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juiVERSITY of CALIFORNIA
LIBRARY OF THE UNIVERSITY OF CALIFORNIA
INIVERSITY OF CALIFORNIA
LIBRARY OF THE UNIVERSITY OF CALIFORNIA
SOLID GEOMETEY.
By the same Author.
AN ELEMENTARY TREATISE
ON
CONIC SECTIONS.
FOURTH EDITION.
Crown 8vo. 7s. 6d.
ALSO
ELEMENTAEY ALGEBEA.
Crown 8vo. 4s. 6d.
" It is a pleasure to come across an Algebra-book which has manifestly
not been written in order merely to prepare students to pass an examination.
Not that we think Mr Smith's book unsuitable for this purpose ; indeed, with
its carefully-worked examples, graduated sets of exercises, and regularly-
recurring miscellaneous examination papers, it compares favourably with the
most approved 'grinders' books Mr Smith shews to great advantage
as a teacher, his style of exposition being most lucid : the average student
ought to find the book easy and pleasant reading." — Nature.
" Beginners will find the subject thoughtfully placed before them, and the
road through the science rendered easy to no small degree." — The School-
viaster.
" There is a logical clearness about his explanations and the order of his
chapters for which both schoolboys and schoolmasters should be, and will
be, very grateful." — The Educational Times.
.* * * • • • • • , ' J • • » • - • '
ELEMENTAKY TREATISE
ON
SOLID GEOMETEY
BY
CHAKLES SMITH, M.A.
FELLOW AND XUTOB OF SIDNEY SUSSEX COLLEGE, CAUBBID6E.
SECOND EDITION,
Honbon :
MACMILLAN AND CO.
1886
[All Eights reserved.]
c r- t ' c
c'o";; ^ ;, PHYSICS DE
PHYSlCS DSPTT
€^ambritjgc :
PRINTED BY C. J. CLAY, M.A. & SON,
AT THE UNIVERSITY PRESS.
PHEFACE.
The following work is intended as an introductory text-
book on Solid Geometry, and I have endeavoured to present
the elementary parts of the subject in as simple a manner as
possible. Those who desire fuller information are referred to
the more complete treatises of Dr Salmon and Dr Frost, to
both of which I am largely indebted.
I have discussed the different surfaces which can be
represented by the general equation of the second degree at
an earlier stage than is sometimes adopted. I think that
this arrangement is for many reasons the most satisfactory,
and I do not believe that beginners will find it difficult.
The examples have been principally taken from recent
University and College Examination papers ; I have also
included many interesting theorems of M. Chasles.
I am indebted to several of my friends, particularly to
Mr S. L. Loney, B.A., and to Mr R. H. Piggott, B.A., Scholars
of Sidney Sussex College, for their kindness in looking over
the proof sheets, and for valuable suggestions.
CHAKLES SMITH.
Sidney Sussex College,
April, 1884.
S€544S
CONTENTS.
CHAPTER I.
Co-ordinates.
PAGE
Co-ordinates 1
Co-ordinates of a point which divides in a given ratio the line joining
two given points 3
Distance between two points 4
Direction-cosines ........... 5
Relation between direction-cosines 5
Projection on a straight line 6
Locus of an equation 7
Polar co-ordinates 8
CHAPTER II.
The Plane.
An equation of the first degree represents a plane ....
Equation of a plane in the form ix-^-7n?/-^ 712=^ ....
Equation of a plane in terms of the intercepts made on the axes .
Equation of the plane through three given points ....
Equation of a plane through the line of intersection of two given
planes . .
Conditions that three planes may have a common line of intersection
Length of perpendicular from a given point on a given plane
Equations of a straight line . . .
Equations of a straight line contain four independent constants .
Symmetrical equations of a straight line
9
9
10
11
11
11
12
14
14
15
Vlll
CONTENTS.
Equations of the straight line through two given points
Angle between two straight lines whose direction-cosines are given
Condition of perpendicularity of two straight lines
Angle between two planes whose equations are given .
Perpendicular distance of a given point from a given straight Hne
Condition that two straight lines may intersect
Shortest distance between two straight hnes .
Projection on a plane
Projection of a plane area on a plane
Volume of a tetrahedron
Equations of two straight lines in their simplest forms
Four planes with a common line of intersection cut any straight line
in a range of constant cross ratio .
