maximum and one for which it is a minimum ; the direc-
tions of these sections are at right angles to each other; and
they are the directions of the axes of the indicatrix. They
plainly bisect the angles between the two inflexional tangents.
We shall call these the principal sections, and the correspond-
ing radii of curvature the priiicipal radii.
If we turn round the axes of x and y so as to coincide
with the directions of maximum and minimum curvature just
determined, it is known that the quantity Ax^ + 'iBxy + Cy'^
will take the form A'x' + By"". Now the formula of Art. 289,
when the coefficient of xy vanishes, gives the following
expression for the half reciprocal of any radius of curvature
-^=^'cos'^+^'sin'6>. But evidently A' and B are the
values of this half reciprocal corresponding to ^ = 0, and 6 = 90°.
Hence any radius of curvature is expressed in terms of the
two principal radii p and p\ and of the angle which the direction
of its plane makes with the principal planes, by the formula
1 cos'-^6' sin"^ ^
— — 1 *
B p p' '
It is plain fas in Conies^ Art. 157) that A' and B'. or — , — ,
are given by a quadratic equation, the sum of these quantities
being A+ C and their product AC— B\
When/3 = /3', all the other radii of curvature are also =/?.
The form of the equation then is z ■\- A[x^ -^ y'^) + &c. = 0, or
the indicatrix is a circle. The origin is then an umhilic.
From the expressions in this article we deduce at once, as
in the theory of central conies, that the sum of the reciprocals
of the radii of curvature of two normal sections at right angles
* This formula (with the inferences drawn from it) is clue to Euler.
CUKVATURE OF SURFACES. 255
to each other is constant ; and again, if tiormal sections he made
through a pair of conjugate tangents (see Art. 268) the sum
oj their radii of curvature is constant.
292. It will be observed that the radius of curvature, being
proportional to the sq^uare of the diameter of a central conic,
does not become imaginary, but only changes sign, if the
quantity A cos^^ 4- 2B cos 6 sin^+ C sin'"'^ becomes negative.
Now if radii of curvature directed on one side of the tangent
plane are considered as positive, those turned the other way
must be considered as negative ; and the sign changes when
the direction is changed in which the concavity of the curve
is turned.
At an elliptic point on a surface ; that is to say, when B'^
is less than AC, the sign of A cos' 6 + 2B cos 6 sin^ + C sin''^
remains the same for all values of 6 ; and therefore at such
a point the concavity of every section through it is turned in
the same direction.
At a hyperbolic point, that is to say, when B'^ is greater
than A C, the radius of curvature twice changes sign, and the
concavity of some sections is turned in an opposite direction
to that of others. The surface, in fact, cuts the tangent plane
in the neighbourhood of the point, and the inflexional tangents
mark the directions in which the surface crosses the tangent
plane and divide the sections whose concavity is turned one
way from those in which it is turned the other way.* And when
we have chosen a hyperbola, the squares of whose diameters
are proportional to one set of radii, then the other set of radii
are proportional to the squares of the diameters of the con-
jugate hyperbola.
293. Having shewn how to find the radius of curvature
of any normal section, we shall next shew how to express,
in terms of this, the radius of curvature of any oblique section,
inclined at an angle <^ to the normal section, but meeting the
* The illustration of the summit of a mountain pass, or of a saddle, will enable
the reader to conceive how a surface may in two directions sink below the tangent
plane, and on the other sides rise above it ; a mountain summit is an instance of an
elliptic point.
256 CUEVATURE OP SURFACES.
tangent plane in the same line. Thus we have seen that the
radius of curvature of the normal section made by the plane
?/ = 0 is half the reciprocal of A. Now let us turn the axes
of j/ and z round in their plane through an angle <^ (which is
done bj substituting z cos0 — ?/ sin^ for s, and z smcfi + y coscfy
for ?/). If we now make the new 3/ = 0, we shall get the
equation (still to rectangular axes) of the section bj a plane
making an angle 0 with the old plane ?/ = 0, but still passing
through the old axis of x 5 and this equation will plainly be
0 = z cos <j> + Ax^ -+2 {B sin (f) + D cos (/>) xz
+ ( C sin''0 -f 2E sin c}> cos^ + F cos^) z^ + &c.
and by the same method as before the radius of curvature is
COS (h
found to be -^rT 1 oi' is =Rcos(f), where B is the radius
of curvature of the corresponding normal section. This is
Meunier'S theorem, that the radius of curvature of an oblique
section is equal to the projection on the plane of this section of
the radius of curvature of a normal section passing through the
same tangent line. Thus we see that of all sections which can
be made through any line drawn in the tangent plane, the
normal section is that whose radius of curvature is greatest ;
that is to say, the normal section is that which is least curved
and which approaches most nearly to a straight line.
Meunier's theorem has been already proved in the case
of a quadric (Art. 194), and we might therefore, if we had
chosen, have dispensed with giving a new proof now ; for
we have seen that the radius of curvature of any section of
M^ + M^ + M3 -f &c. is the same as that of the corresponding
section of the quadric u^ -f u^.
294. It was proved (Art. 203) that if two surfaces Mj4m.^+&c.,
M, + V2 + &c. touch, their curve of intersection has a double point,
the two tangents at which are the intersections of the plane u
with the cone u^ — 'v^. When the plane touches the cone, the
surfaces have what we have called stationary contact. It is
also proved, as at Art. 205, that a sphere has stationary contact
with a surface when the centre is on the normal and the radius
CURVATURE OF SURFACES. 257
equal to one of the principal radii of curvature. In fact, the
condition for stationary contact between
z-\- ax^ + 2hxy + hy' + &c., z-\- aV + '^h'xy -f Vy' + &c.
is {a-a'){h-V) = {h-h')\
which, when Ji and A' both vanish, implies either a = a' or h = }).
The surface therefore z + Ax^ + By'^ + &c. will have stationary
contact wnth the sphere 2rs -\- x^ + y"^ + z^ i^ r = —^ or ^-^ j ^^t
these are the values of the principal radii.
295. The principles laid down in the last article enable
us to lind an expression for the values of the principal radii
at any point ; the axes of coordinates having any position.
If we transform the equation to any point xyz' on the
surface as origin, it becomes
dU' dU dJJ' \ ( d d d\'j, „
or, denoting the first differential coefficients by Z, ili, JV, and
the second by a, &, c, &c.,
2 [Lx + My + Nz) 4- ax' + hy' + cz^ + 2fyz -l- 2gzx + 2hxy + &c. = 0.
The equation then of any sphere having the same tangent
plane is, assuming the axes to be rectangular,
2 [Lx + My -\-Nz) + \ [x' -f / + z') = 0,
and this sphere will have stationary contact with the quadric if
X be determined so as to satisfy the condition that Lx + My + Nz
shall touch the cone
(a -X)x''+{b- X) y' + (c - X) z' + 2fyz -|- 2gzx + 2hxy = 0.
This condition is
a - X,
h, g, L
A,
h-\, /, M
9i
f c-X, N
L,
M, N,
= 0,
which expanded is
[iJb.X){c-\)-f']U^{c-\){a-\)-y']M-'+[{a-\){h-\)-h']N'
+ 2{gh-{a-\)f]3fN+2{hf-{b-\)g]NL + 2{fg-[c-\)h]LM:=0,
LL
258 CURVATURE OF SURFACES.
or X. is given by the quadratic
[U + M' + N')\' - [{h -V c) U ^ [c + a) MU [a^h] N'
- 2fMN- 2gNL - 2hLM] \
+ [he -f) U + (m - /) M-' + {ab - If) N'
+ 2 {gh - a/) MN+ 2 {hf- hg) NL + 2 {f(j - ch) LM=0.
Now if r be the radius of the sphere
\{x' + f + z')+2{Lx+M7/ + Nz) = 0,
we have r"^ = — ^ . We therefore find the principal
radu by substituting — ^ — tor A, in the preceding
quadratic.
The absolute term in the equation for \ may be simplified
by writing for L, M, N their values from the equations
[n- 1) L = ax + hy + gz + hv, &c.,
when the absolute term reduces to r^ where H is the
in -I)
Hessian, written at full length, Art. 285. We might have seen
a 'priori that, for any point on the Hessian, the absolute term
must vanish. For since the directions of the principal sections
bisect the angles between the inflexional tangents ; when the
inflexional tangents coincide, one of the principal sections coin-
cides with their common direction, and the radius of curvature
of this section is infinite, since three consecutive points are
on a right line. Hence one of the values of X (which is
the reciprocal of r) must vanish. By equating to zero the
coefficient of A in the preceding quadratic, we obtain the
equation of a surface of the degree 3n — 4, which intersects
the given surface in all the points where the principal radii
are equal and opposite : that is to say, where the indicatrix
is an equilateral hyperbola.
The quadratic of this article might also have been found
at once by Art 102, which gives the axes of a section of the
quadric
ax^ + by' + cz^ + 2fyz + 2gzx + 2hxy ~ 1
made parallel to the plane Lx -)- My -\- Nz = 0.
CUKVATURE OF SURFACES. 259
296. From the equations of the last article we can find
the radius of curvature of any normal section vicetinrj the
tangent plane in a line whose direction-angles are given.
For the centre of curvature lies on the normal, and if we
describe a sphere with this centre, and radius equal to the
radius of curvature, it must touch the surface, and its equa-
tion is of the form
2 [Lx + My + Nz)+\ {x" + / + z') = 0.
The consecutive point on that section of the surface which we
are considering satisfies this equation, and also the equation
w, -f u.-^ = 0, that is
2 [Lx + 3Iy + Nz) + ax^ + by"" + cz^ + 2fyz + 2gzx + 2hxy = 0.
Subtracting, we find
_ ax' + Inf + cz^ + %fyz + "igzx + 2hxy
x' + ?/' + z'
And since this equation is homogeneous, we may write for
ic, y^ z the direction-cosines of the line joining the consecutive
ponit to the origm. As in the last article A, = .
Hence
a cos'''a+6 cos'^/S+c cos'7+2/coSy8 C0S7+ 2gco%^ cosa+2/t cosacos/3 '
The problem to find the maximum and minimum radius of
curvature is, therefore, to make the quantity
ax^ + hy^ + cz^ + 'ifyz + 2gzx -\- 2hxy
a maximum or minimum, subject to the relations
Lx^- My + Nz = 0^ a;'-f/+2'=l.
And thus we see, again, that this is exactly the same problem
as that of finding the axes of the central section of a quadric
by a plane Lx + ]\[y -H Nz.
297. In like manner the problem to find the directions of
the princi'pal sections at any point is the same as to find the
directions of the axes of the section by the plane Lx + J\fy + Nz
of the quadric ax^ + by'' -t- cz' + 2fyz + 2gzx + 2hxy = 1.
260 CURVATURE OF SURFACES.
Now given any diameter of a quadric, one section can
be drawn through it having that diameter for an axis ; the
other axis being obviously the intersection of the plane perpen-
dicular to the given diameter with the plane conjugate to it.
Thus, if the central quadric be U= 1, and the given diameter
pass through xy'z\ the diameter perpendicular and conjugate
is the intersection of the planes
xx' ^yy ^zz=0^ x'U^+y'U,^^ zU^ = 0.
If the former diameter lie in a plane Lx' + My + Nz\ the
latter diameter traces out the cone which is represented by
the determinant obtained on eliminating x'yz' from the three
preceding equations : viz.
[Mz-Ny] U^ + {Nx-Lz) U^-\-{Ly-Mx) U^ = 0.
And this cone must evidently meet the plane Lx -\- My + Nz
in the axes of the section by that plane. Thus, then, the
directions of the principal sections are determined as the inter-
section of the tangent plane Lx + My + Nz with the cone
[Mz - Ny) [ax + hy + gz) + [Nx - Lz) [hx + hy +fz)
-f [Ly - Mx) [gx +fy + cz) = 0,
or (% - M) x' -f [Nh - Lf) y"" + [Lf- Mg) s"
+ {L{h-c)- Mh + Ng\ yz + [Lh + i/(c - a) - Nf] zx
+ [-Lg + Mf-\- N{a- b)] xy = 0.
298. The methods used In Art. 295 enable us also easily
to find the conditions for an umbilic* If the plane of xy be
* It might be imagined that we could obtain a single condition for an umbilic by
expressing that the quadratic (Art. 295) for the determination of the principal radii of
curvature shall have equal roots. But, as at Art. 83, this quadratic, having its roots
always real, is one of the class discussed Higher Algebra, Art. 44, the discriminant
of which can be expressed as a sum of squares. If we make these squares separately
vanish, we obtain two conditions, which are more easily foxmd as in the text.
In plane geometiy, the problem of finding when ax^ + 2hxy + by- = 1 repre-
sents a circle may be solved by taking the quadratic which gives the maximum
or minimum values of x^ + ?/- = p, viz. [ap — 1) {bp — 1) — h'^p- = 0, and forming the
condition that the quadratic shall have equal roots, viz. (re — b)- + ih- = 0. Now this
single condition is not the condition that the curve shall be a circle, for either of the
factors a — 6 + 2hi separately equated to zero only expresses that the curve passes
through one of the circular points at infinity. But if we have both factors simul-
taneously = 0, that is to say, if we have « — 5 = 0, h = Q, the curve passes through
both circular points and is a circle. And the theory in regard to the umbilics is
CURVATURE OF SURFACES. 261
the tangent plane at an umbllic, the equation of the surface
is of the form
z + A {.v' + f) + 2Dxz 4- 2E1/Z + Fz' 4 &c. = 0 ;
and if we subtract from it the equation of any touching
sphere, viz.
s + \(a;"''+/-f-2;') = 0,
it is evidently possible so to choose X (namely, by taking it
= A) that all the terms in the remainder shall be divisible
by z. We see, thus, that if w, 4 u^-'r &c. represent the surface,
and M, + XUg ^^7 touching sphere, it Is possible, when the
origin Is an umbilic, so to choose \ that u^ — \v^ may contain
w, as a factor. We see, then, by transformation of coordinates
as in Art. 295, that any point xy'z' will be an umbilic if it
is possible so to choose X. that
[a - X) x' +{b-\)f+{c- X) z'^ + 2/7/3 + 2gzx + 2hxy
may contain as a factor Lx + My + Nz. If so, the other factor
must be
a — \ h —X c— X
Multiplying out and comparing the coefficients of yz^ zx^ xy^
we get the conditions
(J-X)^+(c-X)^=2/, {c-X)^+{a-X)^=2g,
[a-X)^^.{h-X)k = 2h.
Eliminating X between these equations, we obtain for an umbilic
the two conditions
IN^ + cM'' - 2fMN _ cU + aN' - 2gLN _ aAP + hU - 2hLM
N-' + ]\P ~ U + N-' ~ JiP + L'
almost identical : the points on the surface for which the two radii of curvature are
equal are points such that for each of them o?ie of the inflexional tangents meets the
imaginary circle at infinity ; an umbilic is a point such that both the inflexional
tangents meet the circle at infinity. The first-mentioned points form on the surface
an imaginary locus having the umbilics for double points.
\
2G2 CURVATURE OF SURFACES.
Since there are only two conditions to be satisfied, a surface
of the n^^ degree has In general a determinate number of
umbilics ; for the two conditions, each of which represents a
surface, combined with the equation of the given surface, de-
termine a certain number of points. It may happen, however,
that the surfaces represented by the two conditions Intersect
in a curve which lies (either wholly or In part) on the given
surface. In such a case there will be on the given surface
a line, every point of which will be an umbilic. Such a
line Is called a line of spherical curvature.
299. Before applying the conditions of the last article, the
form In which we have written them requires that the following
considerations should be attended to.
These equations appear to be satisfied by making X = 0,
hN"" + cM'^ - 2fMN
a = ^TYT, jW ; whence we might conclude that the
surface i = 0 must always pass through umbilics on the given
surface. Now it Is easy to see geometrically that this Is not
the case, for L (or U^ is the polar of the point yzw with
respect to the surface, so that if L necessarily passed through
umbilics it would follow by transformation of coordinates that
the first polar of every point passes through umbilics. On
referring to the last article, however, it will be seen that the
investigation tacitly assumes that none of the quantities Z-, il/, N
vanish ; for If any of them did vanish, some of the equations
which we have used would contain Infinite terms. Supposing
then L to vanish, we must examine directly the condition that
My + Nz may be a factor In
[a - X) x' ■v[h-X)y'^+[c- X) z^ + Ifyz + Igzx + 'llixy.
We must evidently have A, = a, and It is then easily seen that
, . , hN'' + cM'-2fMN ,., .
we must, as beiore, have a= r^^ — ^tft i while m
addition, since the terms 2gzx + 2hxy must be divisible by
My + Nz^ we must have Mg = Nh. Combining then with the
two conditions here found, Z = 0, and the equation of the
surface, there are four conditions which, except in special
cases, cannot be satisfied by the coordinates of any points.
CURVATURE OF SURFACES. 263
If we clear of fractions the conditions given in the last
article, it will be found that they each contain either X, J/,
or iV^ as a factor. And what we have proved in this article
is that these factors may be suppressed as irrelevant to the
question of umbilics.
Again, it can be shown that, introducing homogeneous coor-
dinates as in Art. 295, the numerators of the above fractions
multiplied by (n - 1)"'', are respectively
n{n-l] [be -/') U- [Dx' + Aiv'' - 2Lxw),
n (n - 1) (ca- /) U- {Dif + Bw' - 2Myw),
n (w - 1) [ah- ¥) U- {Dz^ + Cvi' - ^Nzw),
where A^ B^ C, i), X, J/, N are the functions of a, J, c, &c. de-
fined In Art. 67. Hence our equations are satisfied for Z7=0 by
w = 0, D = 0, but these are the points of inflexion of the
intersection of U with the plane at infinity, which are also
irrelevant to the question of umbilics.*
We now proceed to draw some other inferences from what
was proved (Art. 294) ; namely, that the two principal spheres
have stationary contact with the surface.
300. When two surfaces have stationary contact^ they touch
in two consecutive points.
* From wliat has been said we can infer the number of umbilics which a surface
of the re degree will in general possess. We have seen that the umbilics are deter-
mined as the intersection of the given surface with a cmwe whose equations are of
ABC
the form -r, — ^, — -pn • '^^'^ ii A, B, C be of the degree I, and A', B', C of the
ABC
degree m, then AB' — BA', AC — CA' are each of the degree I + m, and intersect in
a curve of the degi'ee {I + mf. But the intersection of these two surfaces includes
the curve AA' of the degree Im which does not lie on the surface BC — CB'. The
degree therefore of the curve common to the three surfaces is l'^ + Im + m^. In the
present case I = 3n — 4, m = 2n — 2, and the degree of the cuiwe would seem to be
19?i- - 46w + 28. But we have seen that the system we are discussing includes three
curves such as
L, a{M^+ N"-) - (6.V2 + cM^- 2fMN)
which do not pass through umbilics. Subtracting therefore from the number just
found 3 [n — 1) (3m — 4), we see that the umbilics are determined as the intersection
of the given surface with a curve of the degree (lOw^ — 25?i + 16), but from the
number of points thus found we must subtract 3« [n - 2) for the inflexions on the
intersection of the given surface with the plane at infinity. Thus the number of
umbiUcs is n {\Qn" — 28« + 22). {Voss, Math. Annalen ix. 1876). In particular, when
w = 2, then the niunber is twelve, viz. there are four umbilics in each of the principal
l^lanes.
264 CURVATUKE OF SURFACES.
The equations of the two surfaces being
z + ax^ + 2hxy + hif + &c. = 0, z-\- aV + 2h'xy + h'y^ + &C.5
the tangent planes at a consecutive point are (Art. 262)
0 + 2 {ax' -f hy) x + 2 [hx' + by') 3/ = 0,
s + 2 [ax" + h'y) x+2 {h'x + Vy) y = 0.
That these may be identical, we must have
ax' + hy = ax' + h'y' ^ hx' + hy' = h'x' + b'y'^
and eliminating x' : y between these equations, we have
[a-a')[h-h') = [h-h')\
which is the condition for stationary contact.
The sphere, therefore, whose radius is equal to one of the
principal radii, touches the surface in two consecutive points ;
or two consecutive normals to the surface are also normals to
the sphere, and consequently intersect in its centre. Now we
know that in plane curves the centre of the circle of curvature
may be regarded as the intersection of two consecutive normals
to the curve. In surfaces the normal at any point will not
meet the normal at a consecutive point taken arbitrarily. But
we see here that if the consecutive point be taken in the
direction of either of the principal sections, the two consecutive
normals will intersect, and their common length will be the
corresponding principal radius. On account of the importance
of this theorem we give a direct investigation of it.
301. To find in what cases the normal at any point on a
surface is intersected hy a consecutive normal. Take the tangent
plane for the plane of xy^ and let the equation of the surface be
z + Ax' + 2Bxy + (7/ -f 2Dxz + 2Eyz + Fz"" + &c. = 0.
Then we have seen (Art. 268) that the equation of a consecutive
tangent plane is
2 + 2 [Ax' + By') x + 2 [Bx' + Cy') y = 0,
and a perpendicular to this through the point x'y' will be
x — x' y — y
Ax' + By' Bx'+Cy
—, — 22!.
CURVATURE OF SURFACES. 265
This will meet the axis of z (which was the original normal) if
^^ y'
Ax' + By' ~ Bx' ^ Cy"
The direction therefore of a consecutive point whose normal
meets the given normal is determined by the equation
Bx"' + {C-A)x'y - By" = 0.
But this is the same equation (Art. 291) which determines the
directions of maximum and minimum curvature. At any point
on a surface therefore there are two directions, at right angles
to each other, such that the normal at a consecutive point
taken on either intersects the original normal. And these
directions are those of the two principal sections at the point.
Taking for greater simplicity the directions of the principal
sections as axes of coordinates; that is to say, making B = 0
in the preceding equations, the equations of a consecutive normal
become —r-r = ^ ;■ = 22, whence it is easy to see that the
Ax Cy ' -^
normals corresponding to the points y' = 0, x' = 0 intersect the
axis of z at distances determined respectively by 2A2 + 1 == 0,
2Cz+l =0. The intercepts thei-efore on a normal by the two
consecutive ones which intersect it are equal to the principal
radii.*
We may also arrive at the same conclusions by seek-
ing the locus of points on a surface, the normals at which meet
a fixed normal which we take for axis of z. Making x = 0,
y = 0 in the equation of any other normal, we see that the
* M. Bertrand, in his theory of the curvature of surfaces, calculates the angle
made by the consecutive normal with the plane containing the original normal
and the consecutive point x'y'. Supposing still the directions of the principal sec-
tions to be axes of coordinates, the direction-cosines of the consecutive normal are
proportional to 2 Ax', 2Cy', while those of a tangent line perpendicular to the radius
vector are proportional to — y', x', 0. Hence the cosine of the angle between these
two lines, or the sine of the angle which the consecutive normal makes with the
normal section, is proportional to 2{C—A) x'y'; or, if a be the angle which the
direction of the consecutive point makes witli one of the principal tangents, is
proportional to (C — A) sin 2a. When a = 0, or = 90°, this angle vanishes, and the
consecutive normal is in the plane of the original normal.
M M
266 CUHVATUEE OF SUKFACES.
point where it meets the surface must satisfy the cotidition
TJ^x= U^y. The curve where this surface meets the given
surface has the extremity of the given normal for a double
point, the two tangents at which are the two principal tangents
to the surface at that point. (See Ex. 9, p. 101).
The special case where the fixed normal is one at an
umbilic deserves notice. The equation of the surface being of
the form z + A {x^ 4- 'if) + &c. = 0, the lowest terms in the equa-
tion xU^ = yU^^ when we make s = 0, will be of the third
degree, and the umbilic is a triple point on the curve locus.
Thus while every normal immediately consecutive to the normal
at the umbilic meets the latter normal, there are three directions
along any of which the next following normal will also meet
the normal at the umbilic*
302. A line of curvature'f on a surface is a line traced on
it, such that the normals at any two consecutive points of it
intersect. Thus, starting with any point 31 on a surface, we
may go on to either of the two consecutive points N, N\ whose
normals were proved to intersect the normal at M. The normal
at Nj again, is intersected by the consecutive normals at two
points, P, P', the element NP being a continuation of the
element MN while the element NP' is approximately per-
pendicular to it. In like manner we might pass from the point
P to another consecutive point Q, and so have a line of curva-
ture MNPQ. But we might evidently have pursued the same
* Sir W. R. Hamilton has pointed out {Elements oj Quaternions, Art. 411) how
this is verified in the case of a quadric. He has proved that the two imaginary
generators (see Art. 139) through any nmbilic are lines of curvature, the third line of
curvature through the umbilic being the principal section in which it lies. In fact,
for a point on a principal section, the cone (Ex. 9, p. 101) breaks up into two planes.
The normal therefore at such a point only meets the normals at the points of the
principal section, and at the points of another plane section. For the umbilic the
latter jilane is a tangent plane and the section reduces to the imaginary generators.
The normals along either lie in the same imaginary plane. At eveiy point on either
generator, distinct from the umbilic, the two directions of curvature coincide with the
line, which is perpendicular to itself {Conies, p. 351). There is, however, some
speciality as regards the theory of the umbilics of a quadric.
t The whole theory of lines of curvature, umbilics, ifec. is due to Monge. See his
"Application de I'Analjse k la Geometrie," p. 124, Liouville's edition.
CURVATURE OF SURFACES. 2G7
process had we started in the direction MN'. Hence, at any
point M on a surface can be drawn two lines of curvature ;
these cut at right angles and are touched by the two " prin-
cipal tangents" at M. A line of curvature will ordinarily not
be a plane curve, and even in the special case where it is
plane it need not coincide with a principal normal section at Ji,
though it must touch such a section. For the principal section
must be normal to the surface, and the line of curvature may be
oblique.
A very good illustration of lines of curvature is afforded
by the case of the surfaces generated by the revolution of any
plane curve round an axis in its plane. At any point P of
such a surface one line of curvature is the plane section passing
through P and through the axis, or, in other words, is the
generating curve which passes through P. For, all the normals
to this curve are also normals to the surface, and, being in
one plane, they intersect. The corresponding principal radius
at P is evidently the radius of curvature of the plane section
at the same point. The other line of curvature at P is the
circle which is the section made by a plane drawn through
P perpendicular to the axis of the surface ; for the normals
at all the points of this section evidently intersect the axis
of the surface at the same point, and therefore intersect each
other. The intercept on the normal between P and the axis
is plainly the second principal radius of the surface.
The generating curve which passes through P is a prin-
cipal section of the surface, since it contains the normal and
touches a line of curvature ; but the section perpendicular to the
axis is, in general, not a principal section because It does not
contain the normal at P. The second principal section at that
point would be the plane section drawn through the normal at
P and through the tangent to the circle described by P. The
example chosen serves also to Illustrate Meunler's theorem ;
for the radius of the circle described by P (which, as we have
seen, is an oblique section of the surface) Is the projection on
that plane of the intercept on the normal between P and the
axis, and we have just proved that this intercept is the radius
of curvature of the corresponding normal section.
268 CUKVATURE OP SUEFACES,
303. It was proved (Art. 297) that the direction-cosines of
the tangent line to a principal section fulfil the relation
(Jlfoos7 — iVcos/3)(a cosa +Acos/3 + ^ cos 7)
-f (iVcosa - L C0S7) [h cosa + h cos/3 +/ C0S7)
+ {L cos/3 — il/cosa) [g cosa +/ cos /3-f c 0037) = 0.
Now the tangent line to a principal section is also the tangent
to the line of curvature ; while, if ds be the element of the
arc of any curve, the projections of that element upon the
three axes being dx, dy^ dz^ it is evident that the cosines of
the angles which ds makes with the axes are -r- , -4 •, -v •
ds ds ds
The differential equation of the lines of curvature is therefore
got bj writing dx^ dy^ dz for cosa, cos/3, cos 7 in the preceding
formula.
This equation may also be found directly as follows (see
Gregory's Solid Geometry^ p. 256) : Let a, /3, 7 be the co-
ordinates of a point common to two consecutive normals.
Then, if xyz be the point where the first normal meets
the surface, by the equations of the normal we have
~^ — = — —■ = ^-xT- ; or, if we call the common value of
these fractions 6^ we have
a = x + Le^ ^ = y + Md, y = z + Nd.
But if the second normal meet the surface in a point x + dx^
y + dy^ z + dz^ then, expressing that a/87 satisfies the equations
of the second normal, we get the same results as if we differen-
tiate the preceding equations, considering a,S7 as constant, or
dx + Ldd + ddL = 0, dy + MdO + OdM^ 0, dz + Ndd + edN= 0,
from which equations eliminating 6^ dd^ we have the same
determinant as in Art. 297, viz.
dx^ dy^ dz
L, M, N
dL, dM, dN =0.
Of course
dL = adx + hdy + gdz^ dM= hdx + bdy +fdzj dN=gdx -\-fdy + cdz.
CURVATURE OF SURFACES. 269
Ex. To find the differential equation of the lines of curvature of the ellipsoid
a;^ ip- 3^
- + p + - = 1.
a- b- c-
Here we have
L = -, M=f-^, N--„, dL= — , dM=-f„, dN= — .
a- 0- c- a- 0- c-
Substituting these values in the preceding equation it becomes, when expanded,
(62 - C-) xdydz+ {c"- - a^) ydzdx+ (a- - b-) z dx dy - 0.
Knowing, as we do, that the lines of curvature are the intersections of the ellipsoid
with a system of concentric quadrics (Art. 196), it would be easy to assume for the
integral of this equation Ax"^ + Bij- + Cz"^ = 0, and to determine the constantsby
actual substitution. If we assume nothing as to the form of the integral we can
eliminate z and dz by the help of the equation of the surface, and so get a differ-
ential equation in two variables which is the equation of the projection of the lines
of curvature on the plane of xy. Thus, in the present case, multiplying by — and
reducing by the equation of the ellipsoid and its differential, we have
W - C') xdy + (c^ - a?) ydx] |^ + ^} = (a^ - 6=) {l - ^^ - g} dx dy,
a? m - c2) , a? {a? - J^)
AxyU) + {x- - Ay-^ - B) -f~ xy = 0,
dy
\dxj ' ^'" "^ "' dx
the integral of which (see Boole's Differential Equations, Ex. 3, p. 135) is, with C an
arbitrary constant,
x^_y^ _ 1
B BC~AC+l'
or the lines of curvature are projected on the principal plane into a series of conies
whose axes a', U are connected by the relation
a'2 (a2 _ c2) yi (^,2 _ c2) _
a? (a2 _ 62) + i-i (^iji _ „2) - 1-
It is not difficult to see that this coincides with the account given of the lines of
curvature in Art. 196.
304. The theorem that confocal quadrics Intersect in lines
of curvature is a particular case of a theorem due to Diipin,
which we shall state as follows : //' tliree surfaces intersect at
right angles^ and if each pair also intersect at ricjht angles at
their next consecutive common point, then the directions of the
intersections are the directions of the lines of curvature on each.
Take the point common to all three surfaces as origin, and
the three rectangular tangent planes as coordinate planes ; then
the equations of the surfaces are of the form
X + ay'^ + 'ihyz + cz^ + ^dzx + &c. = 0,
y + az' -f 2h'zx -\- ex' + 2d'xy-\- &c. = 0,
z + a'x" + 2l"xy + c'y' + &c. = 0.
270 CURVATURE OF SURFACES.
At a consecutive point common to the first and second surfaces,
we must have a; = 0, y = 0, 2; = 2:', where z is very small. The
consecutive tangent planes are
{\-\2dz)x-Y 2hzy +2cz'z=0,
iVzx + (1 + 1cVz)y + 2aV.3 = 0.
Forming the condition that these should be at right angles and
only attending to the terms where z is of the first degree, we
have J + &' = 0.
In like manner, in order that the other pairs of surfaces
may cut at right angles at a consecutive point, we must have
h' + y = 0, h" -^h = 0, and the three equations cannot be ful-
filled unless we have S, h\ h" each separately = 0 ; in which
case the form of the equations shows (Art. 301) that the axes
are the directions of the lines of curvature on each. Hence
follows the theorem in the form given by Dupin ;* namely, that
if there he three systems of surfaces^ such that every surface of one
system is cut at right angles hy all the surfaces of the other two
systems^ then the intersection of two surfaces helonging to different
systems is a line of curvature on each. For, at each point of
it, it is, by hypothesis, possible to draw a third surface cutting
both at right angles.
305. A line of curvature is, by definition, such that the
normals to the surface at two consecutive points of it intersect
each other. If, then, we consider the surface generated by all
the normals along a line of curvature, this will be a developable
surface (Note, p. 89) since two consecutive generating lines in-
tersect. The developable generated by the normals along a line
of curvature manifestly cuts the given surface at right angles.
* Developpements de Geometrie, 1813, p. 330. The demonstration here given
is by Professor W. Thomson : see Gregory's Solid Geometry, p. '2G3. Cambridge
Mathematical Journal, Vol. iv., p. 62. See also the proof by R. L. Ellis, Gregory's
JExam,ples, p. 215. A closely connected theorem is the following :
Jf two surfaces cut at rii/ht angles, and if their intersection is a line qf curvature
on one, it is also a line oj" curvature on the other.
This may be proved as in the text ; viz. taking the origin at any poinf'on the
intersection of the two surfaces, then if they cut at right angles b + b' ='J). Hence if
i = 0, then also b' = 0, which proves the theorem. The theorem is also true if the
Burfacea cut at any constant angle.
CURVATURE OF SURFACES. 271
The locus of points where two consecutive generators of
a developable intersect is a curve whose properties will be
more fully explained in the next chapter, it is called the
cusiridal edge of that developable. Each generator is a tan-
gent to this curve, for it joins two consecutive points of the
curve; namely, the points where the generator in question
is met by the preceding and by the succeeding generator (see
Art. 123).
Consider now the normal at any point M of a surface ;
through that point can be drawn two lines of curvature
3INFQ, &c., MN'FQ\ &c. : let the normals at the points
il/, N^ P, Q^ &c., intersect in C, D, E^ &c,, and those at
3/, N\ P\ Q' in C\ D\ E' ; then it is evident that the curve
CDE^ &c., is the cuspidal edge of the developable generated by
the normals along the first line of curvature, while C'D'E' is
the cuspidal edge of the developable generated by the normals
along the second. The normal at if, as has just been ex-
plained, touches these curves at the points C, G\ which are
the two centres of curvature corresponding to the point M.
What has been proved may be stated as follows. — The
cuspidal edge of the developable generated by the normals
along a line of curvature is the locus of one of the systems of
centres of curvature corresponding to all the points of that line.
306. The assemblage of the centres of curvature C, C
answering to all the points of a surface is a surface of two
sheets, called the surface of centres (see Art. 198). The curve
CDE lies on one sheet while C'D'E' lies on the other sheet.
Every normal to the given surface touches both sheets of the
sui'face of centres : for it has been proved that the normal at
M touches the two curves CBE^ C'D'E'^ and every tangent
line to a curve traced on a surface is also a tangent to the
surface.
Now if from a point, not on a surface, be drawn two con-
secutive tangent lines to the surface, the plane of those lines is
manifestly a tangent plane to the surface; for it is a tangent
plane to the cone which is drawn from the point touching the
surface. But if two consecutive tangent lines intersect on the
272 CUKVATURE OF SURFACES.
surface, it cannot be inferred that their plane touches the
surface. For if we cut the surface by any plane whatever,
any two consecutive tangents to the curve of section (which,
of course, are also tangent lines to the surface) intersect on the
curve, and yet the plane of these lines is supposed not to touch
the surface.
Consider now the two consecutive normals at the points
M^ N, these are both tangents to both sheets of the surface
of centres. And since the point C in which they intersect is on
the first sheet but not necessarily on the second, the plane of
the two normals is the tangent plane to the second sheet of
the surface of centres.
The plane of the normals at the points ilf, N' is the tangent
plane to the other sheet of the surface of centres. But because
the two lines of curvature through M are at right angles to
each other, it follows that these two planes are at right angles
to each other. Hence, the tangent planes to the surface of centres
at the tioo points (7, C", where any normal meets it^ cut each
other at right angles.
307. It is manifest that for every umbilic on the given surface
the two sheets of the surface of centres have a point common ;
or, in other words, the surface of centres has a double point ;
and if the original surface have a line of spherical curvature,
the surface of centres will have a double line. The two sheets
will cut at right angles everywhere along this double line.
This, however, is not the only case where the surface of centres
has a double line. A double point on that surface arises not
only when the two centres which belong to the same normal
coincide, but also when two different normals intersect, and the
point of intersection is a centre of curvature for each. It was
shewn. Arts. 298—9, that a surface of the w'" degree possesses
ordinarily a definite number of umbillcs, and, therefore, in
general not a line of spherical curvature. Hence a double line
of the first kind is not among the ordinary singularities of the
surface of centres. But that surface will in general have a
double line of the second kind. Through any point several
normals can be drawn to a surface : every point on the surface
CURVATIIRE OF SURFACES. 273
of centres is a centre of curvature for one of these normals,
each point of a certain locus on the surface will be a centre of
curvature for two normals, and there will even be a definite
number of points each a centre of curvature for three normals.*
308. It is convenient to define here a geodesic line on a
surface, and to establish the fundamental property of such
aline; namely, that its osculating plane (see Art. 123) at any
point is normal to the surface. A geodesic line is the form
assumed by a strained thread lying on a surface and joining
any two points on the surface. It is plain that the geodesic
is ordinarily the shortest line on the surface by which the two
points can be joined, since, by pulling at the ends of the
thread, we must shorten it as much as the interposition of the
surface will permit. Now the resultant of the tensions along
two consecutive elements of the curve, formed by the thread,
lies in the plane of those elements, and since it must be de-
stroyed by the resistance of the surface, it is normal to the
surface; hence, the plane of two consecutive elements of the geo-
desic contains the normals to the surface.^
* The possibility of double lines of the second kind was overlooked by Monge
and by succeeding geometers ; and, oddly enough, first came to be recognized in con-
sequence of Prof. Kummer's having had a model made of the surface of centres of an
ellipsoid (see Monatsberichte of the Berlin Academy, 1862). Instead of finding the
sheets, as he expected, to meet only in the points corresponding to the umbilics, he
found that they intersected in a curve, and that they did not cut at right angles along
this line. Of course when the existence of the double line was known to be a
fact its mathematical theory was evident. Clebsch had, on purely mathematical
grounds, independently arrived at the same conclusion in an elaborate paper on the
normals to an ellipsoid, of equal date with Kummer's paper, though of later pub-
lication. A discussion of the surface of centres of an ellii^soid, founded on Clebsch'3
paper, will be given in Chapter Xiv.
t I have followed Monge ia giving this proof, the mechanical principles which
it involves being so elementary that it seems pedantic to object to the introduction
them. For the benefit of those who prefer a purely geometrical proof, one or two
are added in the text. For readers familiar with the theory of maxima and minima
it is scarcely necessary to add that a geodesic need not be the absolutely shortest line
by which two points on the surface may be joined. Thus, if we consider two points
on a sphere joined by a great circle, the remaining portion of that great circle, ex-
ceeding 180°, is a geodesic, though not the shortest line connecting the points. The
geodesic, however, will always be the shortest line if the two points considered be
taken sufficiently near.
NN
274 CURVATURE OF SURFACES.
The same thing may also be proved geometrically. In the
first place, if two points A, 0 in different planes be connected
by joining each to a point B in the intersection of the two
planes, the sura of AB and BC will be less than the sum of
any other joining lines AB\ B'C, if AB and BC make equal
angles with TT\ the intersection of the planes. For if one
plane be made to revolve about TT^ until it coincide with the
other, AB and BC become one right line, since the angle TBA
is supposed to be equal to T'BC; and the right line ^C is
the shortest by which the points A and C can be joined.
It follows, that if AB and BC be consecutive elements
of a curve traced on a surface, that curve will be the shortest
line connecting A and C when AB and BC make equal
angles with BT, the intersection of the tangent planes at A
and C.
We see, then, that AB (or its production) and BC are con-
secutive edges of a right cone having B2' for its axis. Now
the plane containing two consecutive edges is a tangent plane
to the cone ; and since every tangent plane to a right cone
is perpendicular to the plane containing the axis and the line
of contact, it follows that the plane ABC (the osculating plane
to the geodesic) is perpendicular to the plane AB^ BT, which
is the tangent plane at A. The theorem of this article is thus
established.
M. Bertrand has remarked {Li'ouviUe, t. xiii., p. 73, cited
by Cayley, Quarterly Journal., vol. I., p. 186) that this funda-
mental property of geodesies follows at once from Meunier's
theorem (see Art. 293). For it is evident, that for an inde-
finitely small arc, the chord of which is given, the excess in
length over the chord is so much the less as the radius of
curvatui'e is greater. The shortest arc, therefore, joining two
indefinitely near points ^, ^, on a surface is that which has
the greatest radius of curvature, and we have seen that this
is the normal section.
309. Eeturning now to the surface of centres, I say that
the curve CDE (Art. 306), which is the locus of points of inter-
section of consecutive normals along a Hue of curvature, is
CURVATURE OF SURFACES. 275
a geodesic on the sheet of the surface of centres on which it
lies. For we saw (Art. 30G) that the plane of two consecutive
normals to the surface (that is to say, the plane of two
consecutive tangents to this curve) is the tangent plane to the
second sheet of the surface of centres and is perpendicular to
the tangent plane at G to that sheet of the surface of centres
3n which C lies. Since, then, the osculating plane of the curve
CDE is always normal to the surface of centres, the curve is
a geodesic on that surface.
310. We have given the equations connected with lines of
curvature on the supposition that the equation of the surface
is presented, as it ordinarily is, in the form <^ [x^ y, z) = 0.
As it is convenient, however, that the reader should be able
to find here the formulae which have been commonly employed,
we conclude this chapter by deriving the principal equations
in the form given by Monge and by most subsequent writers,
viz. when the equation of the surface is in the form z = (^[oc^y).
We use the ordinary notations
dz =pdx + qdy^ dp = rdx + sdy^ dq = sdx + tdy.
We might derive the results in this form from those found
already ; for since U= cfi [x, y) - z = 0, we have
dU_ dD_ ^__7
dx~^'' dy~^' dz ~ '
with corresponding expressions for their second differential
coefficients. We shall, however, repeat the investigations for
this form as they are usually given.
The equation of a tangent plane is
z-z=p{x-x) + q(y-y),
and the equations of the normal are
[x - x) +2}{z-z) = o, y-y +q{^- ^') = o.
If then 0.^'^ be any point on the normal, and xyz the point
where it meets the surface, we have
(a-a;)+^(7-2;) = 0, (/3 -^) + 2^ (7- «) = 0.
276 CURVATURE OF SURFACES.
And If a/37 also satisfy the equations of a second normal, the
differentials of these equations must vanish, or
dx -\-'pdz = (7 — ^) df^ dy -{ qdz = ['y — z)dq]
•whence, eliminating (7 — ^), we have the equation of condition
[dx -\-pdz) dq = [dy + qdz) dp.
Putting in for dz^ dp^ dq their values already given, and
arranging, we have
%A{^-^4')s-pqt] + ^^[[l^q^)r-{i+f)t]-[[l^.f)s-pqr] = ().
This equation determines the projections on the plane of xy of
the two directions in which consecutive normals can be drawn
80 as to intersect the given normal.
311. From the equations of the preceding article we can
also find the lengths of the principal radii. The equations
dx +pdg = (7 — 2) dp^ dy + qdz = {y- z) dq,
when transformed as above become
{1 +p' - {y-z) r] dx + [pq -[y-z) s] dy = 0,
[\+q^-[y-z)t]dy + [pq-[y-z)s]dx = 0,
whence eliminating dx : dy^ we have
{y-z)'\rt-s')-[y-z)[{l + q')r-2pqs-\-{\+py] + [l+f + q^)=0.
Now 7 — s is the projection of the radius of curvature on the
axis of z ; and the cosine of the angle the normal makes with
that radius being -r-, 7. tt we have,
^ ^/[\-\-p)+q) '
R=[y-z)^{l+p-^ + q').
Eliminating then 7 — g; by the help of the last equation, R is
given by the equation
P^ {rt - /} - R {[I + q') r-2pqs + [l +/) t] ^J{l +/ + q')
312. From the preceding results can be deduced Joachim-
Bthal's theorem (see Crelle, vol. xxx., p. 347) that if a line
of curvature be a plane curve, its plane makes a constant
CURVATURE OF SURFACES. 277
angle with the tangent plane to the surface at any of the
points where it meets it. Let the plane be 3 = 0, then the
equation of Art. 310
[dx +pdz) dq = {dy + qdz) dp
becomes dxdq = dydp. But we have also ^c?x + ^f^?/ = 0, con-
sequently pdp + qdq = 0 ; ^ + <f = constant. But p^ + (f is the
square of the tangent of the angle which the tangent plane
makes with the plane xy. since cos 7= -7— 5— — gr .
^ ''^ V(l +>• +2')
Otherwise thus (see Liouville^ vol. xi., p. 87) : Let MM\
M'M" be two consecutive and equal elements of a line of
curvature, then the two consecutive normals are two perpen-
diculars to these lines passing through their middle points /, i',
and C the point of meeting of the normals is equidistant from
the lines MM\ M'M". But if from G we let fall a perpen-
dicular CO on the plane MM'M'\ 0 will be also equidistant
from the same elements; and therefore the angle CIO= CFO.
It is proved then that the inclination of the normal to the plane
of the line of curvature remains unchanged as we pass from
point to point of that line.
More generally let the line of curvature not be plane. Then
as before, the tangent planes through MM' and through M'M"
make equal angles with the plane MM'M". And evidently
the angle which the second tangent plane makes with a second
osculating plane M'M"M'" differs from the angle which it
makes with the first by the angle between the two osculating
planes. Thus we have Lancret's theorem, that along a line
of curvature the variation in the angle between the tange7it plane
to the surface and the osculating plane to the curve is equal to
the angle hetiveen the tivo osculating planes.
For example, if a line of curvature he a geodesic it must
le plane. For then the angle between the tangent plane and
osculating plane does not vary, being always right; therefore
the osculating plane itself does not vary.
313. Finally, to obtain the radius of curvature of any
normal section. Since the centre of curvature a/37 lies on
the normal, we have
(a-a;)42?(7-2;) = 0, {^-y) + q[^-z) = Q,
278 CUKVATURE OF SURFACES.
Further, we have
And since this relation holds for three consecutive points of the
section which is osculated by the circle we are considering,
we have
(a - x)dx + (/3 — y) dy + (7 - z) dz = 0,
(a - X) d'x + (/3 - y) d'y + (7 - «) d'z = dx' + dy' + dz\
Combining this last with the preceding equations, we have
a — x_l3 — y_ <y — z_ R _ dx^ + dy' + dz'
~p ~~q 1 ~ V(l +^ + 9.') ~ pd^x + 9.d'y - d'z "
But differentiating the equation dz=pdx + qdy, we have
d'z — -pd^x — qd'y = rdo^ + Isdxdy + tdy'^
whence 72 = W(l +/ + i) ^-^ -^ ^f ^ [^pdx ^. <idyj
- V ^ ^ t ' 1 ) ^,^^1 ^ ^isdxdy + tdy'
The radius of curvature, therefore, of a normal section whose
projection on the plane of xy is parallel io y = mx is
(1 4-^") + 2pqm + (1 + q') rr^
+ \/(l+/ + 2')
r + 2sw + tni'.
The conditions for an umbillc are got by expressing that this
value is independent of m^ and are
1 -^v^ _pq _ l + g"
r s t '
( 279 )
CHAPTER XII.
CURVES AND DEVELOPABLES.
SECTION I. PEOJECTIVE PROPERTIES.
314. It was proved (p. 13) that two equations represent
a curve in space. Thus the equations Z7= 0, V= 0 represent
the curve of intersection of the surfaces ' Z/, V.
The degree of a curve in space is measured bj the number
of points in which it is met by any plane. Thus, if Uj V be
of the m^^ and 7i^^ degrees respectively, the surfaces which they
represent are met by any plane in curves of the same degrees,
which intersect in m7i points. The curve UV is therefore of
the mn^^ degree.
By eliminating the variables alternately between the two
given equations, we obtain three equations
^(?/, 0) = O, y}r[z,x)^0, % (a;, ?/) = 0,
which are the equations of the projections of the curve on
the three coordinate planes. Any one of the equations taken
separately represents the cylinder whose edges are parallel to
one of the axes, and which passes through the curve (Art. 25).
The theory of elimination shows that the equation (f) (?/, z) —0
obtained by eliminating x between the given equations is of
the mn^^ degree. And it is also geometrically evident that
any cone or cylinder* standing on a curve of the /" degree
is of the 7'^^ degree. For if we draw any plane through the
vertex of the cone [or parallel to the generators of the cylinder]
this plane meets the cone in r lines ; namely, the lines joining
the vertex to the r points where the plane meets the curve.
♦ A cylinder is plainly the limiting case of a cone, whose vertex is at infinity.
280 CUfiVES AND DEVELOPABLES.
315. Now, conversely, if we are given any curve in space
and desire to represent it by equations, we need only take the
three plane curves which are the projections of the curve on
the three coordinate planes ; then any two of the equations
<f) f?/, 2?) = 0, yjr [z, x) =0, ')(^ [x^ 3/) = 0 will represent the given
curve. But ordinarily these will not form the simplest system
of equations by which the curve can be represented. For if
r be the degree of the curve, these cylinders being each of
the r^^ degree, any two intersect in a curve of r^ degree ; that
is to say, not merely in the curve we are considering but in
an extraneous curve of the degree r^ — r. And if we wish
not only to obtain a system of equations satisfied by the
points of the given curve, but also to exclude all extraneous
points, we must preserve the system of three projections ; for
the projection on the third plane of the extraneous curve in
which the first two cylinders intersect will be different from
the projection of the given curve.
It may be possible by combining the equations of the three
projections to arrive at two equations Z7=0, F=0, which shall
be satisfied for the points of the given curve, and for no other.
But it is not generally true that every curve in space is the
complete intersection of two surfaces. To take the simplest
example, consider two quadrics having a right line common,
as, for example, two cones having a common edge. The
intersection of these surfaces, which is in general of the fourth
degree, must consist of the common right line, and of a curve
of the third degree. Now since the only factors of 3 are 1
and 3, a curve of the third degree cannot be the complete
intersection of two surfaces unless it be a plane curve; but
the curve we are considering cannot be a plane curve,* for
if so any arbitrary line in its plane would meet it in three
points, but such a line could not meet either quadric in more
points than two, and therefore could not pass through three
points of their curve of intersection.
* Curves in space which are not plane curves have commonly been called
" curves of double curvature." In what follows, I use the word "curve" to denote
a curve in space, which ordinarily is not a plane ciuve, and I add the adjective
"twisted" when I want to state expressly that the curve is not a plane curve.
PROJECTIVE PROPERTIES. 281
316. The question thus arises how to represent In general a
curve in space, by equations. Several answers may be given.
[A). Generalizing the method at the beginning of the last
article, we may consider a set of surfaces C/"=0, F=0, 1^=0,
&c. (where U, F, W, ... are rational and Integral func-
tions of the coordinates), all passing through the given curve.
This being so, If M, N, P, &c. are also rational and integral
functions of the coordinates, then 3117-^ NV+ PW+...=0 is
a surface passing through the curve. If any one of the original
equations can be thus represented by means of the other
equations, e.g. If we have identically U=NV+ PW-{- ..., we
reject this equation ; and if we have through the curve any
surface whatever 2^=0 which is not thus representable (viz.
if Tk not of the form T=3IU+NV+ PW-i-...), then we
join on the equation T= 0 to the original system ; and so on :
if, as may happen, the adjunction of any new equation renders
a former equation superfluous, such former equation is to be
rejected. We thus arrive at a complete system of surfaces
passing through the given curve, viz. such a system Is ?7=0,
F=0, TF=0, ... where these functions are not connected by
any such equation as U=NV-\- PW+...^ and where every other
surface which passes through the curve Is expressible in the
form ilfC/'+iV'F+PlF+...=0. It Is not easy to prove, but it
may safely be assumed, that for a curve of any given order
whatever, the number of equations In such a complete system Is
finite. And we have thus the representation of a curve In space
by means of a complete system of surfaces passing through it.
[B], Taking as vertex an arbitrary point, the cone passing
through a given curve of the order m Is, as we have seen,
of the order m ; and It is such that each generating line meets
the curve once only. Hence we can on each generating line
of a cone of the order m determine a single point in such-
wise that the locus of these points Is a curve of the order m.
It would at first sight appear that we might thus determine
the curve as the Intersection of the cone by a surface of the
order n, having at the vertex of the cone an (tz— l)-ple
point; for then each generating line of the cone meets the
surface in the vertex counting [n — \) times, and in one other
00
282 CURVES AND DEVELOPABLES.
point. But the curve of intersection is not then in general a
curve of the order w, but is a curve of the order mn having
a singular point at the vertex. To cause this curve to
be of the order ?a, the surface of the order w with the
(n— l)-ple point must be particularised; such a surface has
through the multiple point n[n — l) right lines; and if any
one or more of these lines are on the cone, the complete in"
tersection of the cone and surface will include as part of itself
such line or lines, and there will be a residual curve of an
order less than mn^ and which may reduce itself to m ; viz. the
complete intersection of the cone and surface will then consist
of m {n — 1) lines through the vertex (or rather of lines counting
this number of times), and of a residual curve of the order
m. The analytical representation of the curve (using quad-
riplanar coordinates) is by means of two equations the cone
(a;, ?/, zf — 0, and the monoid [x^ ?/, zf + w (a^, ?/, zY~^ = 0 par*-
ticularised as above.*
(C). The coordinates of any point of a curve in space may
be given as functions of a single parameter 6. They cannot
in general be thus expressed as rational functions of ^, for
this would be a restriction on the generality of the curve in
space (the curve would in fact be unicursal) ; but if we imagine
two parameters ^, <^ connected by an algebraic equation, then
the coordinates of the point of the curve in space may be taken
to be rational functions of 6^ 0. Or, what is the same thing,
. . ? 77 .
writing - and -^ instead of ^, ^, we have between ^, ?;, ^ an
equation (|, 77, ^j"' = 0, and then (using for the curve in space
quadriplanar coordinates) cr, ?/, z^ iv proportional to rational
and integral functions (|, ■?;, tf ; we thus determine the curve
in space, by expressing the coordinates of any point thereof
rationally in terms of the coordinates of a point of the plane
curve(|,'77, ^j'"=0.
[D). A curve in space will be determined if we determine
all the right lines which meet it ; viz. if we establish between
the six coordinates of a right line the relation which expresses
that the line meets the curve. Such relation is expressed by
* See Cayley, Comj>tes liendus, t. Liv. (1862), pp. 65, 396, 672.
PROJECTIVE PROPERTIES. 283
a single equation (j), q^ r, 5, ?, z*)'" = 0 between the coordinates
of a right line. But the difficulty is that, not every such
equation, but only an equation of the proper form, expresses
that the right line meets a determinate curve in space. Thus
the general linear relation (p, 5', ?', s, t. uY = 0 is not the equation
of any line in space ; the particular form
2^s' 4- qt' + ru + sp' + tq' + ur = 0,
where (p', q'^ r', s', t\ u) are constants such that p's-{-q't'-\-ru—(i
is the equation of a right line, viz. of the line the six coordinates
of which are (/>', q\ /, s', t\ u) ; in fact, the equation obviously
expresses that the line (p, q^ ?•, s, t^ u) meets this line.
317. If a curve be either the complete or partial inter-
section of two surfaces C/, F, the tangent to the curve at any
point is evidently the intersection of the tangent planes to the
two surfaces, and is represented by the equations
xv;+yv; + zv;-\-wV{ = o.
When we use rectangular axes, the direction-cosines of the
tangent are plainly proportional to MN' — M'N^ NL' — N'L^
LM' - IJM^ where Z, M^ &c. are the first differential coefficients.
An exceptional case arises when the two surfaces touch, in
which case the point of contact is a double point on their
curve of intersection. All this has been explained before (see
Art. 203). As a particular case of the above, the projection of
the tangent line to any curve is the tangent to its projection ;
and when the curve is given as the intersection of the two
cylinders 3/ = ^ [z\ x=y{r {z)^ the equations of the tangent are
This may be otherwise expressed as follows : Consider any
element of the curve cls-^ it is projected on the axes of co-
ordinates into dx^ dy^ dz. The direction-cosines of this element
are therefore -7- , -f-, -, , and the equations of the tangent are
ds ds ds
/
x — x y -y z — z
dx dy dz
ds ds ds
284 CURVES AND DEVELOPABLES.
Since the sum of the squares of the three cosines is equal to
unity, we have ds^ = da? + dy'' + dz\
We shall postpone to another section the theory of normals,
radii of curvature, and in short everything which involves
the consideration of angles, and in this section we shall
only consider what may be called the projective properties of
curves,
318. The theory of curves is In a great measure identical
with that of developables, on which account it is necessary to
enter more fully into the latter theory. In fact it was proved
(Art. 123) that the reciprocal of a series of points forming a
curve is a series of planes enveloping a developable. We there
showed that the points of a curve regarded as a system of
points 1, 2, 3, &c. give rise to a system of lines; namely, the
lines 12, 23, 34, &c. joining each point to that next consecutive,
these lines being the tangents to the curve ; and that they also
give rise to a system of planes, viz. the planes 123, 234, &c.
containing every three consecutive points of the system, these
planes being the osculating planes of the curve. The as-=
semblage of the lines of the system forms a surface whose
equation can be found when the equation of the curve is given.
For, the two equations of the tangent line to the curve involve
the three coordinates x\ y\ z\ which being connected by two
relations are reducible to a single parameter; and by the
elimination of this parameter from the two equations, we obtain
the equation of the surface. Or, in other words, we must
eliminate xyz between the two equations of the tangent and
the two equations of the curve. We have said (Art, 123)
that the surface generated by the tangents is a developable,
since every two consecutive positions of the generating line
intersect each other. The name given to this kind of surface
is derived from the property that it can be unfolded into a
plane without crumpling or tearing. Thus, imagine any series
of lines Aa^ Bh^ Cc^ Dd^ &c. (which for the moment we take
at finite distances from each other) and such that each inter-*
sects the consecutive in the points a, J, c, &c. ; and suppose
a, surface to be made up of the faces AaB^ BhC^ CcDj &c,^
PROJECTIVE PROPERTIES. 285
then It is evident that such a surface could be developed Into
a plane by turning the face AaB round aB as a hinge until
it formed a continuation of BbC', by turning the two, which
we had thus made into one face, round cG until they formed
a continuation of the next face, and so on. In the limit when
the lines Aa^ Bb^ &c. are indefinitely near, the assemblage of
plane elements forms a developable which, as just explained,
can be unfolded into one plane.
The reader will find no difficulty in conceiving this from
the examples of developables with which he is most familiar,
viz. a cone or a cylinder. There is no difficulty In folding
a sheet of paper into the form of either surface and In un-
folding it again into a plane. But it will easily be seen to
be impossible to fold a sheet of paper into the form of a sphere
(which is not a developable surface) ; or, conversely, if we cut
a sphere in two It is impossible to make the portions of the
surface lie smooth In one plane.
But In order to exhibit better the form of a developable
surface, as also its cuspidal curve afterwards referred to, take
two sheets of paper, and cutting out from these two equal
circular annuli {e.g. let the radii of the two circles be 3 inches
and A^ Inches), and placing these one upon the other, gum
them together along the inside edge by means of short strips
of muslin or thin paper; we have thus a double annulus,
which, so long as it remains complete, can only be bent in the
same way as if it were single ; but cutting through the double
annulus along a radius, and taking hold of the two extremities,
the whole can be opened out Into two sheets of a developable
surface, of which the inner circle, bending into a curve of double
curvature, Is the cuspidal curve or edge of regression.*
It is to be added, that if we draw on each of the two sheets
the tangents to the inner circle, and consider each tangent as
formed of two halves separated by the point of contact, then
when the paper Is bent into a developable surface as above,
a set of half-tangents on the one sheet will unite with a set
* Thomson and Tait (1867), p. 97. Prof. Cayley mentions that he believes the
construction is due to Prof. Blackburn.
286 CURVES AND DEVELOPABLES.
of half-tangents on the other sheet to form the generatuig
lines on the developable surface ; while the remaining two sets
of half-tangents will unite to form on the developable surface
a set of curves of double curvature, each touching a generating
line at a point of the cuspidal curve, in the manner that a plan©
curve touches its tangent at a point of inflexion.
319. The plane AaB containing two consecutive gene-
rating lines is evidently, in the limit, a tangent plane to the
developable. It is obvious that we might consider the surface
as generated by the motion of the plane AaB according to
some assigned law, the envelope of this plane in all its positions
being the developable. Now if we consider the developable
generated by the tangent lines of a curve in space, the equa-
tions of the tangent at any point ccyV are plainly functions
of those coordinates, and the equation of the plane containing
any tangent and the next consecutive (in other words, the
equation of the osculating plane at any point xyz) is also
a function of these coordinates. But since xyz are connected
by two relations, namely, the equations of the curve, we can
eliminate any two of them, and so arrive at this result, that
a developable is the envelope of a plane whose equation contains
a single variable parameter. To make this statement better
understood we shall point out an important difference between
the cases when a plane curve is considered as the envelope of
a moveable line, and when a surface in general is considered as
the envelope of a moveable plane.
320. The equation of the tangent to a plane curve Is a
function of the coordinates of the point of contact ; and these
two coordinates being connected by the equation of the curve,
we can either eliminate one of them, or else express both in
terms of a third variable so as to obtain the equation of the
tangent as a function of a single variable parameter. The
converse problem, to obtain the envelope of a right line whose
equation includes a variable parameter has been discussed,
Higher Plane Curves^ Art. 86. Let the equation of any tan-
gent line be m = 0, where u is of the first degree in x and y^
PROJECTIVE PROPERTIES. 287
and the coustants are functions of a parameter t. Then
the Hne answerhig to the value of the parameter t + h is
du h d'u h' ^ , , . „ . • r ^
w + -r-T -I — 7^ 7-^ + &c. : and the pomt ot intersection 01 these
dt I dt- \.2 ' ^
two Imes IS enven bv the equations u = 0. -7- H ro + &c. = 0.
* -^ ^ ' r/^ 1.2 dt"
And, in the limit, the point of intersection of a line with the
next consecutive (or, in other words, the point of contact of
any line with its envelope) is given by the equations m = 0,
-77 = 0. If from these two equations we eliminate t we obtain
the locus of the points of intersection of each line of the system
•with the next consecutive ; that is to say, the equation of the
envelope of all these lines. It is easy to prove that the result
of this elimination represents a curve to which u is a tangent.
We get that result, if in u we replace t by its value, in terms of
X and ?/, derived from the equation -t- = 0. Now, if we diflPeren-
, du fdu\ du dt , du fdu\ du dt
tiate, we have -—=--)--— -— and -, = , + ~r t- •,
dx \dxj dt dx dy \dy) dt dy '
where (-7-) , \'t\ ^^^ the differentials of u on the supposition
that t is constant. And since -7^ = 0 it is evident that ^- , ^-
dt dx dy
are the same as on the supposition that t is constant. It follows
that the eliminant in question denotes a curve touched by u.
If it be required to draw a tangent to this curve through
any point, we have only to substitute the coordinates of that
point in the equation m = 0, and determine t so as to satisfy
that equation. This problem will have a definite number of
solutions, and the number will plainly be the number of tan-
gents which can be drawn to the curve from an arbitrary
point ; that is to say, the class of the curve. For example,
the envelope of the line
where «, &, c, c7, are linear functions of the coordinates, is
plainly a curve of the third class.
288 CURVES AND DEVELOPABLES.
321. Now let us proceed in like manner with a surface.
The equation of the tangent plane to a surface is a function
of the three coordinates, which being connected by only one
relation (viz. the equation of the surface), the equation of the
tangent plane, when most simplified, contains two variable
parameters. The converse problem is to find the envelope of
a plane whose equation w = 0 contains two variable parameters
s, t. The equation of any other plane answering to the
values a 4-^, t + k will be
" + (*^+*§)-^r2 ('i'|"+&c.) + &c.=o.
Now, in the limit, when h and k are taken indefinitely small,
they may preserve any finite ratio to each other k = \h. We
see thus that the intersection of any plane by a consecutive
one is not a definite line, but may be any line represented by
the equations m = 0, -7- + \-7- = 0, where , \ is indeterminate.
But we see also that all planes consecutive to u pass through
, ..11 • r. du ^ du ^
the ^oint given by the equations w = 0, -7- = 0, -3- = 0.
From these three equations we can eliminate the parameters
s, ^, and so find the locus of all those points where a plane of
the system is met by the series of consecutive planes. It is
proved, as In the last article, that the surface represented by
this eliminant is touched by u. If It be required to draw a
tangent plane to this surface through any point, we have only
to substitute the coordinates of that point in the equation w = 0.
The equation then containing two indeterminates s and t can
be satisfied in an infinity of ways ; or, as we know, through
a given point an infinity of tangent planes can be drawn to
the surface, these planes enveloping a cone.
Suppose, however, that we either consider t as constant,
or as any definite function of s, the equation of the tangent
plane Is reduced to contain a single parameter, and the envelope
of those particular tangent planes which satisfy the assumed con-
dition is a developable. Thus, again, we may see the analogy
between a developable and a curve. When a surface is con-
PROJECTIVE PROPERTIES. 289
sidered as the locus of a number of points connected by a given
relation, if we add another relation connecting the points we
obtain a curve traced on the given surface. So when we con-
sider a surface as the envelope of a series of planes connected
by a single relation, if we add another relation connecting the
planes we obtain a developable enveloping the given surface.
322. Let us now see what properties of developables are to
be deduced from considering the developable as the envelope
of a plane whose equation contains a single variable parameter.
In the first place it appears that through any assumed point
can be drawn, not, as before, an infinity of planes of the system
forming a cone, but a definite number of planes. Thus, if it
be required to find the envelope of af + ^hf -f 3c< + d^ where
a, &, c, d represent planes, it is obvious that only three planes
of the system can be drawn through a given point, since on
substituting the coordinates of any point we get a cubic for t.
Again, any plane of the system is cut by a consecutive plane
in a definite line ; namely, the line ^^ = 0, -tT = 0 j and if we
eliminate t between these two equations, we obtain the sur-
face generated by all those lines, which is the required
developable.
It is proved, as at Art. 320, that the plane u touches the
developable at every point which satisfies the equations m = 0,
(Lit
-T- = 0 ; or, in other words, touches along the whole of the line
ctz
of the system corresponding to u. It was proved (Art. 110)
that in general when a surface contains a right line the tangent
plane at each point of the right line is different. But in the
case of the developable the tangent plane at every point is
the same. If x be the plane which touches all along the line
icy, the equation of the surface can be thrown into the form
xj> + y'^ = () (see Art. 110).*
* It seems unnecessary to enter more fully into the subject of envelopes, in general,
since what is said in the text applies equally if u, instead of representing a plane,
denote any surface whose equation includes a variable parameter. Monge calls the
PP
290 CUEVES AND DEVELOPABLES.
323. Let us now consider three consecutive planes of the
system, and it is evident, as before, that their intersection satisfies
the equations m = 0, -^ =0, -p^ — 0. For any value of f, the
point Is thus determined where any line of the system Is met
by the next consecutive. The locus of these points is got by
eliminating t between these equations. We thus obtain two
equations in a;, ?/, ^, one of them being the equation of the
developable. These two equations represent a curve traced
on the developable. Thus it is evident that, starting with the
definition of a developable as the envelope of a moveable plane,
we are led back to its generation as the locus of tangents to
a curve. For the consecutive Intersections of the planes form
a series of lines, and the consecutive intersections of the lines
are a series of points forming a curve to which the lines are
tangents. We shall presently show that the curve Is a cuspidal
edge* on the developable.
324. Four consecutive planes of the system will not meet
in a point unless the four conditions be fulfilled w = 0, -^ = 0^
-y-j = 0, -jg = 0. It Is in general possible to find certain
curve M = 0, — = 0) in which any surface of the system is intersected hj the con-
secutive, the characteristic of the envelope. For the nature of this curve depends
only on the manner in which the variables a*, y, z enter into the function «, and not
on the manner in which the constants depend on the parameter. Thus, when «
represents a plane, the characteristic is always a right line, and the envelope is the
locus of a system of right lines. When m represents a sphere, the characteristic
being the intersection of two consecutive spheres is a circle, and the envelope is the
locus of a system of circles. And so envelopes in general may be divided into famihes
according to the nature of the characteristic.
* Monge has called this the "arete de rebroussement," or "edge of regression" of
the developable. There is a similar curve on every envelope, namely, the locus of
points in which each " charactenstic" is met by the next consecutive. The part of
the charactenstic on one side of this curve generates one sheet of the envelope, and
that on the other side generates another sheet. The two sheets touch along this
curve which is their common limit, and is a cuspidal edge of the envelope. Thus, in
the case of a cone, the parts of the generating lines on opposite sides of the vertex
generate opposite sheets of the cone, and the cuspidal edge in this case reduces itself
to a single point, namely, the vertex.
PKOJECTIVE PROPEKTIES. 291
values of t, for which these equations will be satisfied. For
if we eliminate ic, y, 2, we get the condition that the four
planes, whose equations have been just written, shall meet in a
point. Since this condition expresses that a function of t is
equal to nothing, we shall in general get a determinate
number of values of t for which it is satisfied. There are
therefore in general a certain number of points of the system
through which four planes of the system pass ; or, in other
words, a certain number of points in which three consecutive
lines of the system intersect. We shall call these, as at Higher
Plane Curves^ p. 25, the stationary points of the system ; since
in this case the point determined as the intersection of two
consecutive lines coincides with that determined as the inter-
section of the next consecutive pair.
Reciprocally, there will be in general a certain number of
planes of the system which may be called stationary planes.
These are the planes which contain four consecutive points
of the system; for, in such a case, the planes 123, 234 evidently
coincide.
325. We proceed to show how, from Pliicker's equations con-
necting the ordinary singularities of plane curves,* Prof. Cayleyf"
has deduced equations connecting the ordinary singularities of
developables. We shall first make an enumeration of these
singularities. We speak of the " points of the system," the
" lines of the system," and the " planes of the system" as
explained (Art. 123).
Let m be the number of points of the system which lie in
any plane ; or, in other words, the degree of the curve which
generates the developable.
* These equations are as follow : see Higher Plane CurveSy p. 65. Let /t be the
degree of a curve, v its class, 6 the number of its double points, t that of its double
tangents, k the number of its cusps, t that of its points of inflexion ; then
V = /x (/x - 1) - 25 - 3/c ; /u = i/ (1/ - 1) - 2t - 3t,
I =3/i (/x - 2) - 65 - 8k ; k= 3i/(i/ - 2) - 6x - 8t.
Whence also t - k = 3 (v - /x) ; 2 {t - 6) = {v - fx) {v + fx - ^).
f See Liouville's Journal, vol. x. p. 245 ; Cambridge and Dublin Mathematical
Journal, vol. V. p. 18.
292 CURVES AND DEVELOPABLES.
Let n be the number of planes of the system which can be
drawn through an arbitrary point. We have proved (Art. 322)
that the number of such planes is definite. We shall call this
number the class of the system.
Let r be the number of lines of the system which intersect
an arbitrary right line. It is plain that if we form the con-
dition that u, -y-, and any assumed right line may intersect,
the result will be an equation in t, which gives a definite
number of values of t. Let r be the number of solutions
of this equation. We shall call this number the rank of
the system, and we shall show that all other singularities
of the system can be expressed in terms of the three just
enumerated.
Let a be the number of stationary planes, and /3 the number
of stationary points (Art. 324).
Two non-consecutive lines of the system may intersect.
When this happens we call the point of meeting a " point
on two lines," and their plane a " plane through two lines."
Let X be the number of " points on two lines" which lie
in a given plane, and y the number of " planes through two
lines" which pass through a given point.
In like manner we shall call the line joining any two points
of the system a " line through two points," and the intersection
of any two planes a " line in two planes." Let g be the number
of " lines in two planes" which lie in a given plane, and h the
number of " lines through two points" which pass through a
given point. The number h may also be called the number of
apparent double points of the curve ; for to an eye placed at
any point, two branches of the curve appear to intersect if any
line drawn through the eye meet both branches.
The developable has other singularities which will be deter-
mined in a subsequent chapter, but these are the singularities
which Pliicker's equations (note, p. 291) enable us to determine.
326. Consider now the section of tJie developahle hy any
plane. It is obvious that the points of this curve are the traces
on its plane of the " lines of the system," while the tangent
PROJECTIVE PROPERTIES. 293
lines of the section are the traces on its plane of the " planes
of the system." The degree of the section is therefore r,
since it is equal to the number of points in which an arbitrary
line drawn in its plane meets the section, and we have such
a point whenever the line meets a " line of the system."
The class of the section is plainly 7i. For the number of
tangent lines to the section drawn through an arbitrary point
is evidently the same as the number of " planes of the system '
drawn through the same point.
A double point on the section will arise whenever two
"lines of the system" meet the plane of section in the same
point. The number of such points by definition is x. The
tangent lines at such a double point are usually distinct, because
the two planes of the system corresponding to the lines of the
system intersecting in any of the points x are commonly different.
The number of double tangents to the section is in like
manner ff ; since a double tangent arises whenever two planes
of the system meet the plane of section in the same line.
The m points of the system which lie in the plane of section
are cusps of the section. For each is a double point as being
the intersection of two lines of the system; and the tangent
planes at these points coincide, since the two consecutive lines,
intersecting in one of the points m, lie in the same plane of
the system. This proves, what we have already stated, that
the curve whose tangents generate the developable is a cuspidal
edge on the developable ; for it is such that every plane meets
that surface in a section which has as cusps the points where
the same plane meets the curve.
Lastly, we get a point of inflexion (or a stationary tangent)
wherever two consecutive planes of the system coincide. The
number of the points of inflexion is therefore a.
We are to substitute, then, in PlUcker's formulae,
yu, = r, V = w, B = Xj T — g^ /c = wz, t = a.
And we have
n= r (r— 1) — 2aj— .3?n; r= n (?i - 1) — 2^- 3a,
a = 3r (r - 2) - 6a; - 8m ; m = 3n [n - 2) - 6.7 - 8a,
294 CURVES IND DEVELOPABLES.
whence also
w — a = 3 (r — w) ; 2 [x — g) = {r — n) {r + n — 9).
327. Another system of equations Is found by considering
the cone whose vertex is any point atid which stands on the
given curve. It appears at once by considering the section
of a cone by any plane that the same equations connect the
double edges, double tangent planes, &c. of cones, which connect
the double points, double tangents, «&c. of plane curves.
The edges of the cone which we are now considering are
the lines joining the vertex to all the points of the system ;
and the tangent planes to the cone are the planes connecting
the vertex with the lines of the system, for evidently the plane
containing two consecutive edges of the cone must contain the
line joining two consecutive points of the system.
The degree of the cone is plainly the same as the degree of
the curve, and is therefore w.
The class of the cone is the same as the number of tangent
planes to the cone which pass through an arbitrary line drawn
through the vertex. Now since each tangent plane contains
a line of the system, it follows that we have as many tangent
planes passing through the arbitrary line as there are lines
of the system which meet that line. The number sought is
therefore r.*
A double edge of the cone arises when the same edge of
the cone passes through two points of the system, or S = A.
The tangent planes along that edge are the planes joining
the vertex to the lines of the system which correspond to
each of these points.
A double tangent plane will arise when the same plane
through the vertex contains two lines of the system, or T=y.
A stationary or cuspidal edge of the cone will only exist
when there is a stationary point in the system, or k — /3.
* It is easy to see that the class of this cone is the same as the degree of the
developable which is the reciprocal of the points of the given system. Hence, the
degree of the developable generated by the tangents to any curve is the same as the degree
of the developable which is the recijwocal of the points of that curve, see note p. 105.
PEOJECTIVE PKOPERTIES. 295
Lastly, a stationary tangent plane will exist when a plane
containing two consecutive lines of the system passes through
the vertex, or t, = n.
Thus we have fj, = m^ v = r, S = ^, t = ^, k — /3j c = n.
Hence, by the formulae (note, p. 291),
r= m{m — l) — 2k- 3^] w = r (r- 1) - 2?/ — Sw,
n = 37n{m-2) -eh-S/3] /3 = 3r (r - 2) - 6y - Sn.
Whence also
(tj — ^) = 3 (7- — m) 5 2 {y — h) = [r — m) [r + m — 9).
And combining these equations with those found in the last
article, we have also
a — j3 = 2 [n — vi) ; x- y = n — m] 2[g — h) = [n — m) {n + m— 7).
Plucker's equations enable us, when three of the singularities
of a plane curve are given, to determine all the rest. Now
three quantities r, m, n are common to the equations of this
and of the last article. Hence, lohen any three of the singu-
larities which we have enumerated^ of a curve in space^ are
given^ all the rest can he found.
328. It is to be observed that, besides the singularities
which we have enumerated, a curve may have others which
may claim to be counted as ordinary singularities. It may,
for example, besides its apparent double points, have H actual
double points or nodes ; viz., considering the curve as generated
by the motion of a variable point, we have a node if ever the
point comes twice into the same position. Reciprocally, the
system may have O double planes ; viz., considering the de-
velopable as the envelope of a plane, if in the course of its
motion the plane comes twice into the same position, we have
a double plane. These singularities will be taken into account
if, ,in the formulae of Art. 326, we write T=g+ G instead of
T = f7, and in the formulee of Art. 327, write h = h-\- H. In
like manner, the system may have v stationary lines, or lines
containing three consecutive points of the system. Such a
29G CURVES AND DEVELOPABLES.
line meets in a cusp the section of the developable by any
m
plane, and accordingly, in Art. 326, instead of having k-
we have K^m-^-v] and, in like manner, in Art. 327, instead
of fc = w, we have t = n + u. Once more, the system may have
G) double lines, or lines containing each two pairs of consecutive
points of the system. Taking these into account we have, in
Art. 326, h = x + a>^ and in Art. 327, t = ^ + to.
329. To illustrate this theory, let us take the developable
which is the envelope of the plane
a/ + Ut''-^ -\-{'k{k- 1) c^"' + &c. = 0,
where ^ is a variable parameter, a, 5, c, &c. represent planes,
and h is any integer.
The class of this system is obviously A;, and the equation
of the developable being the discriminant of the preceding
equation, its degree is 2 (^ — 1) ; hence r = 2 (Z:; - 1).
Also it is easy to see that this developable can have no
stationary planes. For, in general, if we compare coefficients
in the equations of two planes, three conditions must be satisfied
in order that the two planes may be identical. If then we
attempt to determine t so that any plane may be identical
with the consecutive one, we find that we have three conditions
to satisfy, and only one constant t at our disposal.
Having then ?2 = ^, r = 2 (^- 1), a = 0, the equations of the
last two articles enable us to determine the remaining singu-
larities. The result is
m = Z[k-2)] /3 = 4(Z;-3); x^2[k-2) [k-?>);
y = 2{k-\)[k-^); g = \{k-\){k-2); A = ^ (9^^- 53^ + 80).
The greater part of these values can be obtained independently,
see Higher Plane Curves^ p. 71. But in order to economize
space we do not enter into details.
330. The case considered in the last article, which is that
when the variable parameter enters only rationally into the
equation, enables us to verify easily many properties of de-
PROJECTIVE PROPERTIES. 297
velopables. ^ince the system m = 0, -f = 0 is obviously re-
ducible to
ar' + {k-l) bt"" + &c. = 0, hr' + [k - 1) ct'-' + &c. = 0,
and the system w = 0, -^- = 0, -j^ =0 is reducible to
a^'' + {k - 2) bt'^-^ + &c. = 0, bt'-'' -t- (yfc - 2) c^' + &c. = Oj
ct'''^+{k-2) dr'+&c. = 0',
it follows that a is itself a plane of the system (namely, that
corresponding to the value t = cc)^ ab is the corresponding line,
and abc the corresponding point. Now we know from the
theory of discriminants (see Higher Algebra^ Art. Ill) that the
equation of the developable is of the form a(f> + Fyjr = 0, where
i/r is the discriminant of u when in it a is made = 0. Thus we
verify what was stated (Art. 322) that a touches the develop-
able along the whole length of the line ab. Further, ■\|r is
itself of the form b(f>' + c^'^'. If now we consider the section
of the developable by one of the planes of the system (or, in
other words, if we make a = 0 in the equation of the develop-
able), the section consists of the line ab twice and of a curve
of the degree r — 2 ; and this curve (as the form of the equation
shows) touches the line ab at the point abc, and consequently
meets it in r — 4 other points. These are all " points on two
lines," being the points where the line ab meets other lines
of the system. And it is generally true that if r be the rank
of a developable each line of the system meets 7* - 4 other lines
of the system. The locus of these points forms a double curve
on the developable, the degree of this curve is a:, and other
properties of it will be given in a subsequent chapter,
where we shall also determine certain other singularities of
the developable.
We add here a table of the singularities of some special
sections of the developable. The reader, who may care to
examine the subject, will find no great difficulty in establishing
them. I have given the proof of the greater part of them,
Cambridge and Dublin Mathematical Journal^ vol. v., p. 24.
QQ
298 CURVES AND UEVELOPABLES.
See also Prof. Cayley's Paper, Quarterly Journal^ vol. XI.,
p. 295.
Section by a plane of the system
ju = r — 2, V — n— \, I. — a, k = m — 3, t - g - n + 2, 5=:a; — 2;' + 8.
Cone whose vertex is a point of the system
;u = m - 1, v = r-2, i = 7i - 3, k = ft, t - ?/ - 2r + 8, 6 = h - m + 2.
Section by plane passing through a line of the system
H = r ~ 1, V — n, I = a + 1, k: = TO — 2, t — g — \, ^ = a; — r + 4.
Cone whose vertex is on a line of the system
fx — m, i/=r— 1, 1 = n — 2, k = /3 + 1, r — y — r + 4, S = h — 1.
Section by plane through two lines
fx = r — 2, V = n, t = a + 2, k = m — 4, t = g — 2, 5 = a; — 2j' + 9.
Cone whose vertex is a point on two lines
IJL = m, 1/ = r - 2, 1 = « - 4, k = (i + 2, t - y - 2r + 9, 6 = h - 2.
Section by a stationary plane
fx = i — 3, v — 71—2, t = a — 1, K = wj — 4, T = g — 2n + 6, S = x — 3r + l3.
Cone whose vertex is a stationary point
V = w* - 2, v-r-3, t = n - 4, k = /3 - 1, x = j/ - 3r + 13, 6 = h-2m + 6.
In the preceding we have not taken account of the sin-
gularities G, H, V, CO, having shewn in Art. 328 how to modify
the forraulse so as to include them. The . following formulas of
Prof. Cayley's relate to these singularities :
Section by a plane G
H = r-4, v = n-2, i = a, K = m-6 + v, t - g -2n + 6 + G -I, S = x-4r + 20 + lo.
Cone whose vertex is a point R
H = m-2, !/ = ?•- 4, t = n-6+i/, K = [i, T=y - 4r + 20 + w, £- h-2m + 6 + H-l.
Section by plane through stationary line v
fjL=r-2, v = n, i = a + 2, K = m-3 + v~l, T = g -2 + G, i-x-2r + 9 + w.
Cone whose vertex is on stationary line v
fi = m, v-r -2, i = w-3 + u-l, k = /3 + 2, t - y -2r + 9 + u>, £ = h - 2 + H.
Section by tangent plane at contact of line v
fi = r-3, v = ti-l, L — a+l, K = m-4 + v~l, T=g-n+l+ G, d = x-3r + U + to.
Cone whose vertex is contact of line v
/i = m-l, v = r-3, i = n-i + v-l, k = (3 + 1, T = y-3r+U+ w, S = h-m + l + H.
Section by plane through double tangent w
/u = r-2, v = n, L—a+2, «:=:to-4 + u, T = g-2+ G, 6- x - 2r - 10 + w — 1.
Cone whose vertex is on double tangent w
fi = m, i/ = r-2, i-n-4: + v, k = (3 + 2, t = y - 2r + 10 + w - 1, 6-h-2 + H.
CLASSIFICATION OF CURVES. 299
Section by tangent plane at one of the contacts of line to
ju = r — 3, v = n- 1, i = a+l, K = in — 5 + v, -rzig — n+l + G, 6 = x — 3r + l5 + o) — l,
Cone whose vertex is a contact of line to
H = m-1, v = r-3, t=n-5 + v, k = (3+1, t = j^ -3r+ 15 + a)- 1, S = h-mi-l + H.
SECTION II. CLASSIFICATION OF CURVES.
331. The following enumeration rests on the principle that
a curve of the degree r naeets a surface of the degree 2^ in
pr points. This is evident when the curve is the complete
intersection of two surfaces whose degrees are m and n.
For then we have r = inn and the three surfaces intersect in
mtip points. It is true also by definition when the surface
breaks up into p planes.* We shall assume that, in virtue
of the law of continuity, the principle is generally true.
The use we make of the principle is this. Suppose that
we take on a curve of the degree r as many points as are
sufficient to determine a surface of the degree p ; then if the
number of points so assumed be greater than pr^ the surface
described through the points must altogether contain the curve ;
for otherwise the principle would be violated.
We assume in this that the curve is a proper curve of the
degree r, for If we took two curves of the degrees m and n
(where ?« + ?i = r), the two together might be regarded as a
complex curve of the degree r, and if either lay altogether on
any surface of the degree p, of course we could take on that
curve any number of points common to the curve and surface.
All this will be sufficiently illustrated by the examples which
follow.
332. There is no line of the first degree hut the right line^
For through any two points of a line of the first degree and
any assumed point we can describe a plane which must alto-
* Dr. Hart remarks that since every twisted curve of degree r is a partial
intersection of two cones of r — 1 degree, the complete intersection being the twisted
curve together with the line joining vertices of cones and a curve of degree r (?■ — 3) :
this principle is proved for twisted cubics. For, the two quadric cones intersect
any surface of degree )* in in points of which n lie on the line joining vertices so that
3» lie on the twisted cubic.
300 C'UEVES AND DEV.ELOPABLES.
gether contain the line, since otherwise we should have a line
of the first degree meeting the plane in more points than one.
In like manner we can draw a second plane containing the
line, which must therefore be the intersection of two planes ;
that is to say, a right line.
There is no proper line of the second degree hut a conic.
Through any three points of the line we can draw a plane,
which the preceding reasoning shows must altogether contain
the line. The line must therefore be a plane curve of the
second degree.
The exception noted at the end of the last article would
occur if the line of the second degree consisted of two right
lines not in the same plane ; for then the plane through three
points of the system would only contain one of the right lines.
In what follows we shall not think it necessary to notice this again,
but shall speak only of proper curves of their respective orders.
333. A curve of the third degree must either he a plane
cuhic or the partial intersection of two quadrics, as explained,
Art. 315.*
For through seven points of the curve and any two other
points describe a quadric; and, as before, it must altogether
contain the curve. If the quadric break up into two planes,
the curve may be a plane curve lying in one of the planes.
As we may evidently have plane curves of any degree we
shall not think it necessary to notice these in subsequent cases.
If then the quadric do not break up into planes, we can draw
a second quadric through the seven points, and the intersection
of the two quadrics includes the given cubic. The complete
intersection being of the fourth degree, it must be the cubic
together with a right line ; it is proved therefore that the
only non-plane cubic is that explained, Art. 315.
* Non-plane curves of the third degree appear to have been first noticed by
Mbbius in his Barycentric Calculus, 1827. Some of their most important properties
are given by M. Chasles in Note xxxiii. to his Aperqu Uistorique, 1837, and in a
paper in Liouville's Journal for 1857, p. 397. More recently the properties of these
curves have been treated by M. Schrbter, Crelle, vol. LVI., and by Professor Cremona,
of Milan, Crelle, vol. Lviii., p. 138. Considerable use has been made of the latter
pajier in the articles which immediately follow.
>v
CLASSIFICATION OF CURVES. 301
334. The cone containing a curve of the m*^ degree and
whose vertex is a point on the curve, is of the degree vn — I ;
hence the cone containing a cubic, and whose vertex is on the
curve, is of the second degree. We can thus describe a twisted
cubic through six given points. For we can describe a cone
of the second degree of which the vertex and five edges are
given, since evidently we are thus given five points in the
section of the cone by any plane, and can thus determine that
section. If then we are given six points a, 5, c, d^ e, /, we
can describe a cone having the point a for vertex, and the
lines a6, ac, ad^ ae, af for edges ; and in like manner a cone
having b for vertex and the lines ia, 5c, bd^ 5e, bf for edges.
The intersection of these cones consists of the common edge ab
and of a cubic which is the required curve passing through
the six points.
The theorem that the lines joining six points of a cubic
to any seventh are edges of a quadric cone, leads at once to
the following by Pascal's theorem : " The lines of intersection
of the planes 712, 745; 723, 756; 734, 761 lie in one plane."
Or, in other words, " the points where the planes of three con-
secutive angles 567, 671, 712 meet the opposite sides lie in
one plane passing through the vertex 7."* Conversely if this
be true for two vertices of a heptagon it is true for all the
rest ; for then these two vertices are vertices of cones of the
second degree containing the other points, which must there-
fore lie on the cubic which is the intersection of the cones.
335. A cubic traced on a hyperboloid of one sheet meets all its
generators of one system once, and those of the other system twice.
Any generator of a quadric meets in two points its curve
of intersection with any other quadric, namely, in the two points
where the generator meets the other quadric. Now when the
intersection consists of a right line and a cubic, it is evident
that the generators of the same system as the line, since they
do not meet the line, must meet the cubic in the two points;
* M. Cremona adds, that when the six points are fixed and the seventh variable,
this plane passes through a fixed chord of the cubic.
302 CUEVES AND DEVELOPABLES.
while the generators of the opposite system, since they meet
the line in one point, only meet the cubic in one other point.
Conversely we can describe a system of hyperbololds through
a cubic and any chord which meets it twice. For, take
seven points on the curve, and an eighth on the chord joining
any two of them ; then through these eight points an infinity
of quadrics can be described. But since three of these points
are on a right line, that line must be common to all the
quadrics, as must also the cubic on which the seven points lie.
336. The question to find the envelope of at^ — ^bf + Sct — d
(where a, 5, c, d represent planes and f is a variable parameter)
is a particular case of that discussed, Art. 329. We have
r = 4, on = 71 = 3, a = /3 = 0, £c = ?/ = 0, g = h = \.
Thus the system is of the same nature as the reciprocal system^
and all theorems respecting it are consequently two-fold. The
system being of the third degree must be of the kind we are
considering ; and this also appears from the equation of the
envelope
[ad - hcf = 4 [b'' - ac) (c'^ - bd)^
for it is easy to see that any pair of the surfaces ad — bc^ ¥' — ae,
c^ — bd, have a right line common, while there is a cubic
common to all three, which is a double line on the envelope.
It appears from the table just given that every plane con-
tains one " line in two planes," or that the section of the
developable by any plane has one double tangent ; while, re-
ciprocally through any point can be drawn one Hue to meet
the cubic twice ; the cone therefore, whose vertex is that point,
and which stands on the curve, has one double edge ; or, in
other words, the cubic is jprojected on any plane into a cubic
having a double point.
The three points of inflexion of a plane cubic are in one
right line. JSTow it was proved (Art. 327) that the points of in-
flexion correspond to the three planes of the system which can
be drawn through the vertex of the cone. Hence the three
points of the system, which correspond to the three planes which
CLASSIFICATION OF CURVES. 303
can be drawn through any point (9, lie in one plane passing
through that point.*
Further, it is known that when a plane cubic has a conjugate
point, its three poinis of inflexion are real ; but that when the cubic
has a double point, the tangents at which are real, then two of
the points of inflexion are imaginary. Hence, if the chord which
can be drawn through any point 0 meet the cubic in two real
points, then two of the planes of the system which can be drawn
through 0 are imaginary. Reciprocally, if through any line
two real planes of the system can be drawn, then any plane
through that line meets the curve in two imaginary points, and
only one real one.f
337. These theorems can also be easily established alge-
braically ; for the point of contact of the plane af — 3b f ■+ Set — d,
being given by the equations at = h, ht = c, ct = d^ may be denoted
by the coordinates a = l, h = t^ c = i\d=f. Now the three
values of t answering to planes passing through any point are
given by the cubic a'f— 3h't^+ Zc't- d' = 0, whence it is evident,
from the values just found, that the points of contact lie in the
plane a'cZ— 36'c-f 3c'i — c?'a = 0. But this plane passes through
the given point. Hence the intersection of three planes of the system
lies in the plane of the corresponding points. The equation just
written is unaltered if we interchange accented and unaccented
letters. Hence, if a point A he in the plane of points of contact^
corresponding to any point B, B will he in the plane in like
manner corresponding to A. And again, the planes which thus
correspond to all the points of a line AB pass through a fixed
right line, namely, the intersection of the planes corresponding
to A and B. The relation between the lines is evidently reci-
procal. To any plane of the system will correspond in this
sense the corresponding point of the system ; and to a line in
two planes corresponds a chord joining two points.
The three points where any plane Aa + Bh + Cc + Dd
meets the curve have their «'s given by the equation
Df + Cf ■{■ Bt + A = 0^ and when this is a perfect cube, the
* Chas]es, Ziouville, 1857. Sclirbter, Crelle, vol. LVI.
t Joachimsthal, Crelle, vol. LVi. p. 4o. Cremona, Crelle, vol. LVlii. p. HG.
304 CUKVES AND DEVELOPABLES.
plane is a plane of the system. From this it follows at once, as
Joachimsthal has remarked, that any plane drawn through the
intersection of two real planes of the system meets the curve
in but one real point. For, in such a case, the cubic just written
is the sum of two cubes and has but one real factor.
338. We have seen (Art. 134) that a twisted cubic is the
locus of the poles of a fixed plane with regard to a system
of quadrics having a common curve. More generally, such
a curve is expressed by the result of the elimination of \
between the system of equations \a = a', \b = b', Xc = c. Now
since the anharmonic ratio of four planes, whose equations
are of the form Xa = a', \'a = a'^ &c., depends only on the
coefficients X, V, &c. (see Conies, Art. 59), this mode of
obtaining the equation of the cubic may be interpreted as
follows: Let there he a system of planes through any line aa\
a hornographic system through any other line hh\ and a third
through cc , then the locus of the intersection of three corre-
sp)onding jplanes of the systems is a twisted cubic. The lines
aa\ bb'y cc are evidently lines through two points, or chords
of the cubic. Reciprocally, if three right lines be homographically
divided, the plane of three corresponding points envelopes the
developable generated by a twisted cubic, and the three right lines
are " lines in two planes" of the system.
The line joining two corresponding points of two homo-
graphically divided lines touches a conic when the lines are
in one plane, and generates a hyperboloid when they are not.
Hence, given a series of points on a right line and a homo-
graphic series either of tangents to a conic or of generators
of a hyperboloid, the planes joining each point to the corre-
sponding line envelope a developable, as above stated.
Ex. If the four faces of a tetrahedron pass through fixed lines, and three ver-
tices move in fixed lines, the locus of the remaining vertex is a twisted cubic.
Any number of positions of the base form a system of planes which divide homo-
graphically the three lines on which the corners of the base move, whence it
follows that the three planes which intersect in the vertex are corresponding planes
of three homographic systems.
339. From the theorems of the last article it follows, con-
versely, that " the planes joining four fixed points of the system
CLASSIFICATION OF CURVES. 305
to any variable ' line through two points' form a constant anhar-
monic system," and that " four fixed planes of the system divide
any ' line in two planes' in a constant anharmonic ratio." It
is very easy to prove these theorems independently. Thus
we know that the section of the developable by any plane A of
the system,* consists of the corresponding line a of the system
twice, together with a conic to which all other planes of the
system are tangents. Thus, then, the anharmonic property of
the tangents to a conic shows that four of these planes cut
any two lines in two planes, AB, AC in the same anharmonic
ratio ; and, in like manner, AC is cut in the same ratio as CD.
As a particular case of these theorems, since the lines of
the system are both lines in two planes and lines through
two points ; four fixed planes of the system cut all the lines of
the system in the same anharmonic ratio ; and the planes joining
four fixed points of the system to all the lines of the system are
a constant anharmonic system.
Many particular inferences may be drawn from these
theorems, as at Conies^ p. 296, which see.
Thus consider four points a, /S, 7, S; and let us express
that the planes joining them to the lines «, &, and a/8, cut
the line 78 homographically. Let the planes A^ B meet 78 in
points ^, t'. Let the planes joining the line a to /3, and the
line & to a meet 7S in ^, k'. Then we have
[tkr^h] = [k't'ryB] = {kk'ryS}.
If the points t, k' coincide, it follows from the first equation
that the points ^, t' coincide, and from the second that the
points <, t\ 7, S are a harmonic system. Thus we obtain
Prof. Cremona's theorem, that if a series of chords meet the
line of intersection of any plane A with the plane joining the
corresponding point a to any line h of the system, then they
will also meet the line of intersection of the plane B with
the plane joining /8 to a ; and will be cut harmonically where
they meet these two lines and where they meet the curve.
* It is often convenient to denote the planes of the system by capital letters, the
corresponding lines by itahcs, and the corresponding points by Greek letters.
R R
306 CURVES AND DEVELOPABLES.
The reader will have no difficulty in seeing when it will
happen that one of these lines passes to infinity, in which case
the other line becomes a diameter.
340. We have seen that the sections of the developable
by the planes of the system are conies. The line of intersection
of two planes of the system is a common tangent to the two
corresponding conies. Thus the planes touching two conies,
themselves having the line in which their planes intersect
as a common tangent, are osculating planes of a twisted cubic.
We may investigate the locus of the centres of these conies,
or more generally the locus of the poles with respect to these
conies of the intersections of their planes with a fixed plane.
Since in every plane we can draw a " line in two planes"
we may suppose that the fixed plane passes through the inter-
section of two planes of the system A, B.
Now consider the section by any other plane C] the traces
on that plane of A and B are tangents to that section, and
the pole of any line through their intersection lies on their
chord of contact, that is to say, lies on the line joining the
points where the lines of the system a, h meet G. But since
all planes of the system cut the lines «, h homographlcally,
the joining lines generate a hyperbolold of one sheet, of which
a and b are generators. However then the plane be drawn
through the line AB^ the locus of poles is on this hyperboloid.
But further, it is evident that the pole of any plane through
the intersection of A^ B lies in the plane which is the harmonic
conjugate of that plane with respect to those tangent planes.
The locus therefore which we seek is a plane conic. It appears
also from the construction that since the poles when any plane
A + \B is taken for the fixed plane. He on a conic in the
plane-4— A,5; conversely, the locus when the latter is taken
for fixed plane is a conic In the former plane.*
341. In conclusion, it is obvious enough that cubics may
be divided into four species according to the different sections
* The theorems of this article are taken from Prof. Cremona's paper.
CLASSIFICATION OP CURVES. 307
of the curve by the plane at infinity. Thus that plane may
either meet the curve in three real points; In one real and
two imaginary points; in one real and two coincident points,
that is to say, a line of the system may be at infinity ; or
lastly, in three coincident points, that is to say a plane of
the system may be altogether at infinity. These species have
been called the cubical hyperbola, cubical ellipse, cubical hyper-
bolic parabola, and cubical parabola. It is plain that when
the curve has real points at infinity, it has branches proceeding
to infinity, the lines of the system corresponding to the points
at infinity being asymptotes to the curve. But when the
line of the system is itself at infinity, as In the third and fourth
cases, the branches of the curve are of a parabolic form pro-
ceeding to Infinity without tending to approach to any finite
asymptote. Since the quadric cones which contain the curve
become cylinders when their vertices pass to infinity, it is
plain that three quadric cylinders can be described containing
the curve, the edges of the cylinders being parallel to the
asymptotes. Of course in the case of the cubical ellipse two
of these cylinders are imaginary: in the case of the hyper-
bolic parabola there are only two cylinders, one of which is
parabolic, and in the case of the cubical parabola there is
but one cylinder which is parabolic. The cubical ellipse may
be conceived as lying on an elliptic cylinder, one generating
line of which is the asymptote; the curve is a continuous line
winding once round the cylinder, and approaching the asymptote
on opposite sides at its two extremities.
It follows, from Art. 336, that In the case of the cubical
ellipse the plane at infinity contains a real line in two planes,
•which is imaginary in the case of the cubical hyperbola. That
is to say, in the former case, but not in the latter, two planes
of the system can be parallel. From the anharmonic property
we infer that In the case of the cubical parabola three planes
of the system divide in a constant ratio all the lines of the
system. In this case all the planes of the system cut the
developable in parabolas. The system may be regarded as
the envelope of xf — 3i/i^ -\- 3zt — d where d is constant. For
further details we refer to Prof. Cremona's Memoir.
308 CURVES AND DEVELOPABLES.
342. We proceed now to the classification of curves of higher
orders. We have proved (Art. 331) that through any curve
can be described two surfaces, the lowest values of whose
degrees in each case there is no difficulty in determining. It
is evident then, on the other hand, that if commencing with
the simplest values of /a and v we discuss all the different
cases of the intersection of two surfaces whose degrees are
fi and V, we shall include all possible curves up to the r* order,
the value of this limit r being in each case easy to find when
/x and V are given. With a view to such a discussion we
commence by investigating the characteristics of the curve of
intersection of two surfaces.* We have obviously m = /^j/,
and if the surfaces are without multiple lines and do not touch,
as we shall suppose they do not, their curve of intersection has
DO multiple points (Art. 203), and therefore /3 = 0. In order to
determine completely the character of the system, it is necessary
to know one more of its singularities, and we choose to seek
for r, the degree of the developable generated by the tangents.
Now this developable is got by eliminating x'y'z' between the
four equations
C7'=0, F'=0, Z7>+Z7;2^+?7>+C7>=0, F/a:+ F/j/^ F/s+ F'>=0.
These equations are respectively of the degrees /i, v, /^— 1,
j;— 1: and since only the last two contain xi/z, these variables
enter into the result in the degree
fiv (v - 1) + flV {fJ, - 1) = /jLV {/J, + V - 2).
Otherwise thus: the condition that a line of the system
should intersect the arbitrary line
ax + I3y + ryz -]■ Bwj ax + l3'y + <y'z + B'w
13
a>
^, 7, 5
«',iSS 7, S'
U., U., Cr, C7
V V V V
'^1) '^21 '^31 '^4
= 0,
* The theory explained in the remainder of this section is taken from my paper
dated July, 1849, Cambridge and Dublhi Mathematical Journal, vol. v. p. 23,
CLASSIFICATION OF CURVES. 309
which is evidently of the degree /* -f v — 2. This denotes a
surface which is the locus of the points, the intersections of
whose polar planes with respect to U and V meet the arbitrary
line. And the points where this locus meets the curve UV
are the points for which the tangents to that curve meet the
arbitrary line.
Having then m = /jbVj /3 = 0, r = fji,v{fi + v — 2)^ we find, by
Art. 327,
n = 3fxv{fi + v-3), a = 2fH' (3/* + 3u- 10), 2h= fiv {fi - 1) (v- 1)
2g = f^v [fxv (3/A + 3v - 9)' - 22 (/a -f v) + 71},
2x = fMV [/jlv {fj,+ v-2y-A{/M + v)-\-8},
27/ = fiv {fj,v [fJi + v- 2)' - 10 (/* + v) + 28}.
343. We verify this result by determining independently
h the number of "lines through two points" which can pass
through a given point, that is to say, the number of lines
which can be drawn through a given point so as to pass
through two points of the intersection of U and V. For this
purpose it is necessary to remind the reader of the method
employed, p. 101, in order to find the equation of the cone whose
vertex is any point and which passes through the intersection
of U and V. Let us suppose that the vertex of the cone is
taken on the curve, so as to have both U and V= 0 for the co-
ordinates of the vertex. Then it appears, from p. 101, that the
equation of the cone is the result of eliminating \ between
These equations in \ are of the degrees //.— 1, v- 1 ; SU^ S^U,
&c., contain the coordinates xy'z\ xyz in the degrees /"- — 1, 1 ;
/Lt — 2, 2, &c. A specimen term of the result is (St/')''"^F'""\
Thus it appears that the result contains the variables xyz in
the degree j/ — l4-j/(/u. — l)=//,j/- 1; while it contains xyz'
in the degree (^— l)(j/— 1). Every edge of this cone of the
degree ^tv — 1, whose vertex is a point on the curve, is of
310 CURVES AND DEVELOPABLES.
course a " line through two points." If now in this result
we consider the coordinates of any point xyz on the cone
as known and x'y'z as sought, this equation of the degree
[fi — I) V— 1) combined with the equations 6' and V determines
the " points '* belonging to all the *' lines through two points "
which can pass through the assumed point. The total number
of such points is therefore /xv jj — \) v — l]^ and the number of
lines through two points is of course half this. The number
of points thus determined has been called (Art. 325) the number
of apparent double points on the intersection of the two surfaces.
3-44. Let us now consider the case when the curve UV
has also actual double points ; that is to say, when the two
surfaces touch in one or more points. Now, in this case, the
number of apparent double points remains precisely the same
as in the last article, and the cone, standing on the curve
oi intersection and whose vertex is any point, has as double
edges the lines joining the vertex to the points of contact in
addition to the number determined in the last article. It
is easy to see that the investigation of the last article docs
not include the lines joining an arbitrary point to the points
of contact. That investigation determines the number of cases
when the radius vector from any point has two values the
same for both surfaces, but the radius vector to a point of
contact has only one value the same for both, since the point
of contact is not a double point on either surface. Every
point of contact then adds one to the number of double edges
on the cone, and therefore diminishes the degree of the de-
velopable by two. This might also be deduced from Art. 342,
since the surface generated by the tangents to the curve of
intersection must include as a factor the tangent plane at a
point of contact, since every tangent line in that plane touches
the curve of intersection.
If the surfaces have stationary contact at any point (Art. 204)
the line joining this point to the vertex of the cone is a cuspidal
edge of that cone. If, then, the surfaces touch in t points of
ordinary contact and In ,8 of stationary contact, we have
r =fiy[fi-\-v- 2) -2t- 3/S,
CLASSIFICATION OF CURVES. 311
and the reader can calculate without difficulty how the other
numbers in Art. 342 are to be modified.
We can hence obtain a limit to the number of points at
which two surfaces can touch if their intersection do not break
up into curves of lower order; for we have only to subtract the
number of apparent double points from the maximum number of
double points which a plane curve of the degree /u.v can have
[Higher Plane Curves^ Art. 42).
.345. We shall now show that when the curve of inter-
section of two surfaces breaks up into two simpler curves,
the characteristics of these curves are so connected that, when
those of the one are known, those of the other can be found.
It was proved (Art. 343) that the points belonging to the
" lines through two points " which pass through a given point
are the intersection of the curve UV with a surface whose
degree is (/x— 1) (v — 1). Suppose now that the curve of Inter-
section breaks up into two whose degrees are m and ra\ where
ni + m' = ftv, then evidently the " two points " on any of these
lines must either lie both on the curve m^ both on the curve
m\ or one on one curve and the other on the other. Let the
number of lines through two points of the first curve be A,
those for the second curve //, and let H be the number of lines
which pass through a point on each curve, or, in other words,
the number of apparent intersections of the curves. Considering
then the points where each of the curves meets the surface
of the degree (ytt— 1) (v — 1), we have obviously the equations
wi (/i - 1) (v - 1) = 2^ + i/, m (/i - 1) (v - 1) = 27/ + //,
whence 2 {h — Ji) — [m — m) [p, — 1) (v - 1).
Thus when m and h are known m and It can be found. To
take an example which we have already discussed, let the
intersection of two quadrics consist in part of a right line
(for which m = 1, h' — Q)^ then the remaining intersection must
be of the third degree m = 3, and the equation above written
determines /< = 1.
346. In like manner it was proved (Art. 342) that the
locus of points, the intersection of whose polar planes with
312 CURVES AND DEVELOPABLES.
regard to U and V meets an arbitrary line, is a surface of
the degree yu, + v — 2. The first curve meets this surface in
the t points where the curves m and m intersect (since U
and V touch at these points) and in the r points for which
the tangent to the curve meets the arbitrary line. Thus, then,
w (/i + V - 2) = r + «, 7?2' (/A + J/ - 2) = / + «,
(m — m) (yit + V — 2) = r — /,
an equation which can easily be proved to follow from that
in the last article.
The intersection of the cones which stand on the curves
w, m consists of the t lines to the points of actual meeting
of the curves and of the H lines of apparent intersection ; and
the equation H+t = mm''\& easily verified by using the values
just found for H and t^ remembering also that m=iiv — mj
r — m[m- 1) —2h.
347. Having now established the principles which we shall
have occasion to employ, we resume our enumeration of the
different species of curves of the fourth order. Every quartic
curve lies on a quadric. For the quadric determined by nine
points on the curve must altogether contain the curve (Art. 331).
It is not generally true that a second quadric can be described
through the curve ; there are therefore Uvo principal families
of quarticSj viz. tJiose which are the- intersection of two quadricsj
and those through which only one quadric can pass.* We
commence with the curves of the first family. The character-
istics of the intersection of two quadrics which do not touch
are (Art. 342)
m = 4:, w = 12, r = 8, a =16, /3=0, a? =16, ^ = 8, ^=38, h=2.
Several of these results can be established independently.
Thus we have given (Art. 218) the equation of the developable
generated by the tangents to the curve, which is of the eighth
degree. It is there proved also that the developable has in
each of its four principal planes a double line of the fourth
* The existence of this second family of quartics was first pointed out in the
Memoir already referred to.
CLASSIFICATION OF CURVES OF FOURTH DEGREE. 313
order, whence re = 16. It has been mentioned (p. 189) that the
developable circumscribing two quadrics has, as double lines, a
conic in each of the principal planes. The number 3/ = 8 is thus
accounted for. Again, it is shown, p. 191, that the equation
of the osculating plane is Tii = T'v [u and v being the tangent
planes to C/and Fat the point), which contains the coordinates
of the point of contact in the third degree. If, then, it be
required to draw an osculating plane through any assumed
point, the points of contact are determined as the intersections
of the curve UV with a surface of the third degree, the
problem therefore admits of twelve solutions; thus n = 12.
Lastly, every generator of a quadric containing the curve
is evidently a "line through two points" (Art. 345). Since,
then, we can describe through any assumed point a quadric
of the form U-\- \ F, the two generators of that quadric which
pass through the point are two "lines through two points";
or h = '2. The lines through two points may be otherwise found
by the following construction, the truth of which it is easy to
see : Draw a plane through the assumed point 0, and through
the intersection of its polar planes with respect to the two
quadrics, this plane meets the quartic in four points which
lie on two right lines intersecting in 0.
A quartic of this species is determined by eight points
(Art. 130).
348. Secondly, let the two quadrics touchy then (Art. 344)
the cone standing on the curve has a double edge more than
in the former case, and the developable is of a degree less
by two. Hence
jn = 4, w = 6, r = 6; ^ = 6, 7e = 3; a = 4, /3 = 0; a; = 6, y = 4.
Thirdly, the quadrics may have stationary contact at a point,
when we have
wi = 4, w = 4, r = 5; 5r = 2, 7j = 2; a=l, /3=1; x = 2^y = 2.
This system, as noticed by Prof. Cayley, may be expressed
as the envelope of
a<* + 6c«^ + 4.dt + e,
where < is a variable parameter. The envelope is
[ae + Zc'f = 27 {ace -a<f- c^)\
ss
314 CURVES AND DEVELOPABLES.
which expanded contahis a as a factor and so reduces to the
fifth degree. The cuspidal edge is the intersectiou of ae + Zc\
Ace - 3d\
Since a cone of the fourth degree cannot have more than
three double edges, two quadrics cannot touch in more points
than one, unless their curve of intersection break up into
simpler curves. If two quadrics touch at two poirds on the
same generator, this right line is common to the surfaces,
and the intersection breaks up into a right line and a cubic.
If they touch at two points not on the same generator, the
intersection breaks up into two plane conies whose planes
intersect in the line joining the points (see Art. 137).
349. If a quartic curve be not the intersection of two
quadrics it must be the partial intersection of a quadric and
a cubic. We have already seen that the curve must lie on a
quadric, and if through thirteen points on it, and six others which
are not in the same plane,* we describe a cubic surface, it must
contain the given curve. The intersection of this cubic with
the quadric already found must be the given quartic together
with a line of the second degree, and the apparent double
points of the two curves are connected by the relation h — h'= 2,
as appears on substituting in the formula of Art. 345 the values
m = 4, «/= 2, yu. = 3, V — 2, When the line of the second degree
is a plane curve (whether conic or two right lines), we have
A' = 0 ; therefore h — 2, or the quartic is one of the species
already examined having two apparent double points. It is
easy to see otherwise, that if a cubic and quadric have a plane
curve common, through their remaining intersection a second
quadric can be drawn; for the equations of the quadric and
cubic are of the form zw = u,^, zv^ — u^x^ which intersect on
v.^ = xw. If, however, the cubic and quadric have common
two right lines not in the same plane, this is a system having
one apparent double point, since through any point can be
* This limitation is necessary, otherwise the cubic might consist of the quadric
and of a plane. Thus, if a curve of the fifth order lie in a quadric it cannot be proved
that a cubic disthict from the quadric can contaLn the given curve; see Camlridge
and JJublin Muthtmulical Journal, vol. V, p. 27,
CLASSIFICATION OF CURVES OF FOURTH DEGREE. 315
drawn a transversal meeting both lines. Since then // = !,
^ = 3 ; or these quartlcs have three apparent double points, and
are therefore essentially distinct from those already discussed,
which cannot have more than two. The numerical character-
istics of these curves are precisely the same as those of the
first species in Art. 348, the cone standing on either curve
having three double edges, the difference being that one of
the double edges in one case proceeds from an actual double
point, while in the other they all proceed from apparent double
points.
This system of quartlcs is the reciprocal of that given by
the envelope of at^ + -ihf + 6cf + ^dt + e. Moreover, this latter
system has, In addition to its cuspidal curve of the sixth
order, a nodal curve of the fourth, which Is of the kind now
treated of.
It is proved, as in Art. 335, that these quartics are met
in three points by all the generators of the quadric on which
they He, which are of the same system as the lines common
to the cubic and quadric ; and are met once by the generators
of the opposite system. The cone standing on the curve,
whose vertex Is any point of It, is then a cubic having a double
edge, that double edge being one of the generators, passing
through the vertex, of the quadric which contains the curve.
Thus, while any cubic may be the projection of the inter-
section of two quadrlcs, quartlcs of this second family can
only be projected into cublcs having a double point. The
quadric may be considered as the surface generated by all
the " lines through three points" of the curve. It is plain,
from what has been stated, that every quartic^ having three
apparent double points^ may he considered as the intersection
of a quadric with a cone of the third order having one of the
generators of the quadric as a double edge.
350. Prof. Cayley has remarked that it Is possible to
describe through eight points a quartic of this second family.
We want to describe through the eight points a cone of the
third degree having its vertex at one of them, and having
a double edge, which edge shall be a generator of a quadric
316 CURVES AND DEVELOPABLES.
through the eight points. Now it follows, from Art. 347, that
if a system of quadrics be described through eight points, all
the generators at any one of them lie on a cone of the third
degree, which passes through the quartic curve of the first
family determined by the eight points. Further, if >S^, /S', S"
be three cubical cones having a common vertex and passing
through seven other points, \8 + ixS' + vS" is the general
equation of a cone fulfilling the same conditions ; and if it have
a double edge, \S^-\- fi8^ -VvS^^ passes through that edge.
Eliminating then A,, /t, v between the three differentials, the
locus of double edges is the cone of the sixth order
s, {s:^:'- s:s:) + ^, [s:8:'-8:'s;) + ^3 {8:8^- 8:8:) = 0.
The intersection then of this cone of the sixth degree with
the other of the third determines right lines, through any of
which can be described a quadric and a cubic cone fulfilling
the given conditions. It is to be observed, however, that the
lines connecting the assumed vertex with the seven other points
are simple edges on one of these cones and double edges on
the other, and these (equivalent to fourteen intersections) are
irrelevant to the solution of the problem. Four quartzes, there-
fore, can be described through the points.
351. Prof. Cayley has directed my attention to a special case
of this second family of quartics which I had omitted to notice.
It is, when the curve has a linear inflexion of the kind noticed.
Art. 328 ; that is to say, when three consecutive points of
the curve are on a right line. Such a point obviously cannot
exist on a quartic of the first family ; for the line joining the
three points must then be a generator of both quadrics, whose
intersection would therefore break up into a line and a cubic,
and would no longer be a quartic. Let us examine then in what
case three consecutive planes of the system a^*+45^''-f 6c<^+4JHe
can pass through the same line. If such a case occurs, we may
suppose that we have so transformed the equation that the
singular point in question may answer to i = o) ; the three planes
a, &, c, must therefore pass through the same line ; or c must
be of the form \a + jxb. But we may then transform the equation
further by writing for t, t + 6, and determining 6 so that the
CLASSIFICATION OF CURVES OF FOURTH DEGREE. 317
quantity multiplying h in the coefficient of f shall vanish. The
system then is the envelope of a plane af + ^hf + &\af -F 4.dt + e.
A still more special case is when \ vanishes, or when the plane
reduces to at*^ + ^hf + ^dt + e ; it is obvious then, that we have
two points of linear inflexion ; one answering to ^ = go , the other
to ^ = 0. The developable in this latter case is
{ae-Aj)df = 'n {ad' + eVy-,
which has for its edge of regression the intersection of ae — ^hd
with ad'' + elf ; but this consists of a curve of the fourth degree
with the lines ab, de. This system then Is one whose reciprocal
is of the same nature ; for we have m = w = 4, h = k = d^
x = y==4t. And the section of the developable by any plane has
six cusps, viz. the four points where the plane meets the cuspidal
edge, and the two where it meets the double generators aJ, de.
In the case previously noticed where c does not vanish but is equal
to Xa, there is but one point of linear Inflexion ; the envelope in
question is, then, the reciprocal of a system for which ??z = 4,
n = 5, r = 6, A = 3, ^ = 4, x = 6^ y = ^' Another special case
to be considered Is when a curve has a double tangent ; such
a line being doubly a line of the system is a double line on
the developable. But this does not occur In quartic* curves.
• For other properties of curves of the fourth order, see papers by M. Chasles,
Comptes Rendus, vols. nv. and Lv. ; and by M. Cremona, Memoirs of the Bologna
Academy, 1861.
To complete the enumeration of curves up to the fourth order, it would be
necessary to classify, according to their apparent double points, improper systems
made up of simpler curves of lower orders. Thus we have, for m = 2, A = 1, two lines
not in the same plane ; w =; 3, A = 1, a conic and a line once meeting it ; A = 2, a conic
and line not meeting it ; /i = 3, three lines, no two of which are in the same plane ;
»« = 4, A = 2, a plane cubic and line once meeting it, or a twisted cubic and line
twice meeting it, or two conies having two points common ; m = 4, A = 3, a plane
cubic and line not meeting it, or a twisted cubic and line once meeting it, or two
conies having one point common ; m = 4, A = 4, a twisted cubic and non-intersecting
line, or two non-intersecting conies ; A = 5, a conic and two lines meeting neither the
conic nor each other ; A = 6, four lines, no two of which are in the same plane.
An interesting quartic curve, Sylvester's "Twisted Cartesian" (see Phil. Mag.,
1866, pp. 287, 380), may here be mentioned specially : viz. the locus of a point
whose distances from three fixed foci are connected by the relations
Ip + mp' + np" = a, I'p + m'p + n'p" — b.
This curve has an infinity of foci lying in a plane cubic which is the locus of foci
of conies which pass through four points lying on a circle; and may be repre«
sented as the intersection of a sphere and a parabolic cylinder.
318 CURVES AND DEVELOPABLES.
352. The enumeration in regard to curves of the fifth order
is effected in the memoir ab-eady cited. It is easy to see
that besides plane quintics we have, I., quintics which are the
partial intersection of a quadric and a cubic, the remaining
intersection being a right line. These quintics have four ap-
parent double points, and may besides have two actual nodal
or cuspidal pointsl We may have, IL, quintics with five
apparent double points, and which may, besides, have one actual
nodal or cuspidal point ; these curves being the partial inter-
section of two cubics, and the remaining intersection a quartic
of the second class. We may have, TIL, quintics with six
apparent double points being the partial intersection of two
cubics, the remaining intersection being an improper quartic
with four apparent double points. To these may be added,
IV., quintics with six apparent double points which are the
partial intersection of a quadric and a quartic surface ; the
remaining intersection being three lines not in the same plane.
353. Instead of proceeding, as we have done, to enumerate
the species of curves arranged according to their respective
orders, we might have arranged our discussion according to the
order of the developables generated, and have enumerated the
different species of the developables of the fourth, fifth, &c., orders.
This is the method followed by Chasles, who has enumerated
the species of developables up to the sixth order [Comjjtes
Bendus, vol. LIV.), and by Schwarz [Crelle^ vol. LXIV., p. 1)
who has carried on his enumeration to the seventh order.
Schwarz's discussion contains the answer to the following ques-
tion started by Prof. Cayley : the equation considered. Art. 329,
where the parameter enters rationally, denotes a single plane
whose envelope is a class of developables which Prof Cayley
calls 'planar developables ; on the other hand, if the parameter
entered by radicals, the equation rationalized would denote a
system of planes whose envelope would therefore be called a
multiplanar developable : now it is proposed to ascertain con-
cerning each developable, what is, in this sense, the degree of its
planarity. M. Schwarz has answered this question by shewing
that the developables of the first seven orders are all planar.
CLASSIFICATION OF CURVES. 319
In fact when a developable 13 planar, the planes, Hues and
pouits of the system are expressible rationally by means of a
parameter; and therefore every section of the developable is
unicursal [Higher Plane Curves^ Art. 44), as is also the
cuspidal edge and every cone standing on it. It may be
verified by the equations of Arts. 326-7, that
l[r-l)[r-2)-[m + x) = \[r-\)[r-2)-{n^y) =
\[vi-l)[m-2)-[h^^) = \[7i~\){n-2)-{g+a)=^{m^n)-{r--i),
any of these expressions denoting the deficiency either of the
section (Art. 326) or of the cone (Art. 327). When this
deficiency vanishes, the developable is planar ; when it = 1 it
is biplanar, &c. And this number is the same for any curve
in space, and for any other derived from it by linear trans-
formation.
354. The discussion of the possible characteristics of a de-
velopable of given order, depends on the principle (p. 298)
that the section by a plane of the system is a curve of degree
r — 2 having m - 3 cusps. Thus, if the developable be of the
fifth order the section by a plane of the system is a cubic ; and
as this can have no more than one cusp, the edge of regression
is at most of the fourth degree. And it cannot be of lower
degree, since we have already seen that twisted cubics generate
developables only of the fourth order. Hence the only de-
velopables* of the fifth order are those, considered Art. 348,
generated by a curve of the fourth order.
In the same manner the section of a developable of the
sixth order by a plane of the system is a quart ic, which may
have one, two, or three cusps. We have therefore m = 4, 5,
or 6 ; and, in like manner, n is confined within the same limits;
and therefore, p. 298, the section by the plane of the system is
at most of the fifth class. Now a curve of the fourth degree
with one cusp must have two other double points if it is only
of the fifth class: and, if it have two cusps, it must have one
other double point. In any case, therefore, this quartic is
* The properties of these developables are treated of by Professor Cremona,
Comptes Eendus, vol. Liv., p. 604.
320 CURVES AND DEVELOPABLES.
unlcursal and the developable is planar. The case when
the quartic has only one cusp (or m = A) has been already
considered. The edge of regression has a nodal point ; and
the system Is the reciprocal of the system which envelopes
at* + ^If + &cf + ^dt + \a = 0,
where there is a double plane of the system answering to # = 0
and also to ^ = co .
If, again, the quartic section have three cusps, it is of the
third class, and therefore for the developable w = 4. This then
is also a case already discussed. Art. 349, the developable being
the envelope of
at" + ^hf + Qcf + 4(^< + e = 0.
Lastly, when the quartic has two cusps, it must, as we have
seen, also have a double point, and therefore be of the fourth
class. Hence n = 5. From the preceding formulae the charac-
teristics of a system for which 7n = w = 5, r = 6, are g = h = ^^
x = y = 5, a = ^ = 2; and, if we take the two stationary planes
answering to f = ix>^ ^ = 0, the system is the envelope of
at^ + 5Xat* + 10c f + 1 Odf + 5/jLft +/= 0.
M. Schwarz has noticed that the stationary tangent planes
may be replaced by a triple tangent plane ; that is to say, the
system may be the envelope of
at^ + 5\at* -f 10/j,af + lOdf + 5et +/= 0.
I have not examined with any care the theory of the effects
of triple points of the curve of intersection of two surfaces on
the number of its apparent double points. But (considering
the case where X, and fi vanish in the equation last written) if
we make b and e = 0 in the equations which 1 have given
{Cambridge arid Dublin Mathematical Journal^ V. 158) for the
edge of regression of the developable which results as the
envelope of a quintic, the edge of regression is found to be the
intersection of 2e^ — Sdf^ with af'^ — 12d'^e. And this intersection
is the right line e/ with a curve of the fifth order, having the
point def for a triple point. For this being a double point on
each surface is a quadruple point on their curve of intersection ;
and since the right line passes through the point defj the re-
maining curve has a triple point at that point.
CLASSIFICATION OF CURVES. 321
355. We shall conclude this section by applying some of
the results already obtained in it, to the solution of a problem
which occasionally presents itself: "Three surfaces whose
degrees are /u, v, p, have a certain curve common to all three ;
how many of their fivp points of intersection are absorbed
by the curve? In other words, in how many points do the
surfaces intersect in addition to this common curve?" Now
let the first two surfaces intersect in the given curve, whose
degree is m, and in a complementary curve /mv — 7W, then the
points of intersection not on the first curve must be included
in the {/jlv — m) p intersections of the latter curve with the
third surface. But some of these intersections are on the
curve wz, since it was proved (Art. 346) that the latter curve
intersects the complementary curve in 7n(/jt, + v — 2) — r points.
Deducting this number from [fiv — m) p we find that the sur-
faces intersect in ixvp — m (/i-fv + p — 2j + r points which are
not on the curve m ; or that the common curve absorbs
w(/i-f v + p — 2)-r points of intersection.
Ex. Applying this formula to the intersections of three cnbics having a common
curve of degree m, the number of residual points not on the curve m is 27 — Im + r.
Now supposing the surfaces have four right liues common, this at first seems to
give 7?i = 4, A = 6, hence ?• = 0 and the number of residual points — 1. But it is easily
seen that the cubic surfaces iu this case have also common the two transversals
of the four right lines, and these have also an apparent double point ; hence,
the values should have been taken m = 6, h — 7. and these give the number of
remaining points of intersection = 1.
If the common curve be two conies, the line in which their planes intersect is
also contained in the surfaces and thus m — b,h = A give 4 remaining intersections.
In precisely the same way we solve the corresponding
question if the common curve be a double curve on the sur-
face p. We have then to subtract from the number [fiv — m) p,
2 [ra (/x + V — 2) — r} points, and we find that the common curve
diminishes the intersections by on [p + 2/a + 2v — 4) — 2>* points.
These numbers, expressed in terms of the apparent double
points of the curve w, are
771 (/i + V + p - 7?2 - 1) + 27^ and m (p + 2/i + 2v - 2m - 2) + Ah.
356. The last article enables us to answer the question :
" If the intersection of two surfaces is in part a curve of degree
w, which is a double curve on one of the surfaces, in how
TT
322 CURVES AND DEVELOPABLES.
many points does it meet the complementary curve of inter-
section ?" Thus, in the question last considered, the surfaces
/i, p intersect in a double curve m and a complementary curve
fip — 2m ; and the points of intersection of the three surfaces
are got by subtracting from {fip - 2m) v the number of inter-
sections of the double curve with the complementary. Hence
[fip - 2m) V— t = fivp — m{p + 2p, + 2v — 4) -t- 2r,
whence t = w (p + 2/* — 4) — 2r.
We can verify this formula when the curve on is the complete
intersection of two surfaces f/, F, whose degrees are k and I.
Then p is of the form AU' + BUV+CV where A is of the
degree p — 2k, &c., and fx is of the form DIJ -\- EV where D
is of the degree /a — k. The intersections of the double curve
with the complementary are the points for which one of the
tangent planes to one surface at a point on the double curve
coincides with the tangent plane to the other surface. They
are therefore the intersections of the curve UV with the surface
AE' - BDE-[- CD'' which is of the degree p ^ 2ix- 2[k + 1).
The number of intersections is kl [p -V 2 p. - 2 [k + I)] which
coincides with the formula already obtained on putting kl = m^
kl{k+l-2)=r.
357. From the preceding article we can shew how, when
two surfaces partially intersect in a curve which is a double
curve on one of them, the singularities of this curve and its
complementary are connected. The first equation of Art. 346
ceases to be applicable because the surface yu, -f v — 2 altogether
contains the double curve, but the second equation gives us
on {fi + V — 2) = 2i, + r = r -\- 2m (/a -f 2i/ — 4) — 4r,
whence 4>- — / = {2m - on) {p, + v — 2) -{■ 2rn {v — 2).
In like manner we find that the apparent double points of
the two curves are connected by the relation
Sh - 2]i = [2m - on) [p,-\){v-\)- 2m (v - 1).
Thus, when a quadric passes through a double line on a cubic
the remaining intersection is of the fourth degree, of the sixth
rank, and has three apparent double points.
t
NON-PROJECTIVE PROPERTIES OF CURVES. 323
SECTION III. NON-PROJECTIVE PROPERTIES OF CURVES.
358. As we shall more than once in this section have
occasion to consider lines Indetinltely close to each other, it
is convenient to commence by shewing how some of the
formulae obtained in the first chapter are modified when the
lines considered are indefinitely near. We proved (Art. 14)
that the angle of inclination of two lines is given by the
formula
sin''^^ = (co3/3 cos 7' — cos/3' cos 7)''+ (0037 cos a' — cos 7' cosa)"'*
+ (cos a cosyS' - cos a' cos/3)"''.
When the lines are indefinitely near we may substitute for
cosa', cosa-1- 8 cosa, &c., and put sin^ = 8^, when we have
Bd'^ = (cos /3 S cos 7 — cos 7 S cos ^f + (cos 7 8 cos a — cos a 8 cos 7)^^
+ (cosa 8 cos /3 - cos /3 8 cos a)^
If the direction-cosines of any line be -,—,-, where
r + m'' -f ti^ = r\ the preceding formula gives
r'M' = [mhi - nhmf + [nil - ISnf + (IBm - niBlf.
Since we have
cos'"' a + cos'''/3 -I- cos''*7 = 1,
cosa 8 cosa + cos/3 8 cosj3 + COS7 8 C0S7 = 0 ;
if we square the latter equation and add it to the expression
for B6'\ we get another useful form
8^' = (5 cos oL'f -f (8 cosyS)"^ + (8 COS 7)^
It was proved (Art. 15) that cos/3 C0S7' — cos/3' C0S7, &c.
are proportional to the direction-cosines of the perpendicular
to the plane of the two lines. It follows then, that the direc-
tion-cosines of the perpendicular to the plane of the consecutive
lines just considered are proportional to 7nSn — nBm, nBl—lSrij
IBm — mBl^ the common divisor being r'^Bd.
Again, it was proved (Art. 44) that the direction-cosines of
the line bisecting the external angle made with each other by
two lines are proportional to
cos a — cos a', cos /S — cos /3', cos 7 — cos 7', &c.
324 CURVES AND DEVELOPABLES.
Hence, when two lines are indefinitely near, the direction-cosines
of a line drawn in their plane, and perpendicular to their
common direction, are proportional to 8 cos a, S cos^S, S COS7,
the common divisor being Sd.
359. We proved (Art. 317) that the direction-cosines of
a tangrent to a curve are ^- , -, - , -^ , while, if the curve be
^ ds ^ ds ^ ds^ '
given as the intersection of two surfaces, these cosines are
proportional to MN'-M'N, NU-N'L, LM'-L'M, where
Jv, il/, &c. denote the first differential coefficients.
An infinity of normal lines can evidently be drawn at any
point of the curve. Of these, two have been distinguished by
special names; the normal which lies in the osculating plane
is commonly called the principal normal; and the normal
perpendicular to that plane, which being normal to two con-
secutive elements of the curve, has been called by M. Saint-
Venant the hinormal. At any point of the curve, the tangent,
the principal normal, and the binormal form a system of three
rectangular axes.
All the normals lie in the plane perpendicular to the tangent
line, viz.
{x — x) dx-\- {y — y') dy + [z - z) dz = 0
in the one notation ; or in the other
[MN' - M'N) [x - x') + [NU - N'L) [y - y)
+ {LM'-L'M)[z-z') = 0.
360. Let us consider now the equation of the osculating
plane. Since it contains two consecutive tangents of the curve,
its direction-cosines (Art. 358) are proportional to
dyd^z — dzd^y^ dzd'^x — dxd^z^ dxd'^y — dyd^x^
quantities which, for brevity, we shall call JT, F, Z. The equa-
tion of the osculating plane is therefore
X[x-x')+ Y{y-y')-vZ{z-z')=0.
The same equation might have been obtained (by Art. 31)
I
i
NON-PROJECTIVE PROPERTIES OF CIJRVES. 325
by forming the equation of the plane joining the three con-
secutive points
f t r 'iJ' 'i7' ',7'
xy z -^ X ■\- ax ^ y -\- ay ^ z -{■ dz ;
X + 2dx + d'x, y + Idij -f d\j\ z + 2cfe' + cVz.
In applying this formula, we may simplify it by taking one
of the coordinates at pleasure as the independent variable, and
BO making d^x^ d^y or d^z — 0.
361. In order to be able to illustrate by an example the
application of the formulae of this section, it is convenient here
to form the equations and state some of the properties of the
lielix or curve formed by the thread of a screw. The helix may
be defined as the form assumed by a right line traced in any
plane when that plane is wrapped round the surface of a right
cylinder.* From this definition the equations of the helix are
easily obtained. The equation of any right line y = mx ex-
presses that the ordinate is proportional to the intercept which
that ordinate makes on the axis of x. If now the plane of
the right line be wrapped round a right cylinder, so that the
axis of X may coincide with the circular base, the right line
will become a helix, and the ordinate of any point of the
curve will be proportional to the intercept measured along the
circle, which that ordinate makes on the circular base, counting
from the point where the helix cuts the base. Thus the coordi-
nates of the projection on the plane of the base of any point of
the helix are of the form a; = acos^, y = a^\nd^ where a Is
the radius of the circular base. But the height z has been
just proved to be proportional to the arc Q, Hence, the equa-
tions of the helix are
x = a cos -r . y = a sin 7" , whence also cc^ + y' = d^.
k li ''
We plainly get the same values for x and y when the arc in-
creases by 27r, or when z Increases by 27rA ; hence the interval
between the threads of the screw is ^irli.
* Conversely, a helix becomes a right line when the cylinder on which it is
traced is developed into a plane ; and is, therefore, a geodesic on the cylinder
(Art. 308).
326
CURVES AND DEVELOPABLES.
Since we have
, a . z , y , - a z ., a?,
ax = — Y%\nrrdz = —j^ dz^ ay = j- cos j dz — ^ dz^
we have as = — ^^ — dz .
dz
It follows that ~ is constant, or
ds
the angle made by the tangent to the helix with the axis
of z (which is the direction of the generators of the cylinder)
is constant. It is easy to see that this is the same as the
angle made with the generators by the line into which the
helix is developed when the cylinder is developed into a
plane.
The length of the arc of the curve is evidently in a constant
ratio to the height ascended.
The equations of the tangent are (Art. 317)
x — x
y-y
z — z
y X k
If then X and y be the coordinates of the point where the
tangent pierces the plane of the base, we have from the pre-
ceding equations
-'a
ji
[x - xj + {y- y'f = (cc- + y'^) ^ = a'' ^ ,
or the distance between the foot of the tangent and the pro-
jection of the point of contact is equal to the arc which
measures the distance along the circle of that projection from
the initial point. This also can be proved geometrically, for
if we imagine the cylinder developed out on the tangent plane,
the helix will coincide with the tangent .line, and the line
joining the foot of the tangent to the projection of the point
of contact will be the arc of the circle developed into a right
line. Thus, then, the locus of the points where the tangent
meets the base is the involute of the circle.