Oblique axes
Direction-ratios
Kelation between direction-ratios .
Distance between two points in terms of their oblique co-ordinates
Angle between two lines whose direction-ratios are given
Volume of a tetrahedron in terms of three edges which meet
point, and of the angles they make with one another
Transformation of co-ordinates
Examples on Chapter II
m a
PAGE
16
16
17
18
19
19
"20
22
23
24
25
26
26
26
27
28
28
28
29
34
CHAPTER III.
Surfaces of the Second Degree.
Number of constants in the general equation of the second degree . 37
All plane sections of a surface of the second degree are conies . . 38
Tangent plane at any point of a conicoid 38
Polar plane of any point with respect to a conicoid .... 39
Polar lines with respect to a conicoid 40
A chord of a conicoid is cut harmonically by a point and its polar
plane 40
Condition that a given plane may touch a conicoid . . . .41
Equation of a plane which cuts a conicoid in a conic whose centre is
given 43
Locus of middle points of a system of parallel chords of a conicoid . 44
Principal planes 44
CONTENTS. IX
PAGE
Parallel plane sections of a conicoid are similar and similarly situated
conies 45
Classification of conicoids 46
The ellipsoid 49
The hyperboloid of one sheet .50
The hyperboloid of two sheets 51
The cone 51
The asymptotic cone of a conicoid 52
The paraboloids 52
A paraboloid a limiting form of an ellipsoid or of an hyperboloid . . 54
Cylinders 54
The centre of a conicoid 56
Invariants 58
The discriminating cubic ......... 59
Conicoids with given equations ........ 60
Condition for a cone 66
Conditions for a surface of revolution 66
Examples on Chapter III 67
CHAPTER IV.
Conicoids eefekked to their Axes.
The sphere 69
The ellipsoid 71
Director-sphere of a central conicoid 72
Normals to a central conicoid 73
Diametral planes 74
Conjugate diameters 75
Eelations between the co-ordinates of the extremities of three conjugate
diameters 75
Sum of squares of three conjugate diameters is constant ... 76
The parallelopiped three of whose conterminous edges are conjugate
semi-diameters is of constant volume 76
Equation of conicoid referred to conjugate diameters as axes . . 78
The paraboloids 80
Locus of intersection of three tangent planes which are at right angles 80
Normals to a paraboloid 81
Diametral planes of a paraboloid 81
X CONTENTS.
TAGIB
Cones . 83
Tangent plane at any point of a cone 83
Beciprocal cones ^^
Reciprocal cones are co-axial 80
Condition that a cone may have three perpendicular generators . . 85
Condition that a cone may have three perpendicular tangent planes . 86
Equation of tangent cone from any point to a conicoid .... 86
Equation of enveloping cylinder 88
Examples on Chapter IV 90
CHAPTEE V.
Plane Sections of Coxicoids.
Nature of a plane section found by projection . . . . .96
Axes and area of any central plane section of an ellipsoid or of an
hyperboloid ....••••
Area of any plane section of a central conicoid
Area of any plane section of a paraboloid
Area of any plane section of a cone ....
Directions of axes of any central section of a conicoid .
Angle between the asymptotes of a plane section of a central conicoid . 101
Condition that a plane section may be a rectangular hyperbola . . 102
Condition that two straight lines given by two equations may be at
right angles 102
Conicoids which have one plane section in common have also another . 103
Circular sections 103
Two circular sections of opposite systems are on a sphere . . . 105
Circolar sections of a paraboloid 105
Examples on Chapter V 108
CHAPTER VI.
Generating Lines of Conicoids.
Euled surfaces defined 113
Distinction between developable and skew surfaces .... 113
Conditions that all points of a given straight line may be on a surface . 113
The tangent plane to a conicoid at any point on a generating line
contains the generating line 115
Any plane through a generating line of a conicoid touches the surface . 115
97
98
99
99
101
CONTENTS. XI
PAGE
Two generating lines pass througli every point of an hyperboloid of one
sheet, or of an hyperbolic paraboloid 116
Two systems of generating lines 116
All straight lines which meet three fixed non-intersecting straight lines
are generators of the same system of a conicoid, and the three fixed
lines are generators of the opposite system of the same conicoid . 117
Condition that four non-intersecting straight lines may be generators of
the same system of a conicoid 117
The lines through the angular points of a tetrahedron perpendicular to
the opposite faces are generators of the same system of a conicoid 118
If a rectilineal hexagon be traced on a conicoid, the three lines joining
its opposite vertices meet in a point ...... 118
Four fixed generators of a conicoid of the same system cut all generators
of the opposite system in ranges of equal cross-ratio . . .118
Angle between generators 119^
Equations of generating lines through any point of an hyperboloid of
one sheet 120
Equations of the generating lines through any point of an hyperbolic
paraboloid 122
Locus of the point of intersection of perpendicular generators . . 124
Examples on Chapter VI 124
CHAPTER VII.