The equation of the normal plane is
y'x — x'y = h{z — z').
To find the equation of the osculating plane we have
d^x = — ji xdz^^ d^y = - ■yiydz\ d^z = 0,
K'
W
NON-PROJECTIVE PllOPEKTIES OF CURVES. 327
whence the equation of the osculating plane is
h {y'x - xy) + a^{z- z) = 0.
The form of the equation shows that the osculating plane makes
a constant angle with the plane of the base.
We leave it as an exercise to the reader to find the
tangent, normal plane, and osculating plane of the intersection
of two central quadrlcs.
362. We can give the equation of the osculating plane
a form more convenient in practice when the curve is defined
as the intersection of two surfaces U^ V. Since the osculating
plane passes through the tangent line, its equation must be
of the form
\ [Lx + My + Nz ^Pw)=iJ,{ Ux + M'y + N'z + P'w),
where Lx-\-&.c. is the tangent plane to the first surface,
L'x-\-&.c. to the second. Tills equation is identically satisfied
by the coordinates of a point common to the two surfaces, and
by those of a consecutive point ; and, on substituting the coor-
dinates of a second consecutive point, we get
^l = Ld'x^Md:'y\Nd\\Fd\o, X = L'd'x+M'd'y+N'd'z+P'd'w.
But dilFerentiating the equation
Ldx + Mdy + Ndz -\- Pdiu = 0,
we get Ld'x + Md'y + Nd'z + Pd'w = - Z7',
where U' — adx'' + hdy'' + cdz'' + ddw^
+ Ifdydz + 2gdzdx + 2hdxdy 4- 2ldxdio + Imdydw -f 2ndzdw^
where a, 5, &c. are the second differential coeflScients. Now
dx^ &c. satisfy the equations
Ldx + Mdy + Ndz + Pdio = 0, L'dx + M'dy + N'dz + P Ww = 0 ;
and since we may either, as in ordinary Cartesian equations,
take w as constant ; or else x, or ^, or z'^ or, more generally,
must take some linear function of these coordinates as constant ;
we may therefore combine with the two preceding equations
the arbitrary equation
o.dx -I- ^dy + 7C?2! + hdw = 0.
328
CURVES AND DEVELOPABLES.
Now it can easily be verified that if we substitute in the
equation of any quadric, the coordinates of the intersection
of three planes
Lx + My \Nz-\- Pw^ Ux + M'y + N'z + P'lo^ (xx+^y+yz + 8w,
the result U' will be proportional to the determinant (cf. p. 59)
«j ^^ <7, ?, L, ^'5 a
h, b, f, m, if, M\ /3
g, /, c, n, iV, i^^', 7
Z, «2, 71, ^, P, P', 8
Z, if, iV, P
i', i/', i\^', P'
a, /3, 7, S
This determinant may be reduced by subtracting from the
fifth column multiplied by {m — 1) the sum of the first four
columns, multiplied respectively by a;, y, 0, w ; when the whole
of the fifth column vanishes, except the last row, which becomes
— {ax -\- 0y + yz + Sw). In like manner we may then subtract
from the fifth row, multiplied by {m — 1), the sum of the first
four rows multiplied respectively by a;, y, z, w^ when, in like
manner, the whole of the fifth row vanishes, except the last
column, which is —{ax + ^y + yz + Bw), Thus the determinant
, ^ (ax + 0y + <yz + Swf
reduces to ^^ -^ —v, — ^
a.
A,
9i
I, U
/^
h
/,
m, W
9i
y;
Cj
71, N'
h
7n,
/?,
d, P'
L\
M\
N'.
P'
If we call the determinant last written 8, and the corresponding
determinant for the other equation S\ the equation of the
osculating plane is
r-l-ry. {Lx + My^ Nz + Pw)=—^, [L'x + M'y + N'z + P'w).^
* This equation is due to Dr. Hesse, see Crelle's Journal, vol, XLi.
NON-PROJECTIVE PROPERTIES OF CURVES. 329
This equation has been verified In the case of two quadrlcs,
see note, p. 191.
Ex. 1. To find the osculating plane of
ax' + hif + M- + dw^, a'x^ + h'xp- + c'2* + d'w'^.
Am. [ah' - ba') {ac' - ca') {ad' - da') x'^x + {ba' - b'a) {be' - b'c) {bd' - b'd) y'^y
+ {ca' - c'a) {cb' - c'b) {cd' - c'd) z'H + {da' - d'a) {db' - d'b) {dc' - d'c) w'^w - 0.
Ex. 2. To find the osculating plane of the line of curvature
a'^ b^'^ c^~ ' a'^ b'^ c'^
a -xx 0 ~yy c ^zz _
363. The condition that four points should lie In one plane,
or, In other words, that a point on the curve should be the
point of contact of a stationary plane, is got by substituting
in the equation of the plane through three consecutive points,
the coordinates of a fourth consecutive point. Thus, from the
equation of Art. 31, the condition required Is the determinant
d^x [dyd'^z-dzd'^y] + d^y{dzd^x-dxd'z) + d^z{dxd'y - dyd'^x) =0.
If a curve In space be a plane curve, this condition must
be fulfilled by the coordinates of every point of It.
For a curve given as the Intersection of two surfaces
U^ F, Clebsch determined as follows (see Crelle^ LXIII, 1) the
condition for a point of osculation. Writing for brevity
8={m-\Y T, 8'={n- 1)' T\ the equation given in the last
article for the osculating plane is
( TL- TU) x+ ( T'M- TM') y + ( TN- TN') z + ( T'P- TP') m;=0,
and the equation of a consecutive osculating plane differs from
this by terms
( TdL + Ldr- TdL' - L'dT) x + &c. = 0.
Thus, In order that the two planes may coincide, Introducing
an arbitrary differential dt^ we must have the four equations
TdL + LdT - TdL' - L'dT= ( T'L - TU) dt, &c.
If, now, we write
T= AU + BM' 4- CN' + DP", T = A'L + B'M^ C'N-v UP,
u u
330 CURVES AND DEVELOPABLES.
wbere A, B, &c. are proportioual to minors of the determluaut
S, and where In fact
we must have
AL + BM+ CN+ BP= 0, AdL + BdM+ CdN+ DdP= 0,
A'L' + &c. = 0, AdU + &c. = 0 ;
for, if in the determinant S we substitute for the last column
either Z, il/, iV, P, or dL^ dM, dN, dP, it is easy to see that
the determinant vanishes. Multiply then the four equations
last considered hy A, B^ C, I) respectively, and add, and we
have, after dividing by T,
which we may write
dT^\d[T)=Tdf,
where by d[T) we mean the differential of T considered
merely as a function of L\ M\ N\ P' ; a, Z>, &c. being regarded
as constants. Similarly we have dT' -f ^d {T') = T'dt. Let us
now write at full length for dT^ T^dx + T^dy + &c. ; and elimi-
nating dx^ dy^ dz^ dw^ dt between the two equations just obtained,
and the three conditions which connect dx^ dy^ dz^ dw, we
obtain the required condition in the form of a determinant
L, iM, N, P, 0
L\ M\ N\ P', 0
a, ^, 7, ^, 0 I = 0.
Now P is a function of x, y, z^ w of the degree 3?n + 2n - 8,
but when regard is paid only to the xyziv, which enter into
X', JP, &c., (P) is of the degree 2(w-l); if, therefore, we
multiply the first four columns by x, y, z, w respectively, and
subtract them from 3 [m +n-Z) times the last column, the first
four terms of the last column vanish, and the equation just
NON-PROJECTIVE PROPERTIES OF CURVES. 331
written may be reduced by cancelling the fifth row and column
of the determinant. The condition that we have just obtained
is of the degree 6m + 6n — 20 In the variables as might be
inferred from the value of a, Art. 342. If the surfaces U
and V are quadrlcs, and therefore the coefficients a, J, &c.
really constant, (TJ, (TJ, &c. are Identical with T,, T^, &c.,
and the condition that we have obtained is the result of
equating to zero the Jacoblan of the four surfaces T, J",
£/, V.
364. We shall next consider the circle determined by three
consecutive points of the curve, which, as In plane curves, is
called the circle of curvature. It obviously lies in the oscu-
lating plane : its centre is the intersection of the traces on
that plane, by two consecutive normal planes ; and its radius
is commonly called the radius of absolute curvature, to dis-
tinguish it from the radius of spherical curvature, which is
the radius of the sphere determined by four consecutive points
on the curve, and which will be investigated presently. If
through the centre of a circle a line be drawn perpendicular
to its plane, any point on this line is equidistant from all the
points of the circle, and may be called a pole of the circle.
Now the intersection of two consecutive normal planes evidently
passes through the centre of the circle of curvature, and is
perpendicular to its plane. Monge has therefore called the lines
of intersection of pairs of consecutive normal planes the polar
lines of the curve. It is evident that all the normal planes
envelope a developable of which these polar lines are the
generators, and which accordingly has been called the ^'oZar
developable surface. We shall presently state some properties
of this surface. The polar line is evidently parallel to the line
called the BInormal (Art. 359).
365. In order to obtain the radius of curvature, we shall
first calculate the angle of contact^ that is to say, the angle
made with each other by two consecutive tangents to the
„,,-.. . . , , . dx du dz
curve, ihe direction-cosmes or the tangent beuig -^ , -^, -7- ,
332 CURVES AND DEVELOPABLES.
it follows, from Art. 358, that cW^ the angle between two con-
secutive tangents, is given by either of the formulse
''^■=(4:y+(4f)'-(4:)>
or ds'de'' = X''-{- Y' + Z\
where X = dyd^z — dzd'^y^ &c.
The truth of the latter formula may be seen geometrically;
for the right-hand side of the equation denotes the square of
double the triangle formed by three consecutive points (Art. 32) ;
but two sides of this triangle are each ds^ and the angle between
them is dO^ hence double the area is da'^dO.
If now ds be the element of the arc, the tangents at the
extremities of which make with each other the angle d9^ then
since the angle made with each other by two tangents to a
circle is equal to the angle that their points of contact subtend
at its centre, we have pd9 = ds. And the element of the arc
and the two tangents being common to the curve and the
circle of curvature, the radius of curvature is given by the
formula
s
ds . - ds
2 di
or p =
.8
X-'+ Y' + Z''
Ex. To find the radius of curvature of the helix. Using the formuljE of Art. 361,
a' + h^
we find p — ; or the radius of curvature is constant.
* By performing the differentiations indicated, another value for dQ'^ is found
without difficulty,
rf*2(fe2 = {d^xY + [d'^yY + (d'^z)- - {d^s)\
This formula may also be proved geometrically. Let AB, BC be two consecutive
elements of the curve ; AD a line parallel and equal to BC \ then since the projections
of £ Con the axes are dx + d'^x, dy + d-i/, dz + d-z, it is plain that the projections
on the axes of the diagonal BB are d^x, d-y, d-z, whence BD- = {d-xy+ {d-yY+{d-zf.
But BD projected on the element of the arc is d-s, and on a line perpendicular to it ia
ds dB ; whence
{d^'sY + {ds dey ^ {d^xY + {d'yY + {d^y.
A
d':
ds
(Is
ds
or
fds '
" ds '
<• ds
NON-PKOJECTIVE PROPERTIES OF CURVES. 333
366. Having thus determined the magnitude of the radius
of curvature, we are enabled by the formulse of Art. 358 also
to determine its position. For the direction-cosines of a line
drawn in the plane of two consecutive tangents, and perpen-
dicular to their common direction, are, by that article,
\ ^dx \ dy \ ^dz
dd d^' dd is' dd ds'
If x\ y\ z be the coordinates of a point on the curve,
and iCj _y, z those of the centre of curvature, then the projec-
tions of the radius of curvature on the axes are x' — cc, y — y,
z' — z'^ but they are also p cosa, pcos^, p cosy. Putting in
then for cosa, cos/3, C0S7 their values just found, the coordinates
of the centre of curvature are determined by the equations
^dx ,dy ^dz
*
367. When a curve is given as the Intersection of two
surfaces which cut at right angles, an expression for the radius
of curvature can be easily obtained. Let r and / be the
radii of curvature of the normal sections of the two surfaces,
the sections being made along the tangent to the curve; and
let (/) be the angle which the osculating plane makes with
the first normal plane : then, by Meunier's theorem, we have
p = r cos (/), and also p = r sin <^, whence — ^ = ^ 4- -7^ .
The same equations determine the osculating plane by the
r
formula tan <i = — .
r
If the angle which the surfaces make with each other be o),
the corresponding formula is
sin^ft) _ 1 1 2 cos ft)
p r r rr
We can hence obtain an expression for the radius of cur-
vature of a curve given as the Intersection of two surfaces.
334
CURVES AND DEVELOPABLES.
We may write U + IP-vN'' = R% L"' -^ M"' + N"' = R'' ; and
we have
LL' + MM' -f NN'
COSO) =-
. , _ {3fN' - M'Ny + {NU - N'LY + (LM' - L'M)^
^'° *" RR'-'
We must then substitute in the formula of Art. 296,
MN'~M'N ^ NL'-N'L LM'-L'M
RR sm 0) ' RR sm &> ' ' RR siu co
The denominator of that formula becomes
a, 7«, ^, X, X'
A, h, /, if, if
Z, M, N,
r, M\ N'
which reduced, as in Art. 362, becomes — ^ S: giving
(m-iyR'R"&\n'co .... , {n--iYR'R'sm'(o
r = ^ ^ , similarly r = ^^
1 S'
p^" {m- lfR'R"sm'(o
28S' cos ft)
Whence — =
"^ [n - 1)* R*R" sin'ft) " {m - If (?i - l)"^ R'R'sm'a '
In the notation of Art. 363 this may be written
R*R"^m'(o T' r' 2 TT cos CO
— T?i! + 7?/!!
k
E'
RE
368. Let us now consider the angle made with each other
by two consecutive osculating planes, which we shall call the
migle of torsion^ and denote by drj. The direction-cosines of
the osculating plane being proportional to X, F, Z^ the second
formula of Art. 358 gives
(Z'^-f Y'-vZ'Ydrf^[ YdZ-ZdYy-v[ZdX-XdZ)\[XdY-' YdX)\
Now Y= dzd^x — dxd^z^ Z— dxd^y — dyd^x^
dY= dzd^x — dxd^Zj dZ = dxd^y — dyd^x.
NON-PROJECTIVE PROPERTIES OF CURVES. 335
Therefore {Lessons on Higher Algebra^ Art. 31)
YdZ-ZdY=Mdx,
where M is the determinant
Xd'x-\- Ydhj-^Zd'z.
Hence (X ' + Y' + Zy dr,' = ]\Pds\
, _ Mds
"^ ~ X-'+ Y' + Z' '
This formula may be also proved geometrically. For 3f
denotes six times the volume of the pyramid made by four
consecutive points, while X'^ + Y'^ + Z'^ denotes four times the
square of the area of the triangle formed by three consecutive
points. Now if A be the triangular base of a pyramid, A' an
adjacent face making an angle rj with the base, s the side com-
mon to the two faces, andj-j* the perpendicular from the vertex
on s, so that 2A' = sp, then for the volume of the pyramid
we have 3 V= Ajp sin?? and 6 Vs = 'lAps sin 17 = ^AA' sin 17.
Now, in the case considered, the common side is ds^ and in
the limit A — A' \ hence 6 Vds = AA'drj. q.e.D.
Following the analogy of the radius of curvature which is
uS . • (Is
-J- , the later French writers denote the quantity* -7- by the
letter r, and call it the radius of torsion ; but the reader will
observe that this is not, like the radius of curvature, the radius
of a real circle intimately connected with the curve.
369. In the same manner, however, as we have considered
an osculating circle determined by three consecutive points of
the system, we may consider an osculating right cone deter-
mined by three consecutive planes of the system, and we
proceed to determine its vertical angle. Imagine that a
sphere is described having as centre the point of the system
in which the three planes intersect ; let the lines of the system
passing through that point meet the sphere in A and j5;
and let the corresponding planes meet the same sphere in
AT^ BT] then, if we describe a small circle of the same sphere
* The quantity -y is also sometimes called the " second cui-vature" of the curve.
as
336 CUEVES AND DEYELOPABLES.
touching AT, BT, and escribed to AB, the cone whose vertex
is the centre, and which stands on that small circle, will
evidently osculate the given curve. The problem then is, being
given dr] the angle between two consecutive tangents to a
small circle of a sphere, and dd the corresponding arc of the
circle to find H its radius.
Let (j) be the external angle between two tangents to a
circle, s the length of the two tangents, then H the radius of
the circle is given by the formula tan ^0 tan -2^= sin ^5. Now,
taking G the centre of the small circle and t the foot of the
perpendicular from it on AB, we have tan |0 tani7=sln^?,
and tan ^0' tan /Z — sin i?^, where in the limit ^' differs by
an infinitely small quantity from 0.
JSow, since also in the limit AB measures the angle between
consecutive lines of the system and 0 measures that between
consecutive planes of the system, we have then
„ dd r „
tan H=^ — = - *
drj p
370. Imagine that through every line of the system there
is drawn a plane perpendicular to the corresponding osculating
plane, this is called a rectifying flane, and the assemblage of
these planes generates a developable which is called the recti-
fying developable. The reason of the name is, that the given
curve Is obviously a geodesic on this developable, since its
osculating plane is, by construction, everywhere normal to the
surface. If, therefore, the developable be developed Into a
plane, the given curve will become a right line.
The intersection of two consecutive planes of the rectifying
developable Is the rectifying line. Now, since the plane passing
through the edge of a right cone perpendicular to its tangent
plane passes through Its axis, it follows that the rectifying
plane passes through the axis of the osculating cone considered
in the last article; and, therefore, that the rectifying line is
the axis of that osculating cone. The rectifying line may be
* It has been proved by M. Bertrand that when the ratio r : p is constant, the
curve must be a helix traced on a cylinder ; and by Pniseux, that when r and p
are both constant, the cylinder has a circular base. Liouvilles Monge, p. 554.
I
NON-PROJECTIVE PROPERTIES OF CURVES. 337
therefore constructed by drawing in the rectifying plane a
Hue making with the tangent line an angle 11^ where H has
the value determined in the last article.
The rectifying surface is the surface of centres of the original
developable formed by the lines of the system. In fact it was
proved (Art. 306) that the normal planes to a surface along
the two principal tangents touch the surface of centres ; but
the generating line itself is in every point of it one of the
principal tangents ; the rectifying plane, therefore, touches the
surface of centres which is the envelope of all these rectifying
planes. The centre of curvature at any point on a developable
of the other principal section, namely, that perpendicular to the
generating line, is the point where its plane meets the corre-
sponding rectifying line ; for evidently the traces on this plane
of two consecutive rectifying planes are two consecutive normals
to the section. Hence If I be the distance of any point on the
developable from the cuspidal edge measured along the generatoi',
the radius of curvature of the transverse section is l\2inH.
When I vanishes, this radius of curvature vanishes, as it ought,
the point being a cusp.
In the case of the helix the rectifying surface is obviously
the cylinder on which the curve is traced.
371. To find the angle behoeen two successive radii of curvature.'^
Let ABj BC he traces on any
sphere with radius unity, of planes
parallel to the osculating and
normal planes, then the central
radius to B is the direction of the
radius of curvature. If AB\ B' G
be consecutive positions of the os- ^'^ ^~c
culating and normal planes, B' is in the direction of the con-
secutive radius of curvature, and BB' measures the angle
between them. Now the triangle BOB' being a very small
right-angled triangle, we have BB"^ = BO' + OB"'.
* The reader will find simple geometrical investigations of this and other formulae
connected -with curves of double curvature in a paper by Mr. Routh, Quarterly Journal
of Mathematics, vol. vii. p. 37.
XX
338 CURVES AND DEVELOPABLES.
But since the angle ABC la right, BO measures BAB\ which
is ^77, the angle between two consecutive osculating planes,
and OB' measures OCB\ which is dO, the angle between
two consecutive normal planes. The required angle is there-
fore given by the formula BB''^ = drf + d6'\ where drj and
dd have the values already found. The series of radii of
curvature at all the points of a curve generate a surface on
the properties of which we have not space to dwell. It is
evidently a skew surface (see note, p. 89), since two consecutive
radii do not in general intersect (see Art. 374, infra).
Ex. 1. To find the equation of the surface of the radii of curvature in the case
of the hehx.
The radius of curvature being the intersection of the osculating and normal planes
has for its equations (Art. 361) x'y — y'x, z — z', from which we are to eliminate
x'y'z' by the help of the equations of the curve. And writing the equations of the
helix X — a cos nz, y — a sin nz, the requii-ed surface is y cos nz — x sin nz.
Ex. 2. To find the equation of the developable generated by the tangents of
a helix. The equations of the tangent being
X — a cos nz' = — na sin nz' [z — z'), y — a sin nz' = na cos nz' {z — z'),
the result of eliminating z' is found to be
f , {x"- + y"--(r-f] , . r _, (3-2 + ^2 - «2)i-|
X cos - nz + ~ y + y sm i nz + 2 '- } = a.
I - a I "^ L — a J
Since this equation becomes impossible when x- + y- < n-, it is plain that no part of
the surface lies within the cylinder on which the helix is traced.
372. We shall now speak of the pola?- developable generated
by the normal planes to the given curve. Fourier has remarked,
that the " angle of torsion " of the one system is equal to the
" angle of contact" of the other, as is sufficiently obvious since
the planes of this new system are perpendicular to the lines
of the original system, and vice versa. The reader will bear
in mind, however, that it does not follow from this that the
-J- of one system is equal to the -y- of the other, because the
ds is not the same for both.
Since the intersection of the normal planes at two con-
secutive points K, It' of the curve is the axis of a circle of
which K and K' are points (Art. 364), it follows that if any
point D on that line be joined to K and 7i ', the joining lines
are equal and make equal angles with that axis.
I
NON-PROJECTIVE PROPERTIES OF CURVES.
339
It is plain that three consecutive normal planes intersect
in the centre of the osculating sphere ; hence the cu.tpidal edge
of the polar developable is the locus of centres of spherical cur'
vature.
In the case of a plane curve this polar developable reduces
to a cylinder standing on the evolute of the curve.
373. Every curve has an infinity of evolutes lying on the
polar developable;'^ that is to say, the given curve may be
generated In an infinity of ways by the unrolling of a string
wound round a curve traced on that developable. Let MM.\
M'M'\ &c. denote the successive elements of the curve, K^ K\
&c. the middle points of these elements, then the planes drawn
through the points K perpendicular to the elements are the
normal planes. The lines AB^ A'B\ &c. being the lines in
which each normal plane is intersected by the consecutive,
these lines are the generators of the polar developable, and
hence tangents to the cuspidal edge RS of that surface. Draw
now at pleasuref any line KD in the first normal plane,
meeting the first generator in D] join DK' which being in
the second normal plane will meet the second generator A'B\
say in D'. In like manner, let K"D' meet A"B'' in D". We
get thus a curve DD'D" traced on the polar developable which
is an evolute of the given curve. For the lines DK^ U K\ &c.
the tangents to the curve DD'D'\ are normals to the curve
* See Monge, p. 396.
t This figure is taken from Leroy's Geometry of Three Dimensions.
340 CURVES AND DEVELOPABLES.
KICK", and the lengths DK=DK\ B'K' =.B'K'\ &c. (see
Art. 372). If therefore X>^be a part of a thread wound round
DD'D", it is plain that as the thread is unwound the point K
will move along the given curve.
Since the first line DK was arbitrary, the curve has an
infinity of evolutes. A plane curve has thus an infinity of
evolutes lying on the cylinder whose base is the evolute in the
plane of the curve. For example, in the special case where
this evolute reduces to a point ; that is, when the curve is a
circle, the circle can be described by moving round a thread
of constant length fastened to any point on the axis passing
through the centre of the circle
In the general case, all the evolute curves DD'D'\ dec. are
geodesies on the polar developable.
For we have seen (Art. 308) that a curve is a geodesic when
two successive tangents to It make equal angles with the inter-
section of the corresponding tangent planes of the surface ;
and it has just been proved (Art. 372), that DK, DK', which
are two successive tangents to the evolute, make equal angles
with AB which is the intersection of two consecutive tangent
planes of the developable. An evolute may then be found
by drawing a thread as tangent from K to the polar develop-
able, and winding the continuation of that tangent freely round
the developable.
374. The locus of centres of curvature is a curve on the polar
developable, but generally is not one of the system of evolutes.
Let the first osculating plane MM'M" meet the first two normal
planes in KC, K' C, then C is the first centre of curvature;
and, in like manner, the second centre is C", the point of inter-
section of K'C, K" G\ the lines in which the second oscu-
lating plane M'M"M'" is met by the second and third normal
planes. Now the radii K'C, K'C are distinct, since they
are the intersections of the same normal plane by two different
osculating planes, K'C will therefore meet the line AB in a
point / which is distinct from G. Consequently, the two radii
of curvature KG, K'C' situated In the planes P, P' have no
common point In AB the intersection of these planes; two
NON-PEOJECTIVE PROPERTIES OF CURVES. 341
consecutive radii therefore do not Intersect, unless In the case
where two consecutive osculating planes coincide.
The centres of curvature then not being given by the suc-
cessive intersections of consecutive radii, tiiese radii are not
tangents to the locus of centres. Any radius therefore KG
would not be the continuation of a thread wound round CC'C'\
and the unwinding of such a thread would not give the curve
KK'K"^ except in the case where the latter is a plane curve.*
375. To find the radius of the sphere through four con-
secutive 'points. Let R be the radius of any sphere, p the
radius of a section by a plane making an angle 17 with the
normal plane at any point ; then, by Meunier's theorem,
^cos77 = p; and for a consecutive plane making an angle
7] + S77, we have 8p = — B sin rjSi]. Hence i2^ = p'"* + f -^ J .
We have then only to give In this expression to p and drj
the values already found.
-J- Is obviously the length of the perpendicular distance
from the centre of the sphere to the plane of the circle of
curvature.
376. To find the cooi^dinates of the centre of the osculating
sphere.
Let the equation of any normal plane be
(a — x)dx+{^- y) dy + (7 — z) dz = 0,
where xyz is the point on the curve, and a/3y any point on
* The characteristics of the polar developable may be investigated by arguments
similar to those used Higher Plane Curves, Arts. Ill, &c. They are n' = m+r, a' = 0,
r' — 3m + n, m' = bm + a, where m, n, &c., having the same meaning as in Art. 325, are
the characteristics of the given curve, and ?«', n', &c. the coiTCsponding characteristics
of the polar developable. When, as is here supposed, there is nothing special in the
character of the points at infinity of the given curve, the normal planes corresponding
to these points are altogether at infinity ; and the corresponding generators of the
polar developable are common to three consecutive planes. The plane at infinity
meets the polar d(.velopables in m lines, each reckoned three times, and a curve of
the ?i"' order.
342 ruRVES and developables.
the plane ; then the equation of a consecutive normal plane
combined with the preceding gives
(a - x) d'x + (/3 - J/) d''^ + [y - z) d^z = ds^.
And the equation of the third plane gives
(a - x) d'x + {^-y) d^'y + (7 - 2) d^z = Sdsd'^s.
Let us denote, as before, dyd'^z — dzd^y, &c. by X, Y, Z\
dyd^z — dzd^y^ &c. by X\ F', Z\ and the determinant
Xd^x-\- Yd^y -^ Zd^z by M. Then, solving the preceding equa-
tions, we have
M{a.-x) = - X'ds"" + 3Xdsd\ M{/3 -y)=- Tds" + 3 Ydsd%
31 [ry -z) = - Z'ds' + SZdsd's.
By squaring and adding these equations, we obtain another
expression for B'\ which is what the value in the last article
becomes when for p and ^ we substitute their values.
UT]
We add a few other expressions, the greater part of which
admit of simple geometrical proofs, the details of which want
of space obliges us to omit.
Ex. 1. If <r be the arc of the curve which is the locus of centres of absolute
curvature,
d(r- = dp^ + p"dif ; ox d<T - Rdtj.
Ex, 2. If S be the length of the arc of the locus of centres of spherical curvature
dL = ~-r- ; where o =: y- is the distance between the centres of the osculating circle
and osculating sphere. From this expression we immediately get values for the
radii of curvature and of torsion of this locus, remembering that the angle of torsion
is the angle of contact of the original, and vice versa.
Ex. 3. The angle between two consecutive rectifying lines is dlT.
Ex. 4. The angle i|/ between two consecutive H's is given by the formula
i22,/,2 = ds^ + di:"- - dRK*
* The reader will find further details on the subjects treated of in this section in
a Memoir by M. de Saint- Venant, Journal de VEcole Poly technique, Cahier xxx.,
who has also collected into a table about a hundred formulae for the transformation
and reduction of calculations relative to the theory of non-plane curves ; and in a
paper by M. Frenet, Liouville, vol. xvii., p. 437. I abridge the following historical
sketch from M. de Saint-Yenant's Memoir : " Curve lines not contained in the same
plane have been successively studied by Clairaut {Recherches sur les courbes a double
courbure, 1731), who has brought into use the title by which they have been com-
monly known (previously, however, employed by Pitot) and who has given expressions
for the projections of these curves, for their tangents, normals, arc, ifec. ; by Monge
{Memoire sur les developpees, ^e. presented in 1771, and inserted in vol. X., 1785,
of the ' Savants elrangers,' as well as in his ' Application de V Analyse a la Geometrie')
I
CURVES TRACED ON SURFACES. 343
SECTION IV. CURVES TRACED ON SURFACES.
377. The coordinates cc, y^ z of a point on a surface may-
be expressed as functions of two parameters^, q ; and conversely
if the coordinates x, ?/, z are thus expressed as functions of two
parameters, these expressions determine the surface, for by the
elimination of the parameters we obtain between the coordinates
ic, y^ z the equation U= 0 of the surface ; and when a definite
value is assigned to either^; or j, the point xyz is restricted
to a definite curve on the surface. This mode of representation
of a surface is, peculiarly appropriate for the discussion of the
theory of curvature, and it has been used for that purpose
by Gauss.* We proceed to give an account of his investi-
gations, but before doing so must explain his notation and
establish the connexion of this method with that by which
curvature was treated in Chapter xi. We have ic, ?/, z given
functions of /?, q ; and the partial differential coefficients of
cc, ?/, z in regard to these variables are expressed as follows :
dx = adp + a'dq, dy — hdp + h'dq^ dz = cdp + cdq^
d^x = a.df -f 2a.'dpdq + a"dq\
dSj = /3.?y/ + 2^' dp dq + ^"dq\
d'z = 7f/// -f 2^'dpdq + i'dq\
who gave expressions for the normal plane, centre and radius of curvature, evolutes,
polar lines and polar developable, centre of osculating sphere, for the criterion for
'points of simple inflexion' where four consecutive points are in a plane, and for
' points of double inflexion' where three consecutive points are in a right line ; by
Tinseau {Solution de quelques prohlhnes, tj-c. presented in 1774, Savants etrangers
vol. IX., 1780) who was the first to consider the osculating plane and the developable
generated by the tangents; by Lacroix {Calcul DifferentieT) who was the firet to
render the formula3 sj'mmetrical by introducing the differentials of the three co-
ordinates; and by Lancret {Memoire sur les courbes a double courbure, read 1802
and inserted vol. i., 1805, of Savants etrangers de I'lnstitut) who calculated the
angle of torsion, and introduced the consideration of the rectifying lines and rectify-
ing surface." The reader will find some interesting and novel researches respecting
curves of double curvature in Sir Wm. Hamilton's Elements of Quaternions ; as, for
instance, the theoiy of the osculating twisted cubic which passes through six con-
secutive points of the curve.
* See his Memoir '' Disquisitiones circa superficies curvas," Comm. Gott. recent,
t. Yi. (1827), reprinted in the appendix to Liouville's Edition of Monge, and in his
Works, IV. p. 219.
344
CURVES AND DEVELOPABLES.
Gauss also writes
he — cb' = A, ca — ac = B, aV — ha = C,
a' + b'' + 6' = E, aa' + hV + cc = F, a" + h" + c" = G,
which obviously lead to the relation A"" + B' ■\- C = EG- F' ;
and to these notations it is convenient to join V^ = EQ - F'^^
A(x-\-B^+Cy = E\ Aa+B/3' + Cj'=F', Aa" + B/3" + CY =- G' ,
E\ F\ G' denoting respectively the determinants
a, 5, c
1
a'j 5'j c
«, A 7
a, 5, c ' ,
r -It t
a ^ 0 ^ c
a)P)7
t If t
a ^ 0 ^ c
ff o" "
« jP ,7
The identity Adx^ Bdy + Cdz = 0,
replaces the differential equation of the surface, or what is the
same thing, if U=f[x^ y^ z) = 0 is the equation of the surface,
, -r, n • , dU dU dU
then A^ B, u are proportional to ;7^ > ~7~ ? "?" •
Again, since the coordinates are rectangular, if ds be an
element of length on the surface, that is, if it be the distance
between the points (7:), q) and [p + dp^ 9.-^ dq)-, then
ds' = Edf + iFdpdq + Gdc[\
378. The differential equation Art. 303 of the lines of
curvature may be written
dx ^ dy ^ dz = 0.
A, B, C
dA, dB, dC
Repeating the investigation which led to this equation, we
have for the coordinates of an indeterminate point on the normal
^ = x + A\j 7] = y + B'K, ^=z-\-C\,
and if this meets the consecutive normal, then taking ^, 7;, ^ to
be the coordinates of the point of intersection, we have
0 = dx + AdX + \dA, 0 = dy-{ BdX + \dB^ (i = dz+ CdX + \dCj
which equations, by eliminating \ and dX^ give the equation in
question.
CURVES TRACED ON SURFACES.
345
Now this equation may be written [Higher Algebra^ Art. 24)
adx + hdy + cdz^ adx + Vdy + c'dz
ad A + hdB + cdC^ adA + VdB + c'JC
since it is what is denoted by
= 0,
a, 5 , c
= 0.
dA, dB, dC
Calculating the quantity adx + hdy + cdz, by substituting for dx^
adp + adq^ &c., it is found to be Edp + Fdq. Similarly
adx -I- Vdy -f cWs = i^ti^:? + Odq^.
Again, differentiating the identities
a^ + Z)^+c C = 0,
a'A^o'B\dG = ^,
we find adA^rldB-^-cdC^- {Ada + 5J6 + Cdc ),
a'J^ -f &W5 + cdG = - {Ada + BdV + Cdc),
which, substituting for da = adp + a'dq, &c., become respectively
- {E'dp + Fdq) and - [F'dp + G'dq). Whence, finally, the
equation of the lines of curvature is
Edp + Fdq, Fdp + Gdq
Kdp + F'dq, F'dp+0'dq
or, as this may also be written,
d(f, —dpdq, dp''
E, F , a
= 0,
E\ F'
G'
= 0.
379. The equations 0 = Jo; + AdX + \dA, &c., of the last
article may be written, putting dA = A^dp + A^dq^ &c.,
0 = (a + \A^ dp -\- {a + \A^ dq + Ad\
0 = (6 + \B^ dp + (5' + X5J dq + 5J\,
0 = (c + X (7,) fZp + (c' + >^ OJ J^ -F (7JA,,
which equations, by the elimination of dp, dq, d\, give for the
determination of \ a quadratic equation corresponding to that
of Art. 295. Taking p for the radius of curvature, we have
p''=^{^-xr^{v-yf+[^-z)-\ =r'\\ or say \ = p:V;
and writing down the equation in question with this value
Y Y
346
CURVES AND DEVELOPABLES.
= 0.
= 0.
substituted for \, the equation Is
aV+A^p, bV+B^p, cV+C,p
aV+A^p, h'V^-B,p, cV+C^p =0,
A , B , C
a quadratic equation for determining the radius of curvature.
This equation may be treated as before. It becomes
EV+ p [A^a + BJj -f C,c), FV+ p {A^a' + Bfi' + C/)
FV+ p [A^a + BJy + ap), GV+p {A^a! + Bh' + (7/)
In which, by the last article, the coefficients of p are —E\ — F\
— G' ^ whence the equation for the radii of curvature is
E'p-EV, F'p-FV
F'p-FV, G'p- GV
380. By what precedes we have a quadratic equation for the
direction of the lines of curvature, and a quadratic equation
for the value of p ; but it Is obvious that, selecting at pleasure
either of the two lines of curvature, the corresponding value
of p should be linearly determined. The required formula is
at once obtained from the equations 0 = dx + AdX + \dA^ &c.^
of Art. 378, by multiplying them by dx^ dy, dz respectively
and adding; then substituting for A, Its foregoing value p : F,
we have
Y{dx^ + dif + dz'') + p [dxdA + dydB -i dzdC)=0,
where, by what precedes, dx^ -f dy'^ + dz'^ = Fdp^ -f 'iFd'pdq^ + Gdq[\
But, by the equation of the surface Adx+Bdy^ Cdz — 0, we have
dA dx + dBdij + dCdz = - {Ad''x + BdSj -\- Cd'^z),
which, substituting from Art. 377,
= - {F'df + 2F'dj,dq + G'dq'),
whence the equation is
p [E'df -i- 2F'dpdq + G'dq') - V{Edf + 2Fdpdq + Gdq') = 0.
In this, considering dp -=r dq as having at pleasure one
or other of the values given by the differential equation of
the lines of curvature, the equation gives linearly the cor-
responding value of the radius of curvature.
But writing the equation in the form
[pF/ - VE ) df + 2 [pF' - VF) dp dq + {pG'- VG) dq' = 0,
CURVES TRACED ON SURFACES.
347
= 0,
= 0.
and attending to tlie equation for tlie determination of p, it
appears that the equation may be expressed in either of the
forms
{pE\- VE)dp+ {pF'-VF) dq = 0,
[pF' - VF) dp +{pG'- VG) dq = 0 ;
or, which is the same thing, the equations of Arts. 378 and 379
may be expressed in the more complete forms
p , E dp + F' dq^ F dp + Gdq
F, E'dp^F'dq, F'dp+G'dq
dq, pE'- VE, pF' -VF
-dq, pF' - VF, pG' -VG
The first of these gives the quadratic equation for the curves
of curvature, and (linearly) the value of p for each curve ; the
second gives the quadratic equation for the radius of curvature,
and (linearly) the direction of the curvature for each value of
the radius. It also appears that the quadratic equations for p
and for dp -r- dq are linear transformations the one of the other.
381. Returning to the equation
p [E'df + ^F'dpdq + G'dq') = V {Edf + ^Fdpdq + Gdf)
of the preceding article, It Is to be observed that (the ratio
dp -=- dq being arbitrary) this is the equation which deter-
mines the radius of curvature of the normal section through
the consecutive point [p-\-dp, ^.-^d^- The centre of curva-
ture of this section is, in fact, given as the intersection of the
normal at (/>, q) by the plane drawn through the middle point
of the line joining the two points [p^q)^ [P'^^Pi ? + ^2') ^.t
right angles to this line. Taking f , 77, ^ for current coordinates,
the equations of the normal are, as before,
whence (^ - x)'' + (77 - yf + (^- zf = \' V = p%
p being a distance measured along the normal ; the equation of
the plane in question is
[^ - X —Idx — Id'^x - &c.) {dx + Id'^x-r&c.) +...= 0,
or, substituting for | — a:, v —y^ K ~^ ^^^ values ^ , ^^ , ^ ,
348
CURVES AND DEVELOPABLES.
the equation, omitting higher infinitesimals, becomes
^{A{dx-{-ld'x)+B[dy + ^d'y)+C{dz^^d'z)}=^l [dx'+df+dz') ;
which, observing that Adx + Bdy + Cdz = 0, is
p [Ad'x + Bd'ij + Cd'z) - V{dx' + df + dz') = 0,
or, substituting for dx, ..., d^x^ ... their values, it is
p [E'df + 'iF'djpdq + G'dq^) - V[Edf -f 2Fd:pdg^ + Odg") = 0,
the above-mentioned equation.*
The formula explains the meaning of the coefficients
E\ F\ G' ) it shews that the equation
E'df + 2F'dpdq + a'dq" = 0
determines the directions of the inflexional tangents at the
point (p, g). It may be observed that if J5^' = 0, (?' = 0, this
equation becomes dpdq = 0, we then have p = const., q = const.,
as the equations of the " Inflexion curves," or curves which at
each point thereof coincide in direction with an inflexional
tangent.
382. We may imagine the parameters ^, q so determined
that the equations of the two sets of lines of curvature shall
be ^ = const, and q = const, respectively. When this is so the
dlfi*erential equation of the lines of curvature will be d]3dq = Q\
and this will be the case if F= 0, i^' = 0 ; we thus obtain F= 0,
F' = 0 as the conditions in order that the equations of the
lines of curvature may be ^ = const, and q — const. Or, writing
the conditions at full length, they are
dx dx dy dy
djp dq dp dq
dx
dp
dx
dq
d'x
dy_
dp
dy
dq
d'y
dz dz
dp dq '
dz_
dp
d^
dq
d\
dpdq ' dpdq ' dpdq
= 0,
* This equation is obtained geometrically by Mr. Williameon, Quarterly Journal,
vol. XI., p. 3G4(1871).
CURVES TRACED ON SURFACES. 349
where it may be noticed that the first equation merely expresses
that the curves p = const, and q = const, intersect at right
angles.
383. If, as above, F= 0, F' = 0, then the quadratic equation
for p is
{pW-VE){pG'-VG) = Oj
and from the equations of Art. 380, putting successively ^ = 0,
VG
c?2 = 0, it appears that the value p = —^ belongs to the line
VE
of curvature p = const., and the value p = -pr to the line of
curvature q = const.
384. The above determinant-equation F' = 0 may be re-
placed by three equations
dpdq dp dq ' ''
where X, /a, are indeterminate coefiicients ; multiplying first
, dx dy dz
"7 ^ 1 -J- t -J- 1 a^nd addrng, we have an equation containing
only \, and which is
fl'lt* fill (iZ
and similarly multiplying by y- , -^ , — , and adding, we obtain
2d^^^^-''
It thus appears, that ^ = const., 5^ = const., being the equations
of the curves of curvature, the coordinates a-, y, z considered
as functions of p, q satisfy each the partial differential equation
d'u _ I IdEdu _-[_! dGdii _ ^
dpdq 2 Fdq dp~ 2~Gdp d^~^'
385. Entering now upon Gauss's theory of the curvature
of surfaces,! it is to be remembered that in plane curves
* See Lame Leqons sur les coordonnees curvilignes, Paris, lb59, p. 89.
t See hia Memoir referred to in Note to Art. 377.
350 CURVES AND DEVELOPABLES.
we measure the curvature of an arc of given length by
the angle between the tangents, or between the normals, at
its extremities ; in other words, if we take a circle whose radius
is unity, and draw radii parallel to the normals at the ex-
tremities of the arc, the ratio of the intercepted arc of the
circle to the arc of the curve affords a measure of the cur-
vature of the arc. In like manner, if we have a portion of
a surface bounded by any closed curve, and if we draw radii
of a unit sphere parallel to the normals at every point of the
bounding curve, the area of the corresponding portion of the
sphere is called by Gauss the total curvature of the portion
of the surface under consideration. And if at any point of
a surface we divide the total curvature of the superficial element
adjacent to the point by the area of the element itself, the
quotient is called the measure of curvature for that point.
386. We proceed to express the measure of curvature by
a formula. Since the tangent planes at any point on the
surface, and at the corresponding point on the unit sphere,
are by hypothesis parallel, the areas of any elementary portions
of each are proportional to their projections on any of the
coordinate planes. Let us consider, then, their projections on
the plane of a:?/, and let us suppose the equation of the surface
to be given in the form z = (p {x^ y).
If then cr, y, z be the coordinates of any point on the surface,
X^ Y^ Z those of the corresponding point on the unit sphere,
x-^dx^ x-\-hx^ X-{-dX^ X+SA^, &c., the coordinates of two
adjacent points on each, then the areas of the two elementary
triangles formed by the points considered are evidently in the
ratio
dXh Y —dYhX: dxhj — dyhx.
But dX^ dY^ SXj SY are connected with dx, dy^ &c., by
the same linear transformations, viz.
,^ dXj dXj j^r dYj dY,
BX= ',— hx + -^ 5?/, S Y=- -r~Bx -f- -^- Sy ;
ax dy dx dy '' '
CURVES TRACED ON SURFACES. 351
"wlience, by the theory of linear transformations, or by actual
multiplication,
dXSY- ,Y8X= icI.S, - .7,8.) (g^f - ^^'^) ,
^, , .^ dXdY dXdY. . , c ,
thus the quantity -^ = — — is the measure or curvature.
ax ay ay dx
Now X, F, Z^ being the projections on the axes of a unit line
parallel to the normal, are proportional to the cosines of the angles
which the normal makes with the axes. We have, therefore,
V
X= —rr. — ■ n 57- , y=
whence
dX _ (1 + (f) r —pqs dX _ (1 + q^) s -pqt
d^" [1 ^p^'+qj^ ' 'dy ~~(lT/T?ji~ '
dY ^ {l+2f)s-pqr cIY ^ {l + p')t- pqs
dx (1 + / -f q']r ' dy ~ [l +y + q')i '
dXdY dXdY_ (rt-s')
dx dy dy dx (l+i^' + jT'
But from the equation of Art. 311, it appears that the value
just found for tlie measure of curvature is -j^^, , where R and K
are the two principal radii of curvature at the jpoint.
387. It is easy to verify geometrically the value thus found.
For consider the elementary rectangle whose sides are in the
directions of the principal tangents. Let the lengths of the
sides be X, X', and consequently its area XX'. Now the normals
at the extremities of X intersect, and if they make with each
other an angle ^, we have 6 = \\R where R is the corresponding
radius of curvature. But the corresponding normals of the
sphere make with each other, by hypothesis, the same angle,
and their length is unity. Denoting, therefore, by /t the length
of the element en the sphere corresponding to X, we have
352 CUEVES AND DEVELOPABLES.
■\ "x ' ' 1
^ = /ti. In like manner we Lave ^, = /z, and r^-? = ^^7 , which
was to be proved.
388. From the formula of Art. 379, It appears that the value
of the measure of curvature is
~ {EG - FY ^ '^
but Gauss obtains this expression in a very different form, as a
function of only E^ F^ G, and their differential coefficients In
regard to p^ q. To obtain this result we have to express in
this form the function E' G' — F''^ ; that is, the function
a, A 7
«^ ^'\ 1"
«', /3', 7
«, 5, c
X
« , & , c
—
O", h ^ G
a , &', c'
a', h\ c'
a\ h\ d
Now if these products be expanded according to the ordinary
rule for multiplication of determinants, they give the difference
between the two determinants*
aa" + m' + 77", ««" + 5/S" + ci\ aV + V^" + cV'
aa + J/3 + C7 , a""* + i^ + c"'' , act + J5' + cc'
da. + &'^ -f c'7 , aa! + 5&' + cc' , a'^ + &'^ + c'
oT' 4 /S''' + y^ , aa' + W + 07', a'a' + Vff + c'7'
aa' + W + C7' , a' +&"-!- c"'' , aa' + hV + cc'
a'a'-F5'/3'+c'7', aa' -1- J&' + cc' , a" + J'^' + c"
389. Now it is easy to show that the terms In these deter-
minants are functions of E^ F, G and their differentials. Re-
ferring to the definitions of a, J, c, a, a', a", &c. (Art. 377) it is
obvious that
_da , da _ da' „ _ da' „
dp'' dq dp '' dq ^ *'
* I owe to Mr. Williamson the remark, that the application of this rule exhibits
the result in a form which manifests the truth of Gauss's theorem.
CURVES TRACED ON SURFACES.
353
whence, since
aa + h^ + cy =^ j~ , aa' + 5/3' + 07' = | ^ ,
aV + h'/3' + cV = i ^ , a a" + ¥0" + cY = i X '
dG
aa +6/3 + cy = -,- _ (a a + 5 /3 + c 7 ) = ,— -
dp '
a a + 6 /3 + c 7 = J (aa + ftp + 07 j = ^ ^ ^ .
It will be seen that these equations express in terms of E^ F^ G
every term In the preceding determinants except the leading
one in each. To express these, differentiate, with regard to g',
the equation last written, and we have
aa" + m' + 77" =
d''F _i<^_( r da. d^ ,d-y\
dpdq ^ dq^ \ dq dq dq)
Again, differentiate, with regard to p, the equation
and we have
' ' . 7';3'i ' ' , IIKT
''I , o"i , n -> d G f , da!
" +^ +'^ ^^W~V dp^
dp dp dp ) '
Now because -t- —~t- t &c., the quantities within the brackets
in the last two equations are equal. And since the leading
term in each determinant is multiplied by the same factor, in
subtracting the determinants we are only concerned with the
difference of these terms, and the quantity within the brackets
disappears from the result. The function in question is thus
equal to the difference of the determinants
dpdq ^ dq' ^ dq ^ dj) ^ ^ dq
^dE
2 djp '
dF_dE
dp dq ^
E.
F.
F
G
zz
354
CURVES AND DEVELOPABLES.
and
1^
^ dp'
1 ^
^- dq '
2 dp '
dE dG
^ d^' ^ dp
F
K
^;
G
We get the measure of curvature by dividing the quantity
now found by [EG — F'^Y, and the result Is thus a function of
Ej F, G and their differentials. Gauss's theorem Is therefore
proved. It may be remarked that the expression Involves only
second differential coefficients of E, F, 6-', that Is third differ-
ential coefficients of the coordinates; these, however, really
disappear, since the original expression E' G' — F'^ involves only
second differential coefficients of the coordinates.
We add the actual expansion of the determinants, though
not necessary to the proof. Writing the measure of curvature
h^ we have
p{dEdG
\dp) dq
dq dj) dq dq dp dq dp dp )
(d_E dG _^dEdF fdEy"]
(f/^j dp dp dq \dq j J
(Llouvllle's Monge, p. 523).*
d'F d'G\
dp'
390. The foregoing theorem, that the measure of curvature
is a function of E^ F, G and their differentials, shews that if
a surface supposed to be flexible, but not extensible, be trans-
♦ MM. Bertrand, Diguet, and Puiseux (see LiouvUh, vol. Xiii. p. 80 ; Appendix
to Monge, p. 583) have established Gauss's theorem by calculating the perimeter and
area of a geodesic circle on any surface, -whose radius, supposed to be very small, is i.
They find for the peiimeter 27rs •
-, , and for the area irs^ ■
And of course
■dlili" 12^72'"
the supposition that these are unaltered by deformation implies that RR' is constant.
CUKVES TRACED ON SURFACES. 355
formed in any manner ; that Is to say, If the shape of the surface
be changed, yet so that the distance between any two points
measured along the surface remains the same, then the measure
of curvature at every point remains unaltered. We have an
example of this change In the case of a developable surface
which is such a deformation of a plane ; and the measure of
curvature vanishes for the developable, as well as for the
plane, one of the principal radii being Infinite. To see that
the general theorem is true, observe that the expression of an
element of length on the surface Is
els' = Eclf + iFdpdq + Gdq\
Let cc', y\ z denote the point of the deformed surface corre-
sponding to any point cc, y, z of the original surface. Then
x\ y\ z are given functions of x^ ?/, 0, and can therefore also
be expressed in terms of ^:>, q ; and the element of any arc of
the deformed surface can be expressed in the form
ds"' = E^df + ^F^dpdq + G^di'.
But the condition that the length of the arc shall be unaltered
by transformation, manifestly requires that E = E^^ F—F^^
G= G^\ hence, any function of E^ F^ G^ and, in particular the
value of the measure of curvature, Is unaltered by the deformation
in question.
391. We may consider two systems of curves traced on
the surface, for one of which p is constant, and for the other q ;
so that any point on the surface Is the interscctlou of a pair
of curves, one belonging to each system. The expression then
ds'^Edp' -^2Fdi)dq-\- Gd(f shews that ^j{E) dp) Is the element
of the curve, passing through the point, for which q Is constant;
and \j{G) dq is the element of the curve for which jp is
constant. If these two curves intersect at an angle <w, then
since ds is the diagonal of a parallelogram of which \I{E) dp^
\/[G)dq are the sides, we have f^ {E G) cos o) = F, while the
area of the parallelogram is dada' ?\n(jo = \J[EG — F'^) dpdq.
If the curves of the system p cut at right angles those of
the system q^ we must have F= 0.
A particular case of these formulae is when we use geodesic
S5Q CURVES AND DEVELOPABLES.
polar coordinates, in which case, as we shall subsequently shew,
we always have an expression of the form ds'^ = dp" + P^d(o^.
Now if in the formula of article 389 we put F=0, ^= constant,
it becomes
and if we put
^=1, G = F% p = p, A- = ^,, ^ehave^-f^,= 0,
an equation which must be satisfied by the function P on any
surface, if Pdo) expresses the element of the arc of a geodesic
circle. Mr. Roberts verifies [Cambridge and Dublin Mathe-
matical Journal^ vol. III., p. 161) that this equation is satisfied
by the function y cosecw on a quadric.
392. Gauss applies these formulas to find the total curvature,
in his sense of the word, of a geodesic triangle on any surface.
The element of the area being Fdtadp^ and the measure of
1 d'^P . . . d'^P
curvature being — ji~pr i by twice integrating — -^-^ dpdoo
the total curvature is found. Integrating first with respect
( dP\
to p, we get iC — J- \ doi. Now if the radii are measured
from one vertex of the given triangle, the integral is plainly
to vanish for p = 0 ; and it is plain also that for p = 0 we must
dP
have -T— = 1 ; for as p tends to vanish, the length of an element
perpendicular to the radius tends to become pdco. Hence the
first integral la dco (l — p ] •
This may be written in a more convenient form as follows :
Let 6 be the angle which any radius vector makes with the
element of a geodesic arc ah. Now
since aa=Pd(o^ bb'=[P-fdP) doi] and
if cb = aa\ we have cb' = dPdoa^ and
dP
the angle cab' = -7- dco. But cab' is
dp
evidently the diminution of the angle 0
CURVES TKACED ON SURFACES. 357
(IP
6 in passing to a consecutive point ; Lence cW = — -j- dco. The
integral just found is therefore dco + dd, which integrated a
second time is co + 6' — 6"^ where co is the angle between the
two extreme radii vectores which we consider, and ^', 6'^ are
the corresponding values of 0. If we call A, B^ C the internal
angles of the triangle formed by the two extreme radii and
by the base, we have (o = A, 6' = B^ 6" = Tr — C, and the total
curvature is A-\- B+ G —ir. Hence the excess over 180° of
the sum of the angles of a geodesic triangle is measured by
the area of that portion of the unit sphere which corresponds to
the directions of the normals along the sides of the given
triangle.
The portion on the unit sphere corresponding to the area
enclosed by a geodesic returning upon itself is half the sphere.
For if the radius vector travel round so as to return to the
point whence it set out, the extreme values of 6^ and ^" are
equal, while co has increased by 27r. The measure of cur-
vature is therefore 27r, or half the surface of the sphere.*
Gauss elsewhere applies the formulae to the representation of
one surface on another, and in particular to the representation
of a surface on a plane, in such manner that the infinitesimal
elements of the one surface are similar to those of the other;
a condition satisfied in the stereographic projection and in
other representations of the sphere.
393. It remains to say something of the properties of curves
considered as belonging to a particular surface. Thus the
sphere we know has a geometry of its own, where great circles
take the place of lines in a plane ; and, in like manner, each
surface has a geometry of its own, the geodesies on that surface
answering to right lines.")"
* For some other interesting theorems, relative to the deformation of surfaces,
see Mr. Jellett's paper " On the Properties of Inextensible Surfaces," Transactions
of the Royal Irish Academy, vol. XXII. Memoirs have also appeared by MM. Bour
and Bonnet, on the Theory of Surfaces applicable to one another, to one of which
was awarded the Prize of the French Academy in 1860.
t The geometry of curves traced upon the hj-perboloid of one sheet has been
358 CUEVES AND DEVELOPABLES.
We have already by anticipation given the fundamental
property of a geodesic (Art. 308). The differential equation
is immediately obtained from the property there proved, that
the normal lies in the plane of two successive elements of the
curve and bisects the angle between them ; hence Z, if, iV,
which are proportional to the direction-cosines of the normal,
(lOC fill fi '^
must be proportional to d -j- ^ d— ^ d -^ ^ which are the
CIS as G/S
direction-cosines of the bisector (Art. 358). Thus "if the tan-
gents to a geodesic make a constant angle with a fixed plane,
the normals along it will be parallel to that plane, and vice
versa (Dickson, Cambridge and Dublin Mathematical Journal^
vol. v., p. 168). For from the equation
dx , di/ dz
a -^ +o-f +c^r = constant,
as ds ds
which denotes that the tangents make a constant angle with
a fixed plane, we can deduce
aL + hM-\- cN= 0,
which denotes that the normals are parallel to the same plane.
394. If through any point on a surface there he drawyi two
indefinitely near and equal geodesies^ the line joining their ex-
tremities is at right angles to both.^
studied nearly in the same manner by Pliicker, Crelle, vol. XLiii. (1847), and by
Chasles {Comptes Rendus, vol. Lili. 1861, p. 985), the coordinates made use of being
the intercejJts made by the two generators through any point on two fixed generators
taken for axes. It is easy to shew that in this method the most general equation
of a plane section is of the form
Axy + Bx + Cy + D = 0,
and generally that the order of any curve is equal to the sum of the highest powers
of X and y in its equation, whether these highest powers occur in the same term
or not. The curves are distinguished into families according to the number of
intersections of the curve by the generating lines of the two kinds respectively.
Thus, for a quartic curve of the first kind, or quadriquadric, each generating line
of either kind meets the curve in 2 points ; but for a quartic curve of the second
kind, or excubo-quartic, each generating line of the one kind meets the curve in
3 points, and each generating line of the other kind in 1 point.
* This theorem is due to Gauss, who also proves it by the Calculus of Variations ;
Bee the Appendix to Liouville's Edition of Monge, p. 528.
CURVES TRACED ON SURFACES. 359
Let AB = ACj and let us suppose the angle at B not to
be right, but to be =0. Take BD^BCsecB,
and then, because all the sides of the tri-
angle BCD are infinitely small, it may be
treated as a plane triangle and the angle
DCB.is a right angle. We have therefore
DC<DB, AD^ DC<AB, and therefore
<AC. It follows that ^C is not the
shortest path from A to (7, contrary to hypothesis. Or the
proof may be stated thus : The shortest line from a point A
to any curve on a surface meets that curve perpendicularly.
For if not, take a point D on the radius vector from A and
indefinitely near to the curve; and from this point let fall
a perpendicular on the curve, which we can do by taking
along BG a portion =BI) cos6 and joining the point so found
to B. We can pass then from B to the curve more shortly
by going along the perpendicular than by travelling along"
the assumed radius vector, which is therefore not the shortest
path.
Hence, if every geodesic through A meet the curve per-
pendicularly, the length of that geodesic is constant. It is
also evident, mechanically, that the curve described on any
surface by a strained cord from a fixed point is everywhere
perpendicular to the direction of the cord.
395. The theorem just proved is the fundamental theorem
of the method of infinitesimals, applied to right lines {Conies^
pp. 369, &c.). All the theorems therefore which are there
proved by means of this principle will be true if instead of
right lines we consider geodesies traced on any surface. For
example, " if we construct on any surface the curve answering
to an ellipse or hyperbola ; that is to say, the locus of a point
the sum or difference of whose geodesic distances from two
fixed points on the surface is constant ; then the tangent at
any point of the locus bisects the angle between the geodesica
joining the point of contact to the fixed points." The converse
of this theorem is also true. Again, " if two geodesic tangents
to a curve, through any point P, make equal angles with the
360
CURVES AND DEVELOPABLES.
tangent to a curve along which P moves, then the difference
between the sum of these tangents and the intercepted arc of
the curve which they touch is constant" (see Conies^ Art. 399).
Again, " if equal portions be taken on the geodesic normals
to a curve, the line joining their extremities cuts all at right
angles," or, " if two different curves both cut at right angles
a system of geodesies they intercept a constant length on each
vector of the series." We shall presently apply these principles
to the case of geodesies traced on quadrics.
396. As the curvature of a plane curve is measured by the
ratio which the angle between two consecutive tangents bears
to the element of the arc, so the geodesic curvature of a curve
on a surface is measured by the ratio borne to the element
of the arc by the angle between two consecutive geodesic
tangents. The following calculation of the radius of geodesic
curvature, due to M. LlouvIUe,* gives at the same time a proof
of Meunier's theorem.
Let mn^ np be two consecutive and equal elements of the
curve. Produce nt = mn^ and let fall tq perpendicular to the
surface ; join nq and qp. Then, since nt makes an infinitely
small angle with the surface, its projection vq is equal to it. nq
is the second element of the normal
section, and is also the second element
of the geodesic production of mn. If
now 6 be the angle of contact tnp^
and 6' be tnq the angle of contact
of the normal section, we have tp = 6ds,
tq = 6'ds. Now the angle qtp (= ^)
is the angle between the osculating plane of the curve and
the plane of normal section, and since tq = tp cos 0, we have
which is Meunier's theorem
a' Q JL J ^ C0S(^
6 =o cos(p and -^ = ~
R
being the radius of curvature of the normal section and p that
of the given curve.
Now, in like manner, py^q being 6" the geodesic angle of
* Appendix to Monge, p. 57G.
CURVES TRACED ON SURFACES. 361
1 /I// 7 T • . 1 sin (f>
contact, we have pq = o as and pq = tp sin 9, or - = .
The geodesic* radius of curvature is therefore p cosec<^. It is
easy to see that this geodesic radius is the absolute radius of
curvature of the plane curve into which the given curve would
be transformed, by circumscribing a developable to the given
surface along the given curve, and unfolding that developable
into a plane.
397. The theory of geodesies traced on quadrics depends
on Jacobi's first integral of the differential equation of these
lines ; intimately connected herewith we have Joachimsthal's
fundamental theorem, that at every point on such a curve ^;Z)
j is constant, where, as at Art. 166, p is the perpendicular from
the centre on the tangent plane at the point, and D is the
diameter of the quadric parallel to the tangent to the curve
at the same point. This may be proved by the help of the
two following principles: (1) If from any point two tangent
lines be drawn to a quadric, their lengths are proportional to
the parallel diameters. This is evident from Art. 74 ; and (2)
If from each of two points A^ B on the quadric perpendi-
culars be let fall on the tangent plane at the other, these
perpendiculars will be proportional to the perpendiculars from
the centre on the same planes. For the length of the per-
pendicular from x"y"z" on the tangent plane at x'y'z' is
// r //
— J- + '^T^ H i 1 ] , and the perpendicular from xyz
on the tangent plane at x"y"z" is p ( — ^ — |- ~^ — h — ^ — I j .
If now from the points A^ B there be drawn lines AT^ BT
to any point T on the intersection of the tangent planes at
A and i?, and \i AT make an angle i with the intersection
of the planes, the angle between the planes being to, then the
perpendicular from A to the intersection of the planes is ^7^
sini*, and from A on the other plane is -<4 Z' sine sin w. In
* I have not adopted the name "second geodesic curvature" introduced by
M. Bonnet. It is intended to express the ratio borne to the element of the arc
by the angle which the normal at one extremity makes with the plane containing
the element and the normal at the other extremity.
AAA
362 CURVES AND DEVELOPABLES.
like manner the perpendicular from B on the tangent plane at
A is BT s'lni suict). If, therefore, the lines AT, i?7'make equal
angles with the intersection of the planes, the lines AT, BT
are proportional to the perpendiculars fi'om A and B on the two
planes. But A T and BT are proportional to D and i)', and
the perpendiculars are as the perpendiculars from the centre
p' and p. Plence Dp = Up) . But it was proved (Art. 308)
that if ^ J", TB be successive elements of a geodesic, they make
equal angles with the intersection of the tangent planes at
A and B. Hence, the quantity j)D remains unchanged as we
pass from point to point of the geodesic. Q.E. D.*
398. On account of the Importance of the preceding theorem
we wish also to shew how it may be deduced from the differ-
ential equations of a geodesicf Diiferentiating the equation
^ ^p ^2 ^
(where i,7I/,iVare the differential coefficients and/f =Z''+J/"''+iV^'"),
cioc
and then substituting for i, &c., f/y-, &c. (Art. 393), we get
<a<s)^<l)'^(^h'^(S;'(f)--
It is to be remaiked, that this equation is also true for a
line of curvature ; for since L : B, &c. are the direction-cosines of
the normal, the direction-cosines of a line in the same plane
with two consecutive normals, and perpendicular to them, are
(Art. 358) proportional to c? ( y, ) , &c. Hence the y , &c. of
are proportional \.q d\
now we differentiate
a line of curvature are proportional to ^[-^ , &c. But if
doc chi'^ dz^
- — I — - — 1- — = 1
ds' ^ ds' ds' '
* This proof is by Graves, Crelle, vol. XLii. p. 279.
t See Jacobi, Crelle, vol. xix. (1839), p. 309; Joachimsthal, Crelle, vol. xxvi.
p. 155 ; Bonnet, Journal de VEcole PnJy technique, vol. XIX. p. 138 ; Dickson, Cam-
brid</e and Dahlin Mathematical Journ<d, vol. v. p. 168 ; Jacobi, Vorksungen iiber
Jjynamik, p. 212. The theory of geodesic lines on a spheroid of revolution, in
j-articular an oblate spheroid, was considered by Legendre.
CURVES TRACED ON SURFACES. 363
dx
dx
and substitute for -p, &c. the values just given, we have again
the equation
Tf we actually perform the dIfFerentiations, and reduce the result
by the differential equation of the surface Ldx + Mdy -f Ndz = 0,
and its consequence
dLdx + d}[du + dNdz = - [Lfx + Md'y + NcVz),
we get
{dLdx + dMdj f dNdz) [dRds - Rdh)
+ [dLd'x + dMd'y + dNd'z) Rds = 0 *
dLd'x + dJLri/ + dNd'z dB^_<rs _
dLdx ■\-dMdii-\- dNdz B ds ~ '
399. The preceding equation is true for a geodesic or for
a line of curvature on any surface, but when the surface is
only of the second degree, a tirst integral of the equation can
be found. In fact, we have
dLd'x + dMd'y 4- dNd'z = \d [dLdx + dMdy + dNdz).
This may be easily verified by using the general equation of
a quadric, or, more simply, by using the equation
•i 2 'i
X y 2
?+t^ + ? = ''
when i = |„ J/=f,, N=t. a.LJ4^, dM^%, dN^^;
by substituting which values the equation is at once estab-
lished.
* Dr. Gehring has remarked (see Hesse, Vorlesungen, p. 325) that this equation
multipUed by Rds, subject as before to the condition Ldx r Md + Ndz = 0, may be
resolved into the product of the two determinants
So that for quadrics the determinant of the Hues of
dx, dy, dz
L, M, N
curvature is the integrating factor of the geodesies. dL, dM, dN
dx, dy, dz
drx, d-y, d^z
L, M, N
Dr. Hesse shews that the integral so arrived at belongs exclusively to the latter.
364 CURVES AND DEVELOPABLES.
The equation of tlie last article then consists of terms, each
separately integrable. Integrating, we have
E' [dLdx + dMdy 4- dNdz) = Cds\
Now, from the preceding values,
dLdx dMdji dNdz __ 1 dx^ 1^ df ]_ dz^
ds ds ds ds ds ds d^ ds' b' ds'^ c^ ds'^ '
But the right-hand side of the equation denotes the reciprocal
of the square of a central radius whose direction-cosines are
dx dy dz
ds ^ ds ^ ds '
The geometric meaning therefore of the integral we have
found is pD = constant.*
400. The constant pD has the same value for all geodesies
which pass through an umbilic.
For at the umbilic the p is of course common to all, being
=-ac '. b) and, since the central section parallel to the tangent
plane at the umbilic is a circle, the diameter parallel to the
tangent line to the geodesic is constant, being always equal
to the mean axis b. Hence, for a geodesic passing through an
umbilic we have pD = ac.
Let now any point on a quadric be joined by geodesies to
two umbilics, since we have just proved that pD is the same
for both geodesies, and, since at the point of meeting the p is
the same for both, the D for that point must also have the
same value for both ; that is to say, the diameters are equal
* Dr. Hart proves the same theorem as follows: Consider any plane section of
an ellipsoid, let ro be the perpendicular from the centre of the section on the
tangent line, d the diameter of the section parallel to that tangent, i the angle
the plane of the section makes with the tangent plane at any point. Then along the
section vsd is constant, and it is evident that pD is in a fixed ratio to ^d&mi.
Hence along the section pD varies as sin i and will be a maximum where the
plane meets the surface perpendicularly. But a geodesic osculates a series of norma^
sections ; therefore, for such a line pD is constant, its differential always vanishing.
Cambridge and Dublin Mathematical Journal, vol iv. p. 84.
CURVES TRACED ON SURFACES. 365
wliicli are drawn parallel to the tangents to the geodesies at
their point of meeting. But two equal diameters of a conic
make equal angles with its axes ; and we know that the axes
of the central section of a quadric parallel to the tangent plane
at any point are parallel to the directions of the lines of cur-
vature at that point. Hence, the geodesies joining any point
on a quadric to two umbilics make equal angles with the lines
of curvature through that point.'^
It follows that the geodesies joining any point to the two
opposite umbilics, which lie on the same diameter, are con-
tinuations of each other, since the vertically opposite angles
are equal which these geodesies make with either line of
curvature through the point.
It follows also (see Art. 395) that the sum or difference is
constant of the geodesic distances of all the points on the same
line of curvature from two umbilics. The sum is constant when
the two umbilics chosen are interior with respect to the line
of curvature ; the difference, when for one of these umbilics
we substitute that diametrically opposite, so that one of the
umbilics is interior, the other exterior to the line of curvature.
If A^ A' be two opposite umbilics, and B another umbilic,
since the sum PA + PB is constant, and also the difference
PA' - PB^ it follows that PA + PA' is constant ; that is to
say, all the geodesies which connect two opposite umhilics are
of equal length. Jn fact, it is evident that two indefinitely near
geodesies connecting the same two points on any surface must
be equal to each other.
401. The constant pD has the same value for all geodesies
which touch the same line of curvature.
It was proved (Art, 166) that pD has a constant value all
along a line of curvature ; but at the points where either
geodesic touches the line of curvature both p and D have the
same value for the geodesic and the line of curvature.
Hence, then, a system of lines of curvature has properties
completely analogous to those of a system of confocal conies
* This theorem and its consequences developed in the following articles are due
to Mr. Michael Roberts, Liouville, vol. XI. p. 1.
366 CURVES AND DEVELOPABLES.
ill a plane; the umbllics answering to the foci. For example,
two geodesic tangents drawn to one from any 2^oint on another
make equal angles with the tangent at that point. Graves's
theorem for plane conies holds also for lines of curvature, viz.
that the excess of the sum of two tangents to a line of cur-
vature over the intercepted arc is constant, while the intersection
moves along another line of curvature of the same species (see
Conies^ Art. 399).
402. The equation pZ) = constant has been written in another
convenient form.* Let a', a" be the primary semi-axes of two
confocal surfaces through any point on the curve, and let i be
the angle which the tangent to the geodesic makes with one of
the principal tangents. Then, since a!' — a!'\ a'' — a!''' (Art. 164)
are the semi-axes of the central section parallel to the tangent
plane, any other semi-diameter of that section is given by the
equation
1 cosV sin'^'z
+ I^
1 M . 1 (a' - a") (d' - a"')
■while, again, —^ = jy^r-i^ (Art. 165).
The equation, therefore, ^j)Z) = constant is equivalent to
{a^ — «''■*) cos^i + {d^ — a"^) sin'^'i = constant,
or to a'^ cosV+ tt"^ sin'''/ = constant.
403. The locus of the intersection of two geodesic tangents to
a line of curvature^ which cut at right angles, is a sphero-conic.
This is proved as the corresponding theorem for plane conies.
If a', a"' belong to the point of intersection, we have
a
co&^i + a''^ siuV = constant, a^ sinV + a"^ cos'V = constant,
hence a'^ + a'^ = constant ;
and therefore (Art. 161) the distance of the point of intersection
from the centre of the quadric is constant. The locus of inter-
section is therefore the intersection of the given quadric with
a concentric sphere. The demonstration holds if the geodesies
* By Liouvillc, vol. JX. p. 401.
CURVES TRACED OX SURFACES. 367
arc tangents to different lines of curvature ; and, as a par-
ticular case, the locus of the foot of the geodesic perpendicular
from an umbilic on the tangent to a line of curvature is a
sphero-conic.
404. To find the locus of intersection of geodesic tangents
to a line of curvature which cut at a given angle (Besge,
Liouville^ vol. XIV. p. 247).
The tangents from any point whose a', a" arc given, to
a given line of curvature, are determined by the equation
a'^ cosV+ ft"^ sin^i = /3; and since they make equal angles with
either of the principal tangents through that point, i the angle
they make with one of these tangents Is half the angle they
make with each other. We have therefore
^'-^^'^^VR^-^)' '^''^ = a- + a- -2/3 '
(a'2 4 a'" - 2/3y tan'-'^ = 4/3 {a'' + a") - 4a V^ - 4/3^
This is reduced to ordinary coordinates by the equations
(Arts. 160, 161)
a -{ a =x +y +z + h +c —a ] a a = — ^
a'
whence it appears that the locus required is the Intersection
of the quadric with a surface of the fourth degree,*
405. It was proved (Art. 176) that two confocals can be
drawn to touch a given line ; that If the axes of the three
surfaces passing through any point on the line be «, a', a",
and the angles the line makes with the three normals at the
point be a, /3, 7, then the axis-major of the touched coufocal
is determined by the quadratic
cos'^a cos'''/3 cosN
L I 1 — — ' = 0
a - a, a — a a - a:
Let us suppose now that the given line is a tangent to the
* Mr. Michael Roberts lias proved {LioiiviUe, vol. XT. p. 291) by the method
of Art. 188, that the projection of this curve on the plane of circular sections is
the locus of the intersection of tangents, cutting at a constant angle, to the conic
into which the line of curvature is projected.
368 CURVES AND DEVELOPABLES.
quadrlc whose axis is a, we have then cos a = 0, since the line
is of course at right angles to the normal to the first surface ;
and we have cos/3 = sin 7, since the tangent plane to the sur-
face a contains both the line and the other two normals. The
angle 7 is what we have called i in the articles immediately
preceding. The axis then of the second confocal touched by
the given line is determined by the equation
sm 4 cos 1 f,, y . . //■> • •> • '1
—Fi ^ + -773 5 = 0, or a cos i + a sm t = a .
a —a a —a '
If, then, we write the equation of a geodesic (Art. 402)
a'* cosV + a"^ sin'''* = a^, we see from this article that that equa-
tion expresses that all the tangent lines along iJie same geodesic
touch the confocal surface whose primary axis is a*
The geodesic itself will touch the line of curvature in which
this confocal intersects the original surface ; for the tangent
to the geodesic at the point where the geodesic meets the
confocal is, as we have just proved, also the tangent to the
confocal at that point. The geodesic, therefore, and the inter-
section of the confocal with the given surface have a common
tangent.
The osculating planes of the geodesic are obviously tangent
planes to the same confocal, since they are the planes of two
consecutive tangent lines to that confocal.
The value of pD for a geodesic passing through an
umbilic is ac (Art. 400) ; and the corresponding equation
is, therefore, a^ co^^i+a"'^ ^m^i=a^ —h^. Now the confocal,
whose primary axis is \/(«'' — ^')) reduces to the umbilicar focal
conic. Hence, as a particular case of the theorems just proved,
all tangent lines to a geodesic which passes through an umhilic
intersect the umhilicar focal conic.
Conversely, if from any point 0 on that focal conic recti-
linear tangents be drawn to a quadric, and those tangents
produced geodetlcally on the surface, the lines so produced
will pass through the opposite umbilic ; the whole lengths
from 0 to the umbilic being equal.
* The theorems of this article are taken from M. Chasles's Memoir, Liouville,
vol. XI. p. 5.
I
CURVES TRACED ON SURFACES. 369
406. From the fact (proved Art. 176) that tangent planes
drawn tlirough any line to the two confocals which touch it
are at right angles to each other, we might have inferred
directly, precisely as at Art. 309, that tangent lines to a
geodesic touch a confocal. For the plane of two consecutive
tangents to a geodesic being normal to the surface is tangent
to the confocal touched by the first tangent. The second
tangent to the geodesic, therefore, touches the same confocal;
as, in like manner, do all the succeeding tangents. Having
thus established the theorem of the last article, we could, by
reversing the steps of the proof, obtain an independent de-
monstration of the theorem pD = constant.
407. The developable circumscribed to a quadric along a
geodesic has its cuspidal edge on another quadric, which is the
same for all the geodesies touching the same line of curvature.
For any point on the cuspidal edge is the intersection of
three consecutive tangent planes to the given quadric, and
the three points of contact, by hypothesis, determine an oscu-
lating plane of a geodesic which (Art. 405) touches a fixed
confocal. The point on the cuspidal edge is the pole of this
plane with respect to the given quadric ; but the pole with
respect to one quadric of a tangent plane to another lies on
a third fixed quadric.
408. M. Chasles has given the following generalization of
Mr. Roberts' theorem. Art. 400. If a thread fastened at two
fixed points on one quadric A be strained hy a pencil moving
along a confocal B (so that the thread of course lies in geo-
desies where it is in contact with the quadrics and in right
lines in the space between them), then the jJ&ncil ivill trace
a line of curvature on the quadric B. For the two geodesies
on the surface 5, which meet in the locus point P, evidently
make equal angles with the locus of P; but these geodesies
have, as tangents, the rectilinear parts of the thread which
both touch the same confocal; therefore (Art. 405) the pD is
the same for both geodesies, and hence the line bisecting the
angle between them is a line of curvature.
BBB
,370 CURVES AND DEVELOPABLES.
A particular case of this theorem is, that the focal ellipse
of a quadric can be described by means of a thread fastened
to two fixed points on opposite branches of the focal hyperbola.
409. Elliptic Coordinates. The method used (Arts. 403-4)
in which the position of a point on the ellipsoid is defined by
the primary axes of the two hyperboloids intersecting in that
point, is called the method of Elliptic Coordinates (see Art. 188).
As it is more convenient to work with unaccented letters,
I follow M. Liouville* in denoting the quantities which we
have hitherto called a\ a" by the letters ;t, v ; and in this
notation the equations of the lines of curvature of one system
are of the form yu. = constant, and those of the other v = constant.
The equation of a geodesic (Art. 402) becomes
yu,^ C03V+ v^ sm^i= f/'^ ;
and when the geodesic passes through an umbilic, we have
fx'^ = a^- b^=--}i\ It will be remembered (Arts. 159, 160) that
fi hes between the limits k and A, and v between the limits
h and 0.
Throwing the equation of a geodesic Into the form
/J," + v' tanV= /a' '(1 + tan^'),
we see that it is satisfied (whatever be /u-') by the values
fx^zzzv^ tanV= — 1. Hence it follows, that the same pair of
imaginary tangents, drawn from an umbilic, touch all the lines
of curvature,t a further analogy to the foci of plane conies.
* This method is evidently a iDarticular case of that explained Art. 377, In
Prof. Cayley's Memoir on Geodesies {Proceedings of London Mathematical Society,
1872, p. 199) he uses the coordinates in a slightly different form ; viz. if any point
x^ y^ z'-
on the quadric — l-^H — :=lis the intersection with it of the two confocals
^ a b c
+ ^+_^ = l, ^^+^+_^=i;
a+p b ■'rp c +p ' a + q b + q c + q
then ;) and q are the two coordinates : p = const., q = const, denote lines of curvature ;
and we have, by Art. IGO, expressions for x, y, z in terms of p and q. The diffe-
rential equation of the right lines of the surface i.s
dp , ^? 0
J((a +p) [b +p){c+ p)] ^ J((a + q)(fi + q) {c + q)] ' "'
In the ordinarj^ case where the surface is an ellipsoid and a> b> c, the coordinates p
and q may be distinguished by supposing p to range between the limits — a, —i>,
and q between ~ b, — c.
f Mr. Roberts, IJouvilU, yo\. xv. p. 289.
.4
1
CURVES TRACED ON SURFACES. 371
410. To express in elliptic coordinates the element of the
arc of any curve on the surface. Let us consider, first, the
element of any line of curvature, fi = constant. Let that line be
met by the two consecutive hyperbololds, whose axes are v and
v + dv] then, since it cuts them perpendicularly, the intercept
between them is equal to the difference between the central per-
pendiculars on parallel tangent planes to the two hyperbololds.
But (Art. 180) (/' + dp")'' - p"' = [v + dvf - v' or p"dp" = vdv.
Now we have proved that dp" = da^ the element of the arc
we are seeking, and
„, ^ a:"h"'c"'- ^ y-[K^-v'){Jc'-v')
^ (a' - a") [a' - a"') ~ {d' - v') {/m' - v') '
Hence ^^ ^il^iW^)^'-
In like manner, the element of the arc of the line of curva-
ture V = constant is given by the formula
(/x -h){k -fi)
Now, if through the extremities of the element of the arc ds
of any curve we draw lines of curvature of both systems, we
form an elementary rectangle of which d<r, dcr' are the sides
and ds the diagonal. Hence
{fjk' - h') [k' - fjt,') '^ [h' - v') {k' - v')
411. In like manner we can express the area of any portion
of the surface bounded by four lines of curvature ; two lines
/i-j, /Aj, and two Vj, v^. For the element of the area is
the Integral of which is
Ml yu,^ ^J{d' — fj?) djM fi \/{a^ - v') dv
I
V {d' - fi^) dfi pi v' s/{d' - v") dv
CM-i ^J {a' - fji" ) dfjb /""i
*
* The area of the surface of the ellipsoid was thus first expressed by Legendra
Traite des Fonctions Elliptiques, vol. I. p. 352.
372 CURVES AND DEVELOPABLES.
So, in like manner, we can find the differenttal equation of the
orthogonal trajectory of a curve whose differential equation is
Mdii + Ndv = 0. For the orthogonal trajectory to Fda + Qda' is
plainly -p = —^ ; since da-, da are a system of rectangular
coordinates. But Md/j, + Ndv can be thrown without difficulty
into the form Pda + Qda by the equations of the last article.
The equation of the orthogonal trajectory is thus found to be
a'"* — fj^^ d/jb a^ — v'^ dv
[fi' - ¥) {k' - ^') M (K' - v') [k' - v') N
= 0.
n
412. The first integral of a geodesic fx^ co^H + v^ &\v^i = ^i'^
can be thrown into a form in which the variables are separated,
and the second integral can he ohtained. That equation gives
tant =
''-vV'
But tan^-^' - ^/ [[<^' ' ^^') i^^' - v') [^' - v')) ^
c^cr ~ ^J[[a'- v') ifM'-h:') (yt'^-Ac'^)} dy '
whence, equating, we have
^f jd' - fi^) dfi V (g' - v') dv
V (/.'^ - /^") {m:' - h') [k" - fi^ - V{(/^" - y') [1i' - v') {k' - v')} ~ '
the terms of which can be integrated separately.*
If the geodesic passes through the umbilics, we have /jf^ — h'^
(Art. 409), and the equation of the geodesic is
413. To find an expression for the length of any portion of
a geodesic. The element of the geodesic is the hypotenuse of
a right-angled triangle, of which da, da are the sides, and whose
* This is equivalent to Jacobi's first integral of the differential equation of the
geodesic lines, see Art. 397 ; see also Hesse, Vorlesungen, p. 328. The reader is
recommended also to refer to the method of integration employed by Weierstrass,
Monatsberichte der Berliner Akademie, 1861, p. 986, The above equation in the
notation used by Prof. Cayley is
^^ \[{a+p) {b+p)(c+p){Q + p)\ - ^"^ Jl(a + 9) (6 + q) (c + 9) (6 + q)\ ^ ^'
where 6 is the constant of integration. This is nearly the form given by Jacobi
in the Vorlesungen ilber Dijnamik; referred to in note to Art. 398.
CUUVES TRACED ON SURFACES. 373
base angle is t. Hence we have ds = sin ida ± cos ida- ] and
putting in sint=-A_____, cos i = ^ ^^, _ ^.,^ , and giving
da- J da the values of Art. 410, we have
If p be the element of a line through the umbilics, we have
*='''' \/(^") ± * \/(f^') •
It is to be noted, that when we give to the radical in the last
article the sign +, we must give that in this article the sign -.
This appears by forming (Art. 411) the differential equation of
the orthogonal trajectory to a geodesic through an umbilic, an
equation which must be equivalent to dp = 0 (Art. 394).
414. In place of denoting the position of any point on an
ellipsoid by the elliptic coordinates /*, v, we might use geodesic
2)olar coordinates having the pole at an umhilic^ and denote a
point by p its geodesic distance from an umbilic, and by (o the
angle which that radius vector makes with the line joining the
umbilics. Now the equation (Art. 413) of a geodesic passing
through an umbilic gives the sura of two integrals equal to a
constant. This constant cannot be a function of p, since it
remains the same as we go along the same geodesic : it must
therefore be a function of <w only ; and if we pass from any
point to an indefinitely near one, not on the same geodesic
radius vector, we shall have
^{a^-fi']dfj, _ ^/{d'-v')dv ,^, SI
We shall determine the form of the function by calculating its
value for a point indefinitely near the umbilic, for which fi = v = h.
The limit of the left-hand side of the equation then becomes
\/^W^k) ^ ^^"^'^^ "^ {-i^^ + 1^) ' ^"^' '^ ^" P'^^
/M = h-\- 7), y = h-Zj the quantity whose limit we want to find
374 CURVES AND DEVEL0PABLE9.
is — ; ^ — ~i ^, which, as ri and s tend to vanish, becomes
the limit of -^ ( — | or of —y d loa* - .
Now since the angle external to the vertical angle of the
triangle formed by the lines joining any point to two umbilics
Is bisected by the direction of the line of curvature, that external
angle Is double the angle ^ In the formula fi^ cos^i+ v'^ Qn\H = K\
In the limit when the vertex of the triangle approaches the
umbilic, the external angle of the triangle becomes w, and
we have at the umbilic
[h -f iiY cos'"*! &j -f {Ji-zf &\vl^\o3 = h'^
and In the limit tan'''i&) = -.
^ £
Using this value, the limit of the left-hand side of the equation Is
We have therefore
V(a'-At')^At ^J[cl:'-v'')dv _1 /(a'-li\ d(o
{fi"" - K') V(^-' - H^') {h' - v') V(^-' - y") h V ^^' - ^* / sin o) '
And the constant which occurs In the Integrated equation of
a geodesic through an umbilic is of the form
kJ^w^) '°s ■"■■ -^
tan'iw + C.
415. If P, Q be two consecutive points on a curve, and If
PP' be drawn perpendicular to the geodesic radius vector OQ^
it Is evident that PQ' = PP''' ^ P' Q\ Now since (Art. 394)
0P= 0P\ we have P'Q = dp^ while PP' being the element
of an arc of a geodesic circle, for which p Is constant (or
dp — 0), must be of the form Pdw. Hence the element of the
arc of a curve on any surface can be expressed by a formula
d^ — dp-^P'^dw'^. We propose now to examine the form of
the function P for the case of radii vectores drawn through
an umbilic of an ellipsoid. Let us consider the line of cur-
vature /i = p! . We have then (Art. 413)
di' = dv''
{J^^^y^)(j:^-y-^)
CURVES TRACED ON .SURFACES. 375
And
by
the same article
dp^ =
v'
v' '
whence
- ¥) {a
-v1
dv\
But
(Ar
t. 414), when /* is constant,
^J{a:' - v') dv
1
/fa^-
' ^'\
ih' - v') Jik' - v')
~A V
\k'-
-k)
(0)
sin 0)
Putting in this value for dv, we have
^ {a' - ¥) [h' - v") (/g^^ - h') ^ Wy"" f
h^ ik' - h') sin' CO ~ (b'- a') [b' - c') sin^ co ~ sm' w
(Art. 160); therefore P=?/ cosec co.
In this investigation it is not necessary to assume the result
of the last article. If we substitute for the right-hand side of
the equation in the last article an undetermined function of <w,
it is proved in like manner that P=y<^ {w). We determine
then the form of the function by remembering that in the neigh-
bourhood of the umbilic the surface approaches to the form
of a sphere. Now on a sphere the formula of rectification
is ds^ = dp''+ sin^pdco^. Hence P=sinp. But in the sphere
2/ = sinp slnco. The function therefore which multiplies ?/ is
cosecey.
416. Consider now the triangle formed by joining any
point P to the two umbilics 0, 0'. Then for the arc OP we
have the function P = ?/cosec(u, and for the arc O'P, connecting
P with the other umbilic, we have the function P' = ?/ cosecw';
and P '. P' •.'. sinw' : siuco, an equation analogous to that which
expresses that the sines of the sides of a spherical triangle
are proportional to the sines of the opposite angles, since P
and P' in the rectification of arcs on the ellipsoid answer to
sinp, sin/?' on the sphere.
417. Again, if P be any point on a line of curvature we
know (Art. 400) dp ± dp' = 0, where p and p' are the distances
from the two umbilics. Now if 6 be the angle which the
radius vector OP makes with the tangent, the perpendicular
376
CURVES AND DEVELOVABLES.
element Pdoy is evidently dpt^nd. But the radius vector O'P
makes also the angle 6 with the tangent. Hence, we have
Pdo) ± P do3 = 0, or -. — + -. , = 0,
smo) smtw '
whence tan ^co tan |&)' is constant when the sum of sides of the
triangle is given; and tan^o) is to tan | to' in a given ratio
when the difference of sides of the triangle is given. Thus,
then, the distance between two umbilics being taken as the
base of a triangle, when either the product or the ratio of
the tangents of the halves of the base angles is given, the
locus of vertex is a line of curvature.*
From this theorem follow many corollaries : for instance, " if
a geodesic through an umbilic 0 meet a line of curvature in
points P, P' then (according to the species of the line of curva-
ture) either the product or the ratio of tan^PO'O, tan^P'O'O is
constant." Again, " if the geodesies joining to the umbilics
any point P on a line of curvature meet the curve again in P',
P'', the locus of the intersection of the transverse geodesies
O'P', OP'' will be a line of curvature of the same species."
418. Mr. Eoberts's expression for the element of an arc
perpendicular to an
umbilical geodesic has ^^^
been extended as fol-
lows by Dr. Hart:
Let OT, OT' be two
consecutive geodesies
touching the line of
curvature formed by
the intersection of the
surface with a confocal
P, day the angle at
which they intersect ;
then the tangent at
any point T of either
* This theorem, as well as those ou which its proof depends (Art. 414, &c.), is
due to Mr. M. Roberts, to whom this department of Geometry owes so much
{^Liouville, vols. xiii. p, 1, and XT. p. 275).
CURVES TRACED ON SURFACES. 377
geodesic touches B in a, point F (Art. 405) ; and if TT' be taken
conjugate to TF, the tangent plane at J" passes through TF
(Art. 268), and the tangent line to the geodesic at T' touches
the confocal B in the same point P. We want now to express
in the form Fdco the perpendicular distance from J" to TF.
Let the tangents at consecutive points, one on each geodesic,
intersect in F' and make with each other an angle d(p\ Let
normals to the surface on which the geodesies are drawn
at the points T,, T/, meet the tangents FT, FT' at the
points 2^^, 2^', then since the difference between T^T/, T^T^'
is infinitely small of the third order, FT/Icp and F'T^d^' are
equal, to the same degree of approximation. But FT,^, F'T^
are proportional to D and D\ the diameters of the surface
B drawn parallel to the two successive tangents to the geo-
desic. Hence Dd(^ = D'd<p'. This quantity therefore remains
Invariable as we proceed along the geodesic ; but at the point
0, d(p = dw] if therefore D^ be the diameter of B parallel to
the tangent at 0 to the geodesic, Dd^ = D^flw ; and there-
fore the distance we want to express FTd(j> = -~ tdco, where
t(=FT) is the length of the tangent from T to the confocal B]
or -j~t Is a mean between the segments of a chord of B drawn
through T parallel to the tangent at 0. When the geodesic
passes through an umbilic, the surface B reduces to the plane
of the umbilics, and -~ t becomes the line drawn through T
to meet the plane of the umbilics parallel to the tangent at 0,
which is Mr. Roberts's expression.
Hence, If a geodesic polygon circumscribe a line of curva-
ture, and if all the angles hut one move on lines of curvature,
this also loill move on a line of curvature, and the perimeter
of the polygon will he constant when the lines of curvature
are of the same species. The proof Is Identical with that
given for the corresponding property of plane conies [Conies,
Art. 401).*
* See Cambridge and Dublin Mathematical Journal, vol. iv. p. 192.
ccc
378
CURVES AND DEVELOPABLES.
419. If a geodesic joining any umbilic to that diametrically
opposite, and making an angle 6) with the plane of the um-
bilics, be continued so as to return to the first umbilic, it will
not, as in the case of the sphere, then proceed on its former
path, but after its return will make with the plane of the um-
bilics an angle diifereut from w. In order to prove this we
shall investigate an expression for ^, the angle made with
the plane of the umbillcs by the osculating plane at any point
of that geodesic.
It is convenient to prefix the following lemma: In a
spherical triangle let one side and the ad-
jacent angle remain finite while the base
diminishes indefinitely, it is required to find
the limit of the ratio of the base to the
difference of the base angles measured in
the same direction. The formula of spherical
o+do
d<>
trigonometry cos \{^A-\-B^=^ya.\G
cos I
COStVC
gives us In the
Hence
limit dd = cos aclylr. But evidently sin a. dyjr = sin 6 d(l)
dd _ d6
sin 6 tana
Now we know (Art. 405) that the tangent line at any point
of a geodesic passing through an umbilic, if produced, goes to
meet the plane of the umbilics in a point on the focal hyper-
bola; and the osculating plane of the geodesic at that point
will be the plane joining the point to the corresponding tangent
of the focal hyperbola. We know also (Art. 184) that the
cone circumscribing an ellipsoid, and whose vertex is any point
on the focal hyperbola, Is a right cone.
Let now PP' be an element of an umbilical geodesic pro-
duced to meet the focal „
hyperbola in H. Let ^~
P'P" be the consecutive
element meeting the focal
hyperbola in //' ; then
if Wi, U'K be two con-
secutive tangents to the
focal hyperbola, Pllh^
CURVES TRACED ON SURFACES. 379
P'll'K will be two consecutive osculating planes. Imagine
now a sphere round //', and consider the spherical triangle
tornied by radii to the points 7^, //, P'. Then if d(^ be the
angle hHli\ the angle of contact of the focal hyperbola; if 6
be the angle between the osculating plane and liH'h' the plane
of the umbilics, while liH'P' is a the semi-angle of the cone ;
the spherical triangle becomes that considered in our lemma,
, , dd d(b
and we have - — yr = — ^— .
suit; tana
In order to Integrate this equation we must express J^ in
terms of a; and this we may regard as a problem in plane
geometry, for a is half the angle included between the tangents
from H to the principal section in the plane of the umbilics,
while dcj) is the angle of contact of the focal hyperbola at the
same point. Now if a, V ; a\ h" be the axes of an ellipse
and hyperbola passing through H^ confocal to an ellipse whose
axes are «, h ; and if 2a be the angle included between the
tangents from II to the latter ellipse, we have (see Conies.
p. 189) tan*a = —fr^^ ^ . Differentiating, regarding a" as
constant (since we proceed to a consecutive point along the
/ 7 /
same confocal hyperbola\ we have 6?a = — tana -r; 777,. But
a —a
if, jD, p' be the central perpendiculars on the tangents at H
to the ellipse and hyperbola, we have ada =jid(T (Art 410),
where da is the element of the arc of the focal hyperbola, and
if p be the radius of curvature at the same point, da = pd(f).
-r, «'■' — a'^ T-T- , pdd> -, , a'h'dd)
xJut p= ; — . Hence, aa= — tana — roraa = tana — 7777?.
p p a o
But a" =a' + [a' - a") cofa, h"' =h'+ {d' - a"') cof^a.
^ c?(f) a'h"dfx
'^'^ X^o. " V(«''' - a"' + «' tan'^a) V(«' - «"' + ^' tan^a) *
In the case under consideration the axes of the touched
ellipse are a, c ; while the squares of the axes of the confocal
hyperbola are d^ - h\ b^ — c\ Hence we have the equation
d0 ^(d'-h')^/{h''-c')da
smd ~ V(6' + d' tan'a) \/[b' + c' taa'a) *
380 CUKVES AND DEVELOPABLES.
Integrating this, and taking one limit of the Integral at
the umbilic where we have 6 — (o^ and a = ^tt, we have
tan Id r V(«' - h'] ^/iF - c') da
^ tan^eo J ^^ \J[b' + a' tan'^a) ^/{h' + 6' tan'a)
If, then, I be the value of this integral, we have
tan|^^ = k tan I ft), where k = e^.
Now this integral obviously does not change sign between
the limits ±\rr^ that is to say, in passing from one umbilic
to the other. If, then, «' be the value of 6 for the umbilic
opposite to that from which we set out, at this limit / has
a value different from zero, and k a value different from unity ;
and we have tan|a)' = A' tan^w ; &>' is therefore always different
from o). And in like manner the geodesic returns to the original
umbilic, making an angle (o" such that tan|a)" = A;'' tan|ft), and
so it will pass and repass for ever, making a series of angles
the tangents of whose halves are in continued proportion.*
420. If we consider edges belonging to the same tangent
cone, whose vertex is any point H on the focal hyperbola, a
(and therefore /.•) is constant; and the equation tan|^ = Z; tan ^o)
erives —. — >, = -; — • . Now since the osculating plane of the
^ sm^ smo) ^ ^
geodesic is normal to the surface, and therefore also normal
to the tangent cone, it passes through the axis of that cone.
If, then, we cut the cone by a plane perpendicular to the axis,
the section is evidently a circle whose radius is -^^ , and the
^cld idea ^^^
element of the arc is -^r—>. , or -^ — . Now this element, being
sin C7 sm tu > a
the distance at their point of contact of two consecutive sides
of the circumscribing cone, is what we have called (Art. 415)
PcZo), and we have thus, from the investigation of the last
article, an independent proof of the value found for F (Art. 415).
421. Lines of level. The inequalities of level of a country
can be represented on a map by a series of curves marking
* The theorems of this article are Dr. Hart's, Cambridge and Dublin Mathematical
Journal, vol. iv. p. 82; but in the mode of proof I have followed Mr. William
Roberts, Liouville, 1857, p. 213.
i
CURVES TRACED ON SURFACES. 381
the points which are on the same level. If a series of such
curves be drawn, corresponding to equi- different heights, the
places where the curves lie closest together evidently indicate
the places where the level of the country changes most rapidly ;
the curve through the summit of a pass, or at the point
of out-flow of a lake, has this point for a node, &c., &c.*
Generally, the curves of level of any surface are the sections
of that surface by a series of horizontal planes, which we may
suppose all parallel to the plane of xy. The equations of the
horizontal projections of such a series are got by putting 0 = 0
in the equation of the surface ; and a differential equation common
to all these projections is got by putting dz = 0 in the differential
equation of the surface, when we have
U^dx + U^dy = 0.
We can make this a function of x and y only, by eliminating
the 2!, which may enter into the differential coefficients, by the
help of the equation of the surface.
Lines of greatest slope. The line of greatest slope through any
point is the line which cuts all the lines of level perpendicularly ;
and the differential equation of its projection therefore is
U^dy - U,^dx = 0.
The line of greatest slope is often defined as such that the
tangent at every point of it makes the greatest angle with
the horizon. Now it Is evident that the line in any tangent
plane which makes the greatest angle with the horizon is
that which is perpendicular to the horizontal trace of that
plane. And we get the same equation as before by expressing
that the projection of the element of the curve (whose direction-
cosines are proportional to dx^ dy) is perpendicular to the trace
whose equation is U^ {x — x) + U^iy — y) — U^z = O.f
* See Keecli, sur les surfaces fermees, Jour, de VEc. Polyt. t. xxi. (1858), p. 169.
Cayley on Contour and Slope Lines, Plill. Mag., vol. xviii., 1859, p. 264.
t It is evident that the differential equation of the curve, -which is always per-
pendicular to the intersection of the tangent plane, [whose direction-cosines are as
L, M, iV] by a fixed plane whose direction-cosines arc «, b, c, is
dx, dy, dz
L, M, N
a, b , c
382 CURVES AND DEVELOPABLES.
Ex. 1. To find the line of greatest slope on the quadric Ax^ + By^ + Cz^ — B.
The differential equation is Ax dy — By dx, which, integrated, gives f — J = ( -, ) ,
where the constant has been determined by the condition that the Une shall pass
through the point x = x', y — y' . The line of greatest slope is the intersection of
the quadric by the cylinder whose equation has just been written, and will be a curve
of double curvature, except when x'y' lies in one of the principal planes when the
equation just found reduces to a; = 0 or y = 0.
Ex. 2, The coordinates of any point on the hyperboloid of one sheet may be
written - = ■ , ^- — , - = -r^ '^ ; show that if n = ; — , the lines
of curvature are determined by the equations (cf. note p. 370)
dX dfx _
4{l - 2pX^ + X*) - J(l - 2pn'^ + n*)
Ex. 3. Express in the same system of coordinates the differential equation of
geodesies on the surface.
( 383 )
CHAPTER XIII.
FAMILIES OF SURFACES,
SEOTION I. PARTIAL DIFFERENTIAL EQUATIONS.
422. Let the equations of a curve
^ K y, ^j c„ c,...cj = 0, i/r {x, 3/, 2, c„ c,^,..cj = 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
71—1 relations between the parameters be given, the equa-
tions 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 91 -f 1 equations ; viz. the w — 1 relations, and the two
equations of the curve. Thus, for example, if the two equa-
tions above written denote a variable curve, the motion of
which is regulated by the conditions that it shall intersect n— I
fixed directing curves, the problem is of the kind now under
consideration. For, by eliminating x, ?/, 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 w 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 (p and \/r
is the same are said to be of the same family^ though the
equations connecting the parameters may be diflferent. Thus,
if the motion of the same variable curve were regulated by
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
384
FAMILIES OF SURFACES,
to the same family can be included In one equation involving
one or more arbitrary functions, the equation of any individual
surface of the family being then got by particularizing the form
of the functions. If we eliminate the arbitrary functions by
differentiation, we get a partial differential equation, 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 equa-
tion 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„ v = c^ ; where u and v are known functions of
a;, ^, z. In order that this curve may generate a surface, we
must be given one relation connecting Cj, c^, which will be of
the form c^ = <f) (cj ; whence putting for c^ and c.^ their values,
we see that, whatever be the equation of connection, the equa-
tion of the surface generated must be of the form u = (}) [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 Z7=0 represents any such surface, Z7 can only
differ by a constant multiplier from u- ^ [v). Hence, we have
\U=^ u — ^ [v), and differentiating
with two similar equations for the differentials with respect to
y and z. Eliminating then \ and 4>' (u), we get the required
partial differential equation in the form of a determinant
^n ^., ^s
u
1?
'ii
u.
'11
V.
21
u„
v„
0.
In this case ?* and v are supposed to be known functions of the
coordinates; and the equation just written establishes a relation
of the first degree between C/^, U^^ f/.
If the equation of tlie surface were written in the form
* If there were but one constant, the elimination of it would give the equation of
a definite surface, not of a family of surfaces,
PARTIAL DIFFERENTIAL EQUATIONS. 385
z — (f) {x, y] = 0] we should have U^=i, U^ = —p, U^ = — q,
where ^7 and q have the usual slgnitication, 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 (see Boole's Differential Equations^ p. 323) is of the form
u = (f> (u), geometrically represents a surface which can he gene-
rated by the motion of a curve whose equations are of the
form w = c,, v = c^.
The partial differential equation affords the readiest test
whether a given surface belongs to any assigned family. We
have only to give to f/, Z/^, U^^ 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 = (j> (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 m = c,, v = c^^
and eliminating x, y, z between these equations and those of
the fixed curve, we thus find a relation between c, 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 w = c,, 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) (y)
which shall envelope a given surface, we know that at every
point of the curve of contact f^, C^, U^ 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 Z7j, C^, U^ their values derived from the equa-
tion 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, deter-
mines that curve. The problem is, therefore, reduced to that
DDD
386 FAMILIES OF SURFACES.
considered in the first part of this article ; namely, to describe
a surface of the given family tl)rough 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. CyUndrical 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 on 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— h — (f) {y — viz).
More generally, if the right line is to be parallel to the
intersection of the two planes ax + by + cz, ax -f h'y -f- cz^ its P
equations must be of the form '
ax + by + cz = a, ax + Vy 4 cz = /3,
and the equation of the surface generated must be of the form
ax + by -\- cz ^ (j) [ax + b'y + cz).
Writing ax-\-by-\- cz for m, and ax + b'y + cz for v in the
equation of Art. 423, we see that the partial differential equa-
tion of cylindrical surfaces is
{be' - cb') U^ + [ca' - ac) l\ + [aV - ba') U^ = 0,
or (Ex. 3, p. 26) U^ cosa+ C^^ cos/3 -f f/ cos7 = 0, where a, /3, 7
are the direction-angles of the generating line. Eemembering
that f/, t(^, f/ are proportional to the direction-cosines of tlie
normal to the surface, it is obvious that the geometrical meaning
PARTIAL DIFFERENTIAL EQUATIONS. 387
of this equation is, that the tangent plane to the surface is
always parallel to the direction of the generating line.
Ex. L To find the equation of the cylinder whose edges are parallel to a; = h,
y = mz, and which passes through the plane curve z — 0, <p {x, y) = 0.
Ans. <p {x — Iz, y — mz) — 0.
Ex. 2. To find the equation of the cylinder whose sides are parallel to tlie
intersection of ax + by + cz, a'x + b'y + c'z, and which passes through the intersec-
tion of ax + (3y + yz = S, F {x, y, z) = 0. Solve for x, y, z between the equations
ax + by ->r cz = u, a'x + b'y + c'z = v, ax + (iy + yz = c, and substitute the resulting
values in F {x, y, z) = 0.
Ex. 3. To find the equation of a cylinder, the direction-cosines of whose edges
are I, m, n, and which passes through the curve U — 0, F = 0. The elimination
may be conveniently performed as follows : If x', y', z' be the coordinates of the
point where any edge meets the directing curve, x, y, z those of any point on
the edge, we have , =: ~ — =- =: . Calling the common value of these
I m 11
functions 0, we have
x' = x— W, y' —y — md, z' — z ~ nQ.
Substitute these values in the equations U—0, V ~ 0, which x'y'z' must satisfy,
and between the two resulting equations eUminate the unknown 6, 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" -I- By" -I- 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
{AP + Bm' + Cn"-) {Ax'' + By"- + Cz'' - I) - {Alx + Bmy + Cnz)"-.
426. Conical Surfaces. 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 equa-
tions of the right line, we have two relations 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—^ _ z—y
I m n ^
where a, /8, 7 are the known coordinates of the vertex of the
cone, and ?, ?», n are proportional to the direction-cosines of the
generating line ; and where the equations, though apparently
388 FAMILIES OF SURFACES.
containing three undetermined constants, actually contain only-
two, since we are only concerned with the ratios of the quan-
tities /, 7n, n.
Writing the equations then in the form
x — a. I y — ^ tn
z — fy n^ z — ly 71 ^
cone must be of the form = 6 ( | .
Z- ry ^ \z- ry)
we see that the conditions of the problem must establish a
relation between I : n and m : w, and that the equation of the
'y - ^'
7 ' \z- <yj
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 a: - a, ?/ — /3, 2; - 7 ; as may also be seen directly
from the consideration that the conditions of the problem must
establish a relation between the direction-cosines of the gene-
rator; that these cosines being* Z : /\/{(?^ + w^ + w*^)}? &c., any
equation expressing such a relation is a homogeneous function
of ?, w, w, and therefore of a; — a, ?/ — /3, 2 — 7, which are pro-
portional to Z, wi, n.
"When the vertex of the cone is the origin, its equation is
of the form - = 0 [-] ; or, in other words, is a homogeneous
function of ic, ?/, z.
The partial differential equation is found by putting
, V = ^ — ^ , in the equation of Art. 423, and when
u = —
z — y ' z — <y
cleared of fractions is
^ - 7, 0, - (a? - a)
0, ^-7, -iy-^) =0,
or {x-a) U^ + {y-l3) U^+{z-y) U^ = 0.
This equation evidently expresses that the tangent plane at
any point of the surface must always pass through the fixed
point a/37.
We have already given in Ex. 7, p. 101, the method of
forming the equation of the cone standing on a given curve j
= 0,
PARTIAL DIFFERENTIAL EQUATIONS. 389
and (Art. 277) the method of forming tlie 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 remains
parallel to a fixed plane. These two conditions leave two of
the constants in the equations of the line undetermined, so that
these surfaces are of the class considered (Art. 423). If the axis
is the intersection of the planes a, /3, and the generator is to
be parallel to the plane 7, the equations of the generator are
a = Cj/3, 7 = c^, and the general equation of conoidal surfaces
is obviously •;5 = ^ (7)-*
The partial diflferential equation is (Art. 423)
/3a, -a/3„ /3a^-a^„ ^a^-al3.
7i, %i 73
where a = a^x 4- a.^y + cn^z + a^, &c. The left-hand side of the
equation may be expressed as the difference of two deter-
minants /3 ( C/;a^7j - a ( Ufy^) = 0.
This equation may be derived directly by expressing that
the tangent plane at any point on the surface contains the gene-
rator; the tangent plane, therefore, the plane drawn through
the point on the surface, parallel to the directing plane, and
the plane a'/3 — a/3' 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^ ?/, z in the equations of
these three planes.
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 = x4> (2),
and the partial differential equation is xU^+yU^ = 0.
The lines of greatest slope (Art. 421) are in this case always
* In like manner the equation of any surface generated by the motion of a
line meeting two fixed lines a/9, 70 must be of the form .t= <^ (J) •
390 FAMILIES OF SURFACES.
projected into circles. For iu virtue of the partial differential
equation just written, the equation of Art. 421,
U/lx - U^dy = 0,
transforms itself into xdx + ydy = 0, which represents a series
of concentric circles. The same thing is evident geometrically ;
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 concentric
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, z) = 0. EUminating
then X, y. z between these equations and y = CyX, z — c„, we get F {c^a, Co) = 0 ;
or the required equation is F ( — , e J = 0.
WalHs's cono-cuneus is when the fixed curve is a cu'cle \_x = a, y'^ + z^ — r'^].
Its equation is therefore a-y~ + x-z" = r"x-.
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.
428. Surfaces of B evolution. 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
g; a 1/ ^ 8 Z 'Y
— J — = = , then the generating circle In any posi-
6 /to ft
tion may be represented as the intersection of the plane per-
pendicular to the axis Ix + my + nz =Cp 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 - of +{y- ^Y +{z-ryy = cf) [Ix + vxy + nz).
PARTIAL DIFFERENTIAL EQUATIONS.
391
When the axis of z is the axis of revolution, we may take the
origin as the point a/37, and the equation becomes
x^ -^f ^- z^ = ^{z\ or 2; = -v/f (0;"' + /).
The partial differential equation is found by the formula of
Art. 423 to be
I, ??2, n
x-a, y-^, z-y =0,
or {m{z-ry)-7i{y-^)] U^
+ {7i{x-a)-l[z-ry)} U^+[l{y-/3)-m{x-a)} U^ = 0.
When the axis of z is the axis of revolution, this reduces to
yU^-xU,^ = o:
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 3/ — /3 z — y X— x' y — y z —z'
should intersect, we may write the common value of the equal
fractions in each case, Q and B' . Solving then for x^ ?/, ^, and
equating the values derived from the equations of each line,
we have
a.-^ie = x'\V^e\ ^ + m9=y'+U/, y -^ nd = z' -\- U/ ;
whence, eliminating ^, 6\ the result is the determinant already
found
I.
u..
'i1
VI.
n
^' - «) y -^1 ^' - 7
0.
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 - af + [y - jS)" + {z — yf = 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,
p. 99), and we take, as a further example, " to find the surface
392 FAMILIES OF SUEFACES.
generated by the revolution of a circle [7/ = 0, [x- a)'^ + z^ = r^]
round an axis in its plane [the axis of 2;]."
Putting z = If, oc' + y^ = Vj and eliminating between these
equations and those of the circle, we get
y{v)-aY + u' = r\ or y [x' -\- y') - a}' -\- z' = r\
which, cleared of radicals, is
{x' + y' + z' + a' - rj = Aa' {x' + f).
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
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, inter-
secting in points on the axis z = \/{r^ — a^)^ 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' represent circles.
The sections are Cassinians of various kinds (see tig. 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 meets in a continuous curve.
Ex. Verify that x^ + y^ + z^ — 3xyz = r' is a surface of revolution.
Ans. The axis of revolution ia 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 case
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,(f>{c), f (c),&c.|=0, /[a:,y,2,c,</)(c), f (c),&c.}=0.
PARTIAL DIFFERENTIAL EQUATIONS. 393
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 i^ 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 ^, -i/r, &c., it Is not generally
possible to form a single functional equation including all sur-
faces of the same family ; and we can only represent them,
as above written, by a pair of equations from which there
remains a constant to be eliminated. We can, however, elimi-
nate 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 ^, •>/r, &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 equation in-
clude two arbitrary functions <^, i/r, and if we differentiate with
respect to x and ?/, which we take as independent variables,
the differential equations combined with the original one form
system of three equations containing four unknown functions a
0, i/r, ^', -y . The second differentiation (twice with regard
to ic, 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 0, i/r, ^', '>^' ^ 0", -y . We must, therefore, pro-
ceed to a third differentiation before the elimination can be
effected. It is easy to see, in like manner, that to eliminate
n arbitrary functions we must differentiate 2« — 1 times. The
reason why, in the present case, the order of the differential
equation is less, is that the functions eliminated are all functions
of the same quantity.
431. In order to show this, it is convenient to consider first
the special case, where a family of surfaces can be expressed
by a single functional equation. This will happen when it is
E£E
394
FAMILIES OF SURFACES.
possible by combining the equations of the generating curve
to separate one of the constants so as to throw the equations
into the form m = c, ; i^(a:, -?/, 2;, c,, c^.-.c^J = 0. Then express-
ing, by means of the equations of condition, the other constants
in terms of c^, the result of elimination is plainly of the form
F[a', y, z, M, (j) (m), yjr (m), &c.} — 0.
Now, if we denote by F^, the differential with respect to x of
the equation of the surface, on the supposition that u is con-
stant, and similar differentials in j/, z by i'^, F^^ we have
U, = F,+
dF
du "^'
T7 TP dF
u„
F — w
^ du
3*
But, in these equations, the derived functions 0', i/r', &c., only
dW
enter in the term -7- ; they can, therefore, be all eliminated
together, and we can form the equation, homogeneous in
£^., u.. c^s,
F F F
u
1?
M,
2)
M„
= 0,
which contains only the original functions (^, i/r, &c. If we
•write this equation F= 0, we can form from it, in like manner,
the equation
V V V
^ n '^ 2' '^3
u.
u
21
u„
= 0,
•which still contains no arbitrary functions but the original
^, i/r, &c., but which contains the second differential coefficients
of f/, these entering into Fj, F^, Fg. 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 n^ order of differentiation) we can eliminate the n functions
^, t/t, &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 w — 1, each containing one arbitrary function.
PARTIAL DIFFERENTIAL EQUATIONS. 395
These are the first Integrals of the final difFerentlal equation
of the Ji"' order. In like manner we can form ^n{n — l) equa-
tions of the second order, each containing two arbitrary func-
tions, and so on.
432. If we take x and 7/ as the Independent variables, and
as usual write dz=pdx + qdi/, dp = rdx -\- sdy ^ &c., the process
of forming these equations may be more conveniently stated
as follows : " Take the total difFerentlal of the given equation
on the supposition that u is constant,
F^dx + F,^dy + F^ ( pdx + qdy) = 0 ;
put dy = mdx^ and substitute for m its value derived from the
differential of w = 0, viz.
u^dx + u^dy + W3 [pdx -\- qdy) =0."
For, if we differentiate the given equation with respect to
X and 2/, we get
dW
F, + q.F,-\--^[u^+qu^ = %
dF
and the result of eliminating -y- from these two equations is
the same as the result of eliminating m between the equations
F^ -VpF^ + m [F^ + qF^) = 0, w, +pu^ ^m[u^ + qu^) = 0.
It Is convenient in practice to choose for one of the equations
representing the generating curve its projection on the plane
of xy'j then, since this equation does not contain 2, the value
of m derived from It will not contain p or q, and the first
differential equation will be of the form
p + qm = JRj
R being also a function not containing j9 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^ = S,
where S may contain a, y^ Zj p^ q^ but not r, s, or t. If now
396 FAMILIES OF SURFACES.
we had only two functions to eliminate, we should solve for
these constants from the original functional equation of the
surface, and from ][> -f qm = B, ; and then substituting these values
in m and in ^S*, the /onn of the final second differential equation
would still remain
r + Ism + tm"^ = S\
where on' and S' might contain a*, ?/, z, p, q. In like manner
if we had three functions to eliminate, and if we denote the
partial differentials of z of the third order by a, /3, 7, 8, the
partial diflPerential equation would be of the form
a + 3m/3 + 3m'y + w''S = T.
And so on for higher orders. This theory will be Illustrated
by the examples which follow.
433. Surfaces generated hy 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 2 = c^, 3/ = cjx + c^. The functional equa-
tion of the surface is got by substituting in the latter equation
for Cj, ^ (2), and for Cg, i/r [z). Since in forming the partial
difierential equation we are to regard z as constant, we may
as well leave the equations in the form 2 = c^, y = c^x + c^.
These give us
p + qm = 0, m = c^.
According as we eliminate Cg or c,^, these equations give us
P + qc^ = 0, px + qy = qc^. There are, therefore, two equations
of the first order, each containing one arbitrary function, viz.
^ + # (^) = 0, px + qy = qf [z).
To eliminate arbitrary functions completely, differentiate
p + qm = 0, remembering that since m = c^, it is to be regarded
as constant, when we get
r 4 2sm + <w' = 0,
and eliminating m by means of p -^ qm =0, the required equa-
tion is
q\ - 2pqs + p^t = 0,
PARTIAL DIFFERENTIAL EQUATIONS. 897
Next let the generating line be parallel to ax + hi/ -{■ cz 'j its
equations are
and the functional equation of the family of surfaces Is got by
writing for c^ and C3, functions of ax->rhy -{■ cz. Differentiating,
we have
a -\- cp ■\- m [h + cq) = 0, m = c^.
The equations got by eliminating one arbitrary function are
therefore
a -\- cj) -\- {b + cq) (f> [ax -^ hy -\- cz) — 0,
{a -{■ cp) X -{■ ij) + cq) y = {b + cq) yjr [ax + by + cz).
Differentiating a + bm + c[p + mq) = 0^ SLud remembering that
m is to be regarded as constant, we have
r + 2sm + tm^ = 0,
and introducing the value of m already found,
[b + cq)^ r-2[a + cp)[b + cq) s+[a + cpf 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, /3, 7 be the running coordinates, cc, 3/, z those of the point
of contact; then any generator is the intersection of the tan-
gent plane
j-z=p[a-x) + q{0-y)j
with a plane through the point of contact parallel to the fixed
plane
a[oi-x)-\-b[B-y)-\-c[y-z) = 0,
whence [a + cp) [a — x) + [b + cq) (/3 — ?/) = 0.
Now if we pass to the line of intersection of this tangent plane
with a consecutive plane, a, yS, 7 remain the same, while
aj, y, z^ p^ q vary. Differentiating the equation of the tangent
plane, we have
[rdx + sdy) [a- x) + [sdx + tdy) [/3 - y) = 0.
And eliminating oi. — x,^ — y,
[b + cq) {rdx + sdy) = (a + cp) [sdx + tdy).
398 FAMILIES OF SURFACES.
But since the point of contact moves along the generator which
is parallel to the fixed plane, we have
adx + hdy + cdz = 0, or (a + cp) dx + {b + eg) dy = 0.
Eliminating then dx^ dy from the last equation, we have, as before,
[h + cc[Y r - 2 (a + cp) [b + c^) s + (a + cpY 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„x + c^ ; and
the equation of the family of surfaces is got by writing in the
latter equation for c^ and C3, arbitrary functions of y : x. Differ-
entiating, we have ?« = c^, p + mq=c,^^ whence
px + qy = X(l> (^£j , and z- px- qy = ^ {^^ .
Differentiating again, we have r + 2sm + f?^"'' = 0, and putting
for m its value = c^ = — , the required differential equation is
rx^ + 2sxy + ty'^ = 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 consecutive
tangent planes
{rdx -f sdy) [a — x) + {sdx + tdy) (yS — ?/) = 0.
But any generator lies in the plane ay = /3x, or{oi — x)y = [^—y) x.
Eliminating therefore,
X [rdx + sdy) + y [sdx + tdy) = 0.
dv 8 v
But -4- = — = - . Therefore, as before, rx^ + 2sxii + ty^ = 0.
dx a. X
More generally, let the line pass through a fixed axis a/3,
where OL = ax-\-by -{■ cz-\- d^ l3 = ax + b'y + cz + d'. Then the
equations of the generating line are a = c^/S, y = e^x-\- C3, and the
equation of the family of surfaces is ^ = x<^ ^ + '«/^ 3 . Differ-
entiating, we have
m = c^^a + cp-\-m{})-{-cq)=- c^ [ct + cp + m [b' + cq)].
PARTIAL DIFFERENTIAL EQUATIONS. 399
Differentiating again, we have r -f 2sm + tm'' = 0, and putting
in for m from the last equation, the required partial differential
equation is
{{a + cp) /3 - (a' 4 c» a}'^ «+{(& + cq) /3 - (i' + dq) a]'^ r
- 2 {(a -f c/?) /3 - (a + c» a} {(5 + c^) /3 - (i' + c'q) 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 surface of the family which shall pass through
n fixed curves, we write down the equations of the generating
curve M = c,, F{x^ y, z^ c,, c.^, &c.) = 0, and expressing that the
generating curve meets each of the fixed curves, we have a
sufficient number of equations to eliminate c,, c.^, &c. Thus,
to find a surface of the family x + y(f){z) + -v/r (s) = 0 which shall
pass through the fixed curves y = a^ F{x^ 2;)=0; y = — a^
F^ {x^ z) = 0. The equations of the generating line being z = c^,
x==yc^ ^-Cg, We have, by substitution,
^K+C3,cJ = 0, i^, (C3 - ac,^, c,) = 0,
or, replacing for Cj, C3, their values,
F[x-^c^[a-y\z] = Q, F^[x-c^[a + y\ z]=0,
and by eliminating c^ between these the required surface is found.
Ex. Let the directing curves be
x^ z-
63 C2
we eliminate c, between
2' = "' M + :5=^' y = -«, X2 + 22 = ^2,
{X + C, (rt — J/)}^ 2^ , , , ^,„ „ „
Ijl +3"^ = ^' {^ - ^2 (« + «/)}=+ 3- = C-.
Solving for c^ from each, we have
- \{c^ -z^)-x
c _^~ -JC^ ~ ^ )
II — y C' + 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 w^ien 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
400 FAMILIES OF SURFACES.
equation, to substitute for the equation of the surface, 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 coordinates is unknown. If then
differentiating that quantity gives dy = mdx^ we can eliminate
the unknown quantity ?n, between the total differentials of the
two equations of the generating curve, and so obtain the partial
differential equation 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 a right line. The equations of the generating line
are z = c^x-\ Cj, y = c^-\ c^, and the family of surfaces is ex-
pressed by substituting for c^, Cj, c^ arbitrary functions of c^.
Differentiating, we have ^ + W5' = c,, m = c^. Differentiating
the first of these equations, m being proved to be constant by
the second, we have r + 2sm + tni^ = 0. As this equation still
includes m or c^, the expression for which, in terms of the
coordinates is unknown, we must differentiate again, when we
have a + S^m + 3ym^ + hm^ = 0, where a, /S, 7, S are the third
differential coefficients. Eliminating m between the cubic and
quadratic just found, we have the required partial differential
equation. It evidently resolves itself into the two linear equa-
tions 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^, 2, — 1, which are proportional to the direc-
tion-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
2^', 2 vary while the mutual ratios of dx^ dy^ dz remain constant.
PARTIAL DIFFERENTIAL EQUATIONS. 401
This gives rdo^ + 2sdxdy + tdy'^ = 0. To pass to a third tan-
gent plane, we differentiate again, regarding dx : dy constant ;
and thus have adx^ + Z^dx^dy + ?>'^dxdy'^ 4 My^ = 0. Elimi-
nating 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^j from which it appears that one of the integrals is
J) + mq = <f) (m), where m is one of the roots of r + 2sm + tm^ = 0.
The other two first integrals are
y — mx = -^[m)^ and z—px — mqx = x{'>n).
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—\ 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), y^r (c), &c.].
dF
Eliminating c between this equation and — - = 0, which we shall
CtG
write i^'=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 as long as the form of the function F
remains the same, no matter how the forms of the functions
^, a/t, &c. vary. The curve of intersection of the given surface
with F' is the characteristic (see p. 290) 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 6', i|^', &c., it would seem, according to the theory
previously explained, that the partial differential equation of
the family ought to be of the 2/i*^ order. But on examining
the manner in which these functions enter, it is easy to see that
F FF
402 FAMILIES OF SURFACES. |
the order reduces to the n^^. In fact, difFerentiating the
equation s = i^, we get i
dF dF
-^ = ^'+'5c''>' ^ = ^3+-^^2, thatls,2? = i^, + c,i?^;^ = i?;+c,r,
but since F' = 0, we have p = F^^ 2^ = -^2' where, since F^ and F,^
are the differentials on the supposition that c is constant, these
quantities only contain the original functions (^, i/r and not the
derived ^', V^'. From this pair of equations we can form
another, as in the last article, and so on, until we come to
the n^ 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 between
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, 5, connected by an
equation giving J as a function of a, then between the three
equations z = F^ p = F^, q = F^^ we can eliminate a, by and the
form of the result is evidently /(a^, ?/, z^p, q) =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 xi/. This is a particular case of the general class of
tubular surfaces which we shall consider presently.
Now the equation of such a sphere being
{x-aY + {7j-/3r + z' = r\
and the conditions of the problem assigning a locus along which
the point a/S is to move, and therefore determining /3 in terms
of a, the equation of the envelope is got by eliminating a
between
{x - ay + {3/ - </. {a)Y + z' = r\ {x-a)^[y-<i> [a)] <}>' (a) = 0.
Since the elimination cannot be effected until the form of the
function 0 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
surface is generated by a fixed circle, which moves so that
its plane shall be always perpendicular to the path along which
PARTIAL DIFFERENTIAL EQUATIONS. 403
its centre moves. For the equation of the tangent to the
locus of ayS is
2/-/5 = ^(^-a) ov y - (f> [a] = (P' {a) {x - a).
And the plane perpendicular to this is
(a:-«) + [^-<^(a)}f (a)=0,
as already obtained. To obtain the partial differential equa-
tion, differentiate the equation of the sphere, regarding a, /3 as
constant, when we have x — a.+pz = 0, y — ^ + qz = 0. Solving
for a; — a, 7/ — /8 and substituting in the equation of the sphere,
the required equation is
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^,c^...cJ,
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 been just written, touches each of the given curves,
we obtain n — 1 relations between the constants c,, c^, &c.,
which, combined with the two equations of the characteristic,
enable us to eliminate these constants. For example, the
family of surfaces discussed in the last article contains but
two constants and one arbitrary function, and can therefore
404 FAMILIES OF SUEFACES.
be made to pass through one given curve. Let it then be
required to find an envelope of the sphere
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)' + yS' + s' = /, or (1 + m') z' - %nza, + a' + yS' - r' = 0,
the condition that the line should touch the sphere is
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 - ay + {y- ^Y + z'' = r% (1 + m') {a' + /S^ - r') = mV,
{x -a)d(x+{y- /3) d/3 = 0, ada + (1 + m'] /3d^ = 0,
from which last two equations we have
{l+m')^(x-a) = a{y~^).
The result is a surface of the eighth degree.
441. Again, let it be required to determine the arbitrary
function so that the envelope 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{Xj y, Cj, Cg, &c.) which passes through that point, has
also the same tangent plane as the fixed surface. The values
then of 2} and q derived from the equations of the fixed surface
and of the moveable surface must be the same. Thus we have
/j = i^„ f^ = F^, and if between these equations and the two
equations z = F, z =/, 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 re-
quired to determine a tubular surface of the kind discussed
in last article, which shall touch the sphere x^-\-y^+z^=Ii\ This
PARTIAL DIFFERENTIAL EQUATIONS. 405
surface must then touch [x — a)" 4- (?/ — ^Y + z^ = r\ We have
therefore x : y : z = x- a. : y — ^ : z ; conditions which imply
« = 0, ^x = o.y. Eliminating x and y by the help of these
equations, between the equation of the fixed and moveable
sphere, we get ^{<x^ ^ ^')R =^{W -r' ^ d' ^ ^J. This gives
a quadratic for a' + /3''^, whose roots are [R + rf ; showing
that the centre of the moveable sphere moves on one or other
of two circles, the radius being either 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 2!, 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^ = m^ {x^ + y"^) ; then if the vertex be
moved to the point a, /3, the equation of the cone becomes
z^ = rri' {[x — oCf-^- {y — ^Y]^ and if we are given a curve
along which the vertex moves, /3 is given in terms of a.
Differentiating, we have pz = nl' {x- a), qz = 7n^ (y — j3)j and
eliminating, we have p'^ + ^^ = wi*. 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
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/37, the equation of the surface will evidently be
z— y = F{x — a^ y- jS). If we are given a curve along which
the point a/37 is to move, we can express a, /3 in terms of y,
406 FAMILIES OF SURFACES.
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 directing
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 /3 an arbitrary
function of a, the equation of the fixed surface gives 7 as a
known function of a, /3. It is easy to see how to find the partial
dlflferentlal equation In this case. Between the three equations
z-r^ = F[x-a,y-^\2) = F^[x-a,y-^),q=F^{x-a,y-^),
solve for a; — ot, 3/ — /3, 2-7, when we find
x-a=f[p,q), y-l3 = y{p,q), 2-y = y{p,q).
If, then, the equation of the surface along which a^y Is to move
be r (a, /3, 7) = 0, the required partial dlflferentlal equation Is
r [x -f{p, q), y - V{p, q\ z - y[p, q)]=0.
The three functions /, y, y are evidently connected by the
relation d^y=pdf+ qd^f.
It Is easy to see that the partial dlflferentlal 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 con-
stant radius whose centre moves along any curve traced on a fixed equal sphere
The equation of the moveable sphere ig {x — a)- + (y — /3)- + {z — yy = ?•*, whence
X- a+p{s-y) = 0, y~P + g{z-y) = 0,
and we have
■pr -qr
y - /3 = 7 , z-y--
If we write I +p- + q- = p'^ it is easy to see, by actual differentiation, that the
relation is fulfilled
^r-'©-'^©-
The partial differential equation is
{xp + pry + (yp + qr)- + {sp — r)* = pV,
or {z- + 2/^ + z") (1 +i>2 + 5'-')* + 2 {px +qy-z)r-0.
PARTIAL DIFFERENTIAL EQUATIONS. 407
444. We now proceed to investigate the form of the partial
differential equation of the envelope, when the equation of the
moveable surface contains three constants connected by two
relations. If the equation of the surface be z — F{x^ ?/, a, h^ c),
then we have p = F^^ q = F,^. Differentiating again, as in
Art. 432, we have
r -f sm = i^„ + mi^2i s + tm = F^^ + niF,^^ ;
and eliminating ?/?, the required equation* Is
The functions F^^, F^^, F^^ contain a, 5, c, for which we are
to substitute their values In terras of p, q, cc, 2/, z derived from
solving the preceding three equations, when we obtain an equa-
tion of the form
Br + 2Ss+Tt+U{rt- s') = F,
where i?, Sj T, Z7, Fare connected by the relation
BT+ UV=S\
445. The following examples are among the most Important
of the cases where the equation includes three parameters.
Developable Surfaces. These are the envelope of the plane
z = ax + by -\- c^ where for b and c we may write (f) (a) and yjr [a).
Differentiating, we have p = a, q = bj whence q = 4>{p). Any
surface therefore Is a developable surface if p and q are con-
nected by a relation Independent of x, y, z. Thus the family
(Art. 442) for which p^ -\- <t = '^^ Is a family of developable
surfaces. We have also z- px — qy = '>i^ (p), which is the other
first Integral of the final differential equation. This last is
got by differentiating again the equations ^ = a, q = b^ when
we have r + sm = 0, s + tm = 0, and eliminating ??2, rt — / = 0,
which is the required equation.
By comparing Arts. 295, 311, It appears that the condition
rt = s^ is satisfied at every parabolic point on a surface. The
* I owe to Professor Boole my knowledge of the fact, that when the equation
of the moveable surface contains three parameters, the partial differential equation
is of the form stated above. See his Memoh-, F/iil. Trans., 18G2, p. 437.
408 FAMILIES OF SUEFACES.
same thing may be shewn directly by transforming the equation
rt — 5'^ = 0 into a function of the differential coefficients of U^
by the help of the relations
when the equation rt - s^ = 0 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 inflexional
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 w Is tj -^ ~t~,- -=^, — ( - — =- ) I .
^ ^ "^ [dz^ dw' \dzdwj J
This proves that any generator xi/ meets the Pro-Hessian in
the first place, where x^ meets v ; that is to say, twice in the
point on the cuspidal curve (wi), and in r - 4 points on the nodal
curve {x) 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 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 ?• — 4 points on
(a;), and of 2 (r — 5) other points, in all 3r — 8 points.*
* Prof. Cayley has calculated the equation of the Pro-Hessian (Quarterly/ Journal,
vol. VI. p. 108) iu the case of the developables of the fourth and fifth orders, and of
PARTIAL DIFFERENTIAL EQUATIONS. 409
446. Tuhular 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,
a;-a+p(2-7) = 0, 3/-/3 + ^ (s-7) = 0,
whence l+p^ +{z-'y)r + m [pq + (s - 7) s} = 0,
pq + {z -r^)s ^ m[]. + (f + [z - r^)t]=0.
And therefore
[\+f+{z-i)r][l+q' + [z-r^)t] = [pq + [z-^)s]\
Substituting: for s - 7 its value -; 7, ;-, from the first three
equations, this becomes
R'{rt- /) - R[[l+qy-2pqs +[l+f)t] V(l +/+^'') + (1+/+?T=0,
which denotes, Art. 311, that at any point on the required
envelope one of the two principal radii of curvature is equal
to i?, 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^, [r - FJ {t - FJ = (5 - FJ. Let us,
for shortness, write the last equation pr = cr^, and let us write
a - i^„ ^A,(3- i^„, = i?, 7 - F^^^ =C,8-F^^ = B; then, differ-
entiating pr = (t'\ we have
[A + Bm) r + [C+ Dm) p-2 {B-\- Cm) a = 0.
Substituting for m from the equation a- + rm = 0, and remember-
ing that pr = g\ we have
Ar^ - SBar' + 3 Ca\ - Ba^ = 0,
that of the s-'xth 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 ctirve is but a simple curve on the Pro-Hessian,
ajid therefore is only counted twice in the intersection.
GGG
410 FAMILIES OF SURFACES.
in wliicli 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
where m and C/'are functions of x^ ?/, s, 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 de-
rived another differential equation satisfied by every charac-
teristic 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
/(a;, y, 3, p^ q) = 0. Now if this equation belong to the en-
velope 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
a;, y, 2, ^, q are the same for both. Now if x, ?/, z be the
coordinates of any point on the characteristic, since such a
point is the Intersection of two consecutive positions of the
moveable surface, the equation f{x, y, s, ^, q) = 0 will be
satisfied by these values of a;, ?/, 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 equation, regarding p and q as alone variable, then
the points of the characteristic must satisfy the equation
PJ^+ Qdq = 0.