Ststems of Conicoids. Tangential Equations. Reciprocation.
All conicoids through eight given points have a common curve of
intersection 128
Four cones pass through the intersections of two conicoids . . . 129
Self-polar tetrahedron 129
Conicoids which touch at two points 130
AU conicoids through seven fixed points pass through another fixed
point \ 130
Eectangular hyperboloids 131
Locus of centres of conicoids through seven given points . . . 132
Tangential equations 133
Centre of conicoid whose tangential equation is given .... 134
Director-sphere of a conicoid 135
Locus of centres of conicoids which touch eight given planes . . 136
Locus of centres of conicoids which touch seven given planes . . 137
Xll CONTENTS.
PAGE
Director-spheres of conicoids which touch eight given planes, have a
common radical plane 137
The director-spheres of all conicoids which touch six given planes are
cut orthogonally by the same sphere 137
Reciprocation 137
The degree of a surface is the same as the class of its reciprocal . . 138
Reciprocal of a curve is a developable surface 138
Examples of reciprocation 140
Examples on Chapter VII 141
CHAPTER VIII.
CoNFOCAL Conicoids. Concyclic Conicoids. Foci of Conicoids.
Confocal conicoids defined 144
Focal conies. [See also 158] 145
Three conicoids of a confocal system pass through a point . . .145
One conicoid of a confocal system touches a plane .... 146
Two conicoids of a confocal system touch a line 146
Confocals cut at right angles 147
The tangent planes through any line to the two confocals which it
touches are at right angles 148
Axes of central section of a conicoid in terms of axes of two confocals . 149
Corresponding points on conicoids 151
Locus of pole of a given plane with respect to a system of confocals . 152
Axes of enveloping cone of a conicoid 153
Equation of enveloping cone in its simplest form 153
Locus of vertices of right circular enveloping cones .... 155
ConcycHc conicoids 155
Reciprocal properties of confocal and concychc conicoids . . . 156
Foci of conicoids 156
Focal conies 158
Focal hues of cone 159
Examples on Chapter VIII * . .160
CHAPTER IX.
QUADEIPLANAR AND TeTEAHEDRAL Co-OEDINATES.
Definitions of Quadriplanar and of Tetrahedral Co-ordinates . . 164
Equation of plane 165
Length of perpendicular from a point on a plane 167
CONTENTS. Xlll
PAGE
Plane at infinity 167
Symmetrical equations of a straight line 168
General equation of the second degree in tetrahedral co-ordinates . 169
Equation of tangent plane and of polar plane 170
Co-ordinates of the centre 170
Diametral planes 171
Condition for a cone 171
Any two conicoids have a common self-polar tetrahedron . . . 172
The circumscribing conicoid 172
The inscribed conicoid 172
The circumscribing sphere 173
Conditions for a sphere 173
Examples on Chapter IX 175
CHAPTER X.
Surfaces in General.
The tangent plane at any point of a surface 178
Inflexional tangents 179
The Indicatrix 180
Singular points of a surface 180
Envelope of a system of surfaces whose equations involve one arbitrary
parameter 181
Edge of regression of envelope 182
Envelope of a system of surfaces whose equations involve two arbitrary
parameters 183
Functional and differential equations of conical surfaces . . . 184
Functional and differential equations of cylindrical surfaces . . . 185
Conoidal surfaces 186
Differential equation of developable surfaces 188
Equation of developable surface which passes through two given curves 190
A conicoid will touch any skew surface at all points of a generating
line 191
Lines of striction 191
Functional and differential equations of surfaces of revolution . . .192
Examples on Chapter X. , 194
xiv CONTENTS.
CHAPTER XI.
Curves.