Or we might have stated the matter as follows : Let the
equation of the moveable surface be z — F{x^ ?/, a), where
the constants have all been expressed as functions of a single
parameter a. Then (Art. 438) we have p = FJx, y, a),
q = F^{^) 2/j a), which values of ^ and q may be substituted In
the given equation. Now the characteristic is expressed by
PARTIAL DIFFERENTIAL EQUATIONS. 411
combining with the given equation its differential with respect
to a ; and a only enters into the given equation in consequence
of its entering Into the values for p and q. Hence we have,
as before, P^-\- 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 + qdi/ h 8S.iisRed^
whether ^? and q have the values derived from one position of
the moveable surface or from the next consecutive. We have
therefore J- 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-]r Qq = Rf
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 = Rdy. The
reader is aware (see Boole's Differential Equations^ p. 323) of
the use made of those equations in integrating this class of
equations. In fact, if the above system of simultaneous equa-
tions integrated give m = c,, v = c^^ 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 = cp (u).
Ex. Let the equation be that considered (Art. 439), viz. z^ (1 +p- + 5^) = r^, then
any characteristic satisfies the equation pdy — (^dx, which indicates (Art. 421) that
the characteristic is always a line of greatest slope on the surface, as ia geome-
trically evident.
449. The equation just found for the characteristic generally
includes p and q, but we can eliminate these quantities by
combining with the equation just found the given partial dif-
ferential equation and the equation dz =pdz + qdy. Thus, in the
last example, from the equations z^ [1 +jf + q') = 7'\ qdx = pdy ,
we derive
z' [dx' + dy' + dz') = r' [dx' + dy').
The reader Is aware that there are two classes of differential
equations of the first order, one derived from the equation of
412 FAMILIES OF SURFACES.
a single surface, as, for instance, by the elimination of any
constant from an equation f7=0, and its differential
V^dx + TJ^^dy + TJ^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 diflfer-
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 2 : 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, ^S, m^ between
the equations
{x - a)' +{y- 13 f + 2' = r\ x-a + m{y-/3) = 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
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 cuspidal
edge of every surface of the family under consideration. Thus,
in the example chosen, the geometrical property expressed 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 ^, 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
PARTIAL DIFFERENTIAL EQUATIONS. 413
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= 0, -7- = 0, which represents the characteristic when a is re-
garded as constant, represents the cuspidal edge when a is an
unknown function of the variables to be eliminated by means
of the equation -^-5- = 0. But the equations U—0, -7— =0
evidently have the same differentials as if a were constant, when
a is considered to vary, subject to this condition.
Thus, in the example of the last article, if in the equations
(x - ay -\- [y - /3)^ + z' = r\ (cc - a) + m (y" - /S) = 0, we write
/S = 0(a), m = 0'(a), and combine with these the equation
1 + <^' («)"''= (^- /3) 0" (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, yS, 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 we shall
obtain the equations of a curve satisfying this differential
equation by giving any form we please to 0 (a), and then
eliminating a between the equations
{x-aY+{y-4>{a)Y + z^ = r% {x - a) -^ <}>' {a) [y - <f>{a)} = 0,
l + W{a)r=y-cf^{a)]cl>"[a).''
* It is convenient to insert here a remark made by Mr. M. Roberts, viz. that if
in the equation of any surface we substitute for x, x + Xdx, for y, y + Xdy, for z,
z + \dz, and then form the discriminant with respect to \, 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 expresses the
condition that the tangent to the curve represented by it touch the given surface.
Thus the general equation of the cuspidal edge of developables circumscribing a
sphere is
(a;2 + y2 + g2_ a2) [dx'^ + dy^ + dz^) = {xdx + ydy + zdzf,
or {ydz - zdyY + {zdx — xdzf + {xdy - ydxf = a- {dx- + dy- + dz-).
Li 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 distance of that line from
the origin be a, then the area of the triangle formed by joining any element ds to
the origin is half ads, but this is evidently the property expressed by the preceding
equation.
414 FAMILIES OF SURFACES.
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 differential equa-
tion, 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, 2, ^, 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 Bdr + Sds + Tdt = 0. But, since p and q are con-
stant along this line, we have drdx + dsdy = 0, dsdx -\- dtdy = 0.
Eliminating then dr^ ds^ dt^ the required equation for the
characteristic is
Rdij' - Sdxdy + Tdx" - 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, belong 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
PARTIAL DIFFERENTIAL EQUATIONS. 415
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^[{l + q') s -pqt\+m {(1 + q^) r - (1 -^f) t\-[[\ +/) s -iw] = ^'
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, yS between the fol-
lowing equation and its differentials with respect to a and y8 :
{x - ay +{y- iSy +[z-(p{oi + m^)Y = f{0- ma)''.
Differentiating with respect to a and /3, we have
{x — a)+ {z - (^) (J)' = 7n '^'^^
{y-^) + m{z-cf>)c}>'=-^lr'f,
whence [x-a] + m (y - /3) + (1 + m"^) {z - ^) 0' = 0.
But we have also
{x-a)+piz-cf>)=0, (y-^)+q{z-cl>] = 0,
whence (^ — a) + m [y — /3) + {p + mq) [z — (j))= 0.
And, by comparison with the preceding equation, we have
p-\-mq= {!■+ m') (/>' (a + m^). If, then, we call a + «i/S, 7, the
problem is reduced to eliminate 7 between the equations
X -\r my - <y + {p + mq) {z - <f) (7)} = 0, p + mq = (1 4 m') <f)' (7).
Differentiating with regard to x and ?/, we have
(1 + // + mpq) + (?•+ ms) {z-(t>{j)] = {l + {p + mq) (}>'] 7^,
{m (1 + q') +pq] + {s + mt) [z - t^) (7)} = {1 + (^ + mq) (\)'] 7^,
but from the second equation
r -{■ ms '. s + mt : : 7, : y^-
Hence, the result is
(1 4 p' 4 mptq) [s 4 mt) = [m (1 4 q^) -^2^^] (^ + ^"^))
as was to be proved.
416 FAMILIES OF SUKFACES.
SECTION II. COMPLEXES, CONGRUENCIES, RULED SURFACES *
453. The preceding families of cylindrical surfaces, conical
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 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 consisting of a finite
number of right lines, and this may be excluded from con-
sideration ; 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 Plucker has termed a " complex " of lines. As
examples, 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. 80c? and in Art. 316 (Z)), 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 a 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 " congruency" of lines. A well-known example is that
* In Sir W. R. Hamilton's second supplement on Systems of Rays. Transactions
of the Royal Irish Academy, vol. xvi., were first investigated the properties of a
congruency other than that formed by the normals to a suiiace. As to the theory of
complexes and congruences see Pliicker's posthumous work, Neve Geometrie des
Rautnes gegriindet auf die Betrachtung der geruden Linie ah Raumehment, Leipzig,
1868, edited by Dr. Klein ; also Kvimmer's Memoirs, CreUe LVii. p. 189 ; and " Ueber
die algebraischen Strahlensysteme, in's Besondere iiber die der erstea und zweiten
Ordnung," Berl. Abh. 1866, pp. 1 — 120 ; and various Memoirs by Klein and othera
As regards ruled surfaces see M. Chasles's Memoir, Quetelet's Correspondance, t. XI,
p. 50, and Prof. Cayley's paper, Cambridge and Dublin Mathematical Journal, vol. vil.
p. 171 ; also his Memoir, "On Scrolls otherwise Skew Surfaces," Philosophical Tran-
sactions, 1863, p. 453, and later Memoirs.
COMPLEXES, CONGRUEN'CIES, RULED SURFACES. 417
of the normals of a given surface. Each of these touches at
two points (the centres of curvature) a certain surface, the
centro-surface or locus of the centres of curvature of the
given surface, and the normals are thus bitangents of the
centro-surface. And so, in general, we have as a congruency
of lines the system of the bitangents of a given surface. But
more than this, every congruency of lines may be regarded as
the system of the bitangents of a certain surface, for each line
of the congruency 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 congruencles the systems of lines,
(1) the bitangents of a surface,
(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 a congruency, ask how many of the
lines thereof meet each of two given lines? the number in
question Is the " order-class" of the congruency. But Imagine
the two given lines to Intersect ; the lines of the congruency
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 congruency pass through a
given point? the number is the "order" of the congruency.
(2) How many of the lines of the congruency lie in a given
plane ? the number is the " class" of the congruency. The sum
of these numbers is the order-class, as above defined.
C. Relation 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" of the system is the number of lines of the system
which meet a given right line.
HHH
418 FAMILIES OF SUEFACES.
454. In accordance with Pliicker's work on the right line
considered as an element of space, we must therefore first
consider the properties of a complex ; that is to saj, of a system
of lines which satisfy a single relation between the six coordi-
nates. If this relation be of the 7i^^ degree, the complex is of
the n^^ degree ; all the lines of it which pass through a given
point form a cone of the n"^ order, and those which lie in a
given plane, envelope a curve of the n^^ class (see Art. 80c?). If,
for instance, the complex be of the first order, all the lines which
pass through a given point lie in a given plane ; and, reciprocally,
those which lie in a given plane pass through a given point.
To each line in space corresponds a conjugate 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 determioed ; 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 corresponds
a single point, and the locus of these points is therefore a line
of the first order, 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 complex. If the axis and a line of
the complex be given, the complex is determined ; and the
complex in fact consists of the different positions which this
line can assume whether by rotation round the axis or by
translation in a direction parallel to the axis. When the line
meets the axis we have the limiting case of a complex consisting
of all lines which meet a given one. 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 AF+BG+CH=0.
COMPLEXES, CONGRUENCIES, RULED SURFACES. 419
455. We have a congraency of the first order when we
have two equations each of the first degree between the six
coordinates ; or, in other words, the congruency consists of the
lines common to two given complexes. We may evidently for
either of the two given equations A2)+Bq+&c. = 0, A'p+&c. = Oj
substitute any equation of the form {A-\- kA')p + &c. = 0', and
then determine A;, so that this equation shall express that every
line of the congruency meets a given line. We have thus
a quadratic equation for k, and it appears that the con-
gruency consists of the system of lines which meet two fixed
directing lines. Any four lines then determine a congruency
of this kind ; for (see Art, 57c?) we have two transversals which
meet all four lines,* and the congruency 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 congruency, 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 congruency consists of lines each meeting a given
line, and such that considering the common point of the given
line and a line of the congruency, and the common plane of
* 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, aud 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 Prof. 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 expressed 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 Messrs. Sylvester and Cuyley, and by M. Chasles,
in the Comptes Eendus for 1861, Premier Scmeslre,
420 FAMILIES OF SURFACES.
the same two lines, the range of points corresponds homo-
graphlcally with the pencil of planes.
Let us pass now to a complex of the second order; 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 Plucker 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, Plucker's complex
surface. It is easy to see that the given line will be a
double line on the surface, and that the surface will be of
the fourth order, 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 Plucker shows
also that it has eight double points.
456. We here briefly indicate the method by which It is
established, that the lines of a congruency are in general
bitangents of a surface. Let the equations of a right line be
x-x y-y z-z ,,,.,,, , ,
— -7— = - — >— = — -r- J then x ^ y ^ z ^ \ ^ ix ^ v may each be
regarded as functions of two parameters p^ q, as in Gauss's
method (Art. 377). If we take a second line and consider the
line joining a point x' + '^'p\ y + ytiV) ^' + ^' P to a point
x" + X"p", y" + /ti' p", z" + v'V' oil the second line, then the
conditions, that the joining line may be perpendicular to both
lines, give
V [x' - x") + / [y' - y") + / [z' - z") + p' - p" cos ^ = 0,
X" [x' - x") + p!' [y - y") + v" [z' - z") - p" + p' cos ^ = 0,
where Q is the angle between the lines. And if we take the
lines indefinitely near, we can derive from these equations
COMPLEXES, CONGRUENCIES, KULED SURFACES. 421
which deterrnlaes the point where one line is met by the
shortest distance from a consecutive line. If we substitute in
the above for 8x\ aSp + a8q^ &c., we get for p a value of
the form
_ehfj^2fEpSq+jjSq'_ _ ee+2ft-\-g
E8f + 2Fhphq + Ghq^ ' ~ Ef + ^iFt + G '
writing t for the ratio hp : hq. Since the denominator of this
function represents the sum of three squares it cannot change
sign, and p therefore cannot become infinite, but will lie
between a certain maximum and minimum value ; that is to
say, the points on any line of a congruency wliere it is met
by the shortest distance to an adjacent line of the congruency
range on a certain determinate portion of the line, the extreme
points being called by Sir W. Hamilton the virtual foci.* He
has proved also that the planes containing the shortest distances
corresponding to the two extreme values lie at right angles
to each other ; and that if p^, p^ be the extreme values, that
corresponding to another whose shortest distance makes an
angle 6 with one of these is given by the formula
p = p, cos'"*^ -f p.^ ?,W6.
The value of the shortest distance itself between two adjacent
lines is given by an expression similar in form to that already
given for p. It is plain, then, that there are two values of t for
which the shortest distance will vanish, or that each line of the
congruency is in general intersected by two of those adjacent
to it. The locus of the points of intersection will be the surface
to which the lines are bitangent, and is called the " focal
surface" of the congruency ; but this surface may degenerate
into a curve, or it may break up into two surfaces, either or
each of which may degenerate into a curve as already mentioned.
Besides these focal surfaces there are also connected with the
congruency and completely determined by it the surfaces on
which the extreme points of the shortest distances lie and the
surface described by the common centre of both portions of
the ray.
* First "Supplement" Trans. R. I. A. vol. xvi. part I. p. 52.
422 FAMILIES OF SURFACES.
457. For instance, the degeneration whicli has been just
mentioned of necessity takes place when the congrueucy is
of the first order. In this case, since through each point
only one line of the congruency 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 congruency 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 congruency of lines meeting two directing
curves, and if these curves be of the orders ?n, m ^ and have a
common points, the order of the congruency will be mm — a.
The only congruency of the first order of this kind is when
the directing lines are a curve of the vP^ order, and a right
line meeting it w — 1 times.
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 tangents
(Art. 265) at that point. Each different point on the gene-
rator has a different tangent plane (Art. 110), which may be
constructed as follows : 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 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.
COMPLEXES, CONGRUENCIES, RULED SURFACES. 423
There may be on the surface, however, singular generators
which are intersected by a consecutive generator, and in this
case the plane containing 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.
459. The anharmonic ratio of four tangent planes passing
through a generator is equal to that of their four points of con-
tact. Let three fixed lines A^ B, C be intersected by four
transversals in points aa'a!'a'\ l)b'h"h"\ ccc'c". Then the an-
harmonic ratio [hb'h"b"'] = {ccW"}, since either measures the
ratio of the four planes drawn through A and the four trans-
versals. In like manner [ccc'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 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.
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 anharmonic 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 rect-
angle 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
424 FAMILIES OF SUKFACES.
all parallel to the same plane, namely, the plane perpendicular
to the generator. We may speak of the anharmonlc 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 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 paraboloid
intersect a generator of the opposite kind are the points of
contact of the four tangent planes which contain these gene-
rators, 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. III. p. 668), that the shortest distance
between two consecutive generators is not an element of the
line of striction. In fact, if Aa^ Bh^ Cc be three consecutive
generators, ah the shortest distance between the two former,
then h'c the shortest distance between the second and third
will in general meet Bh in a point h' distinct from 5, and
the element of the line of striction will be ah' and not ah.
COMPLEXES, CONGRUENCIES, RULED SURFACES. 425
Ex. L To find the line of striction of the hyperbolic paraboloid
Any pair of generators may be expressed by the equations
X y . X y \
ah a h K
X r^v X •*- y 1
aft)l = '^^' «©! = ;:•
Both being parallel to the plane ^ , their shortest distance is perpendicular to
this plane, and therefore lies in the plane
a?-V^ 1
which intersects the first generator in the point z — — — j„ t— .
"When the two generators approach to coincidence, we have for the coordinates of
the point where either is intersected by their shortest distance
_d?--h'^ \ X y _ a?- 62 1^
^"o^n^X^' o "^ 3 ~ a^ 4- 62 \ '
and hence (a^ + h^) (^ + 1) = (a^ - l^) (| - 1) , or ^ 4- 1, = 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
a^ b^
Ex. 2. To find the line of striction of the hyperboloid
x^ y2 22
— I- '-^ =1.
o^ V^ c^
Ans. It is the intersection of the surface with
o? y"^ z^ '
where -^ = T5 + -i> ^ = -l+-;i <^=a5--5'
463. Given any generator of a ruled surface, we can de-
scribe a hyperboloid of one sheet, which shall have this gene-
rator 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 gene-
rators are given, and consequently that if two ruled surfaces
have three consecutive generators in common, they will touch
III
426 FAMILIES OF SURFACES.
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 tl^ degree, then every terra in its equation must
contain either x qx y\ and that equation arranged according
to the powers of x and y will be of the form
where m^^^, v,,,, denote functions of z of the {n — 1)'" degree, &c.
Then (see Art. 110) the tangent plane at any point on the axis
will be u ,x + v' .it = 0, where xi , denotes the result of sub-
stituting in u^^_^ the coordinates of that point. Conversely, it
follows that any plane y = '"^^ touches the surface in w - 1
points, which are determined by the equation m,^_, + wu,j_j = 0.
If however m^^,, l\^_^ have a common factor i^^, so that the
terms of the first degree in x and y may be written '
u^ (w„_ _iX -f v,j_ _,?/) = 0, then the equation of the tangent plane
will be u\ X + V ^ _,?/ = 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 m,, = 0
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
u^^_^ [ux + vy) + &c. = 0,
so that the tangent plane at any point on the axis will be the
same as that of the hyperboloid ux 4 vy^ viz. ux •+ vy = 0. And
any plane y — mx will touch the surface in but one point. The
factor w ^_2 indicates that there are on each generator n — 2
points which are double points on the surface.
464. We can verify the theorem just stated, for an im-
portant class of ruled surfaces, viz., those of which any
COMPLEXES, CONGKUENCIES, KULED SURFACES. 427
generator can be expressed by two equations of the form
ar + hf-' + cC-' + &c. = 0, a'e + Vr' + cT' + &c. = 0,
where a, a', J, h\ &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 generator (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 (aJ'), (oc'), {ad'), &c., or when m>?2, the first
row is as before and the first column consists of n such consti-
tuents, a and zeros. Now the line aa is a generator, namely,
that answering to f = co ; 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 determi-
nant contains a factor from the first row and a factor from the
first column, the expanded determinant will be a function of,
at least, the second degree in a and a', except that part of it
which is multiplied by (aJ'), 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' — 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
ad 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— 1)'^ degree
meeting the generator In n — \ 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
428 FAMILIES OF SURFACES.
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 w — 2 2^oints.
It may of course happen, that two or more of these ?^ — 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 Algebra^ Lesson xvili.,
on the Order of Restricted Systems of Equations) that the mul-
tiple curve is of the order ^ (m + w — 1) («z -f w — 2), and that
there are on it | [m + w — 2) (m + w — 3) [m -f w — 4) triple points.
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 con-
secutive ; the number of planes through two consecutive gene-
rators 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 note p. 105). 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
I
COMPLEXES, CONGRUENCIES, RULED SURFACES. 429
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.*
467. We shall illustrate the preceding theory by an enume-
ration of some of the singularities of the ruled surface generated
by a line meeting three fixed directing curves, the degrees of
which are ?Wj, m^^ m^.\
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 w?^, m^
and the assumed line ; that is to say, it is 7n^ times the degree
of the latter surface. The degree of this again is, in like
manner, m^ times the degree of the ruled surface whose directing
curves are two right lines and the curve ^3, while by a repe-
tition of the same argument, the degree of this last is 2m^,
It follows that the degree of the ruled surface when the
generators are curves w^, m^^ m^^ is 2vi^m^m^.
The three directing curves are multiple lines on the surface,
whose orders are respectively m^m^^ '^s^d wi^w^- For through
any point on the first curve pass m^m^ generators, the inter-
sections, namely, of the cones having this point for a common
vertex, and resting on the curves tw^, m^.
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 in , m
have a point in common, it is evident that the cone whose
vertex is this point, and base the curve m^ will be included
in the system, and that the order of the ruled surface proper
will be reduced by ??ij, while the curve m^ will be a multiple line
of degree only m.^m^ — 1. And generally if the three pairs made
out of the three directing curves have common respectively
a, /3, 7 points, the order of the ruled surface Avill be reduced
* These theorems are Prof. Cayley's. Cambridge and Dublin MathematicalJournal,
vol. VII., p. 17L
t I published a discussion of this surface, Cambridge and Dublin Mathematical
Journal, vol. Viil., p. 45.
430 FAMILIES OF SUEFACES.
by m^a + m^^ + rn^y^'^ while the order of muUipllcItj of the
directing curves will be reduced respectively by a, /S, 7. Thus,
if the directing lines be two right lines and a twisted cubic,
the surface is in general of the sixth order, but if each of the
lines intersect the cubic, the order 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 order, 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 order consists of six cones of
the second order, 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 tico 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 order of the ruled surface being 27n^m,^ni^, it
follows, from Art. 465, that any generator is intersected by
27n^in^in^- 2 other generators. But we have seen that at
the points where it meets the directing curves, it meets
[m.'in^ — '[) + {m^m^ — l) + {m^m_^ — l) other generators. Conse-
quently it must meet 2m^m^m^ — {m^m^ + m^m^ ■+ m^m,^ + I gene-
rators, in points not on the directing curves. We shall establish
this result independently by seeking the number of generators
* My attention was called by Prof. Cayley to this reduction, which takes place
when the directing curves have jDoints in common.
COMPLEXES, CONGRUENCIES, KULED SURFACES. 431
which can meet a given generator. By the last article, the
degree of the ruled surface whose directing curves are the curves
vHj, ^2, and the given generator, which is a line resting on both,
is 2m^m_^ — m^ — m^. Multiplying this number by m^, we get the
number of points where this new ruled surface is met by the
curve m^. But amongst these will be reckoned (WjW^— 1) times
the point where the given generator meets the curve m^. Sub-
tracting this number, then, there remain
2m mm, — miiK ~ mm, — mm„ + 1
points of the curve m^^ through which can be drawn a line to
meet the curves t??^, ???.^, and the assumed generator. But this
is in other words the thing to be proved.
470. We can examine in the same way the order of the
surface generated by a line meeting a curve ?>z, twice, and
another curve vi^ once. It is proved, as in Art. 467, that the
order is m^ times the order of the surface generated by a line
meeting m^ twice, and meeting any assumed right line. Now
if k^ be the number of apparent double points of the curve w,,
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 h^^ and the section of the ruled surface by a
plane through that line will be that line h^ times, together with
the ^rrtj (»z, — 1) lines joining any pair of the points where the
plane cuts the curve m^. The degree of this ruled surface will
then be k^-\- ^m^{m^— \)^ and, as has been said, the degree
will be wz^ times this number, if the second director be a curve
m^ 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 wj^, m,^ ;
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
A, + ^.^ + wij7n.^ ;* and the order of the surface generated by a
* Where I use h in these formulae Prof. Cayley uses r, the rank of the system,
substituting for h from the formula r = ?re (m - 1 ) - 2k, And when the system is
a complex one, v:e have simply E = i\ + r^.
432 FAMILIES OF SURFACES.
line meeting a right line, and meeting the complex curve
twice, is
^ [m^ + wij {m^ + m^- 1) + A, + \ + m^m^
= ih^i {m^ - 1) + ^il + {h% [^^h -'^) + K] + ^wij^j.
471. The order of the surface generated by a line which meets
a curve three times may be calculated as follows, when the
curve is given as the intersection of two surfaces Z7, V: Let
xy'zw be any point on the curve, xyzw any point on a gene-
rator through xy'zw ; and let us, as in Art. 343, form the two
equations S C/' + iXS' U' + &c. = 0, S F + W 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 V =0, V = 0, eliminate x'yzw\ 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 TJV. But since JJ' and V do not
contain xyzw^ the order of the result of elimination will be the
product of pq the order of Z7', F' by the weight of the other
two equations; (see Higher Algebra^ Lesson xvill.). If, then,
we apply the formulse given in that Lesson for finding the
weight of the system of conditions that two equations shall
have two common roots, putting m = ^— 1, n — q-l, X, = 0,
V = p, /A = 0, /u,' = q, the result is | {pq — 2j {2pq — 3 ( /? + g-) + 4},
and the order of the required surface Is this number mul-
tiplied by ^pq. But the intersection of £/, V is a curve
(see Art. 343), for which m=pq^ 2h =pq [p - I) {q — 1), whence
pq ip + q) = m^ + m — 2h. Substituting these values, the order
of the surface expressed in terms of m and h is
^ (m — 2) (6A + m — 7n^), or (m — 2) h — ^m [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 m^ twice, and also the curves m^ and tWj, it
belongs doubly to the system of lines which meet the curves
COMPl.EXES, CONGRQENCIES, RULED SURFACES. 433
m^, m^^ m^ and is a double generator on the corresponding
surface. But the number of such lines is evidently equal to the
number of intersections of the curve ra^ with the surface gene-
rated by the lines which meet m^ twice, and also w^, that is
to say, is m^m^[\m^{m^—\) + W] the total number of double
generators is therefore
\m^m^i^ (m, + wz,^ + 7n^ — 3) + h^m^^n^ + h^^m^ + h^m^m^.
In like manner the lines which meet w, three times, and also m^
belong triply to the system of lines which meet m^ twice, and also
m^ ; and the number of such triple generators is seen by the last
article to be m^ (m, — 2) A, - \m^m^ {m^ — 1) {m^ — 2). The surface
has also double generators whose nuqaber we shall determine
presently, being the lines which meet both on^ and m^ twice.
Lastly, the lines which meet a curve four times are multiple
lines of the fourth order 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 Z7, V, W contain
xyzw in the degrees respectively X, X', V, and x'y'z'w' in
degrees //,, /*', ii" ^ and let the \W points of intersection of
these surfaces all coincide with x'yz'w \ then it is required to
find the order of the further condition which must be fulfilled
in order that they may have a line in common. When this
is the case, any arbitrary plane ax + /3?/ + 72; + hw 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 f/", F, W and the
arbitrary plane must vanish. This result is of the degree
XW in a/37S, and /iV\" + /i'X"\ + /'XX' in xy'z'io. The first
of these numbers (see Higlier Algebra^ Lesson XVIII.) 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 + /3j/ + 7s + 8i/;, the coordinates
of each of the points of intersection of Z7, F, TF, this re-
sultant must be of the form 11 (a/ -f /S?/' + 72;' + hw'Y^''^". The
KKK
434 FAMILIES OF SURFACES.
condition ax + ^y + 7/ + hw = 0, merely indicates that the
arbitrary plane passes through xyzw\ 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 n = 0; and this must be of the
degree
fi\'\" + [x\"\ -f ijf'Xk' — W'\" 5
that is to say, the degree of the condition is got by subtracting the
order from the weight of the equations f/, F, W.
Al-i. Now let x'yz'w be any point on the curve of inter-
section of two surfaces Z7, F, xyzw any other point ; and, as
in Art. 471, let us form the equations SZ7+ ^/^.5^f^-f &c. = 0,
SF+ ^XS'^F+ifec. = 0. \i x'yz'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 \ 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 xyzio^ 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 xyzio 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'xo .
The order, then, of the condition which x'y'z'id 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 Higher
Algebra^ Lesson xvill.) the weight of this system of con-
ditions is found by making m=p — l^ n = q—\^ ^ =i^) 1^ — 1^
\' = fi' = 0, to be
i WV - ^pY [p + !z) + ¥Y + "^P9. [p + ?)'
+ \5pq [p 4 2) - npq - 66 (;; + q) + 108] ;
while the order of the same system is
I {pY-¥Yip^'i)-^^pY+^P'iip+Qf-^P2 ip+9) + i^pq-^G]-
COMPLEXES, CONGKUENCIES, RULED SURFACES. 435
The order, then, of the condition D = 0 to be fulfilled by
xyz'w\ behig the difference of these numbers, is
B{22>Y-6//(i^+?)+3i?^(i5+2)'''+18p2(p+^)- 262?2-66(^+5)+ 144}.
The intersection of the surface n with the given curve deter-
mines 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 jpq^. As before, putting ^pq^ = ??i,
Vi [p + ?) = "*"' -\- m — 27i, the number of lines meeting in four
points is found to be
2^ {- w* + 18»i' - 71m' + 78m - 48m7i + 132A -I- 127i'| *
From this number can be derived the number of lines which
meet both of two curves twice. For, substitute in the formula
just written ?«, +w,^ for w, and h^-{ h^-V m^m^ for 7«, 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 re-
mainder is the number of lines which meet both curves twice, viz.
W + Im^m^ (m, - 1) [m^^ - 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 j but Prof. 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 we have been
considering, M the degree of that surface, 0 (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 ?«, and w.^, the surface will consist of two surfaces
i)/,, 3/^ having as a double line the intersection of il/, and J/,
* It viay happen, as Prof. Cayley has remarked, that the surface IT may altogether
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 surface by a surface of tho
p^^ order is evidently such that every generator of the ruled surface meets the curve
in p points.
436 FAMILIES OF SURFACES.
in addition to the double lines on each surface. Thus, then,
^ [m] must be such as to satisfy the condition
Using, then, the value already found for M^ in terms of tw^,
solving this functional equation, and determining the constants
involved in it by the help of particular cases in which the
problem can be solved directly. Prof. Cayley arrives at the
conclusion, 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 ?Wj, m^, wz,,
in the case of the surface generated by a line meeting m^ twice
and m^ once, is
^. ilK K - 2) K - 3) + irn^ (m, - 1) {m^ - 2) {m^ - 3)}
+ ^. K- 1) iW-^lK {m;'-m- !) + >, K- l)«-5/«,+ 10)],
and in the case of the surface generated by a line meeting 7n,
three times, is
^A/w, [m^ - 5) - i/i, {7n* - Sm," + Stw/ - A9m^ + 120)
+ T2 (^'^^ - 6"'i' + ^^K - 270mj' + 868m^' - 4087nJ.
SECTION III. ORTHOGONAL SURFACES.
476. We have already given a proof 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 somewhat
more geometrical form as follows. Imagine a given surface,
and on each normal measure off from the surface an in-
finitesimal distance I (varying at pleasure from point to point
of the surface, or say an arbitrary function of the position
of the point on the surface] : the extremities of these distances
form a new surface, which may be called the consecutive
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
ORTHOGONAL SURFACES.
437
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, hut these will not In general Intersect at right angles.
Take A a point on the given surface ; ABj A C elements of
the two curves through
A ; AA', BR, CC the
infinitesimal distances
on the three normals ;
then we have on the
consecutive surface the
point -d', and the ele-
ments A'R, A'C of
the two corresponding
curves; the angles at
A are by hypothesis each of them a right angle ; the angle
B'A'G' 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 GG\ A A' shall
intersect, for it can be shown that if one pair Intersect, the
other pair also intersect. But the normals Intersecting, AB, A C,
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 ic, y, z for the coordinates of the point A\ a, /S, 7
for the direction-cosines of AA' ; a,, ^^, y^ for those of AB^
and a^j, /Sgj 72 ^^^ those of A 0. Write also
^2 = Mx + ^2^!, + 7.A-
Then it will be shown that the condition for the intersection
of the normals AA\ BB' is
a.^S,a + ^,8,^ + 7A7=0,
the condition for the intersection of the normals AA^^ CC is
a,8,a + /3,8,/3 + 7,S^7 = 0,
438
FAMILIES OF SURFACES.
and that these are equivalent to each other, and to the con-
dition for the angle B'A' C being a right angle.
Taking Z, l^^ \ for the lengths AA ^ AB, AC, the coordinates
of -4', J5, C measured from the point A, are respectively
(?a, //3, ?7), (Z,a,, /,^„ ?,7,, (?,a„ Z,/9„ ?,7j.
The equations of the normal at A may be written
where X, Y, Z are current coordinates, and ^ is a variable
parameter. Hence for the normal at B passing from the co-
ordinates X, ?/, z to ic+?^a,, 2/+?i/3,, 2 + ^i7,5 the equations
are
Z=« + ^7+/,7, +Z^S.(^7),
and if the two normals intersect in the point {X, F, Z), then
a, + a8^d + eS^a = 0,
^^ + ^8^6 + 68^0 = 0,
% + 78,^ + 68^y = 0.
Eliminating ^ and 8^6, the condition is
= 0;
«.:
«)
8.a
A,
^,
S./3
%,
7,
8j7
or since a^, ^„ 7^ = I3y^ - ^^7, 7a, - 7,a, a^^ - a^/S,
this is a^8^a + 0,^8^0 + y^8j = 0.
Similarly the condition for the intersection of the normals
AA', CC is
We have next to show that
a,8,a + /3.^8^0 + 7._^8,7 = a,S,a + ^^S,/ + y^8^y.
In fact, this equation is
(a,S, - a,S,) a + (^,S. - ^.S,) /3 + (7,8, - 7,SJ 7 = 0,
which we proceed to verify.
ORTHOGONAL SURFACES. 439
In the first terra the symbol a^S^ — a^8^ is
«2 (a.^x + ^A + yA) - «. («.A + ^A + 7,^J,
this Is (a^/3^ - a^^J d^ + {y^a,^ - 7,^0:,) d^ ;
or, what is the same thing, it is
and the equation to be verified is
(^< - 7^,) a + (7< - a^J /3 + {a.d^ - ^dj y = 0.
Writing «'^''y = S'i2'i2'
where if Z =/(a^, y, 2;) is the equation of the surface, X, F, Z are
the derived functions J^, |^, |^, and R = '^{X'+ Y' + Z'),
(XiJu (aIJ Ct^
the function on the left-hand consists of two parts ; the first is
^ m. - 7^J X+ iad^ - ad,) F+ [ad^ - ^JJ Z],
that Is ^ (a [d^Z- d, Y) + /3 «X- J Z) + 7 (J^ Y- d^X)],
which vanishes ; and the second Is
- ^ (« m - 7^,) + /3 (7^. - «<) + 7 w - ^^ji i?,
which also vanishes ; that is, we have identically
a,S,a + ^,S,^ + 7,^S^7 = a.8,// + /5,S,^^ + 7^5.^7,
and the vanishiog 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
.4' become on substituting in them a; + Z,a„ y+l^^^^ ^+^i7i in
place of x^ 7/, z; that is, these coordinates are
x-i la+ l^a^ + ?jSj (/a), &c.,
or, what Is the same thing, measuring them from A^ as origin,
the coordinates of ^' are
Z, (ttj + B^a + aS/),
h (7. + iKi + 7S/),
440 FAMILIES OF SURFACES.
and similarly those of C measured from the same origin A' are
Hence the condition for the angle to be right is
(a, + ZS,a + aV) K + ^S.a + «¥)
+ {^, -f 18^13 + /3S,?) {/3, + ?S,/3 + ^8J)
+ {% + ^^y + 7S.O (72 + % + 7S.O = 0.
Here the terms independent of ?, 8 J, 8 J, vanish ; and writing
down only the terms which are of the first order in these
quantities, the condition is
a^{l8^a+ a8J) + oiJl8^a + a8J)
+ ^^ [18.^0 + ^8 J) + /3,^{18^0 + ^S J)
+ 7. (% + 7^,0 + 7. (?5,7 + 7^,0 = 0,
where the terms in SJ, 8J vanish ; the remaining terms divide
by Z, and throwing out this factor, the condition is
(a,S,a + ^,8,/3 + 7.8,^7) + (a.S,a + ^A^ + 7,2,7) = 0.
By what precedes, this may be written under either of the
forms
a,8,a + AS,/3 + 7.2,7 = 0,
a3S,a + /3,8,/3-f 7^7 = 0,
and the theorem is thus proved.
Now in any system of 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
ORTHOGONAL 8UKFACES. 441
of curvature on the consecutive surface. The consecutive
surface (as constructed with an arbitrarily varying value of /)
is in fact any surface everywhere indefinitely near to the
given surface ; and since by hypothesis AA' and BB' intersect
and also AA\ GG' intersect, then AB and AB' intersect, and
also AG and A G' \ 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 tangents AB^ AG
of the lines of curvature at A meet respectively the tangents
AB\ AG' 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.
Prof. Cayley has shown that when the position of the point A
on the given surface is determined by the parameters j)^ q^ which
are such that the equations of the curves of curvature are
f = const., g = const, respectively, then the condition is that I
shall satisfy the same partial differential equation as is satisfied
by the coordinates a;, ?/, z considered as functions of 'p^ q^ viz.
the equation (Art. 384)
d'u \ -[ dEda 1 i. ^ ^ _ o
dpdq 'i E dq dp 2 G dp dq
The above conclusion may be differently stated : taking
r=f{x^ y, z) a perfectly arbitrary function of (a:, y, z)^ the
family of surfaces r=f[x^ ?/, s), does not belong to a system
of orthogonal surfaces ; in order that it may do so the foregoing
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 ic, ?/, 0, satisfies a certain partial
differential equation of the third order, Prof. Cayley's inves-
tigation 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 par-
ticular case where r is of the form r —/{x) + (f> (y) + \lf (z), the necessarj' 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. xii.
LLL
442 FAMILIES OF SURFACES.
479. Dnpin's theorem, and the notion of orthogonal surfaces
are the foundation of Lame's theory of curvilinear coordinates.*
Kepresenting the three families of orthogonal surfaces by
P = <^{^) Vi ^\ 2 = ^K 2/? ^) ^=/(^i 2/5 ^\ then conversely
cr, 3/, z are functions of j)^ 5', r which are said to be the
curvilinear coordinates of the point. It will be observed 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=f[x^ ?/, 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. 5- = const, respectively ; whence also (Art. 384)
iBj y, z each satisfy the differential equation
d'u 1 1 dE du 1 1 dG du
dpdq ^ E 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 Lame, but without the
geometrical interpretation.
Conversely we may derive another proof of Dupin's theorem
from these considerations ; taking cc, ?/, z as given functions of
j9, 2, ?', and writing
dx dx dy dy dz dz ^ .,
dp dq dp dq dp dq ^ ^ '
dx d'^x dy d'^y dz d^z ^ -, „
dp> dqdr dp dqdr dp dqdr '- ''
p. 241 (1847). That the same is the case generally was shown by Bonnet {Comptes
rendus, Liv. 556, 1862), and a mode of obtaining this equation is indicated by
Darboux, Ann. de V ccole normale, t. iii. p. 110 (1866), his form of the theorem
is that in the surface r —f (x, //, «), if a, /3, y are tlie 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 + ^dy + ydz — 0 shall be integrable by a factor. The
condition as given in the text is in the form given by Levy, Jour, de V ccole poJyt.,
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
dx dy
(which of course includes as well this result, as the particular case obtained by
MM. Bouquet and Serret) was obtained by Prof. Cayley, Comptes rendiis, t. LXSV.
(1872) ; but in a form which (as he afterwards discovered) was affected with an
extraneous factor.
* Lame, ConijHes rendvs, t. VI. (1838), and Liouv., t. v. (1840), and various later
Memoirs ; also Leqons sur les coordonnees curv'dignes, Paris, 1859.
ORTHOGONAL SURFACES.
443
the conditions for the intersections at right angles may be written
[q,r] = Q, [r,p]=0, [i>, ^J = 0,
and the first two equations give
dx _ (/// dz d?/ dz dz dy dz dx dx dz dx dy 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
to^, 2", r respectively, we find
[rp .q] + [pq. r] = 0, [pq. r] + [qr .j)] = 0, [qi' .p] + [rp .q] = 0,
that is [2r.^] = 0, [rp.q'] = 0, [2?5'.?-] = 0. The last of these
dx dy dz
dr '
dr '
dr
the foregoing
= 0,
equations, substituting in it for
values, becomes
dx dy dz
dp ' dp ' dp
dx dy dz
dq ^ dq ^ dq
d'x d'y ' d'^z
dpdq ' dpdq ' d^dq
and the equation [j^, 3'] = 0 is
dx dx dy dy dz dz
dp dq dp dq dp dq
These equations are therefore satisfied by the values of a:, 3/, z
in terms of j;, q^ r; and regarding in them r as a given constant
but p, q as variable parameters, the values in question represent
a determinate surface of the family r =f{x^ ?/, 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 Prof. Cayley's
differential equation already referred to. Let P be a point
on a surface belonging to an orthogonal system, PiV^ the normal,
PT",, PT^ the principal tangents or directions of curvature,
then, by Dupiu's theorem, the tangent planes to the two
orthotomic surfaces are iVPZ*,, NFT_^. 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 iVPP^, NFT^ must also meet its tangent plane at P'
444
FAMILIES OF SURFACES.
in the two principal tangents P'T^^ ^T^. This is the con-
dition which we are about to express analytically.
Take r —f{x, ?/, z) =0 for the equation of the family of
the orthogonal system, the given surface being that correspond-
ing 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 a;, ?/, z) be X, M^ N of the first order, and
a, J, c, fj 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 determined as the
intersections of T with the cone
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^ + z^ = 0^ 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", h'\ c", /", <j\ h"\x, y, zY = 0, or W= 0,
then the planes NPT^^ NPT,^ must be harmonic conjugates with
these also, so that the resulting condition is obtained by ex-
pressing that the three cones f/, F, W intersect the plane T in
three pairs of lines which form a system in involution.
Now we have here evidently to deal with the same analy-
tical problem as that considered, ]Co7ncs, Art. 388c, viz. to find
the conditions that three conies shall be met by a line in
three pairs of points forming an involution. The general con-
dition there given is applied to the present case by writing
a = y = c = 1 , /' = g' = h' = 0, and in the determinant form is
a
a
y
X, 0,
0 , J/,
0, 0,
0,
2/,
29", 27i'
1g , 27*
0
N
0
L
0
M
L
0
= 0.
ORTHOGONAL SURFACES. 445
We see then that the form of the required condition is
where ^, i3, &c, are the minors of the above written deter-
minant, and it still remains to determine a", h'\ &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 e between T+dr = 0, U+ 2ne -^ SW = 0, where T'
is L' + 3r+N% n is
X [aL + hM+gN) + ij[hL -f hM^fN) + z [gL 4/.¥+ cN\
and JJ' is aL' + bM'' + cN^ + 2yJ/iV+ '2gNL + 2hLM.
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 XZ, XM^ \N^ \ being
an infinitesimal whose square may be neglected, and the cor-
responding differential coefficients for the new point are
L-\-\hL, 3I+\8Mj N-\-XbN, a-^XSaj &c., where 8 denotes
the operation
T d -.r d ,^d
L-r + M^ + N-j- .
ax ay clz
Hence the equation of the tangent plane at P\ referred to that
point as origin, is L'x + M'y + N'z = 0, or T4 A.8 2^=0, where
8 T means xhL + yBM-\- z8N, Siud it is to be observed, that 8 J'
is the same as what we have just called n. And the equation
of the cone which determines the inflexional tangents is
V + XBU=0. The equations of this plane and cone referred
to the original axes are T-\- X [8T-T') = 0, U+X {8U- 20) = 0,
* Professor 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 /o is a given
function of x, y, z, the condition that the new surface shall belong to the same
orthogonal system is
and that this condition is equivalent to that given in the text.
446 FAMILIES OF SURFACES.
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 9
between
T+\{n-r) + d[T' + &c.) = 0,
u^ X [8 u- 2n) + 2^ (n + &c.) ^■d'{U' + &c.) = o.
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 Uhj J' also belongs, we need not attend to any terms
in the result which contain either T or U] nor need we attend
to any terms which contain more than the first power of \.
The terms then, of which alone we need take account, are
- 20 r (n - T') + y (S f/- n) = 0,
or dividing by T' , TB U-2n'' = 0.
We have thus a" = [U + 3P + N') 8a - 2 [8L)% &c,, and the
required condition is
[U -I- AP + N') (^Sa + ^8b + @Sc + 2jfSf+ 2<&8g -[- 2?^?SA)
= 2 (a, 33, er, dF, ©, WL8L, 8M, 8N)\
Prof. Cayley has shewn 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 U ■{■ AP -{■ N'\ 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 u — (a, k, h\x^ ?/)*, V = (a', 7^', h'\x^ yY-, are determined
It zi
by the equation *' ^ =0, Conies, Art. 342 ; if u and v be
given as functions of cc, y, z, where Lx + My + Nz =0, and
therefore u, = -, itt -r » &c., we find immediately that the
' dx N dz^ ^ ■'
ORTHOGONAL SURFACES.
447
= 0.
foci of the involution are given by the equation
L, M, N =0.
Thus then, or as in Art 297, the two principal tangents are de-
termined as the intersections of the tangent plane with the cone
ax + hy i- gz^ lix + hy -{-fz^ gx -\-fy + cz
X ^ y , z
L , M , N
"VVe shall write this equation
|(a, b, c, f, g, hX-»,3/)^r = 0,
that is to say,
^ = 2{3fg-Nk), h = 2{Nh-Lf), c = 2{Lf-2fg),
f=L{b-c)-^ Ng-Mh, g=31 (c- a) +LJi-Nf, h = N[a-h) + 3If- Lg.
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 0 = 0;
that is to say, the two relations are
^a + ^b + Cc + 2i^f + 2 (9g + 2ini = 0,
where A^ B^ &c. are the reciprocal coefficients he -f\ &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
Z7, F, W should be met by a line in three pairs of points form-
ing an involution, it is geometrically evident that if W be a
perfect square (Xa; + yti?/ + vs;)'^, this condition can only be satisfied
if \x-\- /jiy + 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 :
I X, i/, N
(^, 13, cr, iF, ffi, ^}jx, /., vY = - 2 u^, u„ 1.3
where in Mj, &c. we are to write for a;, ?/, z^ fjt,N~ vM^ vL - XiV,
448
FAMILIES OF SURFACES.
"KM- fiL ; that is to saj, in the case we are at present con-
sidering, the determinant is
L, M, N,
fiN- vM, vL - \N, \M- fiL,
where we have written Z', &c. for /nN— vif, &c. This deter-
minant may be otherwise written
L, M, N
r, M\ N'
\, Z, a , h , g
fi, M^ h, h ^ f
J'j N, g , /, c
But in the particular case where X = BL = aL -{■ hM+ gN, &c.^
this determinant may be reduced by subtracting the last three
columns multiplied respectively by L, Jf, N from the first ;
then observing that LL' + MM' -f NN' = 0, we see that, as we
undertook to shew, the determinant is divisible by U+ M^+ N'\
the quotient being
L\ M\ N'
Z, a , A , g
M, h, h, f
N, g, /, c
483. The quotient is obtained in a different and more con-
venient form by the following process given by Professor Cay ley.
The following identities may be verified, ^, &c., a, &c. having
the meaning already explained :
a = a {U + M' + N'') + 2i {MM- MSN),
ii=h{L' + M'' + N') + 2M{UN - ML),
® = c {U + M' + N') + 2N ii¥8L - UM),
jf ={ [L' + M' + N') + M{MBL- UM)+ N{LSN- N8L),
<!5 = g{U + M' + N') + N{ MM- MBN) + L {MBL - LS3I),
^ = ]x{L'' + M' + N') + L {LhN - NBL ) + M{NSM- MSN).
ORTHOGONAL SURFACES. 449
Hence we have
(aSZ + p?Sil/+ ffiSiY) = (aSi -f h8M+ gSN) [U + ]\P + N^)
+ [LhL + MUl+MN) [MM- 3IBN),
with corresponding values for
and hence Immediately
(a, 23, ©, iF, ffi, ^}1SL, SM, BNY
= (i^ + M' + iV^^) (a, b, c, f, g, hXSX, S.l/, S^')^
Hence the equation, Art. 481, omitting the factor L^ + 31'^ + N^j
becomes
^Sa + 23SJ + ©Sc + 2dFy+ 2(SS^ + 2^}Sh
= 2 (a, b, c, f, g, hXSi, BM, BNf.
484. There Is still another form In which the result may be
expressed. Writing, as usual, In the theory of conies, hc—f^=A^
&c., the determinant at which we arrived at the end of Art. 482
is, when expanded,
- [ALL + BMM' + CNW + F{MN' + M'N)
+ G [NU + N'L) + H[LM' + L'M)].
Now, from last article
2LL' = ^ - [U + M' + N') a, &c.,
MN' + M'N=^:S - {L + M'' + A^'"') f, &c.,
and remembering that Aa, + &c. = 0, the expanded determinant
last written is seen to be
^A + 235+ ©6'+ 2dFi^+ 2(&G + 2^^H,
and thus eventually the differential equation Is given In the form
aSa + mBb + ©5c + 2iFS/+ 2<i58g + 2?^a/i
= 2{^A + l5B + &C+2jfFi-2(&G + 2^H].
485. As a particular case of this equation of Prof. Cayley's
may be deduced that which Bouquet had given {Liouville^ XI.,
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'\ b=Y'\ c = Z'\f=g^h = 0'^
MMM
450 FAMILIES OF SURFACES.
A = Y"Z'\ B = Z"X'\ C = X"Y'\ F=G = H=0;
a = ( F'' - Z") X' Y'Z\ 23 = [Z" - X") X' TZ\
© = (Z"- Y")X'Y'Z';
ha = X'X"\ hh=TT'\ U = Z'Z"\
and the differential equation being divisible by X'Y'Z' is
reduced to
X'X'" ( Y" - Z") + Y' Y'" {Z" - X") + Z'Z'" {X" - Y")
+ 2 ( r" - z") [Z" - X') [X" - Y") = 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 M. Bouquet observed that the con-
dition just found is satisfied when the given system is of the
form a;'"?/"^^ = r, but he gave no clue to the discovery of the
conjugate systems. This lacuna was completely supplied by
M. Serret, who has shewn much ingenuity and analytical
power in deducing the equations of the conjugate systems, when
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 U posteriori. He has shewn
that the three equations,
X
represent a triple system of conjugate orthogonal surfaces. The
surfaces (r) are hyperbolic paraboloids. The system [p) is
composed of the closed portions, and the system (g) of the
infinite sheets, of the surfaces of the fourth order,
{z" - y'y - 2f [£' 4 rf + ^x") + / = 0.
M. 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.
ORTHOGONAL SURFACES. 451
He finds also (in a somewhat less simple form) the following
equations for another system of orthogonal surfaces,
q = [x^ + (of + Q)'^z')i + (cc" + 0)''f + o}z'^)i,
r = {x^ + mf + (i>'z^)h - (x" -f oi'if + eu2')l,
where &> is a cube root of unity.
An interesting system of orthogonal surfaces, and very
analogous to the system of confocal quadric surfaces, is given
by M. Darboux in his Memoir above referred to, namely,
the system of bicircular quartics
where a, J, c, d are given constants, and in place of \ we are to
write successively the three parameters p, g-, r. The formulee
for a;, y, z in terms of p, q^ r, are
^ {a — b){a — c) '
^ ^^ {b-c){b-a) '
(c — a) (c - 6) '
where, writing for shortness,
(2d + p)j2d + q){2d + r) {2d - p) {2d - q) {2d - r)
^ ~ ld{2d- a) [2d- h) [2d - c) ' '^~ Id [2d + a)[2d + b)[2d+c) '
4fP
we put M =
y[Adm)±^/[idn)Y'
If c?= CO , the system of surfaces is
^■■2 ^.2 ^'^
^ ^ ^ i — n
^Ta. "^ b + X "*" ^+X "^ * " "'
which Is in effect the system of confocal quadrics: a slight
change of notation would make the constant term become — 1.
Mr. W. Roberts, expressing in elliptic coordinates the con-
dition that two surfaces should cut orthogonally, has sought
for systems orthogonal to L + M+ N^r^ where Lj J/, N are
452 FAMILIES OF SURFACES.
functions of the three elliptic coordinates respectively. He
has thus added some systems of orthogonal surfaces to those
previously known [Comptes rendus, September 23, 1861). Of
these perhaps the most interesting, geometrically, is that whose
equation in elliptic coordinates is ^v = aX, and for it
he has given the following construction : — 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 shewn 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
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 il/, or a line of curvature, is a
circle in a plane perpendicular to the principal plane con-
taining AB.
( ^^^^ )
CHAPTER XIV.
SURFACES DERIVED FROM QUADRICS.
487. Before proceeding to surfaces of the third degree
we think it more simple to treat of surfaces derived from
quadrics, the theory of which Is more closely connected with
that explained in preceding chapters. We begin by defining
and forming the equation of Fresnel's 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 ex-
pressed in terms of the angles which a perpendicular to its
plane makes with the axes of the surface. The same equa-
tion 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
aV by 6'z' ^
a — r 0 —r c —r
where r"^ = x^ + y* + z\ Or, multiplying out,
{x' + if + z^) {a'x' + ly + 6'z')
- [aV {b' + c') + by {6' + d') + dV (a" + 5"]] + d'b'^c' = 0.
♦ See Fresnel, Memoires de V Imtitut, vol. vir., p. 136, published 1827.
454 SURFACES DEEIVED FROM QUADRICS.
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
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 xi/. 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 £c', 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' + 3/?/' - c") {a'xx' + Wyy - a'b') + z' [a' - c') {b' - d') = 0.
The generating quadric being supposed to be an ellipsoid, it
is evident that In the case of the section by the plane Zy the
circle whose radius is c, lies altogether within the ellipse
whose axes are a, 5; and in the case of the section by the
plane a;, the circle whose radius is a, lies altogether without
the ellipse whose axes are &, c. Real double points occur
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 x^ + y'^ + 0^, aV + b''y'^ + c'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 surface
situated at infinity. Thus the surface has in all sixteen nodal
points, only four of which are real.
t
THE WAVE SURFACE. 455
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 sup-
posed to be perpendicular to the
plane of the paper POQ. For
the tangent plane at Q passes
through PQ and is perpendicular to the 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 0^ is perpendicular to the tangent line in the
section Q OR^ it is an apsidal radius of that section.
It follows that OT^ the radius of the apsidal surface corre-
sponding to the point Q^ lies in the plane POQ^ and is per-
pendicular and equal to OQ.
456 SUEFACES DEKIVED FROM QUADKICS.
491. The perpendicular to tlie tangent plane to the apsidal
surface at T lies also in the plane POQ^ and is perpendicular
and equal to OPf^
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 0T\ 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 0T= 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
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^.
492, The polar reciprocal of an apsidal surface, with respect
to the origin 0, is the saine as the apsidal of the reciprocal, 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
* These theorems are due to Prof . Mac Cullagh, Transactions of the Royal Irish
Academy, vol. xvi. in his collected works, p. 4. ic.
THE WAVE SURFACE. 457
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
lengtiis being inversely as OS^ OT ure 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.
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. We shall now
show geometrically that this conic is a circle.*
493. 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
* 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
Mac Cullagh, Ibid. p. 248. See also Pliicker, " Discussion de la forme generale des
ondes lumineuses," Crelle, t, XIX. (1839), pp. 1-44 and 91, 92.
N NN
458
SURFACES DERIVED FROM QUA.DRICS.
J}I
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 0 in which the two lines intersect
is supposed to be above the paper, P being the foot of the
perpendicular 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 OBP. But the plane
OAB intersects the plane of the
paper in a line BT parallel to OA^ and therefore perpendicular
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, p. 100). 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.
Lenma 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 perpendicular
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 OR 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
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
THE WAVE SURFACE. 459
which moves In a fixed plane by Lemma III., therefore OS
generates a cone whose circular sections are parallel to the
planes POR^ QOR. Now T is a fixed point, and TS is
parallel to the plane FOBj 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 POM 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, 6' 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'^^(^-^') ^m^\{e-e')
p' 6' ' d'
and for the other
p"~' c* "'' a* '
while -; n~ {—, r. 1 sin ^ sin 6'.
p' p' W a')
It follows hence also that the intersections of a wave surface
with a series of concentric spheres are a series of confocal
sphero-conlcs. For, in the preceding equations, if p or p' be
constant, we have d ±6' constant.
496. The equation of the wave surface has also been ex-
pressed as follows by Mr. W. Roberta in elliptic coordinates.
The form of the equation
aV
— -. = 0,
460 SURFACES DEKIVED FROM QUADRICS.
shows that the equation inaj be got by eliminating r'' between
the equations
-j^-^ + -7^, , + T^ = 1 } and x' + f + z' = r\
Giving r'"* 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
If = &'■' — c^, Jc^ = a^ — c\ Since r'^ is always less than d^ and
greater than c\ the equation always denotes a hyperboloid, which
will be of one or of two sheets according as r'^ is greater or less
than b'^. The intersections of the hyperboloids 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 \, /i, V denote the primary axes of three confocal surfaces
of the system now under consideration which pass through any
point, then the equation gives us r'"' — c^ = /*''', but (Art. 161)
whence the equation in elliptic coordinates Is
x' + v' = c' + /i' + j<;'=d'-\-b-'-c\
In like manner the equation of the other sheet is
X' + fi' = a' -i- F - c\
The general equation of the wave surface also implies
fj^ + v^ = ci + y^ — c\ but this denotes an imaginary locus.
Since, if \ is constant, fi is constant for one sheet and v
for the other, it follows that if through any point on the
surface be drawn an ellipsoid of the same system, it will meet
one sheet in a line of curvature of one system, and the other
sheet in a line of curvature of the other system.
If the equations of two surfaces expressed in terms of
\, /*, V, when differentiated give
Fd\ + Qdfj, + Bdv = 0, Fd\^Q'dfi + R'dv = 0^
the condition that they should cut at right angles is (cf. Art. 411)
PP'{X'-h'){X'-U') QQ\,M'-Ji'){k'-fj:') RB\h'-v'){U'-v')
which is satisfied if P= 0, ^ = 0, R' = 0. Hence any surface
THE WAVE SQRFACE. 461
v = constant cuts at right angles any surface whose equation is of
the form 0 (X, fx,) = 0. The hyperboloid therefore, v = constant,
cuts at right angles one sheet of the wave surface, while it
meets the other in a line of curvature on the hyperboloid.
497. The 2^^ci,ne of any radius vector of the wave surface and
the corresponding perpendicular on the tangent plane^ makes equal
angles with the 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 QOR of the generating ellipsoid
and the other two planes are perpendicular to the radii of
that section whose lengths are 5, 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 xy'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'"' — a'^, a' — a'"*, which we
shall call p^, p'*. These, then, give the two values of the
radius vector of the wave surface, whose direction-cosines are
"^-i-, 7^1 y • We shall now calculate the length and the
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
462 SURFACES DERIVED FROM QUADRICS.
~^ , ^ , ^-TT • 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
p.^' pJyi PoP±
^^dW ^Ph'h"' ^'^cV^'
where _ = ^y^ |_ + -^^^,, + -^j .
Now if the quantity within the brackets be multiplied by
[a^ — a^)\ we see at once that it will become -^ + -r^ • Hence
p p
-P ^P . andP^- ^^
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
r/ r f r r
jp X p y p z ^ px py p
z
_. Po^ir Pn^l^ Po^—-
"i. 1 ;,'/2 ) „'tx } -^ r „'!.„'% ) -^ r v^u't. > ■*■ r '^ '-.i
^n , ^./. , ^v. , -r^2^.o -r^.^
Forming, by Art. 15, the direction-cosines of a line perpendicular
to these two, we find, after a few reductions,
pp \ a j^ pp \ h ') ^ pp \ c V
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
^x' (i - f^) + yy' (i - f^) + --' (i X) ^pp:
THE WAVE SURFACE. 463
499. If 6 be the angle which the perpendicular on the
tangent plane makes with the radius vector, we have P= p cos^ ;
2 2
but we have, in the last article, proved P* = ./ ,.. . Hence,
' ' ^ p +y
Q,Q's,'-Q = ^- — -^ tan"^=-^-. This expression may be trans-
formed by means of the values given for p and i^' (Art. 165).
We have therefore
, _ a^JV ,, _ (g- - p') [V' - p') jc' - p^)
PP P [P -P )
Whence ,an"« = - ^ " ' ^ ^/^ °^ .
P"
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^ &, or c, and becomes indeterminate when
p = p=l.
r
500. The expression tan^ = — leads to a construction for
P
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 p^ it is
evident from the expression for tan 6^ 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."
464 SUEFACES DERIVED FROM QUADRICS.
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.
Ex. 1. To transform the equation of the surface, as at p. 151, so as to make the
radius vector to any point on the surface the axis of z, and the axes of the corre-
sponding section of the generating ellipsoid the axes of x and y.
Ans. (a;« +y^ + z^) {p^z^ + {p'^ + p") x^ + ( //'^ + p"^) y"^ + 'Ipp'xz + 2pp"yz + 'ip'p"xy\
-pH"" (p- + p'2) - X- {pY- +p'-p'" +p"-p- + |"V')
- / {pV^ +p'y^ + p"'^p'^ + p~p'^) - 1pp'p"'xz - 2p>p"p'^yz +p'^pY^ = 0.
It is easy to see that if we make x and y — 0 in the equation thus transformed,
we get for x- the values p- and p'^ as we ought. If we transform the equation to
parallel axes through the point z — p, the Unear part of the equation becomes
2pp {p^~ - p'^) ipz + 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 surface
obtamed by writmg — for a, &c., in the equation of the wave surface.
Ans. (a;2 + ?/2 + s^) {^2^'2a;2 + p'pY - 2pp'p'"-xz - ^pj/'p^yz + z^ {p'^-p'^ + p"~p^ + p'''p'")]
- X< (p2 +^"2 + p'2) a.2 _ X4 l^jf- +p'2 + ^2) y2 _ X4 (^'2 + p"2 + ^2 + ^'2) ^2
+ 2\Yp"xy + 2\*p2)'xz + 2\*pp"yz + X* = 0.
We know that the surface is touched by the plane pz — X^, and if we put in this
value for z, we find, as we ought, a curve having for a double point the point y — Q,
ppx = p'k^. If in the equation of the curve we make y = 0, we get
(px-^fj{p'^x^^^^ip'^-p^},
P'^-\- / '2 2 ^ ^^
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-y^ +p^ (p'"- - p^)} x^ - 2p'p"p''xy + {p'Y + p' Xp- - r)} /,
a result of which I do not see any geometrical interpretation.*
* I have no space for a discussion what the lines of curvature on the wave
surface are not, though a hasty assertion on this subject in Crelle's Journal has led
to interesting investigations by M. Bertrand, Comptes Rendus, Nov. 1858 ; Combescure
and Brioschi, Tortolini's Annnli di Matematica, vol. ii., pp. 135, 278. It is worth
while to cite an observation of Brioschi, that if in the plane Ix + my + nz = (p ;
I, m, n, <p be functions of two variables p, q, as in Art. 377, then the plane will
envelope a surface in which curves of the families p = constant, q = constant, will,
The surface of centres. 465
501. The Surface of Centres. 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. LXir., 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 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 aa
follows : Let it be required to find a point xt/zw on a quadric
Z7, [ax^ + %^ + cz^ + dw'^), such that the pole, with respect to
another quadric F, {x^ + y'^ + z'^ + iv'^) , of the tangent plane to
U at X7/ZW, 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 —Xx^ 2/'~^^? z' — \z^ lo' — \io^ and
expressing that the polar plane of this point, with regard to F,
shall be identical with the polar plane of xyzw^ with respect
to C7, we get the equations
x' = [a-\- \) X, y = (b A- X) ?/, s' = (c + X) 2, w' = {d + \) w.
And since xyziv is a point on f/, X is determined by the equation
ax' Z»v" c^" dw'^
[a + xy {b + xf {c + xf [d + xy
When X is known, x, ?/, z, w are determined from the preceding
system of equations, and since the equation In X is of the sixth
degree, the problem admits of six solutions. If we form the
at their intersection, be touched by conjugate tangents of the surface, if the condition
be fulfilled,
I, m, n, (f>
If, nil, re„ <pi
h) ^2> '"■2^ 'P2
'121 "^iz* ^m *Pl2 — ^f
where the sufBxes 1, 2, denote differentiation with respect to u and v respectively ;
while the curves will cut at right angles if
{P + m^ + n^) (/,^2 + TO,m2 + n,«2) = {^h + '"'"i + ""i) i^h + ^^2 + '^♦*2)«
000
466 SURFACES DERIVED FROM QUADRICS.
discriminant, with regard to \, of this equation, we get the
locus of points x'yzw for which two values of \ coincide,
and rejecting a factor cc'^y'VW^ (which indicates that two values
coincide for all points on the principal planes), we shall have
a surface of the twelfth degree answering 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
Solve for a*', 3/', z' from these equations, and substitute in the
equations satisfied by xyz\ viz.
X y z X V z ^
now write for a''', d^ - h^^ &c., and we get
aV hy c'z^
(a" - hy '^ [b' - Ky "^ (c^ - Kf ~ '
aV hY c'z' _^
These two equations represent a curve of the foui'th 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 Ji^ be-
tween the equations ; or (since the second equation is the diff'er-
ential of the first with respect to Ji^) by forming the discriminant
of the first equation.
503. I first showed, in 1857 {Quarterly Journal, vol. II.,
p. 218), that the problem of finding the surface of centres was
reducible to elimination between a cubic and a quadratic, and
Clebscli 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 fMU-\-XV] which for the canonical
THE SURFACE OF CENTRES. 467
form (Art. 141), is [ajx + X) [h^x, -I- \) [c/u, + \) [dfjb + X), and let il
denote the reciprocal of fiZ7+\F, viz.
{bfi + X) [cjx + \) [d/jt, + X) x''+ [cfM + X) {d/jb 4- X) (a/t + X) /+ &c.
then we have — = h ^ — - + &c.
A rtyU- -f X o/x + X
Now, if we differentiate the right-hand side of this equation
with respect to /*, and then make yu. = 1, we obtain the equation
(Art. 501) which determines X, which therefore may be written
^A _ da
djj. dfi '
This last equation, which is the Jacobjan of 12 and A, being
the result of eliminating m between A + ?>iXI2 and its differential,*
will be verified when A + 7n\Q. has two equal roots. Its differ-
ential again O, 2 = ^ 'j-n being the result of elimination
between A + m\Q. and its second differential, will be verified
when A + mXil has three equal factors. But both Jacobian and
its differential vanish when both A and 12 vanish. Thus then,
as was stated (Note p. 213), the discriminant of the Jacobian
of two algebraic functions A, i2, contains as a factor the result
of elimination between A and 12 ; and as another factor, the
condition that it shall be possible to determine m, so that
A + 7nXI2 may have three equal factors. In the present case
the elirainant of A, 12, gives the factor x^i/'^zho\ 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 + «iXI2.
504. The discriminant of any algebraic function
ayfr (X) + (X - a)' (f> (X),
must evidently be divisible by a ; and if after the division wo
make a = 0, it can be proved that the remaining factor is "^[a)
^ [af multiplied by the discriminant of ^ (X). Thus, then, the
section of Clebsch's surface by the principal plane w is the conic
* The factor X is introduced to make Q as well as A a biquadratic function in /i ; \.
468 SURFACES DERIA'^ED FROM QUADRICS.
flic' Jw* cz^
, jTi + ,, .... + 7 tt;, taken three times, together with
\a — d) \b — d) {c-d) ' °
the curve of the sixth degree, which is the reduced discriminant of
ax^ hjf cz^
+ Tri^.+
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
aoi? Inl^ cz^
[a + Xf [b + \f (c + X)''
may be regarded as the envelope of all conies which can be
represented by this equation, and therefore touches every parti-
cular conic of the system in the four points where it meets the
conic represented by the differential of the equation with re-
gard to A-, viz.
ax^ hy'^ cz^ _
[a + X)' ^ {b + X)' "^ (^Txf ^
The coordinates of these points are ax^ =[a + Xf [b — c)^
hy^ = [b -Wf [c - a) ^ cz^ = {c + \f [a-b)] and the equations of
the common tangents at them to the conic and its envelope are
In the case under consideration \ = — d. If, then, we use the
abbreviations
(a -h)(a-c)[a-d) = - A\ [b - a)[b-c){b -d) = - B\
(c -a){c-b){c-d) = - C\ [d- a) [d- b) [d- c]=- B\
the equations of the common tangents to the conic, and the
envelope curve, are
A- B- G
The reasoning used in this article can evidently be applied to
other similar cases. Thus, the surface parallel to a quadric
(p. 176, 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, p. 189) touch
the conic in their plane.
i
THE SURFACE OF CENTRES. 469
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 A., we arrive at a sjsteni of
equations reducible to the form
ax'' hii^ cz^ dw''
(a + X)* ^ (6 + X)* (c-fX)* («( + \)
d'x' &y d'z' d'w
aV hY ^z" dho^
+ i, \ 4 + 7-——. + ,, , .,4 = 0.
[a + \f [b + \)* [c + \:* ■ {d+ Xy
The result of eliminating \ between these three equations
will be a pair of equations denoting a curve locus. Now, solving
these equations, we get
r4 ={b-c) [c — d] {d- J), ■ ^ z= (^c— a) (a - d)[c- <f), &c.
whence a + X, & + X, &c. are proportional to 0^3^ Aj^^ &c.
Substituting from these in the equation (Art. 501)
a^i? by'' cz''' duo'
{a + X}' {b + \f {c + X}'' [d + Xf
a^x b-i/ c-z d^w
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.*
* The existence of these eight planes may be also inferred from the consideration
that the reciprocal of the surface of centres has an equation of the form (Art. 199)
U^ — VW, and has therefore as double points the eight points of intersection
of IJ, V, W. The surface of centres then has eight imaginary double tangent
planes, which touch the surface in conies (see Art. 271). The origin of these planes is
accounted for geometrically, as M. 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 centro-surface will arise every
time that a surface contains a right line which meets the circle at infinity.
470 SURFACES DERIVED FROM QUADRICS.
But if we eliminate \ between the three equations
a-\-\ = aix^A^^ h + \ = Uy^B^^ c + \ = cM(7*,
so as to form a homogeneous equation in a;, y^ 0, we get
aU* [b - c) ic* + UB"^ [c - a) 3/* + cW^ [a -h)z^ = 0,
which denotes a cone of the second degree touched by the planes
£c, y, s. 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 condition for
a single pair of equal roots is got by eliminating m between a
quadratic and a cubic equation, namely, the S and T of the
biquadratic A + mXQ.. If we write these equations
a + bm-\- crr^ = 0, ^ + Bm + Grn!' + Drr^ = 0,
it will be found that the degrees in x, y, z, w of these coefficients
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 inde-
terminate '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 ra between two quadratics
a^hm-^ crn^^ a-\- h'm + drri'' is [ac— ca^-\- [ha—db') {be — cb') = 0.
But if we remember that a [be — cb') = b [ac — ca) + c (Ja' — ab')^
this result may be written
a [ac — ca'Y — b [ac — ca') [ha' - ab') + c {ha' — ab'f = 0,
showing that the intersection of ac — ca'^ ha' — 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 + cn^ = 0, ^ + Bm + Cm^ + Dm'' = 0,
THE SURFACE OF CENTRES. 47l
"we may substitute for the second equation that derived by
multiplying the first by A^ the second by o, and subtracting, viz.
(Ba -hA)+{Ca- cA] m + Dara^ = 0,
and thus, as has been just shown, the result of elimination may
be written aP^ - hPQ + c^' = 0, where
P=hcA- acB + d'D, Q = [ac- ¥) A + abB- a' C.
We thus see that the curve PQ is a double curve on the surface
of centres ; but since P is of the sixth degree and Q of the
fourth, the nodal curve PQ is of the twenty-fourth. Further
details will be found in Clebsch's paper already referred to.*
507. It is convenient to give here an investigation of some
of the characteristics of the centro-surface of a surface of the m'"
order.f 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[m — \f (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. II.).
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
* See also a Memoir by Prof. Cayley {Cambridge Philosophical Transactions,
vol. XII.) in which this surface is elaborately discussed. He uses the notation ex-
plained, note, Art. 409, when the equations of Art. 197 become
- /3ya=a;^ («2 + pY (a^ + q), - yahY- = [f'^ + PY (i' + 5), - "^^"2" = (t'' + P? (<^' + ?),
a, P, y having the same meaning as in Art. "206.
t This investigation is derived from a communication by M. Darboux to the
French Academy, Comptes Rendus, t. LXX. (1870), p. 1328.
472 SURFACES DERIVED FROM QUADRICS.
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 extended 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 any ploint is m-\- n + a] or, when the
surface has no multiple points, is m^ — 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 corresponding
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 congruency of lines
(see Art. 453), and the two numbers just determined are the
order and class of that congruency.
508. To find the locus of points on a surface, the normals
at which meet a given line,
ax -\- hy + cz -{■ d = 0^ ax + Vy + cz + J' = 0.
Substituting in these equations the values for the coordinates of
a point on the normal (Art. 273), x = x -^-QTJ^^ y = y+^^ii
z = z' + 6U^^ and eliminating the indeterminate ^, we see that
the point of contact lies on the curve of intersection of the
given surface with
[ax-\-hy + CZ -{■(!) [a U^-\-h'U^-\-c U^
= [a'x + Vy + c'z-^d') {aU, + bU^ + cU^)j
a surface also of the m}^ 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-surface.
A tangent plane to that surface contains two infinitely near
THE SURFACE OF CENTRES. 473
normals to the given surface (Art. 306) ; and therefore the
tangent planes to the centro-surface 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 sui'faces of the m^'^ order, being equal
to the rank of the corresponding developable, is (Arts. 325, 342)
m^ (2m — 2) ; but, since in tliis case the line through which
the tangent planes are drawn meets the curve in m points,
this number must be diminished bj 2m. The class of the
centro-surface therefore is 2m [wb^ — m— 1).
510. Darboux* investigates as follows the order of the
centro-surface. Let /a and v be the two numbers determined
in Art. 507, viz. the order and class of the congruency formed
by the normals ; let M and N be the order 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, yu, — 1 other normals may be drawn ; we see then
that to any position of one plane correspond v (^ — 1) positions
of the other. It follows then, from the general theory of
correspondence, that there will be 2v {fx — 1) cases of coincidence
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
2v(/i- l)=a;+il/.
But in like manner consider the correspondence between
points on the line such that a normal from one is coplanar with
* Similar investigations were also made independently by Lothar Marcks. (See
Math. Annalen, vol. v.). The investigation may be regarded as establishing a general
relation (which seems to be due to Klein) between the order and class of a congruency,
and the order and class of its " focal surface " (see Art. 456).
PPP
474 SURFACES DERIVED FROM QUADRICS.
a normal from the other, and we have
2fi{v-l)=x + N,
whence M-N=2{fjb-v)
and putting in the values already obtained for //., v, K^ we have
M=2m{'m-1) {2m -\).
511. The number thus found for the order of the centro-
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 (Higher Plane Curves^
Art. 112) be a curve of the order 3a + /c. And besides (as in
Art. 198) the centro-surface will include the polar reciprocal
of the section with regard to the circle at infinity. The order
of this will be «, 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 order ty^iu '^ therefore the normals
along the parabolic curve generate a surface whose order is
^.m^ [m — 2). But the section of this surface by the plane
infinity includes the 4m (m — 2) normals at the points where
the parabolic curve itself meets the plane infinity. The curve
locus therefore at infinity answering to finite points on the para-
bolic curve is of the order 4?« {m— 1) (m - 2). The total order
then of the section of the centro-surface by the plane infinity, is
3m [m- 1) + 3??2 {m — 1) + 4?7? [fji — 1) (??z — 2),
or 2m [m - 1) (2m — 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 in a plane section of the reciprocal surface
which is of the fourth degree. Mr. F. Purser* has shown
* Quarterit/ Journal of Mathematics, vol. xiii., p. 338.
PARALLEL SURFAClilS. 475
tbat 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
^'^^ 7,2^,2 „1-i
ax o y c z
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 normo2)olar 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 wliich 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 synnormal.
There are therefore ten synnormals through a point.*
512. Parallel Surfaces. We have discussed, p. 176, the
problem of finding the equation of a surface parallel to a
quadric, and we investigate now the characteristics of the parallel
to a surface of the n'" order. 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, which see Higher Plane Curves^
p. 101. The order of the parallel Is found by making k the
modulus = 0 in its equation, which will not affect the terms of
* In 1862 M. Desboves published his " Theorie nouvelle des normales aux surfaces
du second ordve," in which the locus line and the related surface are discussed under
the names synnormal and normopolar surface. Mr. Purser independently arrived
at the same results {Quarterly Journal, vol. viii., p. G6) and showed the equivalence
of the relation of the fourth order with the invariant 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 j and gave a construction for any synnormai
through a point.
476 SURFACES DERIVED FROM QUADRICS.
highest degree in the equation. The result will represent the
original surface counted twice, together with the developable
enveloped bj 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. xvii. post.) that the
rank of a developable enveloping a surface and a curve is
nm 4- a/, where a, n, are characteristics of the surface and m\ r
of the curve. In the present case vi = / = 2, and the rank of
the developable is 2 [n + a). The order of the parallel surface
is therefore 2 [m + w + a) or 2 [r(t^ — m^ + m) ; 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
CLX + ^y -\- 'yz -\- 8 = 0, and if the surface be given by a tangential
equation between a, /3, 7, S, then the corresponding equation of a
parallel surface is got by writing in this equation for 8^ S -f A'p,
where p^ = a' + /3'^ + 7*. 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 Planes Curves,
Art. 1 17a). Thus then, since (Art. 279 et seq.) we have for the
tangent cone to the primitive,
fi = a = 'm{m — l)j v = n = ni{m-lf,
K = 3m{m- 1) {m — 2), 1 = Am [m — l)[m- 2),
we have for the tangent cone to the parallel {Higher Plane
Curves, 1. e.)
/x = 2 (n + a) = 2m"''(w-l), v = 2n,
K = 2m [m — 1) (4wi — 5), t = 8m [m — 1) (»z - 2).
* 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 aiticle is in
Proceedings of the London Mathematical Society, 1873.
PARALLEL SURFACES. 477
Again, the reciprocal of a parallel surface is of the order 2?«,
having a cuspidal curve of the order 8m (ni — 1) [m — 2), and
a nodal of the order
m{m-l) {2m* - 6?/i'' + 6???' - 16m +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 dis-
criminant 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. Mr. Koberts 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 = co ^ the locus of points on the surface of centres
for which a principal radius of curvature = ^, is the section of
the surface of centres by the plane infinity, counted twice, since
k may be ± co . The singularities of the parallel surface here
assigned are sufficient to determine the remainder by the help
of the general theory of reciprocal surfaces hereafter 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 order, and
having no cuspidal curve, but having a nodal conic. The
parallel itself is of the twelfth order ; 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, the section of the
quadric itself and the circle at infinity. The remainder of the
478 SURFACES DERIVED FROM QUADRICS.
nodal curve consists of 16 right lines, each meeting the circle at
infinity.*
514. Pedals. 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 distinctive 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 Mr. W. Roberts, Liouville, vols. X.
and XII., by Dr. Tortolini, and by Dr. Hirst, Tortolini's Annali^
vol. II., 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 0 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 imme-
diately follows hence, that the perpendicular OR on the tangent
plane at Q lies in the plane POQ, and makes the angle
QOE = POQj so that the right-angled triangle QOB is similar
to POQ] and if we call the angle QOR^ a, so that the first
perpendicular OQ \^ connected with the radius vector by the
equation p = p cosa, then the second perpendicular OB will be
p cos^a, and so on.f
* The parallel to a curve in space might also have been discussed. This is a
tubular surface.
t Thus the radius vector to the n^^ pedal is of length p cos"a, and makes with the
radius vector to the curve the angle na. Using this definition of the method of
INVERSE SURFACES. 479
It is obvious that if we form the polar reciprocals of a curve
or surface A and of its pedal JR, 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,, /S'^, .../S',,, the reciprocals will be
a series S\ /S"_j, S'_^^ ...8'_^^ 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 surface
derived from it by substituting in its equation, for the radius
vector, its reciprocal) ; and that the inverse of the series S^^
S^^^ ..-S^ will be the series S\ /S'_j, ■.•S'_^^_^.
515. Inverse Surfaces. As we may not have the oppor-
tunity to return to the general theory of inversion, we give in
this place the following statement (taken from Hirst, TortoUni^
vol. II., 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 corresponding
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 ab joining any two points on one curve makes the
same angle with the radius vector (?a, that the line joining
the corresponding points ah' 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.
(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.
derivation, Mr. Roberts has considered fractional derived curves and surfaces.
Thus for n — ^, the curve derived from the ellipse is Cassini's oval. Aq
analogous surface may be derived from the ellipsoid.
480 SURFACES DERIVED FROM QUADRIC'S.
(4) It follows immediately from (2), that the angle which two
curves make with each othet 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 corresponding 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 sub-contrary 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
a sphere B which cuts A orthogonally. The plane, therefore,
which is the inverse of A cuts 5' 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 osculating
circle of a. If now a, be the normal section which 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 c^ the oscu-
lating circle of a,. But the normal wi'c, evidently touches the
sphere A at m\ so that c, is the vertex of the cone circum-
scribed 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 mj will
INVEESE SURFACES. 481
evidently correspond two sections enjoying the same pro-
perty ; therefore to the two principal sections on one surface
correspond two principal sections on the 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 Higher Plane
Curves^ p. 106, that the inverse of a surface is of the order 2»i,
and class 3?n + 2a-\- n = nf + 2/n, that It passes m times through
the origin and m times through the circle at Infinity; and
hence that the order and class of the first pedal are 2«,
m + 2a + 3>2, and of the first negative pedal 3m + 2a + n and 2m.
516. The first pedal of the ellipsoid -5 + fr + "5 = 1) being
the inverse of the reciprocal ellipsoid, has for its equation
ax + by + cz — [x + y i- z) .
This surface Is Fresnel's " Surface of Elasticity." The Inverse
of a system of coufocals cutting at right angles is evidently 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 concycllc quadrics.
The origin Is evidently a double point on this surface, and
the Imaginary circle in which any sphere cuts the plane at
infinity Is a double line on the surface.
517. Prof. Cayley first obtained the equation of the first
negative pedal of a quadrlc, that is to say, of the envelope
* Dr. Hart's method of obtaining focal properties by inversion (explained Higher
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 (^Higher Plane Curves, Art. 281) from the focal property of the conic
p + p' = const, is inferred a focal property of the curve in space Ip + 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 <S'm?' une classe remarquable de courbes et de surfaces algi briques). A surface
which is its own inverse with regard to any point has been called an anallagmatic
surface.
QQQ
482 SURFACES DERIVED FROM QUADRTCS.
of planes drawn perpendicnlar 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 xy'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
where t = x^' -{■ y''' + z'\
Solving these equations for x\ y\ z' and substituting their
values in the two equations
'li f'i '1
xx' + yy' + zz' = x" + y'' + z'\ ^ + |^ + ^_=i,
x^ it z"
we get 1- • + = t,
(;-^) ^-^ M\
Now the second of these equations is the differential, with
respect to #, of the first equation ; and the required surface
is therefore represented by the discriminant of that equation,
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 a?, y, s, while
O, i>, 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^\jr • consequently It can contain x, 3/, 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
x' f
a rr
2-3 2-«
NEGATIVE PEDALS. 483
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\ in the last equation. This conic, which is a double curve
on the surftice, touches the curve of the sixth degree in four
points. The double points on the principal planes evidently
answer to points on the ellipsoid, for which t^x"' -\- y"'' + z'' = 2a^
or 2b'^ or 2(f. There is a cuspidal conic at infinity, and, besides,
a finite cuspidal curve of the sixteenth degree.
The reader will find {Philosophical Iransactions, 1858, and
Tortolini^ vol. ii., p. 168) a discussion by Prof. Cayley of the
diiferent forms assumed by the surface and by the cuspidal and
nodal curves according to the different relative values of d\ 6^, c\
518. Mr. W. Roberts has solved the problem discussed
in the last article in another way, by proving that the problem
to find the negative pedal of a surface is identical with that
of forming the equation of the parallel surface. The former
problem is to find the envelope of the plane
XX + yy + zz = X +y + s ,
where ic', y\ z satisfy the equation of the surface. The second
problem, being that of finding the envelope of a sphere whose
centre is on the surface and radius = /o, is to find the envelope of
or Ixx + 2yy + 2zz' = x' + 2/' + ^^ - ^' + ^"' + v' + ^"'•
Now in finding this envelope the unaccented letters are treated
as constants, and it is evident that both problems are particular
cases of the problem to find, under the same conditions, the
envelope of
ax 4 ly + cz = x' + y'^ + z' + d.
It is also evident that if we have the equation of the parallel
surface, we have only to write iu it for U\ xT -f y'' + z\ and
then \x^ \y^ \z for x^ y^ z\ when we have the equation of the
negative pedal. Thus having obtained (p. 176) the equation
of the parallel to a quadric, we can find, by the substitutions
here explained, the equation of the first negative, the origin
484 SURFACES DERIVED FROM QUA.DRICS.
being anywhere, as easily as when the origin is the centre.
Further, if we write for ^, k + k'^ and then make the same
substitution for Z:, we obtain the first negative, the origin being
anywhere, of the parallel to the quadric, a problem which it
would probably not be easy to solve in any other way.
Having found, as above, the equation of the first negative
of a quadric, we have only to form its inverse, when we have
the equation of the second positive pedal of the reciprocal
quadric (Art. 514).
Ex. 1. To find the envelope of planes drawn perpendicularly at the extremities
of the radii vectores to the plane ax + ly + cz + d.
Here the parallel surface consists of a pair of planes, whose equation is
{ax + by + cz + dy — k^, that of the envelope is therefore
{ax + by + cz + 2dy = a;^ + / + z^.
Ex. 2. To find, in like manner, the first negative of the sphere
{x-ay+{y-py + {z-yy = r\
The parallel surface consists of the pair of concentric spheres
{x-ay + {:y- py +{z- yf = {r ± hf.
The envelope is therefore
(a; - 2af + ijj - W? + (^ - Sy)^ = {2r ± ^{x"- + / + z'')Y,
which denotes 51 quadric of revolution.
( 485 )
CHAPTER XV.
SURFACES OF THE THIRD DEGREE.
519. The general theory of surfaces, explained Chap, xi.,
gives the following results, when applied to cubical 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 reciprocal
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
should touch the surface. Multiply the equation of the surface
by 5^, and then eliminate hio by the help of the equation of
the plane. The result is a homogeneous cubic in ic, ?/, s,
containing also a, /3, 7, S 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, /3, 7, S ; but this consists of
the equation of the reciprocal surface multiplied by the
irrelevant factor S^*. The form of the discriminant of a homo-
geneous cubical function in a*, ?/, z is 64>S*+ T^ [Hi(]]ier Plane
Curves^ Art. 224). The same, then, will be the form of the re-
ciprocal of a surface of the third degree, S being of the fourth,
and T of the sixth degree in a, /3, 7, S ; (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.
486 SURFACES OF THE THIKD DEGREE.
520. Surfaces may have either multiple points or multiple
lines. When a surface has a douhle 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 w'" degree that there is
to the number of double points on a curve of the n^ 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 2, the equation of the surface will
be of the form
{ax^ + ^Ix'y + ^cxy' + dy"") + z (aV + 2Vxy + cy"")
■\{a"x^ + 2h"xy + c"f) = ^,
which we may write u^ + zu^ + v^^ = 0. At any point on the
double line there will be a pair of tangent planes z'u,^ + y^ = 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 tivo conjugate planes of a system in involution.
There are two values of z\ real or imaginary, which will
make z'u^ + ■Wj a 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'"' and Y^, it is evident that u^ and v^ can each be expressed
in the form LY* + m Y^. If, then, we turn round the axes so
as to have for coordinate planes the planes X, Y, that is to
say, the tangent planes at the cuspidal points, then every term
* 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 erei-t/ plane
meets the surface in a curve having a double point, and, therefore, tlie plane
ax + Py + yz + dw is to be considered as touching the surface, independently of
any relation between a, /3, y, 6. The reciprocal can be found in this case by
eliminating x, y, z, w between u — 0, a = Wj. (3 = it^, y = u^, i — tii-
SURFACES OF THE THIRD DEGREE. 487
in the equation will be divisible by either d(? or ^^, and the
equation may be reduced to the form zj^ = loif!^
In this form It Is evident that the surface is generated by
lines y = Xx, z = X'lv, Intersecting the two directing lines a;?/,
zio'j and the generators join the points of a system on zw
to the points of a system in involution on cry, homographic
with the first system. Any plane through zw meets the surface
in a pair of right lines, and is to be regarded as touching the
surface in the two points where these lines meet zw. Thus,
then, as the line X7/ 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. The reciprocal of this surface, which is
that considered 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 order. When the point is on the surface the
cone reduces to the second order.
521. There is one case, to which my attention was called
by Prof. Cayley, in which the reduction to the form zx^ = wy''
is not possible. If u^ and v^, in the last article, have a common
factor, then choosing the plane represented by this for one of
the coordinate planes, we can easily throw the equation of
the surface into the form y^ + x {zx 4- ivy) = 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 tlie last
* It is here supposed that the planes X, I', the double planes of the system in
involution, are real. We can always, however, reduce to the form w {x^ + y'^) + 2zxy,
the upper sign corresponding to real, and the lower to imaginary, double planes.
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 meeting for a con-
jugate 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.
488 SURFACES OF THE THIRD DEGREE.
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.*
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.
It is necessary to distinguish the various kinds of node which
the surface may possess. [A] At an ordinary nodef (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 corresponds a plane
section of the reciprocal surface, this double edge evidently
reduces by two the order of the reciprocal, or the class of the
given surface. [B] The quadric cone may degenerate into a
* The reader is referred to an interesting geometrical memoir on cubical ruled
surfaces by Cremona, " Atte del Reale Instituto Lombardo," vol. ii., p. 291.
t Prof. Cayley calls the kind of node here considered a cuie-node, and it is
referred to accordingly as C^.
SURFACES OF THE THIRD DEGREE. 489
pair of planes. Such a node may be called a hinode; 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 arc 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^ the edge does not lie on the
surface ; but if it does, the binode is special B^, and reduces the
class of the surface by four. Thus, let xyz be the binode, a;, y
the biplanes, the general equation of the surface will be of the
form W3 -\- x7/ = 0, where u^ = c^z^ -i Sc^z^x + Sc^z'i/ + &c. Tiie
case where c^ = 0 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 c^x + c^j which
commonly is distinct from one of the biplanes ; but it may
coincide with one of them, that is to say, we may have either
Cj or c^=0. In this case, the binode B^ 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 either a; or ^ a factor in w^, and we have then
a binode B^, 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 C^, which reduces the class of the surface by
six ; the equation then being reducible to the form w^ -f a;* = 0.
* 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 iutersection being of the order n — 2. The line may, however, count three
times, the remaining intersection being only of the order n - 3. Prof. Cayley calls
the line toi-sal in the first case, oscular in the second. He calls it scrolar if the surface
merely contain the right Une, in which case there ia ordinarily a different taugeat
plane at each point of the line.
RRR
490 SURFACES OF THE THIRD DEGREE.
The uniplane x meets the surface in three riglit lines, which
are commonly distinct ; but either, two of these may coincide,
or all three may coincide, when we have special cases of unodes,
Z7^, Z7g which reduce the class of the surface by seven and eight
respectively. JJ^ may be regarded as equivalent to three
conical nodes, TJ^ to two conical and a binode, TJ^ to two binodes
and a conical.
523. Distinguishing cubic surfaces according to the singu-
larities described in the preceding articles, we can enumerate
twenty-three possible forms of cubics, which are exhibited in
the following table :
I, 2, 3, 4, 5, 6, 7, 8, 9, 10,
class 12, 10, 9, 8, 8, 7, 7, 6, 6, 6,
singularities 0, 0„ i?3, 2 (7„ i?„ i?3 + 0,, 5„ 3 (7„ 2^3, 5, + C„
II, 12, 13, 14, 15, 16, 17, 18,
class 6, 6, 5, 5, 5, 4, 4, 4,
singularities B^, U^, B^^ 2 0,, B^^C,, Z7„ 4 0„ 2i?34- C,, 5,+ 2 6;,
19, 20, 21,
class 4, 4, 3,
singularities B^ + C^^ Z7g, 3^3.
These are the various possible combinations of nodal points;
and the number twenty-three is completed by the two kinds of
ruled surfaces or scrolls described Arts. 520, 521, each of
which is of the third class.*
Ex. 1. What is the degree of the reciprocal of xyz — w' ?
Ans. There are three biplanar points in the plane w, and the reciprocal is a cubic.
/ irn try *j
Ex. 2. What is the reciprocal of -+--J h— = 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 effect of the nodes Cj, £3, U^ on the class of the surface was pointed out
by me, Cambridge and Dublin Mathematical Journal, 1847, vol. 11., p. 65 ; and the
twenty-seven right lines on the surface were accovmted for in each case where we
have any combination of these nodes, Cambridge and Dublin Mathematical Journal,
1849, vol. IV., p. 252. The special cases B^, B.^, B^, U., U^ were remarked by Schliifli,
Phil. Trans , 18G3, p. 201. See also Prof. Cayley's Memoir on Cubic Sui-faces,
Phil. Trans., 1869, pp. 231-326.
SURFACES OF THE THIKD DEGREE. 491
, Ix mil nz niu . , ., , „ , , ,. .
form -r; + -j^ + -r, + — r, = Oj whence it follows that the condition that
X- y- z- w 2
The equation of the tangent plane at any point x'y'z'w' can be thrown into the
ence it follows that
ax + (iy + yz + Sw
should be a tangent plane is
(;a)*+(«/3)*+(«y)*+O,£)* = 0,
an equation which, cleared of radicals, is of the fourth degree.* Generally the re-
ciprocal of ax" + %" + C3" + dw" is of the form
n n n fi
A a^i + 5/3^> + Cy»^i + Dd'^^ = 0,
{Higher Plane Curves, p. 73).
The tangent cone to this surface, whose vertex is any point on the surface,
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
aa^ + bfi- + cy- + 2//iy + 2mya + 2na(i,
where a, b, c, I, m, n represent planes ; and a : y, /3 : y are two variable parameters.
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 «a" + i/3" + &c. is of the degree 3 (« — 1)^ and of the
class n^. The tangent cone from any point is of the degree on {ri — 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
n^^ order; or, in other words, is equal to the number of curves of the foim
^7+ \F+ ^JF which can have a cusp. The surface has a nodal curve whose
order is the same as the number of curves of the form 17+ XF + juIF which can
have two double points. For these numbers, see Sir/her Algebra, Lesson iviii.
524. The equation of a cubic having no multiple point may
be thrown into the form aas' + bt/^ + cz^ + dv^ + ew^ = 0, where
Xj 2/, z, V, w represent planes, and where for simplicity we
suppose that the constants implicitly involved in cc, ?/, &c. have
been so chosen, that the identical relation connecting the equa-
tions of any tive planes (Art. 38) may be written in the form
x+y + z + v-\-w = 0. In fact, the general equation of the third
degree contains twenty terms, and therefore nineteen independent
* Writing a-, y, z, w in place of la, mji, ny, i)^ respectively, the equation of the
reciprocal surface is
\{A + \^) + ■!{-) + J('^) = 0,
which rationalised is
{x- + y'^ + z- + W-- 2yz — 2zx — 2xy - 2xio - 2yio — 2zw)- — Giryi'.u = 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, «Sjc. : its properties have been
studied by Kummer, Weierstrass, Schrbter, Cremona (see Crelle, vols. 63, 64), and
more recently in a memoir by F, Gcrbaldi, Tiuin, 1881.
492 SUKFACES OF THE THIRD DEGREE.
constants, but the form just written contains five terras and,
therefore, four expressed independent constants, 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 Mr. Sylvester in 1851 [Cambridge and Dublin Mathematical
Journal, vol. VI., p. 199) is very convenient for the investi-
gation of the properties of cubical surfaces in general.*
525. If we write the equation of the first polar of any point
with regard to a surface of the w " order,
then, if it have a double point, that point will satisfy the
equations
ax' + Tiy 4 gz + Iw = 0, hx + by + fz -{- mvo = 0,
gx' -\-fy' + cz' + nw = 0, Ix' + 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'yzw, we obtain the locus of all points
which are double points on first polars. This is of the degree
4 (w — 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 xy'z'vf
will be of the degree 4 (w — 2)^, and will represent the locus of
• It wa,s observed {Higher Plane, Curves, Art. 25) that two forms may apparently
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 demonstrated 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 lUath. Annalcn, v. 341 ; and on the general theory of cubic
surfaces Cremona, CVeZfe, vol. 68 ; Sturm, Synthetlsche Untersuclmngen iiber Fldchen
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.
SURFACES OF THE THIRD DEGREE, 493
points whose first polars have double points. Or, again, II 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'yz'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 he a cone lohose 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 loith 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 inter-
section passes indefinitely near B^ the vertex of either cone;
therefore the polar plane of B passes through 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 surface of any
degree is the tangent plane of the corresponding point B on the
Steinerian. In particular, the tangent p)lanes to U along the para-
bolic curve are tangent planes to the Steinerian ; that Is to say,
in the case of a cubic the developable circumscribing a cubic
along the parabolic curve also circumscribes the Hessian. If
any line meet the Hessian in two corresponding points A^ Bj
and in two other points C, D, the tangent planes nt A, B Inter-
sect along the line joining the two points corresponding to (7, B.
527. We shall also investigate the preceding theorems by
means of the canonical form. 'J^he polar quadric of any point
with regard to ax^ + by^ + cz^ -f dv^ + ew^ is got by substituting
for w Its value — (a; + ?/ + 2 + u) , when we can proceed according
to the ordinary rules, the equation being then expressed in
terms of four variables. We tiius find for the polar quadric
494 SURFACES OF THE THIRD DEGREE.
axx^ + hy'y^ + cz'z'' -f- dv'v'' + eww"' = 0. If we differentiate this
equation with respect to x^ remembering that dw = - dx^ we
get ax'x = ew'xc ; and since the vertex of the cone must satisfy
the four differentials with respect to a*, ?/, 0, v, we find that
the coordinates x', y\ z\ y', w of any point A on the Hessian
are connected with the coordinates ic, 3/, ^, y, w of i?, the
vertex of the corresponding cone, by the relations
ax'x = hy'y — cz'z = dv'v = e?/;'?^.
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 i>,
11111
ax ' hy ' cz' ' dv '
eio
Since the coordinates of B must satisfy the identical relation
a; +y + s + y + ly = 0, we thus get the equation of the Hessian
11111^
ax by cz dv etc
or hcdeyzvw + cdeazvivx + deahvwxy + eabcwxyz + ahcdxyzv = 0.
This form of the equation shows that the line vio lies altogether
in the Hessian, and that the point xyz is a double point on the
Hessian ; and since the five planes x, ?/, s, y, lo give rise to
ten combinations, whether taken by twos or by threes, we have
Sylvester's theorem that the five planes form a iMntahedron
whose 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"^^ which resolves itself into two planes inter-
secting along vio^ any point on which line may be regarded
as the point B corresponding to xyz'^ thus, then, there ai'e ten
points whose polar quadrics hreak ujy into pairs of lAanes ; 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 reso-
lution of the given equation into the sum of five cubes cau
be completely established.
* It appears from Iligher Alr/ebva, Lesson sviii., that a S3'mmetric clcterminant
of p rows and columns, each constituent of which is a function of the n order in
the variables, represents a surface of the n;j degree having ^p (p^ — 1) n^ double
points; and thus that the Hessian of a surface of the n^ degree always has
10 (n — 2)' double points.
SURFACES OF THE THIRD DEGREE. 495
The equation of the tangent plane at any point of the
Hessian may be written
X y z V '^ c\
ax by cz dv eiv
which, if we substitute for x. — , , &c., becomes
ax
ax'^x + hy'^y -\- cz''^z -f dv''^v + ew'\io = 0,
but this is the polar plane of the corresponding point with
regard to U.
528. If we consider all the points of a fixed plane, their
polar planes envelope a surface, which (as at Higher Plane
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 (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 B, 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
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 lx-\- my -\-nz+pv-\ qto, the corresponding points lie on
t 7iX 11 ly O
the surface of the fourth order 1-^ — ! V ■— -\- ~ . Also
ax by cz dv eio
the intersection of this surface with the Hessian is of the
sixteenth order, and includes the ten right lines xy^ zwj &e.
496 SURFACES OF THE THIRD DEGREE.
The remaining curve of the sixth order is the curve along
which the polar cubic of the given plane touches the Hessian.
The four double points He on this curve; they are the points
whose polar quadrics are cones touching the given plane.
529. If on the line joining any two points xy'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^, + "iXP^^ -f Vi'^^i where Pj„ P^^
are the polar planes of the two given points, and P^^ is the
polar plane of either point with regard to the polar quadric
of the other. The envelope of this plane, considering \
variable, is evidently a quadric cone whose vertex is the inter-
section 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^^
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 on the Hessian passes through the intersection of the tangent
planes to the Hessian at these points.'^ 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 corre-
sponding 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.
530. We said, note, p. 29, that a cubical surface necessarily
contains right lines, and we now enquire how many in general
lie on the surface.f In the first place it is to be observed that
* 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.
f 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 Matheiiiatical Journal, vol. IV., pp. 118, 252.
Prof. Cayley first observed that a definite number of right lines must lie on the
Burfacc ; the determination of that number as above, and the discussions in Art, 533
were supplied by me.
\
SURFACES OF THE THIRD DEGREE. 497
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
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 w V= z F; let
us substitute w = fjiz, divide out by z, and then form the dis-
criminant of the resulting quadric in x, 3/, z. Now in this
quadric it is seen without difficulty that the coefficients of
a;*, XT/j and y'^ only contain ^ in the first degree ; that those of
zz and yz contain fi in the second degree, and that of z^ in
the third degree. It follows hence that the equation obtained
by equating the discriminant to nothing is of the fifth degree
in fi ; and therefore that through any right line on a cubical
surface can he draion Jive planes^ each of which meets the surface
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 referred 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 inter-
sected 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, tioenty-seven in all.
We shall hereafter show how to form the equation of a
surface of the ninth order 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
SSS
498 SURFACES OF THE THIRD DEGREE.
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 surface.
We have seen that through each of the twenty-seven 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 5 there are therefore
in all forty-Jive triple tangent planes.
532. Every plane through a right line on a cubic is obviously
a double tangent plane ; and the pairs of p)oints 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^ + bz + c)x-\- [az'^ + b'z ■\-c)y] then the two
points of contact of the plane y = ixx are determined by the
equation
[az' -\-hz-Yc) + ^l [az^ -f 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
contact 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 constants,
and therefore is one into which the general equation of a
cubic may be thrown.* This surface obviously contains nine
* It will be found in one hundred and twenty ways.
SURFACES OP THE THIRD DEGREE. 499
lines (a5, cd^ &c.). Any plane then a = fji,h which meets the
surface m right lines meets it in the same lines in which It
meets the hyperboloid fjbce = df. The two lines are therefore
generators of different species of that hyperboloid. One meets
the lines cd, e/, and the other the lines c/, de. And, since
fi has three values, there are three lines which meet ai, cd^ ef.
The same thing follows from the consideration that the hyper-
boloid determined by these lines must meet the surface in
three more lines (Art. 345).
Now there are clearly six hyperbololds, ab, cd^ efj db^ 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 follows :
there are the original nine ab^ ad^ af^ cb^ cd, cf^ eb, ed, efj
together with [ab.cd.ef)^^ [ab.cd.ef),^^ [ah.cd.ef)^^ 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 5,
meeting respectively in the pairs of lines ac?, q/"; Z>c, be ; and
the three planes which meet In {ab.cd.ef)^, {ab.cf.de)^\
{ab.cd.ef)^ {ab.cf.de}^] (ab.cd.ef)^, {ab.cf.de)^. The five
planes which can be drawn through any of the lines {ab.cd.ef)^^
cut in the pairs of lines, a6, {ab.cf.de)^] cd, {af.cd.he\^
ef, {ad.bc.ef)^; and In {ad.be. cf)^, [af.hc.de]^] [ad.be. cf)^,
{af.bc.de)^.
534. Prof. Schlafli has made a new arrangement of the
lines [Quarterly Journal of Mathematics, yo\. II. p. 116), which
leads to a simpler notation, and gives a clearer conception
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)^,
cf, be, ad, [ab.cd.ef)^, [ab.cd.ef)^, [ab.cd.ef)^,
it is easy to see that each line of one system does not Intersect
the line of the other system, which is written in the same
500 SURFACES OF THE THIRD DEGREE.
vertical line, but that it intersects the five other lines of the
second system. We may write then these two systems
«U «2? «35 «4J ^5J «67
^» h K K^ K^ \->
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 Oj, h^^ is the same as that which lies in the plane of
flj, h^. Hence the fifteen other lines may be represented by
the notation 0,^, Og^, &c., where c,^ lies in the plane of a,, J^,
and there are evidently fifteen combinations in pairs of the
six numbers 1, 2, &c. The five planes which can be drawn
through c,2 are the two which meet in the pairs of lines
«i^.) «2^i) and those which meet in c^^c^, c^f^^ c^^c^^. 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."
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, and five others
which intersect it, b^, Jg, b^, 5^, b^. These determine a cubical
surface, for if we describe such a surface through four of the
points where «, 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 Hues, we can in general draw two
transversals which shall intersect them all ; for the hyperboloid
determined by any three meets the fourth in two points through
which the transversals pass (see Art. 57 c? and note p. 419).
Through any four then of the lines b^^ J^, b^, b^ we can draw
in addition to the line a, another transversal a^^ which must also
lie on the surface since it meets it in four points. In this
manner we construct the five new lines a.^, Og, a^, a^, a^. If we
then take another transversal meeting the four first of these
lines, the theory already explained shows that it will be a line b^
which will also meet the fifth. We have thus constructed a
SURFACES OF THE THIRD DEGREE. 501
"double-six." We can then immediately construct the remain-
ing lines by taking the plane of any pair a^&.^, which will be
met by the lines 6,, a^ in points which lie on the line c,.^.
536. M. Schliifli 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
Jourtialj 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.
Thns 2 x6 + 15 = 27; 2x15 + 15 = 45.
Again, if the surface have four double points, the lines are
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 Schliifli in the Philo-
so^liical Transactions for 1863.
537. It is known that in a plane cubic the polar line, with
respect to the Hessian, of any point on the curve, meets on
502 SUEFACES OF THE THIRD DEGREE.
the curve the tangent at that point. Clebsch has given as
the corresponding theorem for surfaces, The jjolar plane^ loith
respect to the Hessian^ ^/"^'^J/ point on the cubic, meets the tangent
plane at that point, in the line which joins the three points of
inflexion of the section hy the tangent plane. It will be re-
membered 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^ + y^ + Gxyiv = 0.
"Writing the equation of the surface (the tangent plane being s),
x^ + y^ + Gxyio + zu = 0, where m is a complete function of the
second degree u = dz^ -\- Glxtv + 6myw + 'Snzio + &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'^w^ + c? (w — 2Z«i) ziiJ^.
The polar plane then of the Hessian with respect to the point
xyz is 4c?^<;^- (n— 2???i) 2, which passes through the intersection
of zw, as was to be proved.
If the tangent plane ^ = 0 pass through one of the right lines
on the cubic, the section by it consists of the right line x and
a conic, and may be written x^ -f ^xyw = 0 ; and, as before, the
polar plane of the point xyz with respect to the Hessian passes
through the line w, a theorem which may be geometrically
stated as follows : When the section hy 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 xyio, and the polar plane still passes through w, 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 inflexion 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 surtacc with which we have
INVARIANTS AND COVARIANTS OF A CUBIC. 503
worked, terras 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 a;, ?/, 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 tangent
plane, of the polar cubic of the same point.
INVARIANTS AND COVARIANTS OF 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 CDvariant, as for
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, /S, 7, S,
expressing the condition that the plane ax + ^g + iyz + Sw 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, yS, &c., ;j- 5 7- , &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- 1 -ir\i &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
504 SURFACES OF THE THIRD DEGREE.
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 a, /3,
7, h ; 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. io = — {x-^ y -{- z-\- v). To find then the
condition that the plane aaj + /3?/ + 7s + 8v + zw may have any
assigned relation to the given surface, is the same problem as
to find that the plane (a — e) ic -f (yS - s) y + (7 - s) z + (S — 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 — s, /3 — e, 7 — e, 8 — e respectively for a, /8, 7, S. Every
contravariant in five letters is therefore a function of the
difi'erences between a, /9, 7, S, e. This method will be better
understood from the following example :
Ex. The equation of a quadric is given in the form
ax- + by- + ez^ + dv- + eio- = 0,
where x-i-y + z + v + w = 0; to find the condition that ax -\- ^y + yz + Sv -i- tw
may touch the surface. If we reduce the equation of the quadric to a function of
four variables by substituting for ?y its value in terms of the others, the coefficients
of a;', y''', z^, v'- are respectively « + e, 6 + e, <r + e, (Z + e, while every other coefficient
becomes e. If now we substitute these values in the equation of Art. 79, the con-
dition that the plane aa; + /32/ + ya + ^v may touch, becomes
a^ {bed + hce + cde + dht) + ^ {cda + cde, -V due + ace) + y^ {dab + dae + aLe + bde)
+ ^ (abc + abe + bee + cue) - 2e {ad^y + bdya + ccZa/3 + hcah + ca^h + abyK) - 0.
Lastly, if we write in the above for a, /3, Ac, a — £, /? — «, &c., it becomes
bed (a - £)2 + cda (fi - iY + dab (y - t)^ + abc {6 - i)^ + bee (a - d)"- + cae (/3 - 5)2
+ abe (y - 5)2 + ade {J3 - y)- + bde [a - yf + cde {a - /3)2 = 0,
a contravariant which may be briefly written "Lcde (a — p)- = 0.
539. We have referred to the theorem that when a con-
travariant in four letters is given, we may substitute for
a, y8, 7, S differential symbols with respect to a;, y^z^w\ and
that then by operating with the function so obtained on any
covariant wc get a new covariant. Suppose now that we operate
INVAEIANTS AND COVARIANTS OF A CUBIC. 505
on a function expressed in terms of five letters rr, 3/, z^ v, w.
Since X appears in this function both explicitly and also
where it is introduced in w, the differential with respect to
d dw d . . . , , .
X \& - ~ -\- - — ^ , or, in vu-tue ot the relation connecting w
with the other variables, -^ . Hence, a contravariant in
ax aw
four letters is turned into an operating symbol in five by
substituting for
rt^dddddddd
' ' ' c?ic dio ' dij dw ' dz dw ' dv dw '
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 we substitute for a, /3, 7, S, s,
d d d d d 7, . ,. viu-L
■7—) -7-) -v ) -J" 5 —^ — , Vie obtain an operating symboL^ with.
which operating on the original function^ or on any covariantj
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 Zcde (a - ji)~ is a contravariant of the form
ax- + hif- + cz^ + dv^ + evfl.
If then we operate on the quadric with Zcde (-^ ^ j , the result, which only differs
by a numerical factor from
hcde + cdea + deah + eahc + abed,
13 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, « + e, 6 + c,
c + e, (i + e for a, b, c, d, and putting all the other coeflacients equal to e.
540. In like manner it is proved that we may substitute
in any covariant function for cr, ?/, z, y, w, differential symbols
with regard to a, /?, 7, S, s, and that operating with the function
so obtained on any contravariant we get a new contravariant.
In fact if we first reduce the function to one of four variables,
and then make the diflerential substitution, which we have a
TTT
506 SURFACES OF THE THIRD DEGREE,
right to do, we have substituted for
d d d d , f d d d d\
ar, J, <v, <^,
But since the contravariant In five letters was obtained from
that in four by writing a — s for a, &c., it is evident that the
differentials of both with regard to a, /3, 7, 8 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 or, /3, 7, S. 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 with-
out limit.
541. The polar quadric of any point with regard to the
cubic ax^ + hif' + cz^ + dv^ + eiJ" is
axx^ + I'y'y^ + czz^ + dv'v^ + eww'' = 0.
Now the Hessian is the discriminant of the polar quadric.
Its equation therefore, by Ex., Art. 539, is 'S.hcdeyzvw = 0, as
was already proved. Art. 527. Again, what we have called
(Art. 528) the polar cubic of a plane
aa; -f /Sy -f- 7^ + Su + £w,
being the condition that this plane should touch the polar
quadric is (by Ex., Art 538) ^cdezvw {a- ^f = 0. This is
what is called a mixed concomitant, since it contains both
sets of variables ic, ?/, &c., and a, /3, &c.
A I.
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 4>,
4> = ahcde^abx^y^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 tlic contravariant cr, which occurs in the equation
If now we substitute in this for ot, ^, &c., ^ , — , &c.,
INVARIANTS AND COVARIANTS OF A CUBIC. 507
of the reciprocal surface, which, as we have already seen, is
of the form 6-io-' = r\ The contravarlant a expresses the
condition that any phane oax + ^j/ + &c. should meet the surface
in a cubic for which Aronhold's invariant S vanishes. It is
of the fourth degree both in a, /3, &c., and in the coefficients
of the cubic. In the case of four variables the leading term
is a* multiplied by the S of the ternary cubic got by making
x = 0 in the equation of the surface. The remaining terms
are calculated from this by means of the differential equation
[Lessons on Higher Algehra^ Art. 150j. The form being found
for four variables, that for five is calculated from it as in
Art. 538. t suppress the details of the calculation, which,
though tedious, present no difficulty. The result is
o-=2ak^(a-£)(/3-s)(7-e)(S-£) [1].
For facility of reference I mark the contravariants with
numbers between brackets, and the covariants by numbers
between parentheses, the cubic itself and the Hessian being
numbered (1) and (12). We can now, as already explained,
from any given covariant and contravarlant, 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 in-
different with which we operate. The result in this case is
an invai-iant of the sum of their degrees in the coefficients.
The results of this process are given in the next article.
542. (a) Combining (1) and [1], we expect to find a con-
travarlant of the first degree in the variables, and the fifth
in the coefficients ; but this vanishes identically.
(?) (2) on [1] gives an invariant to which we shall refer
as invariant A^
A = Wc^d'^e' - 2abcde2ahc.
If A be expressed by the symbolical method explained
[Lessons on Higher Algebra^ XIV., Xix), its expression is
(1235) (124G) (1347) (2348) (5G78)'.
508 SURFACES OF THE THIRD DEGREE.
(c) Combining [1] with tlie square of (1) we get a covariant
quadric of the sixth order In the coefficients
abode [ax^ -V hy'' -\- cz^ -^ dv' + ew^) (3),
which expressed symbolically is (1234) (1235) (1456) (2456).
{d) (3) on [1] gives a contravariant quadric
a'h'ed\^^ (a - ^Y [2].
(e) [2] on (1) gives a covariant plane of the eleventh order
in the coefficients
dVc'd'^e^ {ax + hy-^cz-^dv + eio) (4) .
(/) (3) on [2] gives an invariant B^
a^b^c^d'^e^ (a + J -f c + c? + e).
[g] Combining with (3) the mixed concomitant (Art. 541)
we get a covariant cubic of the ninth order in the coefficients
abcde^cde (« + b) zvw (5).
(A) Combining (5) and [ij we have a linear contravariant
of the thirteenth order in the coefficients
abcde^ [a -b) [a- /3) {[a + b) c'd'e^ - abode [cd +de-\- ec)] . . .[3].
It seems unnecessary to give further details as to the steps
by which particular concomitants are found, and we may there-
fore 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
sura of these quantities, of their products in pairs, &c., by
p, 2) '^ ^j ^5 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'-M, B^fp, G=t\ D = t\ E=f;
whence also C^ — AE= Afr.
We can, however, form skew Invariants which cannot be
rationally expressed In terms of the five fundamental invariants,
although their squares can be rationally expressed in terms of
these quantities. The simplest invariant of this kind is got
INVAEIANTS AND COVARIANTS OF A CUBIC. 509
by expressing in terms of its coefficients the discriminant
of the equation whose roots arc a, Z>, c, (?, e. This, it will
be found, gives in terms of the fundamental invariants
A, B, C, B, E an expression for t^ multiplied by the product
of the squares of the differences of all the quantities «, 5, &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 ^,2/) ^j ■y? that is to say,
ax^ — hy'^ = cz^ — cW = evf.
Hence x\ y'\ &c. are proportional to hcde^ cdea^ &c. Sub-
stituting then in the equation a; + 2/ + s + v + w = 0, we get the
discriminant
\/{bcde] + \/{cdea) + \/[deah) -f \/{eabc) + \/[ahcd) = 0.
Clearing of radicals, the result, expressed in terms of the
principal invariants, is
[A' - UBf = 16384 [D + 2 A C).
544. The cubic has four fundamental covariant planes of
the orders 11, 19, 27, 43 in the coefficients, viz.
L = f2ax, L'^e^hcdex^ L" = e^d'x^ L'" = f2a''x.
Every other covariant, including the cubic itself, can, in
general, be expressed in terras 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, i*2 (a — ^) (a — /3) ; the next of the twenty-
ninth, t^cde [a — b) [a — /S), &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.
510 SURFACES OF THE THIRD DEGREE.
There are covarlant cubics of the ninth order t'2.cde[a-\-h)zvw^
and of the seventeenth, f^d^x^^ &c.
If we call the original cubic U^ and this last covariant F,
since if we form a covariant or invariant of Z7+XF, 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
A ( -i ^ T> d .,d ,.. d ., d\
V da do dc dd dej
Of higher covariants we only think it necessary here to mention
one of the fifth order, and fifteenth in the coefficients fxyzvio^
which gives the five fundamental planes ; and one of the ninth
order, 0 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. Art. 79, using a, ;8, &c.
to denote the first differential coefficients of the Hessian with
respect to the variables, and «, 5, &c. the second differential
coefficients of the cubic.
The equation of a covariant, whose intersection with the
given cubic determines the twenty-seven lineg, is 0 = 4H4>,
where 4> 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, s,
z^y + w'x -f "ixyz -f 'ixyw + ax^y + hy^x + cx'z + dy'^w = 0.
The part of the Hessian then which does not contain either
a; or ^ is z^w"^-^ the corresponding part of <l> is — '2, [cz^ -\- dw^) ^
and of 0 is - 8m;V (cs*+ cZw^). The surface & - AU^ has
therefore no part which does not contain either x or y, and
the line xy lies altogether on the surfticc, as in like manner
INVAEIANTS AND COVARIANTS OF A CUBIC. 511
do the rest of the twenty-seven lines* Clebsch obtained the
same formula directly, by the symbolical method of calculation,
for which we refer to the Lessons on Higher Algebra.
* This section is abridged from a paper which I contributed to the Philosophical
Transactions, 1800, p. 229. Shortly after the reading of my memoir, and before its
pubhcation, there appeared two papers in Crelle's Journal, vol. Lviir., by Professor
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, and the exptression
given above for the surface passing through the twenty-seven lines. The method,
however, which I pursued was different from that of Professor 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 fovir invariants as functions of the coefficients of
the Hessian. Thus the second is the invariant (1234)* of the Hessian, &c.
( 512 )
. CHAPTER XVI.
SURFACES OF THE FOURTH ORDER.
545. The theory of quartic surfaces In general has hitherto
been little studied. 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,
and Cremona ; and quartics with a nodal conic which
have been studied, in their general form, by Kummer,t
Clebsch, Korndorfer, and others; and In the case where the
nodal curve is the circle at Infinity (under the names of cyclldes
and auallagmatic surfaces) by Casey, Darboux, Moutard, and
others. In fact, in the classification of surfaces according to
their order, the extent of the subject Increases so rapidly
with the order, that the theory for example of the particular
kind of quartics last mentioned may be regarded as co-extenslve
with the entire theory of cubics.
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^ + wv^^ where
^4? %i ^3 ^^^ functions of the fourth and third orders respectively
* See his memoirs on Scrolls, Phil. Trans., 1864, p. 559; and 18G9, p. Ill, and
the references there given.
t Kummer, Berlin Monatsherichte, July, 1863 ; Crelle, LXIV. (18G4) ; Clebsch,
Crelle, LXIX. (1868) ; Korndorfer, Math. Annalen, III. ; Casey and Darboux, as cited,
p. 481. See also the list of memoirs on the same subject given in Darboux's work.
SURFACES OF THE FOURTH ORDER. 513
in X and ?/, and xy denotes the triple line. The three tangent
planes at any point on the triple line are given by the equa-
tion z'u^-\-wv^ = ^. Forming the discriminant of this equation,
we see that there are in general four 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 u^ = z [ax^ + hx^y) -^ w {cxy"^ + dy^). Further,
by substituting for 0, 2: + ax + /3_y, and for w^ lo + <yx + By, we
can evidently determine a, /3, 7, 5, so as to destroy the terms
x*j x^y, y^x, y* in m^; and so, finally, reduce the equation
to the form mx'y^ = z{ax^ + bx''y) -]■ to {cxy^ + dy'^). The planes
2!, w evidently touch the surface along the whole lengths of the
lines zy, wx, respectively ; and we see that the surface has four
tarsal generators, see note, p. 489. The surface may be gene-
rated 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 also having common the points in which each is
met by the intersection of their planes. 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 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,
Xaj + ?/ = 0, \^u + Vu + Xmj + s = 0 ;
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 torsal points already
considered.
UUU
514 SUEFACES OF THE FOURTH ORDER.
547. Returning to the equation
mx^'if' = z [ax^ + hx^y) + w [cxf + dy")
there Is an important distinction according as m does or
does not vanish ; or, In the form first given, according as
u^ is or Is not capable of being expressed in the form
[a.x-^By)u^-\r[nx-^ly)v^. 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 gene-
rator; 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 inter-
sect. Every plane through the right line therefore being a
triple tangent plane, there will 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 re-
ciprocally 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 confirm
this result by actually forming the equation of the reciprocal
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
SURFACES OF THE FOURTH ORDER. 515
of the scroll, we see that any generator is jjjiven by the
equations
y = \x, viX^x = z [a + b\) + w {cX' + c/A,'),
and joins the points
x = a + b\ y = \[a+ hX) , z = mX\ ro = 0,
x = c -\- dXj y = X[c + dX)j 2 = 0, w = m.
The reciprocal line is therefore the intersection of
[x + Xt/) [a + bX) -f mX^z = 0, {x + Xy) (c + dX) + mw = 0,
and the equation of the reciprocal is got by eliminating X
between these equations. JJut if we consider the scroll gene-
rated by the intersection of corresponding tangent planes to
two cones
X^x + X7/ + z = Oj Xhi + Xv + w = 0,
this will be a quartlc {xiv — uzf = {i/w— zv][xv — yu) which has
a twisted cubic for a nodal line, since the three quadrics
represented by the members of this equation have common a
twisted cubic, as is evident by writing their equations in the
form - = -= — . In the case actually under consideration,
X y z •' '
the equation of the reciprocal is
[m^zio + mczx + mbyw + [be — ad) xyY
= [mdzx + mczy + {be — ad) 7/] [mbxw + amy 10 + {be — 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 = Xa;, (a + bX) z + X^ (c + dX) w = 0.
The generator of the reciprocal scroll will be Xy + x = Oj
V (c + dX) z = [a + bX) w, and the reciprocal is obviously of like
nature with the original.
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 torsal
points coincide.
516 SUEFACES OF THE FOURTH ORDER.
548. Besides the two classes of quartic scrolls with a triple
line, already mentioned, we count the following :
III. Mg and v^ may have a common factor, which answers to
the case ad— be in the equation already given: which is then
reducible to the form
mx^^'^ = {ax + hy) [zx^ + wy\
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 fixed line, ax + hy being the corresponding
tangent plane which osculates along its whole length. Also
the equation of the reciprocal scroll being
{mzw + axz + hywf = zw {ay + Sa^)",
■we see that it has as nodal lines the plane conic ay + hxj
mzw + axz-^byw^ and the right line zw which intersects that conic.
This class contains as subform, the case where u^ + Xv^ includes
a perfect cube. The equation may then be reduced to the form
my* = x {zx^ -\- wy^]^ the reciprocal of which Is [xz — mw''f = y'^zw.
IV. Again, u^ and v^ may have a pair of common factors and
the equation is reducible to the form x^y'^={ax^-\^hxy-\-cy^){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, Mg and v^ may have common a square factor, the
equation then taking the form'
x^y^ = {ax + lyY {xz + yiv),
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^ and u, have three common
factors need not be considered, as the surface would then be
a cone.
549. We come now to quartic scrolls with only double lines.
If a quartic have a non-plane nodal line, it will ordinarily 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
* 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 ORDER. 517
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 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 Hues, each counted twice, and there Is
no Intersecting line lying In the surface. This is Stelner's
quartlc mentioned note p. 491. We consider now the other cases
of quartlcs 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
quartlc is nothing else than the quadric, counted twice, gene-
rated by a line meeting these three director lines.
Let us commence with the case where the nodal line Is a
twisted cubic (VI and YII). Such a cubic may be represented
by the three equations xz — y^ = 0, xw — yz = Q^ yio — z^ = 0]
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^ : X '. \. If the three quantities xz — y^^
xw-yz^ yio — z^ are called a, yS, 7 respectively, any quartlc which
has the cubic for a nodal line will be represented by a quad-
ratic function of a, /3, 7, say
aa^ + 5/3' + erf -f '2//37 + 2g^oi + 2Aa/S = 0.
Now consider the line joining two points on the cubic \, fi ;
the coordinates of any point on It will be of the form \^ -l- Oim ^
X'+d/M^ X+Ofi, 1-^6. If we substitute these values in a, /3, 7,
they become, after dividing by the common factor 6 (X — fij^
X/x, \ + /A, 1. Consequently the condition that the line should
He on the surface Is
a^V + b{X + fj,Y + c + 2fiX + /a) + 2y\fM + 2hXfjL (\ + yti) = 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 coordi-
518 SURFACES OF THE FOUETH ORDER.
nates (Art. 57a) of the line joining the points X., fi are easily
seen to be (omitting a common factor X — /*)
X'' + X/x, + A*''') l^+A'')) 1) X/z, — X/x (X + /I.), X'/i**,
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 an
" involution of six lines " (see note, p. 419), which join two points
on a twisted cubic.
In fact, if p^ 2', r, s, ^, u be the six coordinates, we have
the relation
hp -f 2fq + cr -\-[h + 2fj)s- 2ht + au = 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 fine ; that
is to say (VII), provided the condition be fulfilled,
b{b + 2g) - ifh + ac = 0.
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^, X'', X, 1 ; yu,'', /i'^, fi^ 1, will cor-
respond on the reciprocal scroll the generator whose equations are
£CX^ + yX* + Z\ + W = 0^ XfJb^ + 1/fx' +2/4 + ^ = 0,
and the equation of the reciprocal is got by eliminating X, /j,
between these equations and the relation already given con-
necting X, fi. This elimination has been performed by Prof.
Cayley ; 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 j so that the
SUEFACES OF THE FOURTH OKDER. 519
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
of a skew cubic. Thus then the fundamental division of scrolls
with nodal skew cubic is into scrolls whose reciprocals are of like
form (VI), and scrolls whose reciprocals have a triple line (V^H).
It is to be noted that the general form of the equation of the
reciprocal contains as a factor the quantity h'' + 2hg — \fh + 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 \ = ix'va. the equation just given, 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 4a7 — /3"^ = 0,
of which the cubic is the cuspidal edge, is made up of the cubic
together with four common generators. There will be four
points on the cubic, at which the two tangent planes to the
scroll coincIde,t these points being obtained by arranging the
condition already obtained
/*■' (aV + 2A\ + h) + 2/A [hX' + [h^-g)\ +/} + hX' + 2/\ + c = 0,
and forming the discriminant
[dX + 2h\ + h) {bV + 2f\ + c) = [JiX' +{h+rj)X +f]\
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 yu. = co , and
in this case /=0, b = 0; or the equation of the scroll may
always be transformed to the form
aoi' + cy' + 2c/ya + 2Aa/3 = 0.
Or, again, by choosing the planes of reference so that two of
* These classes, my sixth and seventh, answer to Cayley's tenth and eighth,
f Points on a double line at which the two tangent planes coincide are called by
Prof. C&jley pinch 2}0vUs.
520 SURFACES OF THE FOURTH ORDER.
the four points may be X = 0, X = oc , the equation may be
changed to the form [aa. + b^ + cy)'^ = Avfrya.
We have a subform of the scroll, if either a or c = 0 in this
equation ; for in this case two of the four cuspidal 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 5 = w? ;
and if along with this condition we have both a and c = 0, the
surface is the torse /3* — 47a = 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. It is
easy to see that the most general equation of the scroll can be
reduced to the form
[xz — y'^y + myw [xz - y'^) -\- w"^ {axy + hy^) = 0,
where xz — y'^^ 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*, X, 1, 0; and any point
z = fiio on the line xy^ and the line joining these points will lie
altogether on the surface if
X'V + wiX/A + aX 4 5 = 0.
Thus two generators pass through any point of either nodal
line or nodal conic. The reciprocal is got by eliminating be-
tween X^x + \y + z = 0, /jbZ-{- w = Oj and the preceding equation,
and is
{bxz — w^Y ~ y (^^^ ~ *^^) (% + ^"^ ~ ^^) "I" ^^ (% + ^^^ - a^Y = 0)
which for h not equal 0 is a scroll of the same kind having the
nodal conic, hxz - w^, hy + mw — as, and the nodal line ziv^ this
is VIII. If, however, J = 0, we have the case IX ; the reci-
procal quartic has here a triple line, and is of the third class
SURFACES OF THE FOURTH ORDER. 521
already considered.* There is one pinch point on the conic
and two on the line. There is a subform when m^ = ibj that
is to say, when the equation is of the form
{xz - 2/^ + myioY = aw^xy^
in which case there is but one pinch-point, and that on the line.
553. The next case is where the conic degenerates into
a pair of lines, in other words, where there are two non-inter-
secting 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 order
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 ac-
cordingly a third double line on the scroll. The general
equation may be written as in last article.
x^z^ + mxzyw + ui'' [axy + hy"^) =■ 0 ;
the line x = Xy, z = /juw will be a generator if
X^fi^ + raX/j, + aX + b = Oj
and the reciprocal is
y'^v)^ -f mxzyw + xz'' [hx — ay) = 0,
that is to say, is of the same nature as the original. This is
Cayley's second species. As before, the form [xz — yiof =■ axyxo^
may be regarded as special.
554. Next let us take the general case (Cayley's first species)
where there are two non-intersecting double lines. This scroll
may be generated by a line meeting a plane binodal quartic,
and two lines, one through each node. When the quartic has
* These two species, my eighth aad ninth, are Cayley's seventh and eleventh
respectively.
XXX
522 SURFACES OF THE FOURTH ORDER.
a third node we have the species of last article. The most
general equation is
a;" (as;' + 2hzw + hw') + 'Ixy [az' + 2}izw + Z»V)
+ y"" [a"z' + 2li"zw + Vw"") = 0,
the reciprocal of which is easily shown to be of like form.
There are obviously four pinch-points on each line, and subforms ^
may be enumerated according to the coincidence of two or more
of these points.
But again, 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
where ?^^, u^^ are a binary quartic and quadratic in x and y ;
and the reciprocal is of like form. Once more this class of
scrolls also admits of a double generator. This will be the
case if any factor y — ax of u^ enters twice into u^. In that
case it is obvious that the line y — ax, aiv — z is &, 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.
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 W
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 parameters at our disposal
we can satisfy the two conditions in a finite number of ways.*
* The same argument proves that if a surface of the n^^ order have a multiple
line of the (« — 2)'" order of multiplicity, the surface will contain right lines. If the
SURFACES OF THE FOURTH ORDER. 523
In the case where the quartic has a nodal right line a;y, sub-
stituting ?/—A,aj in the equation, and 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.
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 would need to be considered in a complete enumeration.*
The general equation of a quartic with a nodal right line
may be written
u^ + zu^ + wv^ + z^t^ + zwu^ + lO^V^ = 0,
where ?t^, Wg, &c. are functions in x and i/ 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,^, u^j v.^
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 fiictor not containing x or 3/, and so be
reducible to the form [az + hw) [zu^ -|- wv^ ; (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-\- bi/) [az + h'lo) [xz ^-yw) ; (5) we may
have ^21 ^iij ^2 OJ^^J 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 2« 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 {bb)
reducible to the form xyz^ ; (6) the three terms may break up
multiple line be a right line it is easily proved, as in Art. 530, that the number of other
right lines is 2 (3jj — 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. Annalen, t. iv. (1871).
* On the subject of multiple right lines on a suiiace the reader may consult a
memou- bj' Zeuthen, Math, Annalm, iv. (1871).
524 SURFACES OF THE FOURTH ORDER.
into the factors {xz — 7/w){zu,+ivv^)', (7) the terms may form
a perfect square {ocz + yio)\ 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'zio^ or (8Z>), ajV"*. This enumeration 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 -f yw divide
not only the last three terms but also the terms zu^ + lov^.
From the theory of reciprocal surfaces afterwards 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 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.
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 C/, F, W represent three quadrics having
a right line common and consequently four common points,
then any quadratic function of Z7, F, 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 Pllicker's Complex Surface (Art. -455), and its
SURFACES OF THE FOURTH ORDER. 525
equation Is
X,
y^
1
X,
a,
h,
9
y^
^
h
f
h
9i
/,
c
= 0,
where a, 5, h are of form (a, lof ; /, ^ of form (s, ?o)\ 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.
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^ U^ = 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 F contains the line 12, and
therefore F^ contains it twice ; hence, U^ should contain the
remaining six lines each twice, that Is, it breaks up Into four
planes ABCD which intersect in pairs In the six lines. Taking
in like manner P'^A'B' CD' = 0 for the sextic cone belonging
to the node 2, the eight nodes lie by fours in the eight
planes A^ B^ 0, -0, A\ B\ G\ 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 6gure 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*-\-hxy+ cx^y^-\-dxy^[y — mw) -|- ey'^ {y-mwY-\-{Ax^+ Bx^y ■\- Cxy'^) z
+ Dy^z [y — mw) + [A'x^ + B'x^y) w + C'xyw {y — mw)
+ [oLx' + I5xy + t/) ^' + (a'aj' + ^'^y) ^w + a'xW = 0,
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
wi = 0, the point will lie on xy. The kind of nodal line here
indicated appears to be different from any of those previously
considered.
558. Let us take next the case whei'e there are two Inter-
secting nodal lines. The equation then is
x'^y'^ + 2mxyzio + w'hi^ = 0,
526 SURFACES OF THE FOURTH ORDER.
where u.^ Is a quadratic functloa of a;, ?/, 2, w. Proceeding
as before we find immediately that four planes, besides the
plane w^ can be drawn through each of the nodal lines to
meet the surface In right lines ; and thus that there are sixteen
lines on the surface, eight meeting each nodal line. It is easy
also to see that each line of one system meets four lines of
the other system. Besides the nodal lines, the surfaces may
have four nodal points. The theory of this case is included
in that which we have next to consider, namely, where the
nodal line is a 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 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 Gelser and Darboux, with
the 27 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 honiographic
transformation. The inverse of such a quartic, the centre of
* It was from this point of view these surfaces were studied by Kummcr, viz. as
quartics on which lie an infinity of conies.
SURFACES OF THE FOURTH ORDER. 527
inversion being any point on the surface, is a cubic also passing
through the circle at infinity. 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 therefore
entering into ten of them.
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 generalize the results, it is only necessary in the equations
of the nodal line, iv = 0, x' + y' + s'^ = 0, to suppose ic, y, ^,
w to be any four planes ; while in the special case xo is at infinity,
and ar, ?/, z are ordinary rectangular coordinates. The properties
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 quartic
whose equation may be written (A', F, Z, Wf = 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^ + h Y' + cZ"^ + f?TF^=0,
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
* It has been shown by Dr. Valentiner, Zeuthen Tuhskri/t (4), III., 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 w"^ degree which contains the complete curve of inter-
section 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.
528 SURFACES OF THE FOURTH ORDER.
\fj,X-\Z— fiW+ Y=0, touching the surface In the eight
points where these curves intersect. And, generally, the quadric
aX+ ^Y+yZ-{ ST'Fwill touch the quartic, provided a, ^8, 7, 8
satisfy the familiar relation of Art. 79. All quadrics included
in this form have a common Jacoblan on which will lie all
possible vertices of cones involved in the system. Thus,
through each of the quadriquadrlc curves just spoken of, can
be drawn four cones whose vertices lie on the Jacoblan.