PAOB
Equations of tangent at any point of a curve 197
Lines of greatest slope 198
Equation of osculating plane at any point of a curve . . . . 201
Equations of the principal normal 202
Radius of curvature at any point of a curve 202
Direction-cosines of the binormal 203
Measure of torsion at any point of a curve 203
Condition that a curve may be plane 204
Centre and radius of spherical curvature 206
Radius of curvature of the edge of regression of the polar developable . 207
Curvature and torsion of a heHx 208
Examples on Chapter XL 210
CHAPTER XII.
Curvature op Surfaces.
Curvatures of normal sections of a surface 213
Principal radii of curvature 214
Euler's Theorem 214
Meunier's Theorem 215
Definition of lines of curvature 217
The normals to any surface at consecutive points of a line of curvature
intersect 217
Differential equations of lines of curvature 217
Lines of curvature on a surface of revolution 218
Lines of curvature on a developable surface 218
Lines of curvature on a cone 219
If the curve of intersection of two surfaces is a line of curvature on both
the surfaces cut at a constant angle 220
Dupin's Theorem 221
To find the principal radii of curvature at any point of a surface . . 222
Umbilics 223
Principal radii of curvature of the surface z=f{x, y) . . , , 224
Gauss' measure of curvature 225
Geodesic lines . . • . , » 226
CONTENTS. XV
PAGE
Lines of curvature of a conicoid are its curves of intersection with con-
focal conicoids 227
Curvature of any normal section of an ellipsoid 228
The rectangle contained by the diameter parallel to the tangent at any
point of a line of curvature of a conicoid, and the perpendicular
from the centre on the tangent plane at the point is constant . 228
The rectangle contained by the diameter parallel to the tangent at any
point of a geodesic on a conicoid, and the perpendicular from the
centre on the tangent plane, is constant . . - . ,. . 223
Properties of lines of curvature of conicoids analogous to properties of
confocal conies 229
Examples on Chapter XII 230
Miscellaneous Examples 237
SOLID GEOMETKY.
CHAPTER I.
CO-ORDINATES.
1. The position of a point in space is usually determined
by referring it to three fixed planes. The point of inter-
section of the planes is called the origin, the fixed planes are
called the co-ordinate 'planes, and their lines of intersection
the co-ordinate axes. The three co-or^dinates of a point are
its distances from each of the three co-ordinate planes,
measured parallel to the lines of intersection of the other
two. When the three co-ordinate planes, and therefore the
three co-ordinate axes, are at right angles to each other, the
axes are said to be rectangular.
2. The position of a point is completely determined when
its co-ordinates are known. For, let YOZ, ZOX, XOY be
the co-ordinate planes, and X' OX, Y'OY, Z'OZhe the axes,
and let LP, MP, NP, be the co-ordinates of P. The planes
MPN, NPL, XPJ/are parallel respectively to YOZ, ZOX,
XOY; if therefore they meet the axes in Q, B, S, as in the
figure, we have a parallelepiped of which OP is a diagonal;
and, since parallel edges of a parallelepiped are equal,
LP=OQ, MP = OR, and XP = OS.
Hence, to find a point whose co-ordinates are given, we have
only to take OQ, OP, OS equal to the given co-ordinates,
S. s. G. 1
CO-ORDINATES.
3,nd drai^ tbt-ee planes tliroiigh Q, R, S parallel respectively
to the co-ordinate planes ; then the point of intersection of
these planes will be the point required.
Z
M
r
s
^Y'
^
^
P
0
/^ ^
y
^
z'
N
If the co-ordinates of P parallel to OX, OY, 0^ respec-
tively be a, h, c, then P is said to be the point (a, h, c).
3. To determine the position of any point P it is not
sufficient merely to know the absolute lengths of the lines
LP, MP, NP, we must also know the directions in which
they are drawn. If lines drawn in one direction be con-
sidered as positive, those drawn in the opposite direction
must be considered as negative.
We shall consider that the directions OX, OY, OZ are
positive.
The whole of space is divided by the co-ordinate planes
into eight compartments, namely OXYZ, OX'YZ, OXYZ,
OXYZ', OXY'Z, OX'YZ, OX'YZ, and OX'YZ.