A special case is when the equation of the quartic can be
expressed in terms of three quadrics only (X, Y, Zf = 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^-\- cZ'^=Q^
or XZ= Y'\ Such a quartic is evidently the locus of the system
of curves 7= XX, Z=\Y, and the quadric X'X-2XY-\- 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, F, Z, Whe four spheres, the equation (X, F, Z, WY = 0
is general enough to represent any cyclide. Since the Jacoblan
of four spheres is the sphere which cuts them at right angles,
all spheres of the system 01.X + ^Y-\- yZ+ SW 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, /3, 7, 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, yS, 7, 8, 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 quartlcs {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 /; for any sphere
SURFACES OF THE FOURTH ORi)ER. 529
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, p. 481. Secondly, the intersection of i^ and J
is a focal curve of the cyclide ; for the Jacobian / is the locus of
all point-spheres belonging to the system aX+ ^Y+ryZ+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, together 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
asymptote 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.
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 quartic having for
nodal line the intersection of the quadric U by the plane P,
may obviously be written U'^ = P'^V. Or, again, if we write
down the following most general equation of a quartic, having
as a nodal line the intersection of x^ + ?/* + z^-, and w^
[x' + y' + z'Y + 2wu^ [x' + y' + z') + id'u,^ = 0 ;
this can obviously at once be written in the above form as,
[x^ + 3/^ + z^ + wu^^ = lo^v^.
We can simplify this equation by transformation to parallel
axes through a new origin, so as to make the j/, disappear,
and we may suppose the axes of coordinates to be parallel to the
axes of the quadric v.^, so that v^ does not contain the terras
yz^ zx^ xy. It appears then from what has been said, that the
cyclide, the general equation being reduced to the form
[x" -f / + zy = ax' + hy' -\- cz^ + 2lx + 2???^ + 2nz + cZ = F,
YYY
530 SURFACES OF THE FOURTH ORDER.
is the envelope of the quadrlc V+ 2\ [x^ +3/^4- z^) + X'^ = 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
m n 7^2
a-\-2\ b + 2X C+2X
and this equation being a quintic in X, we see that there are
five values of \ 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
of the last article, it may be inferred that there are five spheres J,
each of which combined with a corresponding quadric i^ gives a
mode of generating the cyclide. And this may be shown directly
by investigating the condition that the sphere x'^ -\- if ■\- z^ — u^
should have double contact with the cyclide, or meet it in
two circles. For, substituting in the equation of the cyclide
we get u^ = F, and if we add this to X [x^ 4 y'^ -f ^'^ — wj and
determine X by the condition that the sum shall represent two
planes, we get the same quintic as before for X; and we find
also that the centre of the sphere must satisfy the equation
+ ^^^ + .-^ = l>
X — « X— 5 X — c
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 / itself counted twice. Again, if
we have two cyclides both expressed in the form (A', F, Z^ W f=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 tfX'' + bY'' + cZ'' + dW'\ Thus then it
SURFACES OF THE FOURTH ORDER.
531
Is possible to express the equation of any cycllde In the form
aX'' + b'Y''-]-cZ'' + d'W% while at the same time we have
an Identical equation J^ = aX'' + bY'' + cZ'' + dW-. For the
actual transformation we refer to Casey, p. 599, Darboux, p. 135,
but we can show in another way what this identical equation
is. JMultiply by the ordinary rule the two determinants
p^
-X,
-y^
-^J
1, 2aj,
2.y,
2^, p'
d,
-h
-m,
-'h
1, 2/,
2?«,
2/2, t?
d\
-l\
— m^
-n,
1, 2/',
2»i',
2«', ^'
d:\
-l'\
-m ,
rr
I, 2r,
2?;i ,
2n", fZ"
d"\
- n
rf/
-VI ,
///
-n ,
1, 2r,
2/U ,
2w'", iZ'"
(where we have written for brevity p'^ Instead of x^-\-y^-\-^^
and where either determinant equated to zero gives the equation
of the sphere cutting orthogonally four spheres), and the product Is
X, r,
w
0,
X, -%r\ (12), (13), (14)
F, (12), -2r-, (23), (24)
Z, (13), (23), -2r-, (34)
W, (14), (24), (34), - 2/-
where (12) is d-\-d' — 211' — 2mm —Inn^ 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, F, Z, W
X, -2r''', 0,
F, 0,
Z, 0, 0,
0,
-2r"\ 0,
0
0
- 2r"\ 0
IF, 0,
0,
0,
-2r
//'2
whence It Immediately follows that If five spheres cut each
other orthogonally, the identical relation subsists
X^ Y' Z
V
2 1 /2 ' "'i "T ''ft '
r r r r r
W2_
532 SURFACES OF THE FOUETH ORDER.
It may be noted in passing, that In virtue of this Identity, the
equation W= 0 may be written in the form
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
and that it may be generated as the envelope of a sphere cutting
PF 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' F" -f b'Z^ + cV'^ + 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, F, Z^ F, W should
cut each other orthogonally, admits of being simply expressed.
It is in the first instance
This equation is reduced by the two following identities, which
are easily verified,
(dX\' fdX\' fdX\' , ^ , , 2
dXdY dXdY d^dY^^ y.
dx dx dy dy dz dz
The condition may then be written
SURFACES OF THE FOURTH ORDER. 533
The first two groups of terms vanish, because (f> and i^, which are
satisfied by the coordinates of the point in question, are homo-
geneous functions of X, F, &c. The condition therefore is
dXdX^ dYdY^^^-~^-
We may simplify the equations by writing X instead of X : ?', &c.,
80 that the identity connecting the five spheres becomes
and the condition for orthogonal section
d^df Mdjr
dXdX^dYdY- '
a condition exactly similar in form to that for ordinary co-
ordinates.
565. We can now immediately, after the analogy of quadrlcs,
form the equation of an orthogonal system of cyclides. For
write down the equation
X^ Y' Z^ V W ^
+ ^ T + ^ + ^ 1 + ^ = 0,
%— a X—b \—c \—d X—e
in which A, 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
form of the fourth degree, is in reality only of the third, the
coefficient of X* 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.* These 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.
566. The mode of generating cyclides as the envelope of
a sphere admits of being stated in another useful form. All
* Casey and Darboux seem to have independently made this beautiful extension
to three dimensions of Dr. Hart's theorem for the corresponding plane cui'ves,
Higher Plane Curves, Art, 278.
534 SURFACES OF THE FOUETH OEDER.
spheres whose centres He 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 moveable 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.
Pr. Casey has easily proved, p. 598, that the results of sub-
stituting 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 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 0 (X, /u, v, p) = 0, the derived surface is
0 (X, F, Z^ W) = 0 ; and if the first be the equation of a
quadric, the second will be the corresponding cyclide.
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 which has for its anti-points the intersections 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 quadric, there being,
I
SURFACES OF THE FOUKTH ORDER. 535
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 sphero-quartic
curve. The properties of these sphero-quartics have been in-
vestigated in detail by Casey and Darboux. These curves
may be inverted into plane bicircular quartics, and therefore
(see note, p. 481) 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 intinity. 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 originally 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
in a given plane ; and we have thus a more precise definition,
that the cyclide is the envelope of a series of spheres each
having its centre in a given plane and touching two given
536 SURFACES OF THE FOURTH ORDER.
spheres. But all such spheres have their centres on a conic;
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 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
confocal 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.
SURFACES OF THE FOURTH ORDER. 537
568. Passing now to quartic surfaces without singular
lines, they may have any number of nodes (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*16, =4.
Some of the nodes may be replaced by, or may coalesce into,
binodes or u nodes, but the theory has not 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 =4 conditions.
It would, therefore, at first sight appear that we could with
eight given points as nodes determine a quartic surface con-
taining two constants ; but this is not so. We have through
the eight points two quadric surfaces Z7=0, V=0 (every other
quadric surface through the eight points being in general of the
form U+ A, V= 0) and the form with two constants is in fact
Zr^ + aUV+ ^V^ = 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.
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 reference
to those which follow, it becomes interesting to consider how the
zzz
538 SURFACES OF THE FOURTH ORDER.
equation can be built up with quadrlc functions representing
surfaces which pass through the given nodes. In the case of
4 given nodes we have six such surfaces -P=0, ^ = 0, i? = 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 equation (P, Q, i?, 8j 7', U)'^=0,
containing apparently twenty constants. The explanation is
that the six functions, although linearly independent, are con-
nected by two quadric equations, and the number of constants
is thereby reduced to 20 — 2, = 18, which is right.
In the case of 6 given nodes we have through these the
five quadric surfaces P= 0, ^ = 0, P = 0, S=0, T=0, and we
have the quartic surface (P, Q^ Bj S^ Tf = 0^ containing, as it
should do, 14 constants.
571. In the case of 6 given nodes, we have through these
the four quadric surfaces P— 0, ^ = 0, i? = 0, S—0, and the
quartic surface (P, Q, P, 8^ — 0 contains only 9 constants;
there is in fact through the six points a quartic surface,
the Jacobian of the four functions, /(P, Q^ P, aS) = 0, not
included in the foregoing form, and the general quartic surface
with the six given nodes is
(P, Q,R,Sf+ej{P, Q,R,S) = 0,
containing, as it should do, 10 constants.
The foregoing surface J(P, Q, P, /S) =0, where P=0, ^=0,
P = 0, S=0 are any quadric surfaces through the six given
points, or 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 iiitersection of the
plane through three of the points with the plane through the
remaining three points.
In the case of 7 given nodes we have through these three
quadric surfaces P=0, (> = 0, P = 0; but forming herewith the
equation (P, Q, Iif = 0, this contains only five constants ; that
it is not the general surface with the seven given nodes appeal's
SURFACES OF THE FOURTH ORDER. 539
also by the consideration that It has, in fact, an eighth node,
for each of the intersections of the three quadrlc surfaces Is a
node on the surface. We can without difficulty find a quartic
surface not iuckided In the form, but having the seven given
nodes: for Instance, this may be taken to be v = 0, where v
is 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, Rf +ev = 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 quadrlc surfiices through the seven
points, and we thus have a form of surface
with eight nodes, the common intersection of three quadrlc
surfaces ; this Is the octadic eight-nodal quartic surface.
Among the surfaces of the form in question are included the
reciprocals of several Interesting surfaces, for example, 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.
There is, however, another kind of 8-nodal surface for
which the eighth node is any point whatever on a certain
surface determined by means of the seven given points ; and
this is called the octo-dianome.
The last-mentioned surface may be made to have another
node, which is any point whatever on a certain curve determined
by means of the eight nodes ; we have thus the ennea-diauome ;
and finally this may be made to have a new node, one of a
certain system of twenty-two points determined by means of
the nine nodes ; this Is the deca-dlanomc. But starting with
540 SURFACES OF THE FOUKTH ORDER.
seven given points as nodes, the number of nodes of the quartic
surface is at most =10.
A kind of 10-nodal surface is the svrametroid, which is
represented by means of a symmetrical determinant
= 0,
a,
^
f7,
I
^
h
/,
m
Oi
/,
c,
n
I,
m,
n,
d
where the several letters represent linear functions of the cO"
ordinates ; such a surface has ten nodes, for 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. E-everting to the consideration of the circumscribed
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
sixteen-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 fourteen-nodal 8 (3, 1, 1, 1) + 6 (2, 2, 1, 1).
In the case of 13 nodes, the cones are (4^, 1, 1), (3,, 2, 1),
(3, 1, 1, 1), or (2, 2, 2), viz. (4^, 1,1) is a three-nodal quartic
SURFACES OF THE FOURTH ORDER. £41
cone and two planes, and so (3,, 2, 1) is a nodal cublcone, a
quadricone, and a plane. It is found that there are two forms
of surface, the 13-(a)-nodal
3(43, 1, 1) + 1(3, 1,1, 1) + 9(3,, 2,1),
and the 13-(/S)-nodal 13 (2, 2, 2).
The like principles apply to the cases of twelve, eleven, &c.
nodes, but the number of kinds has not been completely
ascertained.
573. We only consider the 16-nodal quartic, the equation
of which in general can be exhibited. Write for shortness
-p_x y z ^, _x y z p//_ ^ y ^
where a+yS + 7 = 0, a'+/3' + 7' = 0, a"+/3" + y' = 0,
X =a {yyy-^'l3"z), Y =/3 {aa"z-ry'y"x), Z =7 (/3'/S"a.-a'a"^),
X =a' (773/ -^"/S^ ), ^' = /3' (a"a^-7"7^ ), Z' =7' (/3"/3^ -a"a^),
X"=o:\riiy - ^^'z ), r"= yS" {aa'z - 77'aj}, Z"=y" (/3/3'a; -aa'y ),
A = x'' + y'^ + z^ - 2yz - 2zx - 2xyj
5= aaV {y'z - z"'y) + /3y8'/3" {z'x-zx') +777" {x'y - xf)\Mxyz,
C= aaayz + ^/3'^"zx + yy'y'xy
where i/= (^ - 7)aV'+(7 - a)W + (a - ^)y'y"
= (;8' - 7') a"a + (7' - a ] /3",S + (a' - /3') 7'V
= (/3" - 7") aa' + (7" - a") /S/3' + (a" - ^") 77'
= -i{(/S-7)(/3-70(^"-70 + (7-«)(7-a)(7-a'0
+ (a-^)(a'-/3')(a"-/3'0},
values which give identically
AC- B' = 4aa'a'W/3"77'7"a;?/«PP'P'' ;
then the equation of the surface may be written in the irrational
form
^/{x{X-w)}+'^{y{Y-w)}+^/[z{Z-w)}=Oj
which rationalized is Aio"^ + 2 Bio + C,
and is one of four hundred and eighty like forms.
542
SURFACES OF THE FOURTH ORDER.
For each node the sextic cone is made up of six planes, but
we thus obtain in all only sixteen planes; for each of these
planes is a singular plane touching the surface along a conic,
on which conic are contained six nodes of the surface. The
coordinates of the sixteen nodes and the equations of the sixteen
planes can easily be obtained. For instance, the planes are
X, y, Z, W, P, P', P", X-w, X' - w, X" - w, Y- w, &c.
574. The 16-nodal quartic includes as a particular case
Prof. Cayley's tetrahedroid, obtained by him as a mere homo-
graphic transformation of the wave surface. In this case the
sixteen planes pass in fours through the summits of a tetrahedron.
To obtain its equation independently of the general case, write
down the general equation of a quartic met by each of the
four coordinate planes in two conies having for common con^
jugate points the vertices of the tetrahedron of reference which
lie in that plane. The equation so formed contains in general
a terra xyzw and represents a surface without nodes : but if
we add the further condition that this term shall vanish, the
surface at once acquires sixteen nodes, each of the intersections
of the two conies in each of the four planes becoming a node.
The equation may be written
0 2 2 2 'i
, x\ y ^ z\ lo'
y\
w
A, 0,
I
m.
0,
m
n
0
= 0*
= 0,
or, what is the same thing,
{A, P, C, P, F, a, //, P, il/, NJx% y% z\ xo\
where the coefficients are those of the reciprocal of a quadric
wanting the terms a;'"', y^ z\ w\ The equation expanded is
(see Art. 208)
mnfx^ + nlgy'*' + Imhz^ +fg]iw*
+ \ {hfz' +fxW) + fi {mz'x' + Oy'io^) + V [iix^ + hz'W) = 0,
where \ = If— mcj — nli^ A* = — ^'+ ma — nh, v = — If— mg + nh.
* The deduction of this form from that of the general 16-nodal is a process
of some difficulty ; and it is to be noted that the x, y, &c. here used are not the same
coordinates as those used in that equation.
SURFACES OF THE FOURTH ORDER. 543
And the nodes may be exliibitcd by writing the equation in
the following or one of the three corresponding forms
= y (1, 1, 1, - 1, - 1, - ij^j-'n, z^m, ivyy-,
where y = T'f -f m\f + nV - 27nngh - 2n Jhf - 2 hnfg.
These last equations serve to show that the sections by a
plane of the tetrahedron are two conies as above mentioned ;
thus writing in the first of them lo — O it becomes
i^mnfx^ + nvif + miiz^)^ = A [ifn - z'm]'\
a pair of conies.
To deduce the ordinary form of the equation of the wave-
surface write
I = a/3y [by - cj3)j m = ajSy {ecu. — ay), n = a^y (rt/3 — ha),
f= kaoL [by - c^), g = kb/S [ca — ay), h = key [a(3 - ba),
equations which serve to determine the ratios a:b : c: a: ^ : y : k
in terms of I, m, n,f, g, h. The equation of the surface then
becomes
a/3y [ax"" -f bf + cz') [ax"" + ^f + yz') + Ic'abcw''
- kaa. [by + c/S) jk'V - kb^ [ca. + ay) ?/W - key (a/3 + la) zW= 0,
*eh putting in X,r,Zf„.^^(|),y(f),£y(|)
respectively, and m\ ^b'\ ye' for a, b, c, becomes
{X' -f YH Z') {a'X' + b'Y' + c'Z') + a'i;'c'
- [b' + c') a'X' - {c' + a') b'' Y' - [a' + ¥] c'Z' = 0,
the equation of the wave-surface.
{ 544 )
CHAPTER XVII.
GENERAL THEORY OP 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 pointy is a surface of the degree m + n-\-p + q — A]
the Jacobian of the system. For its equation is evidently
got by equating to nothing the determinant whose consti-
tuents are the four differential coefficients of each of the four
surfaces. If a surface of the form \U + fiV-\- vW touch Ty
the point of contact is evidently a point on the Jacobian, and
must lie somewhere on the curve of the degree q (m+w+p-f g— 4)
where the Jacobian meets T. In like ina.nnerj pq{7n-{-n-\-p+q—4:)
surfaces of the form \U+ fiV can be drawn so as to touch
the curve of intersection of T, IF; 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^ F, W (that is to say, the condition that two of the
m7ip points of intersection may coincide), contains the coefficients
of the first in the degree 7ip [2m -f w +p — 4) ; and in like manner
for the other two surfaces. For, if in this condition we sub-
stitute for each coefficient a of Z7, a + Xa, where a is the
corresponding coefficient of another surface U' of the same
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+ W\ which
can be drawn to touch the curve of intersection of F, TF.*
* Moutard, Terquem's Annahs, vol. xix. p, 58,
GENERAL THEOEY OF SURFACES. 545
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 UW. 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 + 07/ + yz + Biv. Thus the point of contact annuls the
determinant, which has for one row, a, /5, 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, F, W. The result will contain a, /3, 7, S
in the degree miiij ; and the coefficients of U in the degree
np [m + n-\-p — 3) + mnp. But this result of elimination contains
as a factor the condition that the plane ax -{■ 0y + <yz -'r Bw
may pass through one of the points of intersection of Z7, F, W.
And this latter condition contains a, /3, 7, S in the degree mnjy^
and the coefficients of U in the degree rq). Dividing out this
factor, the quotient, as already seen, contains the coefficients of
U in the degree
np[2m + n +2^ — 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
£^., u,. u,. u,
V,, n, F3, F,
H^„ W^, TF3, IF, = 0.
But this curve (see Higher AJgehro^ Art. 257) is of the order
[m' + n^ + p"^ + mil + np +pin)^
where vi is the order of Z7,, &c., that is to say, ni =in~ I, &c.
If a surface of the form XZ7+/aF touch IF, 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 IF,
that is to say, is p times the degree of that curve. Reasoning
AAAA
546 GENEEAL THEORY OF SURFACES.
then, as In the last article, we see that the tact-invariant of two
surfaces t/, F, that is to say, the condition that they should
touch, contains the coefficients of U in the degree
or n {n^ -t- 2mn + 3m^ — An — 8?n + 6).
This number may be otherwise expressed as follows: if the
order and class of V be M and iV, and the order of the tangent
cone from any point be E, then the degree in which the coeffi-
cients of U enter into the tact-invariant is
N-i 2R{m-l) + ^M{m-\)\
We add, in the form of examples, a few theorems to which
it does not seem worth while to devote a separate article.
Ex. I. Two surfaces U, V of the degrees m, n intersect ; the mimber of tangents
to their curve of intersection, which are also inflexional tangents of the first surface,
is mn [am + 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 T' in the second degree, we have the equa-
tion of a surface of the degree (3??i+2w — 8) which meets the curve of intersection
in the points, the tangents at which are uiflexional tangents on U,
Ex. 2. In the same case to find the degree of the surface genei-ated 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
f7' = 0, F'=0, AC/' = 0, A=f/' = 0,
which are in x'y'z'w' of the degrees respectively m, n, m — \, m — 2, and in xyzio of
the degrees 0, 0, 1, 2. The result is therefore of the degree mn {3in — 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(w — 2), the degree of the surface generated by these inflexional
tangents is 4ni{m — 2){Sm — 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 im (»i — 2) {om — 4).
Ex. 4. To find the characteristics, as at p. 298, of the Tievelopable 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
/i = 6/»(w-2), v = m{m-l), r = m{3m-5), a = 0, (3 = 2m {5m - U), &c.
GENERAL THEORY OF SURFACES. 547
Ex. 5. To find the characteristics of the developable which touches a surface of
the degree m along its intersection with a surface of degree ti.
Ans. V = mil {m — 1), a = 0, r = vin {3m + u — G), whence the other singularities
are found as at p. 298.
Ex 6. To find the characteristics of the developable touching two given surfaces,
neither of which has multiple lines.
Ans. V = mn {m — 1)- (« — 1)^ ; a = 0, ?■ = mn (m — 1) {n — 1) {m + n - 2).
Ex. 7. To find the characteristics of the curve of intersection of two developables.
The surfaces are of degi-ees r and ?■', 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. We have as there \t. = rr' ; but the degree of the reciprocal of this cone is
p = rr' (»• + )•' - 2) - r {2x' + 3m') - r' {2x + 3m),
or, by the formulae of Art. 327, p = rii' + nr'. In like manner
V = ar' + a'r + 3n-'.
Ex. 8. To find the characteristics of the developable generated by a line meeting
two given cm-ves. This is the reciprocal of the last example. We have therefore
i; = rr', p = rm' + mr', p. = /3r' + (i'r + 3rr'.
Ex. 9. To find the characteristics of the curve of intersection of a surface and
a developable. The letters 31, N, R relate to the surface as in the present article ;
m, 11, r to the developable. Ans. jj. = Mr, p tz rB + iiM, v = aJil + orR.
Ex. 10. To find the characteristics of a developable touching a surface and also
a given curve. Ans. p. = [Sy + 3rR, p = rR + mX, v — 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 the n^ order ;
the surfaces satisfying these conditions form a system whose
characteristics are yu., v, p\ where [x, 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
system of surfaces by any plane form a system of curves
whose characteristics are /a, p ; and the tangent cones drawn
from any point form a system whose characteristics are p, v.
Several of the following theorems answer to theorems already
proved for curves.
(1) The locus of the poles of a fixed plane loith regard to
surfaces of the system is a curve of double curvature of the
548 GENERAL THEORY OF SURFACES.
order 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 conditions.
Thus, when eight points are given, the locus Is a curve of the
third order ; when eight planes, it is a right line.
(2) The envelope of the polar planes of a fixed pointy with
regard to all the surfaces of the system^ is a developable of the
class yt6.
(3) The locus of the poles with regard to surfaces of the system^
of all the planes which pass through a fixed right line^ is a surface
of the 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 the degree p.
(4) Reciprocally, The polar planes of all the points of a line^
with respect to surfaces of the system^ envelope a surface of the
class p.
(5) The locus of the points of contact of lilies drawn from a
fixed point to surfaces of the system is a surface of the order
//- + /?, having the fixed j^oint as a midtipile point of order fi.
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
orthogonally by surfaces of the system. It Is the curve In which
the plane is cut by the locus surface yu, + p, answering to the
point at infinity on a perpendicular to the given plane.
(6) The locus of p)oints of contact^ with surfaces of the system^
of planes passing through a fixed line^ is a curve of the order
v + p meeting the fixed line in p points. This also may be stated
as the locus of points, the tangent planes at which to surfaces of
the system passing through It contain a given line.
GENERAL THEORY OF SURFACES. 549
(7) The locus of a point such that its polar ^:>?c<??e with regard
to a given surface 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 linCj is a surface of the degree
mjji 4- 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 the order yu,.
(8) If in the preceding case the line of intersection is to lie in a
given plane, the locus will be a curve of the order m[m—\)p,+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 fxm {in — 1) + mp.
The locus just considered meets the fixed surface In
m [m [m — 1) ya + mp + v] points. But it is plain that these must
either be ihe, fjim {m — \) -\- mp points just mentioned, or else
points where surfaces of the system touch the fixed surface.
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 txm [m — l)"* + pm [rn — 1) + vm ; or, more gener-
ally, if n be the class of the surface, and r the order of the
tangent cone from any point, the number is ixn + r/j + vm.
We can hence determine the number of surfaces of the form
XZ7+ V which can touch a given surface. For if U and V
are of the degree m, these surfaces form a system for which
/u.= l, v = 3 (?«— l)'^, p = 2(?n— 1). If, then, n be the degree
of the touched surface, the value is
n{n- 1)' -1- 2?i [n - 1) (m - 1) + 3n (w - l)"',
the same value as that given, Art 576. This conclusion may
otherwise be arrived at by the following process.
578. If there he in a plane tivo systems of points having
a (/?, m) correspondence, that is, such that to any point
of the first system correspond in of the second, and to any
550 GENERAL THEORY OF SURFACES.
point of the second correspond n of the first: ayid^ moreover^
if any right line contains r pairs of corresponding poi7its, then
the number of points of either systera which coincide with points
corresponding to them is m + w + r. Let us suppose that the co-
ordinates of two corresponding points xy^ x'y\ are connected by
a relation of the degrees /z, im in xy^ x y respectively ; and
by another relation of the degrees v, v ; then if x'y be given,
there are evidently /xv values of xy^ hence n = fiv. In like
manner m = yi!v . If we eliminate a:, y between the two equa-
tions, and an arbitrary equation aa; + % + c = 0, we obtain a
result of the degree /x/ 4- /aV in x'y ; showing that if one point
describe a right line, the other will describe a curve of the degree
IxV + /u,V, which will, of course, intersect the right line in the
same number of points, hence r = /xv' + /tV. But if we suppose
X and y respectively equal to x and ?/, we have (/i -|- /*') (v + /)
values of x and y ; a number obviously equal to m-\-n^- r.
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
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 tiieorera (]), on a curve of the order v. This
curve will meet the assumed plane in the points a which corre-
spond to A^ whose number therefore is v. On the other hand, if
a be given, its polar planes with respect to surfaces of the systera
envelope, by theorem (2), a developable whose class isyu,; 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)'^;*
this surface and the developable have common /u. {m — \f tangent
planes, which will be the number of points A corresponding to a.
* It was mentioned (p. 491) that if the equation of a plane contain two
parameters in the degree n, its envelope will be of the class «-.
GENERAL THEORY OF SURFACES. 551
Lastly, let A describe <a rif^ht line, tlicn Its polar planes witli
respect to the fixed surface envelope a developable of the class
on— I ] but with respect to the surfaces of the system, by theorem
(3), envelope a surface of the class p. There may, therefore, be
p{m — l) 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 fi (m - l)'^ + 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 Jonquicres.
(10) The locus of a point such that the line joining it to a
Jixed pointy and the tan (jent plane at it to one of the surfaces of the
system which pass through it, meet the plane of a fixed curve in a
point and line which are pole and polar with respect to that curve^
is a curve of the degree fim [m — 1 ) + pm -f v. This is ])roved as
theorem (8). Let the fixed curve be the imaginary circle at
infinity, and the theorem becomes the locus of the feet of the
normals drawn from a fixed point to the surfaces of the system is
a curve of the degree 2//, + 2p + v.
(11) If there be a system of plane curves, whose characteristics
are /t, v, the locus of a point such that its polar with regard 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 m/j, -\- 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, whose polars with respect to the fixed
curve divide the given line harmonically. And since these are
points on the locus for each of the fi curves which pass through
them, the degree of the locus is mfjb+ v. Taking for the finite
line the line joining the two Imaginary circular points at Infinity,
It follows that there are m [mp, -f v) curves of the system which
cut a given curve orthogonally. De Jonquicres finds that in
like manner tlte locus of a point such that its j^olar plane with
552 GENERAL THEORY OF SURFACES.
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 surface
of the order my. + p. And consequently that a surface of this
order meets the fixed surface in points where it is cut orthogo-
nally 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 the n^^ class, the locus of
the intersections of the tangents of one system with those of the
other is a curve of the order v {2n - 1). For consider any curve
touching the line QQ', then one point of the locus will be the
point of contact, and ?i - 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 the order {n - 1) v,
and the line QQ meets the locus in v{^n-\) 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 (2n - 1). If we investigate, in like manner, the locus of
the intersection of cones drawn to a system of surfaces from two
fixed points QQ\ it is evident, from what has been said, that any
plane through QQ' meets the locus in a curve whose order is
p(2»-l); but the line QQ' is a multiple line of degree p,
being common to both cones in every case where the line
QQ' touches a surface of the system. The order of the locus
therefore is 2np ; and accordingly, 4/> is the order of the locus
of foci of sections of a system of quadrlcs by planes parallel to
a fixed plane.*
* Cliasles has given the theorem that if there be a system of conies whose
characteristics are ju, v, then 2v — fi conies of the system reduce to a pair of Hnes,
and 2ft. — V to a pair of points. It immediately follows hence, as Cremona has
remarked, that if there be a system of quadrics, whose characteristics are w, v, p,
of which o- reduce to cones and <r' to plane conies, then considering the section
of the system by any plane, we have v = 2p — n, c' = 2/x - /o, and, reciprocally,
a = 2u — p. These theorems, however, are obviously subject to modifications if it
can ever happen that a sui-face 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 arc 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
GENERAL THEORY OF SURFACES. 553
581. The theory of the transformation of curves and of the
correspondence of points on curves (explained Higher Plane
Curves^ Chap. Vill.) 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. The reader
may consult Cremona, Memoire de geometric pure sur les
surfaces du troisi^me ordre, Crelle, LXVIIT, pp. 1-96 (18G8) ;
Clebsch, Ueber die Abbildung algebraischer Flachen insbeson-
dere der vierten und funften Ordnung, Math. Annalen^ i. pp. 253
— 316 (1868) ; Cayley, On the rational transformation between
two spaces, Proc. Lond. Math. Soc, III. pp. 127—180 (1870);
and other papers by the same authors, and by Darboux, Klein,
Korndorfer, Nother, Zeuthen, and others.
It will be recollected that a unicursal curve is a curve, the
points of which have a (1, 1) correspondence with those of a line ;
or, analytically, we can express the coordinates x, ?/, 2; of a point
of it as proportional to homogeneous functions, of the same
order m, of two parameters X, /j,. 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 order ?«, of three parameters X, /t, 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 0
on the surface, we obtain at once a (1, 1) correspondence
between the points of the quadric and of the plane. In the
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 con-
ditions, Quarterly Journal, vm. 1 (1866). The same problem has been more com-
pletely dealt with by Chasles and Zeuthen (see Comptes Eendus, Feb. 1866, p. 405).
BBBB
554 GENERAL THEORY OF SURFACES.
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 construction in the
case of the quadric can easily be derived analytical expressions
giving a;, 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 p. 358) 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) at once suggests a similar expression for
the coordinates of a point (see p. 382) on the surface. Thus,
on the quadric xw = yz^ the systems of generators are \x = fiy^
fiw = \2 ; \x = vz^ vw = Xj/, whence the coordinates of any point
on the quadric may be taken /tv, \v, X/a, X\ 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 corresponding
planes, each of which passes through a fixed point. It A, B^ C]
A' J B\ C ; A'\ B'\ C" represent planes, we evidently obtain the
equation of a cubic by eliminating X,, /x, v between the equations
\A^lj.B+vC=0, \A'-\-fiB'-{-vC' = 0, \A" + fj.B''+vC'' = 0',
and if we take X-, yu., v as parameters, we can evidently, by
solving these three equations for ic, ?/, s, w^ which they implicitly
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 : E : S, where P, Q, E, S are
functions, of the lu^ order, of three parameters \, /a, v. This
system of equations evidently represents a surface, the equation
of which can be found by eliminating \, ft, v from the equations,
when there results a single equation in ic, ?/, z^ iv. If \, /tt, v
GENERAL THEORY OF SURFACES. 555
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 the m^ order in that plane. Let us first examine
the order of the surface represented by the system of equations,
or the number of points in which it is met by an arbiti-ary line
ax-^hy-\- cz + dw, ax-\-h'y-'tcz-\-d'w. To these points evidently
correspond in the plane the intersections of the two curves
aP^ hQ-VcR + dS= 0, a'F+ ¥Q + c'R + d'S=0,
whence it follows that the order of the surface is in general
7n^ If, however, the curves P, Q^ B, S have a common points,*
the two curves have besides these only vf — a. other points of
intersection, and accordingly this is the order 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 double points, that is to say,
whether the surface contains multiple lines. To the section
of the surface, by a surface of the /c^ order, ax'' + &c.= 0 cor-
responds in the plane a curve aP'' + &c. = 0 of the order wA-, and
on this each of the a points is a multiple point of the 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 order 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 -{■ cP + dS. This
number will be, in general, ??«/:>, but it will be reduced one
* For simplicity, we only notice the case where the common points are ordinary
points, but of course some of them may be multiple points.
556 GENERAL THEORY OF SURFACES.
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^ i?, S be quadratic functions of A., /i, 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^ M, 8 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
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 -4, 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^ P, 8 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^ P, 8 have no common point, the surface is
a quartic ; but every plane section being unicursal, the quartic
has a nodal curve of the third order ; this is Steiner's surface
already referred to.
586. Again, let P, Q^ P, 8 be cubic functions of X, //., v; in
order that the surface represented should be a cubic, the curves
P, Q^ P, 8 must have six common points. Then the deficiency
X)f 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-
GENERAL THEORY OF SURFACES. 557
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,
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, E, 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 nodal 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 [Giselle, vol. 69).
587. The " deficiency" of a plane curve of the order n with
B double points and k cusps is =^ (w — 1) (n — 2) — 8- /c, it is
equal to the number of arbitrary constants contained (homo-
geneously) in the equation of a curve of the order 7i — 3, which
passes through the 8 + k double points and cusps ; and it was
found by Clebsch that there is a like expression for the
" deficiency " of a surface of the order n having a nodal and
a cuspidal curve ; it is equal to the number of arbitrary con-
stants contained (homogeneously) in the equation of a surface
of the order (w — 4), which passes through the nodal and cuspidal
curves of the given surface.* Prof. Cayley thence deduced the
* More generally, if the surface has an t'-ple curve and also ^'-ple points, then
it is found by Dr. Kother that the deficiency is equal to the number of constants,
558 GENERAL THEORY OF SURFACES.
expression
Z>=i(n-l)(n-2)(w-3)-(n-3)(^>+c)+l((?+r)+2^ + ^/3-f |7 + t-i^,
where J, q are the order and class of the nodal curve, c, r those
of the cuspidal curve, t the number of triple points on the nodal
curve, yS, 7, i the number of intersections of the two curves
(/3 of those which are stationary points on the nodal curve,
7 stationary points on the cuspidal curve, i not stationary on
either curve), and 6 the number of singularities of a certain
other kind. In the case where there Is only a double curve
without triple points the formula is
Z) = 1 (w - I) (n - 2) (n - 3) - (n - 3) 5 + \q.
Thus in the several cases,
Quadric surface w = 2, 5 = 0, ^ = 0.
General cubic surface w = 3, 5 = 0, q = 0.
Quartic with nodal right line w = 4, 5 = 1, ([ = 0.
„ „ nodal conic % = 4, 5 = 2, q = %.
Quintlc with nodal curve,
a pair of non-intersecting right lines « = 5, 5 = 2, q = (^..
„ „ nodal skew cubic n = 5, 5 = 3, 5' = 4,
and in all these cases we find D = 0 or the surface is unicursal.
CONTACT OF LINES WITH SURFACES.
588. We now return to the class of problems proposed in
Art. 272, viz. to find the degree of the curve traced on a surface
by the points of contact of a line which satisfies three conditions.
The cases we shall consider are : [A) to find the curve traced
by the points of contact of lines which meet in four con-
secutive points ; [B) when a line is an inflexional tangent at
one point, and an ordinary tangent at another, to find the
degree of the curve formed by the former points, and (C) that
of the curve formed by the latter; {D) to find the curve
traced by the points of contact of triple tangent lines. To
as above, in the equation of a surface of the order n — A, which passes {i— 1) times
through the i-ple curve (has this for an {i — l)ple line), and {j — 2) times through
each ^-ple point (has this for a (J - 2) pie pointj.
COM TACT OF LINES WITH SURFACES. 559
these may be added : (a) to find the degree of the surface
formed by the Tines A ; [b] to find the degree of that formed
by the lines considered in [B] and ((7) ; (c) to find the degree
of that generated by the triple tangents.
Now to commence with problem ^ : if a line meet a surface
in four consecutive points we must at the point of contact not
only have Z7'=0, but also AZ7'=0, A''U'=0, A'Z7'=0. The
tangent 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'yzw\ the problem is a case of that treated
in Art. 473. Since then, by that article, the condition n = 0,
that the three surfaces should have a common line, is of the degree
substituting
X = l, V = 2, X" = 3; /i = ??-!, /i' = w-2, /' = 7i-3;
we find that n is of the degree (lln- 24). The points of con-
tact then of lines which meet the surface in four consecutive
points lie on the intersection of the surface with a derived surface
S of the degree 1 In — 24.*
The intersection of this surface S with the given surface U
is a curve of the order n (llw — 24), "the flecnodal curve" of Z7.
at any point of this curve the tangent plane of U meets U
in a curve having at the point a flecnode, or double point
having there an inflexion on one branch ; the tangent to this
inflected branch is of course the osculating (4-pointic] tangent.
589. We proceed to give Clebsch's calculation, determining
the equation of this surface S which meets the given surface
* I gave this theorem in 1849 {Cambridge and Dublin Journal, vol. iv. p. 260).
I obtained the equation in an inconvenient form {Quarterly Journal, vol. I. p. 33(5) ;
and in one more convenient {rhUosophical Transactions, 1860, p. 229) which I shall
presently give. But I substitute for my own investigation the very beautiful piece
of analysis by which Professor Clebsch performed the eUmiaation indicated in the
text, Crelle, vol. Lviii, p. 93. Prof. Cayley has observed that exactly in the same
manner as the equation of the Hessian is the transformation of the equation ?-t — s^
which is satisfied for every point of a developable, so the equation <S = 0 is the
transformation of the equation (Art. 437) which is satisfied for eveiy point on a ruled
surface.
560
GENERAL THEORY OF SURFACES.
at the points of contact of lines which meet it in four consecu-
tive points. It was proved, in 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 AZ7', A'^U'^ A^U\
This elimination is performed by solving for the coordinates of
the two points of intersection of the arbitrary plane, the tangent
plane AZ7', and the polar quadric A^U'', substituting these
coordinates successively in A^f/', and multiplying the results
together. Let the four coordinates of the point of contact be
£c„ 0*2, rr,, x^ ; the running coordinates ?/„ j/^, 3/3, 7/^ ; the differ-
ential coefficients m,, w^, M3, u^ ', the second and third differential
coefficients being denoted in like manner by suffixes, as
«i2, M,23. Through each of the lines of intersection of A Z7',
A*Z7', we can draw a plane, so that by suitably determining
^» ^2' ^35 ^4' ^® ^^^' ^^ ^^ infinity of ways, form an equation
identically satisfied
= ( i?, y, + p,i/2 + Psi/s + P4I/,) (?. y. + ^,^2 + ?32^3 + ^4^4) • • • (I) •
We shall suppose this transformation effected ; but it is not
necessary to determine the actual values of <,, &c., for it
will be found that these quantities disappear from the result.
Let the arbitrary plane be Cj7/^+c^i/^-\- 0^7/^ + cj/^, then it is
evident that the coordinates of the intersections of the arbitrary
plane, the tangent plane u^y^-\-u^^y^ + u^y^-\- u^y^^ and A^U\
are the four determinants of the two systems
c,,
C^J
^35
^4
^t»
%,
^31
u.
Pri
^2?
P,^
V,
c„
^2.
^35
C4
Wi,
«2)
«^3,
^4
<!.->
^25
^35
?4
These coordinates have now to be substituted in A'Z7', which
we write in the symbolical form («,2/, +^2^2 + o^?/, + a^yj';
where a, means -^ — ,
&c., so that, after expansion, we may
substitute for any term a^a^a^y^y^y^, '^^nzVyV^Vz^ &c. It is evi-
dent then that the result of substituting the coordinates of
the first point in A^U' may be written as the cube of the
symbolical determinant SajC^Wgp^, where, after cubing, we are
to substitute third differential coefficients, for the powers of the
CONTACT OF LINES WITH SURFACES. 561
a's as has been just explained. In like manner, we write the
result of substituting the coordinates of the second point
(SJjCgMg^'^)*, where 5, is a symbol used in the same manner
as a^. The eliminant required may therefore be written
The above result may be written in the more symmetrical form
For, since the quantities a, b are after expansion replaced by
diflferentlals, it is immaterial whether the symbol used originally
were a or b] and the left-hand side of this equation when
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 p and q.
590, Let us write
The eliminant is F'+G' = Q, or (i^+ Gf-^FG{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^p,+'m^p^+ 7n^p^+ m^p^) [n^q^ + n.^q^ + n^q^ + n^q^,
G = {n^p^ + n^p,^ + n^p^ + n^p^) {m^q^ + m^q^ + rti^q^ + m^q^\
where, for example, m^ is the determinant '2a^cjti^^ and n^ is
SJjCjMg. If then «, j be any two suffixes, the coefficient of
miUj in F+ G is [piqj +Pj(li)- And we may write
F+G = 2^m,nj{piqj + pjqi),
where both { andj are to be given every value from 1 to 4.
♦ The reason why we use a different symbol for ,-"» <S!rc. in the second deter-
minant is because if we employed the same symbol, the expanded result would
evidently contain sixth powers of a, that is to say, sixth differential coefficients.
We avoid this by the employment of different symbols, as in Prof. Cayley's " Hyper-
determinant Calculus" {Lessons on Higher Algebra, Lesson Xiv.), with which the
method here used is substantially identical.
cccc
562
GENERAL THEORY OF SURFACES.
But, by comparing coefficients In equation (I), we have
whence F-\- G = 22'2minju-j 4 I,I,m,nj {t^Uj + tjUi).
Now It Is plain that if for every term of the form ^,5'y -{-p^qi
we substitute tiUj + tju-^ 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 ^a^c^u^q^^ Ih^c^n^^ we alter q Into m, the
determinants would vanish as having two columns the same.
The latter set of terms therefore in F-V G disappears, and we
have ^ {F+ G) = ^^minjUij.
Now, if we remember what is meant by ?n,, ??y, this double
sum may be written In the form of a determinant
^ij W12) ^35 ^.45 o-,) ^1) «,
^21? %Z^ "231 ^24? «2J C„ W^
*^8H ^32? ^33: ^^34? «3? ^3^ ^3
**4.5 ^2> ^3? ^447 «45 C„ W4
^? ^2? ^3 5 K
^1 J '^S ? ^3 ) ^4
«^,) ^2? ^3? ^4
For since this determinant must contain a constituent from each
of the last three rows and columns it is of the first degree In
z^,„ &c., and the coefficient of any term u^^ is
In the determinant just written the matrix of the Hessian
is bordered vertically with «, 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
a, c, ?«
5, c, u
591. The quantity FG is transformed In like manner. It
is evidently the product of
{m,p^ + m^^p^ + m^p^ + m^p^) {m^q^ + on^^q^ + m^q^ + m^q^^
and (r?,p, + w j7^ 4 n^p^ 4 w,pj (9?,2, + '^S-, + ^'3*73 + '>\n^'
CONTACT OF LINES WITH SURFACES.
563
Now if the first line be multiplied out, and for every term
[P\^-i'^P-/l\) ^® substitute its value derived from equation (I),
it appears, as before, that the terms including t vanish, and it
becomes '2^^m,mjU-j^ which, as before, is equivalent to ( ' ' ) ,
where the notation indicates the determinant formed by border-
ing the matrix of the Hessian both vertically and horizontally
with a, c, u. The second line is transformed in like manner ;
and we thus find that (i^+ Gf -^FG[F+ G) = 0 transforms
into
/a, c, u\ f fa, c, uV _ ^ fa, c, u\ fb, c, m\
\h. c, u) \ \b, c, It) \a, c, uj \b, c, uj
vo. c, uJ ( \b, c, UJ \a, c, UJ \d>, c,
It remains to complete the expansion of this symbolical ex-
pression, and to throw it into such a form that we may be
able to divide out c,a;, + c^x^ -V c^x^ + c^x^. We shall for short-
ness write a, J, c^ instead of a^x^ + a^x^ + a^x^ -f a^x^, b^x^ + &c.,
CjCCj + &c.
592. On inspection of the determinant, Art. 590, which we
have called ( r' ' ) , it appears that since
w„a;, + u^,^x.^ + u^^x^ + u^^x^ = (w - 1) m„ &c.,
this determinant may be reduced by multiplying the first four
columns by a?,, cc^, x^^ x^, and subtracting their sum from the
last column multiplied by (n — 1), and similarly for the rows ;
when it becomes
[n-\f
u,o
«^2)
^13?
^.47
«^.)
«!»
0
%r,
^'■2.J
'^..s)
^4?
«2>
^2,
0
^H
^32?
^33?
^41
«3»
^3)
0
«^4I)
^42)
^43?
^^4»
«47
^45
0
\.
K,
K.
Z'4,
0,
0,
-b
^1)
CaJ
^3,
^4,
0,
0,
— c
0,
0,
0,
0,
-a,
-c,
0
which partially expanded is
(n-ir
a
ac
:)-k:)-k:)j
564
GENEEAL THEORY OF SURFACES.
where (,) denotes the matrix of the Hessian bordered with
a single line, vertically of a's and horizontally of 5's.
In like manner we have
(:::::)=-,A^H:)—(:) -'(:)}.
\5, c, u)
L
-H>^'0]
Now as it win be our first object to get rid of the letter a,
we may make these expressions a little more compact by
writing ch^ — hc^ = d^^ &c., when it is easy to see that
(^<D-K^^'(:)'
Thus
O-O' G)-(:)-k:)-
Uc,'J- [n-lY\d)'\h\c,u)~ (n-irfW ""Wr
and the equation of the surface, as given at the end of last
article, may be altered Into
'Kdi-^Kd
i\c
a<:.)F-K:;)H:)-(:)
4- a'
593. We proceed now to expand and substitute for each term
^1^2^35 <^c., the corresponding differential coefficient. Then, in
the first place, it is evident that
a^ = w (w - 1) (n - 2) M = 0 ; a^a^ = [n-\)[n-2) m„ &c.
Hence
a
(:)=(«-i)(-^)(:).
But the last determinant is reduced, as In many similar cases,
by subtracting the first four columns multiplied respectively by
aj„ ajg, ajg, x^ from the fifth column, and so causing it to vanish,
except the last row. Thus we have
a« {^\ = - (n - 2) He.
CONTACT OF LINES WITH SURFACES. 565
Again, f j is {see Lessons on Higher Algebra^ Art. 34)=— 2-^ — ^„,^„'
We have therefore
al''\=-(n-2)2^u =-A(n-2)H.
Lastly, it is necessary to calculate a( )[t)' Now if ?7,,„
denote the minor obtained from the matrix of the Hessian by
erasing the line and column which contain w^^,^, it is easy to see
that a ( j f,j = — (?? — 2) ^UmpUqnUmnCpdq^ where the numbers
m, n^ Pj q are each to receive In turn all the values 1, 2, 3, 4.
But (see Lessons on Higher Algebra, Art. 33)
77 TT — TJ TJ ^Tf ^^g
UUjfin
Substituting this, and remembering that S C^„,„m„,„ = 4Zr, we have
''(:)Q=-("-)^C)"
Making then these substitutions we have
tC)-«G)}H:)— O-'C)}
But attending to the meaning of the symbols c?j, &c., we see
that d or d^x^ + d^x.^ + d^x^ + d^x^ vanishes Identically. If then
we substitute in the equation which we are reducing the values
just obtained, it becomes divisible by c", and is then brought
to the form
^C)'-3(:)(:)d)=o-
594. To simplify this further we put for d its value, when It
becomes
4 \c
\a
566 GENERAL THEORY OF SURFACES.
Now tills Is exactly the form reduced in the last article,
except that we have h instead of a, and a in place of d. We
can then write down
AOOHth^-'H:) (:)-("-)^k:)} >
while the remaining part of the equation becomes
But (last article) the last terra in both these can be reduced to
12 {n— 2y Wc { j . Subtracting, then, the factor c^ divides out
again, and we have the final result cleared of irrelevant factors^
expressed in the symbolical form
oh:) AD Oh-
595. It remains to show how to express this result in tho
ordinary notation. In the first place we may transform it by
the identity (see Lessons on Higher Algebra^ Art. 33)
'a, b\ (a\ fb'
E
a,b\ _(a\ lb\_ /ay
a,b) \a} Kb) \bj '
whereby the equation becomes
000-<)i::l)--
Now ( ) I if/) expresses the covariant which we have before
called 0. For giving to U^^^^ the same meaning as before, the sym-
bolical expression expanded may be written 2 Z7,„„ Upg UrsUmnrUp^s-,
where each of the suffixes is to receive every value from 1
to 4. But the differential coefficient of H with respect to x^
can easily be seen to be 2 UmnUmnrj so that 0 is 2 f/^, -r— -j— ,
which is, in another notation, what we have called 0, p. 510.
The covariant S is then reduced to the form 0 — 4/f<f>, where
CONTACT OF LINES WITH SURFACES. 567
where Up,j^rs 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^ I860, p. 239.
For surfaces of the third degree Clebsch has observed that ^
reduces, as was mentioned before, to 2 ZJ,,,^!/,^^, where ZT,,,,
denotes a second differential coefficient of H.
596. The surface S touches the surface H along a certain
curve. Since the equation 8 is of the form 0 - 4jE?4> = 0,
it is sufficient to prove that 0 touches II. But since 0 is got
by bordering the matrix of the Hessian with the dIflFerentlals
of the Hessian, 0 = 0 Is equivalent to the symbolical expression
(TJ \
1 = 0. But, by an identical equation already made use of,
we have
rjfc,H\ [H\ (c\ (H
M 1 — 1
^c, H J \II J \c/ \ c
where c is arbitrary. Hence 0 touches H along Its Intersection
/ 7-I\
with the surface of the degree 7« — 15, f j . It is proved
then that S touches H, and that through tlie curve of contact
an infinity of surfaces can pass of the degree In — 15.
597. The equation of the surface generated by the 4-pointIc
tangents is got by eliminating x'y'z'w between Z7' = 0, A Z7' = 0,
^ U' = 0y A^ Z7' = 0 ; which result, by the ordinary rule, Is of
the degree
n{n-2){n-S) + 2n{n-l){n-3) + Sn{n-l)[n-2) = Gn^-22n^+lSn.
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^3I=0. The
degree of M is therefore
2n(w-3)(3?i-2).
568 GENEEAL THEORY OF SURFACES.
598. We can in like manner solve problem B of article 577.
For the point of contact of an inflexional tangent we have
U' = 0, A Z7' = 0, A'"* C/"' = 0 ; and if it touch the surface again,
we have besides W' = 0^ where W is the discriminant of the
equation of the degree n — 3 in X : /a, which remains when the
first three terms of the equation, p. 242, vanish. For W^ then
we have V= (w + 3) (n - 4), /*" = (w -3) (w — 4) ; and having,
as in Art. 577 and last article, X = l, /t = n— 1 ; V = 2, /x' = 7i-2,
we find for the degree of n
2 (w - 3) (n - 4) + (w - 2) [n + 3){n- 4)
-\-2{n-l)[n + S){n-4:)-2{n + S){n- 4).
The degree, then, of the surface which passes through the
points B is {n- 4) {Sii^ + 5n- 24).
The equation of the surface generated by the lines [h)
which are in one place inflexional and In another ordinary
tangents. Is found by eliminating x'y'z'w between the four
equations Z7' = 0, AZ7' = 0, A'''Z7' = 0, TF' = 0; and, from what
has been just stated as to the degree of the variables In each
of these equations, the degree of the resultant Is
n [n - 2) (n - 3) (w - 4) + 2n [n - 1) (n - 3) {n - 4)
+ w (w - 1) (w - 2) (n + 3) (w - 4) = w (n - 4) [n^ + Sri" - 20n + 18).
But it appears, as in the last article, that this resultant contains
as a factor U in the power 2 (n + 3) (w — 4). Dividing out
this factor, the degree of the surface {b) remains
n {n - 3) (n - 4) [n' + 6n- 4).
599. In order that a tangent at the point xyzw' may
elsewhere be an inflexional tangent, we must have AU' = 0^
(an equation for which \=], fi = n-l), and, besides, we must
have satisfied the system of two conditions, that the equation
of the degree n -2 in X : /a, which remains when the first
two terms vanish of the equation, p. 242, may have three
roots all equal to each other. If then X', /u,' ; X'', fxf^ be the
degrees in which the variables enter into these two conditions,
the order of the surface which passes through the points (C)
is, by Art. 473, X>" + X"/ + {n - 2) X'X". But (see Higher
CONTACT OF LINES WITH SURFACES. 501)
Algebra on the order of restricted systems of equations)
W = [n - 4) (n* + w + 6), XV" + Vy = [n-2)[n- 4) [n -f 6).
The order of the surface C is, therefore,
[n -2) in- 4) {n^ + 2w + 12).
The locus of the points of contact of triple tangent lines
is investigated in like manner, except that for the conditions
that the equation just considered 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) for this system of conditions we have
XX' = -Kn - 4) {n - 5) [n' + 3n + 6),
Xy + \V = (w - 2) {11 - 4) (w - 5) [n + 3).
The order of the surface which determines the points [D]
is, therefore, \{n — 2){n — 4) {n - 5) [yi^ + 5n 4 12).
To find the surface generated by the triple tangents we
are to eliminate xyz'w between U' = 0, AU' = 0, and the two
conditions, the order of the result being
n/jb'/j," + 71 (n — ]) [Xfj,'' + X'im) ;
but since this result contains as a factor U^"^'\ in order to find
the degree of the surface (c) we have to subtract n\'\" from the
number just written. Substituting the values last given for
VX", XV" + X"/x' : and for /^V", i (^ - 2) (n - 3) (w - 4) [n - 5),
we get, for the order of the surface (c), after dividing by three,
^n (n - 3) (w - 4) (n - 5) (/i' + 3w - 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-ixjint tangents.
The complete curve of intersection with U of the ruled surface M whose degi-ee
is a consists of the curve of points of simple intersection, whose order we call a„ and
of the curve of fourfold points, whose order we call a^. We have manifestly
4«4 + n, = na. Putting in their values a = 2n {n - 3) (3« - 2), a^ = n (Ibi — 24), we
find a, = '2n {n - 4) (S?*^ + n~ 12).
Ex. 2. To find the degree of the curve formed by the points of simple intersection
of inflexional tangents which touch the suiiace again.
DDDD
570 GENERAL THEORY OF SURFACES.
The complete curve of intersection of the ruled surface h with U consists of tlie
curve of points at which tlie tangents are inflexional, of order ^3 ; of that of the
ordinary contacts, of order b^ ; and of that of the simple intersections, of order 6,.
Among these we have the obvious relation nb = 8^3 + 2^2 + ^1 ; putting in their
values
b = n {n - 3) {n - 4) (?i2 + 6n - 4), 63 = w {n - 4) (Sw* + 5n - 24),
*2 = n (« - 2) (n - 4) («2 + 2n + 12),
we find bi = n {n — 4) (« — 5) {n^ + Gn~ — 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 = ICo + Cj, whence as
c = |«(w-3)(M-4)(n-5)(?j2 + 3,j_2) and c, = !«(«- 2) (w - 4) (?i - 5) (ji^ + 5n + 12),
we have c, = ^n {n - 4) (11 -h){n- 6) («3 + 3,^2 _ 2w - 12).
600. 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 : (a) the number of points
at which both the inflexional tangents meet in four con-
secutive points ; (/3) the number of lines which meet in five
consecutive points ; (7) the number of lines which are doubly
inflexional (fourpoint) tangents in one place, and ordinary
tangents in another ; (S) of lines inflexional in two places ;
(s) inflexional in one place and ordinary tangents in two others ;
(^) of lines which touch in four places.
The first of these problems has been solved, as follows,
by Clebsch, Crelle^ vol. LXiil. p. 14, but with an erroneous
result, as has been shown by Dr. Schubert, Math. Ann.^
vol. XI. p. 375. 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^ + yll^ + zli^ + wH^. Let it be required now to find
the locus of points xy'z'w on a surface such that the line
joining xy'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' + /ia?, \y' + fJi-y, &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 A//', and also of the polar cubic A'"'f/'. JS'ow
CONTACT or LINES WITH SURFACES. 571
the result of substitution in A//' is 4 (w - 2) XIT' + /i A//' = 0.
When we substitute in A^C/', the coefficient of A,* vanishes
because x'yz'w is on the surface, and that of A,'^ vanishes
because xyzuo is in the tangent plane. The result is then
3 (?i-2)A,A'J/'+yuA't^'=0. Eliminating \: /i between these two
equations, we have 4.H' ^' U' = ^ ^IF /^' U\ where in A'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
w„ w„ 1*3, M,
a, ^, 7, S
a? /^'j 7', S'
By this substitution A^ U' becomes in x'yzw of the degree
w — 3 + 3 {ii - 1) = 4/i — 6, and H' being of the degree 4 (n — 2),
the equation is of the degree 8?i - 14. This, then, is the degree
of the locus required.
Now the points at which two fourpoint tangents can be
drawn belong to this locus. At any one of these points
the doubly inflexional tangents evidently both lie on the
polar cubic of that point, and their plane will therefore inter-
sect that cubic in a third line which, as we saw (Art. 537),
lies in the plane A//'. Every point on that line is to be con-
sidered as a point of inflexion of the polar cubic ; and therefore
the plane through the point x'yz'w and any arbitrary line must
pass through a point of inflexion. The points then, whose
number we are investigating, and which are evidently double
points on the curve U8^ are counted doubly among the
w (llw — 24) (8n- 14) intersections of the curve TJ8 with the
locus determined in this article. Let us examine now what
other points of the curve TJ8 can belong to the locus. At
any point on this curve the fourpoint tangent lies in the polar
cubic, the section of which by the tangent plane consists
of this line and a conic ; and since all the points of inflexion of
such a system lie in the line, the fourpoint tangent itself is,
in this case, the only line joining x'yz'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 2n (w — 3) (3n — 2).
572 GENERAL THEORY OF SURFACES.
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) (3n - 2) = w (1 bi - 24) (8n - 14),
whence a = 57i {In^ - 28w + 30),
which is the solution of the problem proposed.
601. To find the points on a surface where a line can be
drawn to meet in five consecutive points, we have to form the
condition that the intersection of A U\ A'^ U\ and an arbitrary
plane should satisfy A^f/"', as well as A^CT"'. Clebsch
applied to A*Z7' the same symbolical method of elimination
which has been already applied to A^f/'. He succeeded in
dividing out the factor c^ 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 the degree 14n — 30 in the variables, all that can be
concluded from it is that through the points which I have
called /3 (Art. 600) an infinity of surfaces can be drawn of the
degree 14?i — 30. We can say, therefore, that the number of
such points does not exceed w (1 1 n - 24) (14n — 30).
602. The numerical solution 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 arbitrary
right line, then the number of planes of the system which
* Giitt. Nach:, Feb. 1876 ; Math. Ann., X. p. 102, xi. pp. 348-378. See also his
Kulkid dcr ab<:dMe>uku Gwmtlvin (1879), pp. 236-7, 246.
CONTACT OF LINES WITH SURFACES. 573
contain a pair of corresponding points Is p + q', but since of
these there are q whose connecting lines meet the arbitrary
line, the remaining- p-\- q — g contain coinciding pairs of points
of the systems.
We proceed In the first place to establish the value already
stated for a. The points of contact of the inflexional tangents
which meet an arbitrary given right line I are easily shown
as in p. 546, to lie on the intersection of U with a surface
of the degree 3n — 4. This surface meets the flecnodal curve
(see notation in Examples, Art. 599) in (3w — 4) a^ points, which
consist of the a points of contact of. fourpoint tangents which
meet the line ?, and the J=(3«-4)rt^-a flecnodes, whose
ordinary inflexional tangent meets I.
Accordingly, we may suppose a pencil of rays in a plane
such that to each ray which meets a fourpoint tangent corresponds
one which meets the other inflexional tangent at the same
flecnode. In such a pencil there will be a + c? = (3n — 4) a^
rays meeting as well a fourpoint tangent as also the other
inflexional tangent at its flecnode. But these rays include
the a^ rays to the points of the flecnodal curve in the plane
of the pencil and {n — \)a^ which He in the tangent planes
through the vertex of the pencil to U at flecnodes. Thus
there remain
a ■'r d—a^- (n — 1) a^ = 2 (w — 2) a^
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
2n[n-2) (lln-24)
points on a surface of the degree n in which coincident
injiexional tangents ham a fourpoint contact.
The d tangent lines generate a ruled surface intersecting
Z7 in a curve of degree nd which consists of the curve of
threefold points whose degree is a^ and of that of ordinary
intersections of degree «,'. These give
a^ + 3(7^ = nd.
574 GENERAL THEORY OF SURFACES.
Now applying the principle of correspondence, to each of the
a^ 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^. Hence putting (w — 3) c?
fory,
a/ + [n — B)a^- {n — S)d
is the number of coincidences of a flecnode and one of the
simple points on the ordinary Inflexional tangent. Now we
saw that in 2[n — 2)a^ fourfold points the two osculating
tangents coincide, hence the difference
a/ + (w - 3) a, - (?i - 3) t? - 2 (w - 2) a^ = [Sn - 14) a^ - Ba
is double the number of biflecnodal points, as in Art. 600.
603. Next to determine ^. A fivepolnt contact arises
from a fourpolnt contact by the coincidence of one additional
simple point of intersection. To each of the a^ points in
a plane correspond w — 4 simple Intersections of the osculating
tangents at them with Z7; and to each of the points a^
in the plane corresponds a single fourfold point. Hence
the number p + q for these two systems is [n — 4) a^ + a^.
But the surface If meets any right line In a points through
each of wlilch passes a line connecting the ??— 4 points a, to
the corresponding o.^ • hence in this case ^ is (n — 4) a. Ac-
cordingly, the number of coincidences of a point a, with a
point a^ is
fi = {n - i) a^ + a^~ [n - A) a = {n- 8) a^ + ia = 6n (n-4) [In- 12).
The same number Is found from the analogous relation
fi = K + K-b,
since the union of a threepoint with an ordinary contact also
leads to a fivepolnt one.
Again, fourpolnt tangents having another ordinary contact
may arise either through coincidence of two simple intersections
CONTACT OF LINES WITH SURFACES. 575
ou a fourpo'mt tangent, giving in a similar manner by the
principle of correspondence
7 = 2 (« — 5) «, — (w — 5) (n — 4) a ;
or, through the coincidence of a simple Intersection with the
threepoint contact of an inflexional tangent which touches else-
where, giving
y=(^n-5)h^ + b^ — {n-5)h',
or, lastly, by the coincidence of two contacts of a triple ordinary
tangent, giving
7 = 4c.^ — 6ci
Each method leads to
7 = 2n {n - 4) (w - 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)b^ + b^-{n-5)b = 2S,
which gives
S = ln{n-A){n- 5) {rf + 3n' + 2dn - 60).
Inflexional tangents having two further ordinary contacts
arise from coincidences of two simple Intersections among those
on inflexional tangents having one other ordinary contact, thus
2s = 2 (w - 6) &. - (n - 5) (n - 6) 5 ;
or, from coincidence of a simple Intersection with one of the
ordinary contacts among those ou tangents having three such,
whence
£ = (n - 6) A^ + 3c, - 3 (?? - 6) c
= In (« - 4) {n -5){n- 6) (n' + dn' + 20n - 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 (w - 7) c, - [n - 6) (n - 7) c ;
^= iij7i [n - 4) [n - 5) (n - 6) {n - 7) {ii' + On' + 7?i - 30).
576 GENERAL THEORY OF SURFACES.
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 tan-
gent 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 additional
condition is given. Of the latter class of problems we shall
consider but two, the discussion of the case when the plane
meets the surface in a section having a cusp, and that when
it meets it in a section having two double points. Other cases
have been considered by anticipation in the last section, as
for example, the case when a plane meets In a section having
a double point, one of the tangents at which meets in four
consecutive points.
605. Let the coordinates of three points be xyzio\
x'y"z"w\ ocyziv ; then those of any point on the plane through
the points will be \x + fjux" + vx^ \ij' + [xy" -+ v?/, &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 [f^] =0, then [Z/J may be written
r u' + vv^.. u' + a."-va u' + \x'~' (/i A,, + v^Y u' + &c. = 0,
where A,. = a: ^^, + 2/ ^, + ^ ^. + t^ ^,
d d d d
dx dy dz' dw '
The plane will touch the surface if the discriminant of this
equation in X, yti, v vanish. If we suppose two of the points
CONTACT OF PLANES WITH SURFACES. 577
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 xy'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 V = 0, A^^ V = 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
that line. If the tangent plane at x'y'zw 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 [Z7] as follows, the
coefiicients of \", \""'/x being supposed to vanish,
T^-\ + i V'-^ {Aix,' + 2 S/ij/ + Cv^) + &c. = 0.
T represents the tangent plane at the point we are considering,
G its polar quadric, while ^ = 0 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
TA (5' -AG)'<^\T^ = ^^
where ^ is the discriminant when T vanishes as well as TJ'
and Ajjy ' In order that the discriminant may be divisible
by y, some one of the factors which multiply T'' must either
vanish or be divisible by T.
606. First, then, let A vanish. This only denotes that the
point x"y"z"io" lies on the polar quadric of x'y'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 x'y'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 ^ — 2 distinct from that plane ; but
that if the line be an inflexional tangent, three will coincide
EEEE
578 GENERAL THEOKY OF SURFACES.
with that tangent plane, and there can be drawn only p - 3
distuict 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 discussion.
We may examine, at the same time, the conditions that T
should be a factor in B'^ - AGj and in 0.
The problem which arises in both these cases is the fol-
lowing : Suppose that we are given a function F, whose degrees
in xyzw\ in z"y"z"w\ and in xyzw are respectively (X., yu., /*).
Suppose that this represents a surface, having as a multiple
line of the order yu, 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 [ii- 1, 0, 1). If so,
any arbitrary line which meets T will meet F, and therefore
if we eliminate between the equations T= 0, F= 0, and the
equations of an arbitrary line
ax^hy -\r cz + dw = 0, ax + h'y + cz + d\o = 0,
the resultant R must vanish. This is of the degree jx in ahcd^
in ah'c'd\ and in x'y"z"w'\ and of the degree fjb{n—V)-\-\
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 F. The condition (il/= 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 the
degree fi {ti — 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 <^ both represent
planes through the same line. The first is of the degree
{2 [n - 2), 2, 2), while 4> is of the degrees [n - 2) [n^ - 6),
?i'' — 2?t* -\-n — Q, n^ — '2n^ + ?^ - 6, as appears by subtracting the
sum of the degrees of 7''% A, and {B" - ACf from the degrees
of the discriminant of [?7], which is of the degree n[n—\Y
in all the variables. It follows then from the last article that
the condition (// = 0) that T should be a factor in B'- AC
CONTACT OF PLANES WITH SURFACES. 579
is of the degree 4(n — 2), and the condition (^=0) that T
should be a factor in 0 is of the degree {n — 2) [ti^ — 1^ ■\-n— 12).
At all points then of the intersection of TJ and // the tangent
plane must be considered double. H is no other than the
Hessian ; the tangent plane at every point of the curve TJH.
meets the surface in a section having a cusp, and is to be
counted as double (Art. 269). The curve TJK is the locus of
points of contact of planes which touch the surface in two
distinct 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 along the curve UH. They form a developable
whose degree is /j = 2/? (71 - 2) (Sy? - 4), Ex. 3, Art. 576. The
class of the same developable, or the number of planes of the
system which can be drawn through an assigned point, is
j/ = 4« (/z — I) (n — 2). For the points of contact are evidently
the intersections of the curve TJH with the first polar of
the assigned point. We can also determine the number of
stationary planes of the system. If the equation of Z7, the
plane z being the tangent plane at any point on the curve C///,
be 2 + y'' -I- W3 + &c. = 0, it is easy to show that the direction
of the tangent to UR is in the line -— ._f = 0. Now the tan-
gent planes to TJ are the same at two consecutive points
proceeding along the inflexional tangent y. If then ii^ do
not contain any term o;^ (that is to say, if the inflexional tan-
gent meet the surface in four consecutive points), the direction
of the tangent to the curve TJH. is the same as that of the
inflexional tangent ; and the tangent planes at two consecutive
points on the curve TjH will be the same. The number of
stationary tangent planes is then equal to the number of inter-
sections of the curve TJH with the surface 8. But since the
curve touches the surface, Art. 596, we have
a = 2?«(n-2)(ll?i-24).
From these data all the singularites of the developable which
touches along TJH can be determined, p being the r, v the «,
580 GENERAL THEORY OF SURFACES.
and a the same as at p. 292, we have
yLi = w(w-2)(28?i-60), v = 4w(w-l)(n-2), p = 2??(w-2)(3n-4),
a = 2w (n - 2) (llw - 24), /3 = w (n - 2) (70w - 160) ;
2g=^n{n-2) (16n* - 64w'' + 80n' - 108n + 156),
2A = w (n - 2) (784n* - 4928n' + 10320n' - 7444^2 + 548).
The developable here considered answers to a cuspidal line
on the reciprocal surface, whose singularities are got by inter-
changing yu. and V, a and yS, &c. in the above formulge.
The class of the developable touching along UK^ which is
the degree of a double curve on the reciprocal surface, is seen
as above to be n(?2 — 1) (n — 2) (?i^ — n''^ + w — 12). Its other
singularities will be obtained In the next section, where we
shall also determine the number of solutions in some cases where
a tangent plane is required to fulfil two other conditions.
THEORY OF RECIPROCAL SURFACES.
609. Understanding by ordinary singularities of a sur-
face, those which in general exist either on the surface or
its reciprocal, we may make the following enumeration of
them. A surface may have a double curve of degree h and
a cuspidal of degree c. The tangent cone, determined as in
Art. 277, includes doubly the cone standing on the double
curve and trebly that standing on the cuspidal curve, so that
if the degree of the tangent cone proper be a, we have
a + 2j + 3c = 7i(n-l).
The class of the cone a is the same as the degree of the
reciprocal. Let a have S double and k cuspidal edges. Let
h 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 h and c intersect in 7 points,
which are stationary points on the former, in /3 which are
stationary points on the latter, and in i which are singular
points on neither. Let the curve of contact a meet b m p
points, and c in cr points. Let the same letters accented denote
singularities of the reciprocal surface.
THEORY OF RECIPROCAL SURFACES. 581
610. We saw (Art. 279) that the points where the curve
of contact meets ^^U, give rise to cuspidal edges on the
tangent cone. But when the line of contact consists of the
complex curve a +25 + 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^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, J, c, with the surface /^^U,
a{n-2) = K + p + 2a- "l
b{n-2)=p + 2l3 + 3y + 3t[ (A).
c(n-2) = 2o- + 4/3 + 7 i
The reader can see without difficulty that the points indicated
in these formulae are included in the intersections of A*Z7
with a, 5, 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.*
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 618^= T'\
where S is of the fourth and T of the sixth degree (p. 485).
Each of the twenty-seven lines (p. 497) on the surface 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=G, b' = 27, c' = 24, n'=12, a + 2b' + Sc = 12.11. The
* The first attempt to explain the effect of nodal and cuspidal lines on the degi'ee
of the reciprocal surface was made in the year 1847, in two papers which I con-
tributed to the Cainbridge and Dublin Mathematical Journal, vol. II. p. 65, and
IV. p. 188. It was not till the close of the year 1849, however, that the discovery
of the twenty-seven right lines on a cubic, by enabling 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 Acudewj, vol, XXIII. p. 4G1.
582 GENERAL THEORY OF SURFACES.
intersections of the curves c and h' 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 Z7i7,
and with the twenty-seven lines. Consequently there are
twelve points o-' and twenty-seven points p ; one of the latter
points lying on each of the lines, of which the nodal 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
9a + 27&+ 12c= 60, by any integer values of a, Z», c except
1, 1, 2. Thus we are led to the first of the equations {A).
Consider now the points wdiere any of the twenty-seven
lines h meets the same surface of the tenth order. The points
/S' 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 « -f 2& + 5c = 10,
can have only the systems of Integer solutions (1, 2, 1) or
(3, 1, 1), the ten points of intersection of one of the lines
with the second polar must be made up either p -f 2/3' + 1\ or
3p' + /3' + i', and the latter form is manifestly to be rejected.
But, considering the curve V 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 V {n -2)=p+ 2/3' + 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 o-',
and the fifty-four points /3'. Now the equation 12a+ bib=^2i0
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.
THEORY OF RECIPKOCAL SURFACES. 583
612. Let us now examine in the same way the reciprocal
of a surface of the n" order, Avhich has no multiple points.
AVe have then n=n {n - 1)', n- 2 = [n - 2) (w''+ 1), «'= n{n-l)]
and for the nodal and cuspidal curves we have (Art 286)
b' = ^n [n - 1) {71 - 2) (n' - ?«' + n- 12), c=An{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^ - n' -i n-1 2), cr'.= 4/i [n - 2). Substitute
these values in the formula a [n —2) = k + p' -\-2a\ and it is
satisfied identically, thus verifying the first of formulee {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 /3' is 2n [n — 2) (Ibz — 24). Now the intersections of the
nodal and cuspidal curves on the reciprocal surface answer to
the planes which [touch at the points of meeting of the curves
UH, and UK on the original surface. If a plane meet the
surface In a section having an ordinary double 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 /3' are
to be Included in the intersections of the nodal and cuspidal
curve, the points U, H, K must either answer to points /8'
or points 7'. Assuming, as It Is natural to do, that the
points /8 count double among the Intersections of UHK^
"we have
7' = n {4 [n - 2)] . {[n - 2) (n' - if + n- 12)} - 4;^ [n - 2) (11m - 24)
= 4n [n - 2) [n - 3) (n' + 3n - 16).
But If we substitute the values already found for c', n\ <t', /3',
the quantity c {n' - 2) — 2cr' — 4/3' becomes also equal to the
value just assigned for 7'. Thus the third of the formulae A
Is verified. It would have been sutficlent to assume that the
points /3 count p, times and that the points 7 count p, times
among the intersections of UHK, and to have written that
584 GENERAL THEORY OF SURFACED.
formula provisionally c{n — 2) = 2a + fi0 + ^7, when, proceeding
as above, it would have been found that the formula could not
be satisfied unless X = 1, /a = 4.
It only remains to examine the second of the formulce [A).
We have just assigned the values of all the quantities involved
in it except f. Substituting then these values, we find that the
number of triple tangent planes to a surface of the n^" degree
is given by the formula
6^' = n{n- 2) (w' - 4w' + 7n' - 45w* + 114w' - 11 In" + 548w - 960),
which verifies, as it gnves 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 in two distinct
points lie on a certain surface of the degree (w — 2)(w-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 aJ, which
meet the curves a, h in distinct points ; and, similarly, for the
other pairs of cones. If then we denote by \ab] the number
of the apparent intersections of the curves a and ?>, 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, &, c, with the surface of
the degree (w — 2) [n - 3) :
a[n-2) {n - 3) = 28 -f 3 [ac] + 2 [ah],
b{n-2)[n- 3) = 4:k+ [ab] + 3 [be],
c{n-2){n-3)=6h+ [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,
*[a&] = a5-2p, [ac] = ac-Sa, [be] = be -3/3 - 2y- i.
* If the surface have a nodal curve, bat no cuspidal, there ■will still be a deter-
miriate number i of cuspidal points on the nodal curve, and the above equation
receives the modification [ab] = ab — 2p — i. In determining, however, the degree of
the reciprocal surface the quantity [ab] is eliminated.
THEORY OF RECIPROCAL SURFACES. 585
Substituting these values, we have
a{n-2)[n-3)=28 + 2ab + Sac - 4p - 9o- -j
b{n-2){n-3) = U + ab + Sbc - 9/3 - 67 - Si- 2p [...(5).
c{n-2){n-S) = Qh + ac + 2bc - 6/3 - 47 - 2t- Sa J
The first and third of these equations are satisfied identically
if we substitute for ^S, 7, p, o-, &c., the values used In the last
article, to which we are to add 28' = n[?i- 2) [if — d), / = 0,
and the value of // got from (Art. 608),
2h' = w (w - 2) (1 6n* - 64n' + 80n' - 108n + 156).
The second equation enables us to determine k' by the equation
Sk' = n{n~2) (n'" - 6n' + 16/*' - 54n'
+ lUn" - 288n' + 547n* - 1058?i' + 1068n' - 1214w ■{- 1464) ;
from this expression the rank of the developable, of which h' is
the cuspidal edge, can be calculated by the formula
R = b''-b'-2k'-6t'-Sy\
Putting in the values already obtained for these quantities
we find
R = n [n -2){n- 3) [n' -\-2n- 4).
This is then the rank of the developable formed by the planes
which have double contact with the given surface.
614. From formulse A and B we can calculate the diminu-
tion in the degree of the reciprocal caused by the singularities
on the original surface enumerated Art. 609. If the degree of
a cone diminish from m to m — Z, that of its reciprocal diminishes
from m {m — 1) to {m- I) [in- l—l) ] that is to say, is reduced
by I [2m- ?— 1). Now the tangent cone to a surface is in
general of the degree n[n — 1), and we have seen that when
the surface has nodal and cuspidal lines this degree is reduced
by 2b + 3c. There is a consequent diminution in the degree
of the reciprocal surface
D = [2b + 3c) [2n' -2n-2b-Sc- 1).
But the existence of nodal 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
FFFF
586 GENERAL THEORY OF SURFACES.
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 A^ we have
« = (a - 5 - c) (?z - 2) + 6yS + 47 + 3^
But, since if the surface had no multiple lines, the number of
cuspidal edges on the tangent cone would be {a + 2b + 3c) (n - 2),
the diminution of the number of cuspidal edges is
K= {3b + 4c) (n - 2) - 6/3 - 47 - 3t.
Again, from the first system of equations in last article, we have
{a -2b- 3c) (n - 2) [n -3) = 2B-8k- I8h - 12 [bc]j
and putting for [be] its value
28={a-2b-Sc){n-2)(n-3) + 8k + I8h-+ \2bc - 36/3-247-12«.
But if the surface had no multiple lines, 2h would
= (a + 2Z> + 3c) [n -2){n- 3).
The diminution then in the number of double edges is given
by the formula
2H= {Ab + 6c) {n - 2) {n - 3) - 8^- - 18^ - 12Jc + 36/3+ 247 + 12i.
Thus the entire diminution in the degree of the reciprocal
D — 3K— 2H is, when reduced,
n (75+ 12c) -45'-9c''- 8h-l5c + 8k + 18^ - 18/S - I27 - 12/ + 9^.
615. The formulae B^ reduced by the formula
a + 26+ 3c = n[n- 1),
become a (- 4w + 6) = 28 - a^ - 4/j - 9o- "j
& (- 4/1 + 6) = 4A; - 2^*" - 9^ - 67 - 3/ - 2p I . . . ( C).
c (- 4?i + 6) = 6A - 3c' - 6/8 - 47 - 2i-3a)
To each of these formulae we add now four times the corre-
sponding formula A ; and we simplify the results by writing
for a^ — a— 28- 3k, n^ the degree of the reciprocal surface, by
giving a the same meaning as in Art. 613, and by writing for
c'"' — c — 2^ — 3/3, S the order of the developable generated by
the curve c; we thus obtain the formula in the more convenient
shape,
n - a = K — a "j
2R = 2p- /3-3i{ {D).
3S+c = i3 + 5(7-2{\
THEORY OF RECIPROCAL SURFACES. 587
From the first of equations A and D we may also obtain
the equation
{n — l)a = n+p-\- Str,
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 the p points where a meets &, or the a points
where it meets c, since every first polar passes through the
curves 5, c.
616. The effect of multiple lines 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 (ti — iy^ 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 pass each through
this curve, so that their intersection breaks up into this curve
h 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 6 + c? meets the surface, we are only to take those in
which d meets it, which causes a reduction 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
2b[n — 2) — 7' (Art. 346) where r is the rank of the system b.
Now, these points consist of the r points on the curve Z*,
the tangents at which meet the line through which we are
seeking to draw tangent planes to the given surface, and of
2b{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 line
588 GENERAL THEOEY OF SURFACES.
only count for two. The total reduction then is
nb + 2r + 3 {2b {n-2)- 2r] =h[ln- 12) - 4r,
which agrees with the preceding theory.
If the curve Z*, instead of being merely a double curve,
were a multiple curve on the surface of the order p of mul-
tiplicity, I have found for the reduction of the degree of the
reciprocal (see Transactions of the Royal Irish Academy^ vol.
XXIII. p. 485)
h{p-l)[Zp-\-\)n- 2hp {f - 1) -f {p - 1) r,
for the reduction in the number of cuspidal edges of the cone
of simple contact
h[S{p-iyn-p{p-l){2p-l)]-p{p-l){p-2)r,
and for twice the reduction in the number of its double edges
2hp{p-l)n'-h{p- 1) (142? - 8) n
■^hp{p-i){8p-2)-p'{p-iyb''+p{p-l)(ip-e)r.'^
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 determine 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 line of contact
(a) consists of n lines of the system each of which meets the
cuspidal edge m once, and the double line x in (r — 4) points
(see Art. 330). The lines 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 line a^.f
* 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.
t It is only on account of their occurrence in this example that I was led to
include the points i in the theory.
THEORY OF RECIPROCAL SURFACES.
589
,{E].
We have then the following table. The letters on the left-
hand side of the equations refer to the notation of this Chapter
and those on the right to that of Chapter xii. :
n = r, a = n, b = x^ c = /« ;
p = 7i{r-A), (T = n, /c = 0, /3 = /8, h = hj t = a; n=0, S = r'j
and the quantities t, 7, R remain to be determined. On sub-
stituting these values in formulae A and Z), Arts. 610, 615, we
get the system of equations
71 (r- 2)=w{2+ (r-4)},
a; (r - 2) = w (r - 4) + 2/3 -t- 37 + 3<,
m (r - 2) = 2n + 4/3 + 7,
2R = 2n (?• - 4) - ^ - 3a,
3r + m = 5?2 — 2a + /3,
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 deter-
mined, 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 *i the number of apparent double points on b, Art. 609, itc.
An$. y = 6 (/t - 3) (/fc - 4), 3(! = 4 (A; - 3) {k - 4) (^ - 5),
k^ = {k- 3) (2i' - 18A2 + 67^• - 65), R=2{k-l) {k - 3).
And for the reciprocal singularities
7' = 2 (A - 2) {k - 3), 3t' = 4 (^- - 2) {k - 3) {k - 4),
k^' = (/t - 2) {k - 3) (2^-2 - lok + 11), R' = e {k - zy.
Ex. 2. Two surfaces intersect the sum of whose degrees is p and their product q.
Ans. y = q {pq -2q — Qq+ 16).
This follows from the table, p. 309, but can be proved directly by the method used
(Arts. 343, 471), see Transactions nfthe Roijnl Irish Academy, vol. XXUI. p. 469,
li = 32 {p - 2) [q {p-3)- 1).
590 GENERAL THEORY OF SURFACES.
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 order of the
developable, is given by the formula
2R' - 2m (r - 4) - a - 3/3.
618. Since the degree of the reciprocal of a ruled surface
reduces always to the degree of the original surface (p. 105)
the theory of reciprocal surfaces ought to account for this re-
duction. I have not obtained this explanation for ruled surfaces
in general, but some particular cases are examined and ac-
counted for in the Memoir In the Transactions of the Royal
Irish Academy already cited. I give only one example here.
Let the equation of the surface be derived, as in Art. 464, from
the elimination of t between the equations
at!' + &«*"' + &c. = 0, at' + h't'-^ + &c. = 0,
where a, a', &c. are any linear functions of the coordinates.
Then if we write k+ 1 = /ju, the degree of the surface is ytt,
having a double line of the order ^ {/j, - 1) {/u, — 2) , on which
are ^ (yu, — 2) (/i - 3) [fi - 4) triple points. For the apparent
double points of this double curve we have
2k = i{fjL-2)[fM-3) (ya'^ - 5/x + 8) ;
and the developable generated by that curve Is of the order
2 (/Lt - 2) (yu. — 3). It will be found then that we have
a = 2(/x-l), & = l(y[^-l)(yCi-2), « = 3(/i-2), S = 2{fi-2){fi -3)
values which agree with what was proved, Art. 614, that
the number of cuspidal edges In the tangent cone Is diminished
by 3b (yu, — 2) — 3^, while the double edges are diminished by
2b [fi — 2) (/A - 3) — 4/i. In verifying the separate formulae B
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 «a" + Z)/S" -f c7"-f &c., where
a, y8, 7 are arbitrary parameters, but have only succeeded when
n = 3. We have here (see Art. 523, Ex. 2) n = 12, w' = 9, a = 18 ;
b being the number of cublcs 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
THEORY OF RECIPROCAL SURFACES. 591
of the surfaces of the fourth and sixth order represented by
the two Invariants of the given cubic equation ; for the same
reason A =180 and /S = c' - c- 2^ - 3/3= 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 con-
sidering can have no cuspidal curve. This consideration gives
/c = 27, 8 = 108. The formulae A and D then give
180 = 27 + p + 2o-, 210 = p + 2/8 + 37 + 45, 240 = 2o- + 4;8 + 7,
9-18 = 27-0-, 2R=2p-^, 3(192-3/8) + 24 = 5o- + /3.
These six equations determine the five unknowns and give one
equation of verification. We have
p = 81, o- = 36, y8 = 42, 7 = 0, i2 = 60.
619. It may be mentioned here that the Hessian of a ruled
surface meets the surface only in its multiple lines, and in the
generators each of which is intersected by one consecutive.
For, Art. 463, if xi/ 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)4>. Then, Art. 287, the part of the Hessian which
does not contain x and 1/ Is
* ' -^ ' "■• ' wz '
(^-f)(*-"2)
dz div
which reduces to <^*. But xy intersects (f) only in the points
where it meets multiple lines. But if the equation be of the
form ux + vy^{AYt. 287) the Hessian passes through ccy. Thus
in the case considered In the last article, the number of lines
which meet one consecutive are easily seen to be 2 (/ti — 2) ;
and the curve UH, whose order is 4/<t (yu, — 2), consists of these
lines, each counting for two and therefore equivalent to 4 [fi - 2)
in the Intersection, together with the double line equivalent
to A {/J, — 1) [fjL — 2). Again, if a surface have a multiple line
whose degree Is w, and order of multiplicity p, it will be a
line of order A{p- 1) on the Hessian, and will be equivalent
to imp {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
592 GENERAL THEORY OF SURFACES.
order 2m, having the right lines as multiples of order w,
having ^m {)7i — 1) -\- h double generators, and 2r generators
which meet a consecutive one. Comparing then the order of
the curve UH with the sum of the orders of the curves of
which it is made up, we have
16m [m - 1) = 8m (w - 1 ) + 4m (w - 1) + 8^ -f 4r,
an equation which is identically true.
ADDITION BY PROF. CAYLEY ON THE THEORY OF RECIPROCAL
SURFACES.
620. In further developing the theory of reciprocal surfaces
it has been found necessary to take account of other singula-
rities, some of which are as yet only imperfectly understood.
It will be convenient to give the following complete list of
the quantities which present themselves :
w, order of the surface.
a, order of the tangent cone drawn from any point to the
surface.
S, number of nodal edges of the cone.
/c, number of its cuspidal edges.
Pj class of nodal torse.
0-, class of cuspidal torse.
b, order of nodal curve.
k, number of its apparent double points.
y, number of its actual double points.
t, number of its triple points.
j\ number of its pinch-points.
5, its class.
c, order of cuspidal curve.
h, number of its apparent double points.
6, number of its points of an unexplained singularity.
%, number of its close-points.
o), number of its off-points,
r, its class.
/3, number of intersections of nodal and cuspidal curves,
stationary points on cuspidal curve.
THEORY OF EECIPROCAL SURFACES. 593
7, number of intersections, stationary points on nodal curve.
4, number of intersections, not stationary points on either
curve.
(7, number of cnicnodes of surface.
jB, number of binodes.
And corresponding reciprocally to these :
n\ class of surface.
a', class of section by arbitrary plane.
8', number of double tangents of section.
/c', number of its inflexions.
p', order of node-couple curve.
o-', order of spinode curve.
})\ class of node-couple torse.
A:', number of its apparent double planes.
/', number of its actual double planes.
t\ number of its triple planes.
/, number of its pinch-planes.
2"', its order.
c', class of spinode torse.
A', number of its apparent double planes.
d\ number of its planes of a certain unexplained singularity.
%', number of its close-planes.
o)', number of its off-planes.
r\ its order.
/S', number of common planes of node-couple and spinode
torse, stationary planes of spinode torse.
7', number of common planes, stationary planes of node-
couple torse.
^', number of common planes, not stationary planes of either
torse.
C\ number of cnictropes of surface.
B\ number of its bitropes.
In all 46 quantities.
621. In part explanation, observe that the definitions of p
and o- agree with those given, Art. 609 : the nodal torse is the
torse enveloped by the tangent planes along the nodal curve ; if
GGGG
59-4 GENERAL THEORY OF SURFACES.
the nodal curve meets the curve of contact a, 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 a 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 a : 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
paraholic planes of the surface, and the spinode curve is the
locus of the points of contact of these planes; viz. it is the
curve UR of 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\ a correspond to those of
p and a.
622. In regard to the nodal curve 5, we consider h the number
of its apparent double points (excluding actual double points) ; /
the number of its actual double points (each of these is a point
of contact of two sheets of the surface, and there is thus at the
point a single tangent plane, viz. this is a plane /', and we
thus havey =/) ; t the number of its triple points; and y the
number of its pinch-points — these last are not singular points of
the nodal curve per se, but are singular in regard to the curve
as nodal curve of the surface ; viz. a pinch-point is a point at
which the two tangent planes are coincident. The curve is
considered as not having any stationary points other than the
points 7, which lie also on the cuspidal curve ; and the expres-
sion for the class consequently is q = l)' -l> — 2k — 2/- 87 — 6<.
623. In regard to the cuspidal curve c we consider Ji the
number of its apparent double points ; and upon the curve,
not singular points in regard to the curve p)^'"' ^6, but only in
regard to it as cuspidal curve of the surface, certain points in
number ^, ;i^, w respectively. The curve is considered as not
having any actual double or other multiple points, and as not
having any stationary points except the points ^, which lie also
THEORY OF RECIPROCAL SURFACES. 595
on the nodal curve ; and the expression for the class consequently
hr = c'-c-2h- 3/5.
624. The pouits y 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 inter-
section with such other sheet belonging to the nodal curve of
the surface has evidently a stationary (cuspidal) point at the
point of intersection.
As to the points yS, to facilitate the conception, imagine the
cuspidal curve to be a semi-cubical parabola, and the nodal
curve a right line (not in the plane of the curve) passing-
through the cusp ; then intersecting the two curves by a series
of parallel planes, any plane which is, say, above the cusp, meets
the parabola in two real points and the line 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 ap-
proach 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 nodal line changes from crunodal to acnodal.
625. At a point t the nodal 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 crunodal, viz.
the two half-sheets Intersect each other along this portion of
the nodal curve. There is at the point a single tangent plane,
which is a plane «"; and we thus have t = i.
626. As already mentioned, a cnlc-node (7 is a point where,
instead of a tangent plane, we have a tangent quadrl-cone ;
and at a binode B the quadri-cone degenerates into a pair
of planes. A cnictrope C" is a plane touching the surface along
a conic ; in the case of a bitrope B', the conic degenerates into
a flat conic or pair of points.
627. In the original formulae for a [n - 2), b (n - 2), c {n - 2),
we have to write k- B instead of k, and the formula are further
596 GENERAL THEORY OF SURFACES.
modified by reason of the singularities 6 and w. So in the
original formulae for a (w — 2) [n- 3), h (n — 2)(w — 3), c(?i — 2)(n — 3),
we have instead of S to write 8— (7— 3ft); and to substitute
new expressions for [«J], [<^c], [Jc], viz. these are
[ab~\ = ab — 2p —j\
[acj = ac — 3<x — p^ — o),
[5c] = ?>c - 3yS - 27 - «.
The whole series of equations thus is
(1) a' = a.
(2) r=/
(3) i" = t.
(4) a = w (r? - 1) - 25 - 3c.
(5) K = 3n (n - 2) - 65 - 8c.
(6) S' = i w (n - 2) (/i'^ - 9) - (n* - w - 6) (25 + 3c)
+ 25(5-l) + 65c + |c(c-l).
(7) a[n-2) = K-B+p + 2a + 3oi.
(8) 5(w-2)= p + 2/3 + 37+3<.
(9) c{n-2)= 2a- + 4^+ y + 0 +(o.
(10) a(n-2)(w-3)
= 2{B-C-So)) + 3{ac-B(T-x-3»}) + 2{ah- 2p -J).
(11) 5 {n-2){n-3)
= 4^ + {ah-2p-j ) + 3 (5c- 3/5-27-/).
(12) c (n-2)(n-3)
= 6A + (ac-3(r-;)j;-3ft)) + 2(5c-3/3-27-i).
(13) q = ¥-b-2k-2f-3y-6t.
(14) r = c'-c -2;«-3/S.
Also, reciprocal to these
(15) a =?/(«' -1)- 25' -3c'.
(16) K =3n{ii-2)-Gb'-8c\
(17) S = In' (7/ - 2) {7i"' - 9) - (w"^ - ?i' - 6) (25' + 3c')
4 25'(5'-1) + 65'c't|c'(c'-1).
THEORY OF RECIPROCAL SURFACES. 597
(18) a'(rt'-2) = /c'-^' + p' + 2cr'+3a)'.
(19) &'(«'- 2)= p'+2/3' + 37'+3«'.
(20) c' {n' - 2) = 2o-' + 4/3' + 7 + 61' + o)'.
(21) a'(7/-2)(w'-3)
= 2 (S' -C" - 3a)') + 3 [a'c' - 3o-' - / - 3ft)') +2 (a'Z>'-2^'-J').
(22) &'(?z'-2)(w'-3)
= 4A' + (a'6'-2/-/) +3(J'c'-3/3'-27'-0.
(23) c' (w - 2) (»/ - 3)
= 6A' + (a'c'-3(7'-x'-3'a)') + 2(5'c'-3/3'-27'-0.
(24) q' = V^ -V- 2k' - 2/' - 37' - 6«'.
(25) r' = c'^-c'- 27/ -3/3',
together with one other independent relation, in all 26 relations
between the 46 quantities.
628. The new relation may be presented under several
different forms, equivalent to each other in virtue of the
foregoing 25 relations ; these are
(26) 2 (n - 1) (n - 2) (n - 3) - 12 [n - 3) (5 + c)
+ 6^ 4-6r-f 24«+42yS+ 3O7 - |^ = 2;
(27) 26?i- 12c-4a-10J5+yS-7j-8x + ^^-4ft) = S,
in each of which two equations 2 is used to denote the same
function of the accented letters that the left-hand side is of
the unaccented letters.
(28) /3' + i^'= 2n(?i-2)(llw-24)
+ (-66n + 184)&
+ (-93n + 252)c
4 22 (2/3 + 37 + 3«)
+ 27 (4/3 + 7 + 6)
+ /3 + i^
- 24 0 - 285 - 21 j - 38x - 73ft)
+ 4(7' + 105' + If + 8x' - 4ft>'.
598
GENERAL THEORY OF SURFACES.
Or, reciprocally,
(29) 13 + 16= 2n'(n'-2) (lb/-24)
+ (- 66?/ + 184) b'
+ (- 93// + 252) c
+ 22 (2/3' f 37 + 30
+ 27(4;8' + 7 +^')
+ /3'-fi^'
- 24 C - 285' - 27 j" - B8x - 73a)'
+ 40 +105+ 7j + 8;^ -4ft).
Where the equation (26) in fact expresses that the surface
and its reciprocal have the same deficiency ; viz. the expression
for the deficiency is
(30) Deficiency = ^ (n - 1) (?i - 2) [n - 3) - (n - 3) (5 + c)
+ l(^ + r) + 2^+^^ + f7 + i-i^,
= i{n- l){n-2){n'-S)-&c.
629. The equation (28) (due to Prof. Cayley) is the correct
form of an expression for /S', first obtained by him (with some
errors in the numerical coefficients) from independent considera-
tions, but which is best obtained by means of the equation (26) ;
and (27) is a relation presenting itself in the investigation. In
fact, considering a as standing for its value n [n — l)—2h- 3c,
we have from the first 25 equations
6 a =S
+ 2 3w-c-« =2
-2 a{n-2)- K + B-p-2(T-3Q) =2
-4 h{n-2)- p-2/3- 3y-m =2
-6 c{n- 2)-2a-A^-y -6-0) =2
+ 2 ?i + /c-o--2(7-45-2j-3x-3ft)=S
- 3 2q-2p + ff+j =2
-2 3r + c-5a-/3- A9 + x-(o =2
and multiplying these equations by the numbers set opposite to
them respectively, and adding, we find
- 2/i' + 12?i' + 4n + J (12?i - 36) + c (12/i - 48)
- 62 - 6r - 4(7- 105- 41/3 - 3O7 - 2U - 7j- 8% + 2^ - 4g) = 2,
THEORY OF RECIPROCAL SURFACES. 599
and adding hereto (26) we have the equation (27) ; and from
this (28), or by a like process, (29), is obtained without much
difficulty. As to the 8 2-equations or symmetries, observe that
the first, third, fourth, and fifth are in fact included among
the original equations (for an expression which vanishes is in
fact = 2j ; we have from them moreover Sn — c = Sa — k\ and
thence "6n — c — k = ^a — k — k\ which is = 2, or we have thus
the second equation ; but the sixth, seventh, and eighth equations
have yet to be obtained.
630. The equations (15), (16), (17) give
w' = a (a — 1) — 28 — 3/c,
c = 3a (a - 2) - 68 - 8/c,
V = \a[a- 2) [a"" - 9) - (a* - a - 6) (28 + 3/c)
-f 28(8- l) + 6SA; + |/c(ytf- 1);
from (7), (8), (9) we have
[a-h-c) (?i-2) =/c-5-6/3-47-3^-5' + 2«,
(a - 25 - 3c) [n - 2) (n - 3) =
2 (8 - C) - 8^• - 18A - 65c + 18yS + I27 + 6t - 6<w,
and substituting these values for k and 8, and for a its value
= n [n — 1) — 2b — 3c we obtain the values of ?/, c', b' ; viz. the
value of n is
n' = n {n - If-n [lb + 12c) + 45' + 85 + 9c' + 15c
-8A;-]8^+18;8+ Ur^+VIi-^t
-2C-3B-Sd.
Observe that the effect of a cnicnode C is to reduce the class
by 2, and that of a binode B to reduce it by 3.
631. We have
(n - 2) {n - 3) = n' - n + (- 4n + 6) = a + 25 + 3c + (- 4n + 6),
and making this substitution in the equations (10), (11), (12),
which contain (n- 2) (w- 3), these become
a (- 4n + 6) = 2 (8 - (7) - a' - 4p - 90- - 2;- 3^ - 15a),
5 (- in + Q)=ik- W - 9/3 - 67 - 3i - 2p -j,
c (- 4n -I 6) = 6^ - 3c' - 6/3 - 47 - 2t - 3o- - ^ - 3&),
600 GENERAL THEORY OF SURFACES.
(the foregoing equations (C)) ; and adding to each equation four
times the corresponding equation with the factor (n — 2), these
become
.a'~2a = 2{S- C) + 4l{k-B)-(7-2j-3x- 3«,
2&' - 2 J = 4Z; - /3 + 67 + 12« - 3i + 2p -j\
3c' -2c= 6h + 10/3 + id-2i-\-5(T-x + (^'
Writing in the first of these a^ -2a = n +2B + Sk- a, and
reducing the other two by means of the values of g-, r, the
equations become
w - a = - 2 0- 45+ /c - o- - 2;'- 3;)^- 3a),
2q + l3 + 3i+j=2p,
3r + c + 2i + ;)j; = So- + /3 + 4^ + 0),
which give at once the last three of the 8 2-equations.
The reciprocal of the first of these is
o-' = a-w + /c'-2/-3x'-2C"-45'-3a,',
viz. writing herein
a = n{n-l)-2b-Sc and k =Sn{n-2) - &b- Sc,
this is 0-' = 4w (w - 2) - 8^. - lie - 2/ _ 3^' - 2 0' - 45' - 3a)',
giving the order of the splnode curve ; viz. for a surface of the
order 71 without singularities this is = An {71 — 2), the product of
the orders of the surface and its Hessian.
632. Instead of obtaining the second and third equations as
above, we may to the value of & (— 4n + 6) add twice the value
of b (n - 2) ; and to twice the value of c (— 4n + 6) add three
times the value of c (n — 2), thus obtaining equations free from
p and cr respectively ; these equations are
b[-2n-\-2)=4:k- 2h' - 5/8 - 3/+ U -j,
c(-5n + 6) = 12A-6c"''-57-4t- 2;^4 3^-3a),
equations which, introducing therein the values of q and r, may
also be written
J (271 -4) =22+ 5/9+67+6< + 3^+J + 4/,
c [hn - 12) + 3^ = 6r + 18/3 +57 + \i + 2^ + 3a).
THEORY OF RECIPROCAL SURFACES. 601
Considering as given, n the order of the surface ; the nodal curve
with its singularities J, k, /, t ; the cuspidal curve and its sin-
gularities c, h ; and the quantities /S, 7, i which relate to the
intersections of the nodal and cuspidal curves ; the first of the
two equations gives j\ the number of pinch-points, being sin-
gularities of the nodal curve quoad the surface ; and the second
equation establishes a relation between 6, x-> ^t ^^^ numbers
of singular points of the cuspidal curve quoad the surface.
In the case of a nodal curve only, if this be a com-
plete intersection P=0, ^ = 0, the equation of the surface is
(vl, 5, C\P^ Q? = ^i and the first equation is
J (- 2n + 2) = 4A; - l¥ + U -^ ;
or, assuming ^ = 0, say y=2 (n — 1) & — 26'' + 4^-, which may be
verified ; and so in the case of a cuspidal curve only, when this
is a complete intersection P=0, ^ = 0, the equation of the
surface is [A, B, G\P, QY = % where AC-B' = MP+NQ'j
and the second equation is
c (- 5n + 6) = I2h - Gc" ^ 2^ + 3^ - 3w,
or, say 2;^ + 3o) = {5n - 6) c — 6c'* + I2h + 3^, which may also be
verified.
633. We may in the first instance out of the 46 quantities
consider as given the 14 quantities
n: bjkjf^t :c,h,0,x '^,%^'0,Bj
then of the 26 relations, 17 determine the 17 quantities
and there remain the 9 equations
(18), (19), (20), (21), (22), (23), (24), (25), (28),
connecting the 15 quantities
p% a' : A', t'J\ i : A', e\ x', «', r : ^, 7' : G\ B\
Taking then further as given the 5 quantities y^, ;^', w', C\ B\
equations (18) and (21) give p", o-',
equation (19) gives 2/3' + 37' + 3/',
„ (20) „ 4^'+ 7' + ^,
„ (28) „ ^' + he\
UHHH
602 GENERAL THEORY OF SURFACES.
SO that taking also t' as given, these last three equations deter-
mine yS', 7', 6' \ and finally
equation (22) gives A;',
„ (23) „ //,
» (24) „ ?',
„ (25) „ /,
viz. taking as given in all 20 quantities, the remaining 26 will
be determined.
614. In the case of the general surface of the order ??,
without singularities, we have as follow ;
n = w,
a = 7? (n — 1),
8 =|72(n-l)(«-2)(n-3),
K = n[n — l){n — 2),
7/ = n[n — !)'•',
a = n {n — 1),
S'=in(7i-2) (n"'-9),
K =3n[n— 2),
h' =ln{n- 1) {?} -2)(n^-n^ + n- 12),
k' =^n{n- 2) {n'° - Gw" + 16n' - 54n' + 164n" - 288n'
+ 547 w^ - 1058w" + 1068n' - 1214w + 1464),
t' = |n (n - 2) (m' - 4«^ + 7w' - 45n^ + 1 Un^ -Mht'-^ 548n- 960),
2' = w (n - 2) (w - 3) (w'' + 27? - 4),
p = n{n-2) {rf-n'-\-n-12)j
c =4n (n — 1) (w — 2),
A' = |n (« - 2) (16n* - 64n' + 80m' - 108« + 156),
/ = 2n (n - 2) (3/j - 4),
0-' =4n (w — 2),
^8' =2n(?i-2)(lln-24),
7 = 4n (n - 2) (n - 3) (n' - 3n + 16),
the remaining quantities vanishing.
I
I
THEORY OF RECIPROCAL SURFACES. 603
615. The question of singularities has been considered
under a more general point of view bj Zeuthen, in the memoir
" Recherche des singularit^s qui ont rapport a une droite
multiple d'une surface," Math. Annalen^ t. IV. pp. 1-20, 1871.
He attributes to the surface :
A number of singular points, viz. points at any one of which
the tangents form a cone of the order /u., and class v, with
y + r] double lines, of which y are tangents to branches of the
nodal curve through the point, and z -r ^ stationary lines, whereof
z are tangents to branches of the cuspidal curve through the
point, and with u double planes and v stationary planes ;
moreover, these points have only the properties which are
the most general in the case of a surface regarded as a
locus of points; and 2 denotes a sum extending to all such
points. [The foregoing general definition includes the cnic-
nodes (ya = j/ = 2, y = r} = z = !^=u=v = 0)j and the binodes
(/i = 2, 17 = 1, v = y = &c. = 0)]. ^
And, further, a number of singular planes, viz. planes any
one of which touches along a curve of the class /i' and order /,
with ^' + 77' double tangents, of which 2/' are generating lines of
the node-couple torse, z' + ^' stationary tangents, of which s'
are generating lines of the splnode torse, u double points and
y' cusps ; it is, moreover, supposed that these planes have only
the properties which are the most general in the case of a
surface regarded as an envelope of its tangent planes ; and S'
denotes a sum extending to all such planes. [The definition
includes the cnictropes [f/ = / = 2, ?/' = 77' = / = ^' = w' = v' = 0),
and the bitropes (/*' = 2, 17' = 1, v' = 2/' = &c. = 0)J.
616. This being so, and writing
x = v + 2r} + S^, x' = / + 27)' + S^\
the equations (7), (8), (9), (10), (11), (12), contain In respect of
the new singularities additional terms, viz. these are
a (n - 2) =...-1- 2 [u; (/* - 2) - 77 - 2^],
b{n-2)=...+ 2[y{fM-2]],
c (n-2)=...+ 2[s (/A -2)],
a (w - 2) {n _ 3) =...+ 2 [a; (- 4/i -f 7) + 2»; + 4^],
b {n - 2) [n - 3) =...+ 2 [y (- 4/* + 8) J - 2' {Au + 3y'),
c{n-2) (?i-3)=...-l-2[2(-4/x+9)J-2'(2y'),
604 GENERAL THEORY OF SURFACES.
and there are of course the reciprocal terms in the reciprocal
equations (18), (19), (20), (21), (22), (23). These formulae are
given without demonstration in the memoir just referred to:
the principal object of the memoir, as shown by its title, is the
consideration not of such singular points and planes, but of the
multiple right lines of a surface; and in regard to these, the
memoir should be consulted.
INDEX.
Abbildung, 553.
AUman, on paraboloids, 93
Amiot, on non-modular foci of quadrics,
136.
Anallagmatic surfaces, 481, 512, 529.
Anchor ring, its properties. 392, 404, 539.
Anharmonic ratio, of four planes, 24, 86, 88.
Of four generators of a quadric, 92.
Of sphero-conics, 220.
Of four fixed tangent planes of quar-
tic developable, 305.
Of four tangent planes to any ruled
surface, 423.
Apparent, double points, 292.
Intersection of curves, 311.
Apsidal surfaces, 455.
Area of surface of ellipsoid, 371.
Asymptotic lines on surfaces, 238.
Axes, of a quadric found, 66.
How found when three conjugate
diameters are given, 157.
Of central section of a quadric, 80,
82, 144.
Of tangent cone to a quadric, 147.
Bedetti, on section of surface by its tan-
gent plane, 234.
Bertrand, his theory of the curvature of
surfaces, 265.
On fundamental property of geo-
desies, 274.
On curves of double curvature, 336.
On the proof of a theorem of Grauss',354.
On the lines of curvature of the
wave surface, 464.
Besge, on geodesic tangents to a line of
curvature, 367,
Biflecnodes, number of, 574.
Binodes, 489.
Binormal, 324.
Biplanes, 489.
Bitangent, lines, 244, 246, 420, 474.
Planes. 251, 579.
Blackburn on representation of curves, 285.
Bonnet, on surfaces applicable to one
another, 357.
On orthogonal surfaces, 442.
On second geodesic curvature, 361,
362.
Boole, his method of finding axes of a
quadric, 66.
On integration of equation of lines
of curvature of an ellipsoid, 269.
Boole, on the envelope of surfaces whose
equations contain parameters, 407.
Booth, on centro-surface of quadric, 172.
Bouquet, on the condition that a surface
should belong to a triple orthogonal
system, 441, 449.
Bour, on surfaces applicable to one
another, 357.
Brian chon-hexagon and point, 123.
Brioshci, on lines of curvature of wave
surface, 464.
Canonical form, reduction of equations of
two quadrics to their, 206.
Of equation of a cubic, 491.
Cartesian surfaces, 535.
Cartesians, twisted, 317.
Casey, on obtaining focal properties by
inversion, 481.
On cyclides, 481, 527, &c.
Cauchy's proof that discriminating cubic
has only real roots, 67.
Cayley, on moment of two lines, 34.
On equality of degree of ruled surface
with that of reciprocal, 105.
On developable of tangents to curve
common to two quadrics, 190.
On tact-invariants, 211.
On discriminants of discriminants, 213.
On the section of a surface by its
tangent plane, 234.
On contact of Hessian with surf ace,252.
On the fundamental property of
geodesies, 274.
On differential equation of orthogonal
systems, 441.
On representation of curves, 282.
On singularities of developables, 291,
On singularities of curves, 298.
On quintic developables, 313.
On description of quartic curves
through eight points, 315.
Distinguishes planar and multiplanar
developables, 318.
On geodesies, 370.
On contour and slope lines, 381.
On equations of Pro-Hessians, 408.
On ruled surfaces, 416, 429, 430, 431,
435, 512.
On centro-surface, 471.
Obtains equation of first negative
pedal of a quadric, 481.
On cubical ruled surface, 487.
606
INDEX.
Cayley, on scrolar and oscular lines, 489.
On species of cubics, 490.
On right lines on a cubic, 496.
On involution of six lines, 419.
On generalization of wave surface, 542.
On differential equation of ruled
surfaces, 559.
On reciprocal surfaces, 592.
On transfoiination and correspond-
ence, 563.
On deficiency of surfaces, 557.
Centro-surface of a quadric, 170, 465.
Its reciprocal, 172.
Its equation formed, 178.
Sections by principal planes, 171, 468.
Its cuspidal and nodal lines, 469, 470.
Extension of problem by Clebsch, 465.
Centro-surface, in general, 271.
Tangent planes at points where nor-
mal meets cut at right angles, 272.
When has double lines, 272,
Its characteristics, 471,
Of a developable, 337.
Characteristic, of envelopes, 290, 401.
Their differential equations, 410.
Characteristics, of curves which to-
gether make up intersection of tv/o
surfaces, how connected, 322,
of developable, 319.
of systems of surfaces, 547.
Chasles, on lines joining corresponding
vertices of con j ugate tetrahedra, 119.
On analogues to Pascal's theorem, 122.
On foci and confocal quadrics, 126, &c.
On focal lines of tangent cones to a
quadric, 152.
On the axes of these tangent cones, 155.
On finding the axes of a quadric, 157,
On sphero-conics, 215.
On curves of third order, 300.
On curves of fourth order, 317.
On enumeration of developables, 318.
On curves on a hj^erboloid, 358, 554.
On geodesies of ellipsoids, 368, &c.
On ruled sm-faces, 416, 512.
On involution of six lines, 419,
On systems of surfaces, 552.
Circular sections of a quadric, 82.
The problem considered geometri-
caUy, 116.
Sum or difference of angles made with
Vjy any plane depends on axes of sec-
tion, 2Z4.
Clairaut, on name " curves of double cur-
vature," 342.
Clebsch, on double lines of surface of
centres, 273.
On condition that four consecutive
points of a curve should lie in a
plane, 329.
On surface of centres, and normals
from any point to a quadric, 465, i'c.
On reduction of a cubic to its ca-
nonical form, 492.
On intersection of tangent plane and
polar with respect to liessian, 602.
Clebsch, on surface passing through 27 lines
of a cubic, 511.
Its equation calculated, 559.
On quartics with nodal conies, 512, 557,
On doubly inflexional tangents, 559.
On number of points at which two
doubly inflexional tangents can be
drawn, 570.
On representation of curves on sur-
faces, 553.
On generation of cubic surfaces, 554.
Cnicnodes, 488, 240, 248.
Combescure, on lines of cuivature of
wave surface, 464.
Combinants of quadrics, 209.
Complexes, 63, 190, 214, 416.
Complex surface, 420, 524.
Condition, that two planes cut at right
angles, 17.
That right lines should lie completely
in surface, 29, 64.
That two lines intersect, in terms of
six coordinates, 43.
That a plane or line should touch
a quadric, 58, 59.
That a tetrahedron self-conjugate
with respect to one quadric may
be inscribed in another, 175.
That two quadrics should touch, 175.
That a tetrahedron may be inscribed
in one quadric having two pairs of
opposite edges on another, 180.
That three asymptotic lines or planes
should be rectangular, 183.
That line should pass through in-
tersection of two quadrics, 189.
That equation in quadriplanar co-
ordinates represent a sphere, 202.
That section of quadric be a parabola
or equilateral hyperbola, 202.
That three quadrics may be polars to
same cubic, 209.
That two intersections of three quadrics
may coincide, 211.
That four points of intersection of
three quadrics be coplanar, 211.
How many necessary to determine a
siu'face, 233.
That three quadrics should meet a
line in involution, 214.
That four consecutive points of a
curve should lie in a plane, 329.
That intersecting surfaces should have
a common line, 433.
That four hues should be met by
only one transversal, 419.
That five lines may have a common
transversal, 419.
That two surfaces should touch, 546.
Cone, defined, 48.
Equation of a, with given vertex and
restmg on a given curve, 101.
Properties of, 215, &c., 387.
Equation of right cone. 227.
Confocal quadrics, surfaces inscribed in a
common developable, 128,
INDEX.
607
Confocal quadncs, properties of, hence de-
rived, 197.
Cut at right angles, 143.
And also appear to do so, 153.
General form of equation, 196.
Congruencies, 416.
Order and class how connected, 473.
Formed by normals to a surface, 417.
Of bitangents to focal surface, 421.
Conical points on surfaces, 240, 248, 4^8.
Conicoids, 45.
Conjugate tangents, 239.
Lines of quadric, 107.
Contact, of two surfaces a double point on
their intersection, 177, 234, 283, 310.
Of lines with surfaces, 558.
Of planes with surfaces, 576.
Contravariants of systems of quadrics, 185.
Of cubics, 485, 504.
Corresponding points on confocals, 161.
Correspondence, 473, 549, 553, 572.
Covariants of quadrics, 187.
Cremona, on section of a surface by its
tangent plane, 234.
On curves of third order, 300, &c.
On curves of fourth order, 317.
On developables of fifth order, 319.
On cubical ruled surfaces, 488.
On Steiner's quartic, 491.
On cubics, 492.
On ruled qnartics, 512.
On transformation and correspondr
ence, 553.
Cubic twisted, 300, &c.
Different species of, 306.
Curvature of quadrics, 167.
Of surfaces in general, 252.
Lines of curvature, 167, 170, 266.
their differential equations, 268, 344.
their property, if plane, 277.
the same for two orthogonal sur-
faces, 270.
their differential equation integrated
for qua'lrics, 269.
if geodesic is plane, 277.
Gauss's theory of curvature, 350, <fcc.
Second curvature of curves, 335.
Geodesic curvature, 360.
Lines of wave surface, 464.
Curve in space how represented by equa-
tions, 281.
Cuspidal edge, of developables and
envelopes, 271, 290.
Of polar developable, 339.
Its difierential equation, 413.
Cyclic planes of cone, 220.
Cyclides, 481, 512, 527, &c.
Cylinders, defined, 15.
Limiting case of cones, 48, 279.
Their differential equation, 386.
Darboux, on orthogonal surfaces, 442, 451.
On centro-surface of quadric, 46'J.
On centro-surface in general, 471,473.
On cycUdes, 481, 527, &c.
On transformation of surfaces, 553.
De Jonquieres. on s}-stenis of surfaces, 5,51.
Deficiency of curve in space, 319, of sur-
face, 557.
Desboves on normals to quadrics, 475.
Developable defined, 89, 104.
Circumscribing two quadrics, 188, 208.
Generated by tangent lines of their
common curve, 190.
How these developables meet the
quadric, 191.
Imaginary, which touches a system
of confocals, 195.
Generated by normals along a line of
curvature, 271.
General theory of, 284, &c.
Pliicker-Cayley equations of, 293, 295.
Of same degree, as developable gene-
rated by reciprocal curve, 294.
Planar and multiplanar, 318.
Polar of curves, its singularities, 341.
Differential equation of, 407.
Which touches along parabolic curve,
its degree and singularities, 546, 579.
Which touches a surface along a
given curve, 546.
Grenerated by a line meeting two
given curves, 547.
By a line meeting a given curve
twice, 590.
Generated by curve of intersection of
two given surfaces, 308.
Enveloping two given surfaces, 547.
Enveloped by bitangent planes, 580.
Theory of their reciprocals, 588.
Dickson, on geodesies, 358, 362.
Diguet, on the proof of a theorem of
Gauss's, 354.
Distance, between centres of inscribed and
circumscribing circles of spherical
triangles, 228.
Discriminant, of a quadric, 51.
Of a surface in general, 249.
Of discriminants, 213.
Double, points on surfaces, 240, 457,488,595.
On curves, 310.
Curves are ordinary singularities, 249.
on developables, 297.
on surface of centres, 272, 470.
on ruled surfaces, 428.
Generators on ruled surfaces, 432.
Points, apparent, on common curve
of two surfaces, 292, 310.
Tangent lines, how many pass through
a point, 244, 246.
Tangent planes, locus of their points
of contact. 251, 579.
Sixes, Schljifli's, 500.
Dupin, on indicatrix and elliptic, (fee,
points, 234, 236.
On cyclide, 535.
On conjugate tangents, 239.
On orthogonal surt'aces, 269, 436, A'C.
Elasticity, surface of, 481.
Elliptic coordinates, 162, 370. 460.
Elhs, on Dupiu's theorem, 270.
60S
INDEX.
Envelope of a plane containing one para-
meter, 286.
entering rationally, 296.
Of a plane containing two parameters,
288.
entering rationally, 491, 590.
General theory of, 401.
Equilateral hyperboloids, 102, 120, 183.
Euler, on curvature of surfaces, 254.
Evolutes of curves, 339, &c.
Families of surfaces, 383, &c.
Eaure, extension of his theorem on self-
conjugate triangles, 175.
Eerrer's proof of theorem of Chasles, 119.
Feuerbach's theorem on circles touching
sides of a triangle, 229.
Elecnodal curve, 559.
Focal conies of quadrics, 129, 139.
tangential equation in general, 199.
Curves, general definition of, 128.
Lines of cones, 106, 133.
Properties obtained by inversion, 481.
Foci, general definition of, 127.
Of section normal to focal conic, 132.
Of plane section of a quadric, co-
ordinates of, 203.
Fourier, on polar developable of curves,
338.
Frenet, on curves of double curvature, 342.
Fresnel, on wave surface, 453.
On surface of elasticity, 481.
Frost and Wolstenholme's treatise on Solid
Geometry, 45.
Gauss's theorems on geodesies, 358.
On curvature of surfaces, 343, &c., 356.
Gehring, on difiEerential equation of geo-
desies, 363.
Geiser, on right lines of nodal quartics, .526.
Geodesies, fundamental property, 273, 358.
Their differential equation, 362.
On centro-surface, 275.
On ellipsoid, 364, &c.
Curvature, 360.
Polar coordinates. 356, 373.
Gerbaldi, on Steiner's quartic, 491.
Gordan, on cubics, 492.
Graves, his translation of Chasles on
sphero-conics, 215.
Theorem on arcs of sphero-conics, 225.
extended to geodesies, 366.
Proof of Joachimsthal's theorem, 362.
Gregory's solid geometry, 234, 268, 270.
Gudermann, on spherical coordinates, 216.
Hamilton, Sir Wm. R., his method of
generating quadrics, 102.
His theorem that umbilics lie in threes
on eight lines, 116.
On circles which touch three great
circles, 232.
On lines of curvature at umbilics, 266.
On curves of double curvature, 343.
On nodal points of wave surface, 457.
On congruencie?. 116,
Hart, his extension of Feuerbach's theo-
rem, 229.
On twisted cubics, 299.
Proof of Joachimsthal's theorem, 364.
On geodesies, 376, 380.
On obtaining focal properties by in-
version, 481.
Theorem that confocal plane circular
cubics cut orthogonally, 533.
HeUx and Helicoid, 325, 332, 338, 390.
Hesse, on the construction of a quadric
through nine points, 113.
On Brianchon's hexagon, 123.
Theorem as to the vertices of two
self-conjugate tetrahedra, 175.
On osculating plane of curves, 328.
On integration of equation of geodesic
on ellipsoid, 372.
On geodesies, 363.
Hessian of a surface, 250.
Touched by every right line on the
surface, 251.
Has double points, 494.
Of a developable, 408.
Of cubic identical with Steinerian,
493.
Of a ruled surface, 591.
Hirst, on pedal surfaces, 478.
On inverse surfaces, 479.
nomographic correspondence ; surface
generated by line joining corre-
sponding points on two Unes, or
enveloped by plane joining corre-
sponding points on three, 92, 304.
Locus of intersection of three cor-
responding planes, 554.
Imaginary circle at infinity, its equation,
184, 199, 227.
Generators of quadric, 116, 469.
Indicatrix, 236.
Inextensible surfaces, 357.
Inflexion linear on curves, 295.
On quartics, 316.
Inflexional tangents of surfaces, 236.
How many pass through a point, 245.
How many tangents to a given curve
on a surface are inflexional, 546.
Intersection of two surfaces, its singu-
larities, 308.
Of three surfaces, common curve equi-
valent to how many points, 321.
Invariants and covariants of quadrics,
173, &c.
Of a cone and quadric, 182.
Of sections of quadrics, 202.
Of a system of three quadrics, 208.
Of circles on a sphere, 228.
Of a cubic, 503.
Inverse surfaces, 479.
Inversion applied to obtaining focal pro-
perties, 481.
To study of cyclides, .528.
Involution of tangent and normal planes
to a ruled surface, 423.
Of six lines, 419, 518.
7
'UH
INDEX.
Ivory's theorem on distance between cor-
responding points of confocals, IGJ.
His mode of generating quadrics, 164.
Integrates equation of geodesies on
,.anellipsoid, 361, 302,372
J acobian of four quadrics, 205
Curve, 213.
Of four surfaces, 544.
Jellett, on inextensible surfaces, 357.
Joacliimsthal, his method of findm- intei-
5?2°42,522? ^^^ ^^^^^ ^ ^^'^^'
On tetrahedra, 120.
On normal to ellipsoid, 165.
His theorem on plane lines of curva-
ture, 2/6.
On curves of the third order, 303
On geodesies of an ellipsoid, 361.
Klein^ edits Pliicker's work on lines, 416
On relation between order and class
ot congruency, 473.
On transformation and correspond-
ence, 553.
Komdorfer on quartics with nodal lines, 5 1 2
On representation of curves, 553
Kummer, on double lines of surfaces of
centres, 273.
On Steiner's quartic, 491.
^'sr^'^S'''' ^^°' "^'"^ """^^^ '=°"*^'
On congi-uencies, 416.
GOD
Locus of vortices of riglit cones circum-
Rcriljuig a qiiadric, loS.
Of intersection of rectangular gene-
rators of a hyperboloid, 160.
Of points of contact of parallel tan-
gent planes to confocals, 1(!6
Of centres of spheres circumscribine
self-conjugate tetrahedron, 175
Of^foci of central sections of a quadric,
°Hn°e';'204''°"°°' ^^'^"^^ *° ^ ^^^"
Of^vertices of cones through six points,
Of intersection of rectangular tan-
gents to a sphero-conic, 225
Of points of contact of double tangent
plaaes to a surface, 579.
Of curves of contact from points on
axes to system of confocal ellipsoids,
Of intersection of three homogranhi-
cally corresijonding planes, 551
Lacroix, contributions of to the theory of
curves of double curvatm-e, 343
Un lines of striction, 424.
Lame, curvilinear coordinates" 349 442
Lancrefs theorem, 277.
On curves of double cui-vature, 343
Legendre, on area of ellipsoid, 371
Level, lines of, 380.
Levy, on orthogonal surfaces, 442
Line, SIX coordinates of, 33, CO, 1 93, 28' 518
Liouville, his calculation of radius of
geodesic curvature, SCO.
His mode of writing equation of geo-
desics of an ellipsoid, 3C6.
On elhjitic coordinates, 370.
Lloyd, on conical refraction, 457
Locus of intersection of three rectangular
tangent Unes to a quadric, 100, 160
Of^ three rectangular tangent pknes,
If the planes each touch one of three
confocals, 155.
Of poincs on quadric whose normals
meet a fixed normal, 101, 167- on
any surface, 265.
Of centres of quadrics satisfi-inff eio-ht
conditions, 112, 518. •' " »
Of pole of plane with regard to a
series of confocals, 146.
Mac Culkgh, on foci and confocal surfaces,
§nSSrKi^^;r^-w35.
'"face';f56;r''^^'^^'^"^™--
Marcks, on order of centro-surface, 473
M Gay s proof of tlieorem of Chasks, 120
Un foci of sections of quadric '^04 *
Meumer s theorem, 168, 256, 267''360
''ofgrodS:ij^f^'^^"'^^p-p-^y
''''":ph:L,^s^""°^ "' --^ -^
On twisted cubics, 300
Modular property of foci, 135
Monge, on lines of curvature, 266, 273
On geodesies, 273.
On tetrahedron, 120,
On envelopes, 289, 290.
On polar lines of curves, 331
On evolutes, 339.
On curv.;s of double curvature, 342
On families of surfaces, 410, 4U 415
Monoid, defined, 28> "' ^^ ', liO.
Moutard ,on anallagmatic surfaces, 51->
^ touchTSr54t^^ ^^^-^^^^ ^'-"^^^
Node couple curve. 579,
Normal to a surface, its equations, 244
Plane to a curve, 324
To confocals through given line gene-
rate paraboloid, 153 ^
^ nuo?; 'm^ ''^^'''' ^'°°e a gene-
When intersects consecutive, 264
Lxtension of notion of, 465
To a quadric, Clebsch on, 465
How many can be drawn "from a
pomt to a surface, 47'
Aormopolar sui-facc, 475
nil
610
INDEX.
Nbther, on deficiency of surfaces, 553, 557.
Order of condition that three surfaces
should have a common line, 434.
Orthogonal hyperboloid, 100, 183.
Surfaces, Dupin's theorem on, 269,
436, &c.
On systems of, 441, &c.
Cayley's differential equation of, 443.
Confocal quadrics are, 143.
Confocal cyclides are, 533.
Osculation of two surfaces, condition for,
329.
Osculating plane, 104, 28o 324.
Sphere of a curve, 331, 341.
Right cone of a curve, 335.
Oscular lines on a surface, 489.
Painvin, on foci of sections of a quadric,
204.
Parabolic, points defined, 237.
Tangent planes at, count double, 240.
Polar quadrics of, are cones, 249.
Paraboloid, its equation reduced, 71.
Parallel to a quadric, 176, 477.
Its sections by principal planes, 468.
To a surface in general, 475.
To a curve, 478.
Pascal, theorem of, 122.
Plane, 123.
Pedal surfaces, 478,
Perpendicularity, generalization of the re-
lation, 200, 465.
Condition of for two circles on sphere,
219.
Pinch points, 519, 594,
Pliicker's relations between singularities
of plane curves, 291.
On curves on a hyperboloid, 358, 554.
On complexes, &c., 63, 416.
On wave surface, 457.
On complex surface, 420, 524.
Polar, of points on a sui-face, 243.
Of line to a quadric, 49, 60.
Developable of a curve, 331, 338.
Curve of a line, 548.
Pole of plane with regard to quadric,
coordinates of, 58.
Principal planes of quadric, equation of,
55.
Pro-Hessians, 408.
Projections of lines of curvature on planes
of circular sections, 163.
Puiseux, on curves of double curvature,
330,
On the proof of Gauss's theorem, 354.
Purser, F., envelope of face of tetrahedron,
180.
On intersecting normals to quadric,
167.
On bitangejits to centro-surface, 474.
Quadrics, 45.
Having double contact, 115, 314.
Touching four planes or going through
four points, 18i.
Quadriplanar coordinates, 23, 199.
Conditions general equation in, may
represent a sphere, 202.
Quartic curves, two families of, 312.
Quartic surfaces, 512, &c.
Nodal quartics, 537.
Quintic curves, species of, 318.
Eadii of curvature, principal, their lengths,
257, 276.
Of any normal section, 259, 278.
Of a curve of double curvature, 33 1, 333.
Rank of system, 292.
Reciprocal surfaces, 103.
Cones, their sections, 101, 105.
Of double points on surfaces, 241.
Of a surface, its degree, 248.
Of ruled surface and of developable,
of same degree, 105, 294, 590.
Of apsidal surface, 456.
Of cubic surfaces, 485.
Of cubic surface with double line,
487.
General theory of, 580.
How affected by double and multiple
lines, 587.
Rectilineal generators of a quadric, 85.
Rectifying developable of curves, 336.
Reech, on closed sui'faces, 381.
Revolution, surface of, conditions quadric
should be, 96.
This problem considered geometri'
cally, 116.
Reciprocal of quadric, when a, 137.
Generated by revolution of right line,
99.
Differential equation of family of
surfaces of, 390.
Right lines on a cubic, 29, 496.
On a surface touch the Hessian, 251.
On quartic with nodal lines, 523, 527.
Roberts, M., his theorems on geodesies on
an ellipsoid, 356, 365, 376.
On differential equation of cuspidal
edge of enveloping developable, 413.
Roberts, S., on parallel surfaces, 476.
Roberts, W., on geodesies of an ellipsoid,
380.
On orthogonal surfaces, 451.
On equation of wave surface in elliptic
coordinates, 459.
On pedal surfaces, 478.
On negative pedals, 483.
Roberts, W. R., on curve of intersection
of two quadrics, 193.
Routh, on curves of double curvature, 337.
Ruled surfaces, 89, 422, &c., 590.
Their differential equation, 400, 559.
Reciprocals of same degree, 105, 428.
Generated by a line meetmg three
directing curves, 429.
By a line meeting a curve three
times, 432,
Double generators on, 433,
Cubical, 486.
Quartic, 512—522,
INDEX.
611
Saint Venant, on curves, 324, 342,
Schlafli, on reduction of degree of reci-
procal by nodal points, -190.
On right lines on a cubic, 499.
Analysis of different species of cubics,
490, 501.
Schroter, on orthogonal and equilateral
hyperboloids, 100, 102, 120.
On curves of the third order, 300.
On Steiner's quartic surface, 491.
Schubert, on fourpoint inflexional tan-
gents, 570.
On fivepoint contact, 574.
Schwarz, on developables, 318, 320.
Scrolls, 89, 512.
Serret, on orthogonal surfaces, 441, 450.
Slope, line of greatest, 381, 389, 411.
Sphere circumscribing tetrahedron, its
radius, 37.
Its equation, 201.
Inscribed in a tetrahedron, 201, 227.
Cutting four spheres at right angles,
206.
Principal spheres, have stationary
contact, 264.
Spherical curvature, line of, 262.
Sphero-conics, 215.
Sphero-quartics, 535.
Spinode torse and curve, 594.
Stationary contact, 178.
Implies contact in two consecutive
points, 263.
Principal spheres, have stationary
contact, 178, 257.
Points on twisted curve, 292.
Conditions for stationary contact of
two surfaces, 329.
Tangent planes to a surface, 240.
How many pass through a point, 250.
V. Staudt, sine of solid angle, 37.
Steiner, on perpendiculars in tetrahedron,
120.
Quartic surface cut by every tangent
plane in two conies, 491, 517, 556.
On cubical surfaces, 492, 495, 496.
Steinerians, 493.
Striction, lines of, 424.
Sturm, on cubics, 492.
On multiple lines, 523.
Sylvester, on canonical form of a cubic,
492, 494.
On twisted cartesians, 317.
On involution of six lines, 419.
Symmetroid, 540.
Syunormal explained, 475.
Systems of quadrics through a common
curve. 111.
Inscribed in a common developable,
111.
Of surfaces whose equations include
one indeterminate, 547.
Tact-invariant of two quadrics, 175 ; of
three quadrics, 211.
Of any two surfaces, 546 ; of any
three surfaces, 544.
Tait on curves, 285.
Tangent to a curve, 283.
Tangent cone to a quadric, its equation,
57, 148.
To any surface, its equation and
singularities, 245, 247.
Tangential equation, of quadric, 58, 109.
Of imaginary circle at infinity, 184,
199.
Of a curve in space, 184.
Of a sphere, 200.
Of the centro-surface of a quadric,
172.
Tetrahedroid, 542.
Tetrahedron, intersection of lines joining
middle points of sides, 6.
Volume of, formed by four points or
four planes, 21, 22.
in terms of edges, 35.
in quadriplanar coordinates, 39.
Sphere circumscribed to, 37.
Relation between perpendiculars in,
120.
Self-conjugate with regard to a
quadric, 117, 175.
Lines joining corresponding vertices
of two conjugates, how connected,
118.
Thomson's proof of Dupin's theorem, 270.
On curves, 285.
Tinseau, on curves of double curvature,
343.
Toeplitz, on a combinant of three quadrics,
210, 214.
Torsal lines on a surface, 489.
Torse, 89.
Torsion, angle of, 334.
Tortohni, on pedal surfaces, 478.
Torus, 539.
Townsend, on quadric through nine points,
113.
On foci of quadric, 136.
On Jacobi's mode of generating quad-
rics, 165.
Triple tangent lines to a surface, 500.
Planes, an ordinary singularity of sm'-
faces, 235.
For cubic, 498.
Their number in general, 584,
Tubular surfaces, 402, 409, 478.
Umbilics of quadric defined, 85.
Their coordinates, 142.
Lie in threes on right lines, 116.
Section of enveloping quadric by tan-
gent plane at, 117.
Conditions for, 260, 278.
Their number in general, 263.
Three lines of curvature pass through,
266.
Umbilicar foci, 131.
Uuicursal curve, 282,
Surface, 553,
Unodes, 489.
Valentincr, on general quartic, 527.
612
INDEX.
Vo88, on umbilics, 2C3.
Wallis's cono-cuncus, 390.
Wave surf.ioe, 4.')3, A-c.
Generalization of, 512.
Weicrstrass, on integration of equation of
geodesies, 372,
T\* ~s on Stciner's qnartic, 401.
^^ 11, on GaiiKs's iiivt-stiguiion of
liues of cnrvuture, 348, 352.
Zeuthcn, ■ 'p23.
Oni^i; , ices, 603.
Ou systems ul surfaces, 553.
THi: END.
•W. METCALFE A>D SON, PIUSTEUS. CAMDRIDGE.
QA Salmon, George
553 A treatise on tlie analytic
S25 geometry of three dimensions
1882 4.th ed.
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