If P be any point in the first compartment, there is a
point in each of the other compartments whose absolute
distances from the co-ordinate planes are equal to those of P ;
and, if P be (a, h, c) the other points are (— a, h, c), {a, — h, c),
{a, h, — c), (a, — b, — c), (— a, b, — c), (— a, - b, c) and (— a,—b,—c)
respectively.
CO-ORDINATES. 3
4. To find the co-ordinates of the point which divides the
straight line joining two given points in a given ratio.
Let P, Q be the given points, and R the point which
divides PQ in the given ratio m^ : m^ .
Let Pbe (a^^, y^, z^, Q be (a-,, y,, z.^, and R be [x, y, z).
Draw PL, QM, B^'' parallel to OZ meeting XO Y in Z, i/,
iV^. Then the points P, Q, R, L, M, N are clearly all in one
plane, and a line through P parallel to LM will be in that
plane, and will therefore meet QM, RN, in the points K, H
suppose.
^, ER PR m,
Then -777^ = -jT7\ = — •
KQ PQ m^ + ??i2
But LP = z^, MQ = z.^,NR = z)
z — z
I —
m.
■^2 1
Similarly
^ =
57 =
and ?/ =
m^z,^ + m./j
7?i^ + rti.^
When PQ is divided externally, m^ is negative.
1—2
4 CO-OKDINATES.
The most useful case is where the line PQ is bisected : the
co-ordinates of the point of bisection are
The above results are true whatever the angles between
the co-ordinate axes may be.
We shall in future consider the axes to he rectangular in
all cases except ivhen the contrary is expressly stated.
5. To express the distance between two points in terms of
their co-ordinates.
Let Pbe the point {x^, y^, z^ and Q the point {x.^, 7/^, z^).
Draw through P and Q planes parallel to the co-ordinate
planes, forming a parallelepiped whose diagonal is FQ.
_V
a
/
^^-^-^
Ti
/■
P
/
3^
£
J
KT
:k
Let the edges PZ, LK, KQ be parallel respectively to
OX, OY, OZ. Then since PL is perpendicular to the plane
QKL, the angle PLQ is a right angle,
.-.PQ'^PU + QU
= PL'-\-LK' + KQ\
Now PL is the difference of the distances of P and Q
from the plane YOZ, so that we have PL = x^ — x^, and
similarly for LK smd KQ.
Hence PQ'' = (^, - a>,y + {y, - y.^ + {z., - z^\ . .. . .(i).
The distance of P from the origin can be obtained from
the above by putting x^ = {)^ y^ = ^, z^ = 0. The result is
CO-ORDINATES. - 5
Ex. 1 . The co-ordinates of the centre of gravity of the triangle whose angular J
points are (Xj, y^, z-^, (ojg, y^ Zg). (a^' 2/3' ^3) ^re \ (x^ + ar.^ + ar^), \ {yi + y^ + Vz)^
and i (% + -22 + ^3)-
Ex. 2. Shew that the three lines joining the middle points of opposite
edges of a tetrahedron meet in a point. Shew also that this point is on the J
line joining any angular point to the centre of gravity of the opposite face,
and divides that line in the ratio of 3:1.
Ex. 3. Eind the locus of points which are equidistant from the points
(1, 2, 3) and (3, 2, - 1). Aiis. x - 2z = 0.
Ex. 4. Shew that the point (|, 0, |) is the centre of the sphere which /
passes through the four points (1, 2, 3), (3, 2, - 1), ( - 1, 1, 2) and (1, - 1, - 2). ^
6. Let Of, fi, 7 be the angles which the line PQ makes
with lines through P parallel to the axes of co-ordinates.
Then, since in the figure to Art. 5 the angles PLQ, PMQy PKQ
are right angles, we have
PQ cos a = PL,
PQcos/3 = Pil/,
and PQ cos 7 = PN.
Square and add, then
PQ' {cos^a + cos'^yS + cos^} = PL' + PJ\P + PF' = PQ\
Hence cos^a + cos^yS + cos^7 = 1.
The cosines of the angles which a straight line makes
with the positive directions of the co-ordinate axes are called
its direction-cosines, and we shall in future denote these
cosines by the letters I, m, n.
From the above we see that any three direction-cosines
are connected by the relation l^ -\- m' -h n' = 1. If the
direction-cosines of PQ be Z, m, n, it is easily seen that those
of QP will be — Z, — ??i, — n ; and it is immaterial whether we
consider I, m, n, or the same quantities with all the signs
changed, as direction-cosines.
If we know that a, 6, c are proportional to the direction-
cosines of some line, we can at once find those direction-
cosines. For we have - = -^ = - : hence each is equal to
a b c
V(f + m' + »') ■ . 1 ■;_ « &C
V(a'+ b'+c') ' ^{a' + b' + c') ' ' ' -Jia' + b' + r) "
G CO-ORDINATES.
Ex. Th« direction-cosines of a line are proportional to 3, - 4, 12, find
their actual values. Am. x\, — tV> tI«
7. The projection of a point on any line is the point
where the line is met by a plane through the point per-
pendicular to the line. Thus, in the figure to Art. 2, Q, R, S
are the projections of P on the lines OX, OY, OZ re-
spectively.
The jDrojection of a straight line of limited length on
another straight line is the length intercepted between
the projections of its extremities. If we have any number of
points P, Q, R, S... whose projections on a straight line are
p, q, r, s..., then the projections of PQ, QR, RS... on the
line, are pq, qr, vs....
In estimating these projections we must consider the
same direction as positive throughout, so that we shall
always have pq + qi^ + rs = ps, that is the projection of
PS on any line is equal to the algebraic sum of the pro-
jections of PQy QR and RS. This result may be stated in a
more general form as follows: — The algebraic sum of the
projections of any number of sides of a polygon beginning at
P and ending at Q is equal to the projection of PQ.
8. If we have any number of parallel straight lines, the
projections of any other line PQ on them are the intercepts
between planes through P and Q perpendicular to their
directions. These intercepts are clearly all equal ; hence the
projections of any line on a series of parallel straight lines
are all equal. And, since the projection of a straight line on
an intersecting straight line is found by multiplying its
length by the cosine of the angle between the lines, we have
the following proposition : —
The projection of a finite straight line on any other
straight line is equal to its length multiplied hy the cosine of
the angle between the lines.
9. In the figure to Art. 2, let OQ = a,OR = h, 0S= c.
Then it is clear that a) = a for all points on the plane
PMQN, and that y — hioi all points on the plane PNRL,
CO-ORDINATES. 7
and that ^ = c for all points on the plane PLSM. Also
along the line NP we have x = a, and y = h; and at the
point P we have the three relations x = a, y = h, z = c.
So that a plane is determined by one equation, a straight
line by two equations, and a point by three equations.
In general, any single equation of the form F (cc, y, z) = 0,
in which the variables are the co-ordinates of a point,
represents a surface of some kind ; two equations represent a
curve, and three equations represent one or more points. This
we proceed to prove.
10. Let two of the variables be absent, so that the
equation of the surface is of the form F (x) = 0. Then the
equation is equivalent to {x — a) (x — h) (x — c) = 0, where
a, h, c,... are the roots of F{x) = 0', hence all the points
whose co-ordinates satisfy the equation F{x) — 0 are on one
or other of the _/9?a?ie.9 x — a — 0, a? — 6 = 0, x — c= 0,
Let one of the variables be absent, so that the equation
is of the form F {x, y) = 0. Let P be any point in the plane
z = 0 whose co-ordinates satisfy the equation F {x,y) =0 ;
then the co-ordinates of all points in the line through P
parallel to the axis of z, are the same as those of P, so far as
X and y are concerned ; it therefore follows that all such
points are on the surface. Hence the surface represented by
the equation F (x, y) = 0 is traced out by a line which is
always parallel to the axis of z, and which moves along the
curve in the plane z~0 defined by the equation F{x, y) = 0.
Such a surface is called a cylindrical surface, or cylinder.
Next let the equation of the surface be F(x,y,z) = 0.
We have seen that all points for which x = a, and y=h
lie on a straight line parallel to the axis of z. Hence, if in
the equation F(x, y, z) = 0, we put x = a, and y = 'b, the roots
of the resulting equation in z will give the points in which
the locus is met by a hne through {a, b, 0) parallel to the axis
oi z.
Since the number of roots is finite, the straight line will
meet the locus in a finite number of points, and therefore the
locus, which is the assemblage of all such points for different
values of a and 6, must be a surface and not a solid figure.
8
CO-ORDINATES.
11. The points whose co-ordinates satisfy two equations
must be on both the surfaces which those equations represent
and therefore the locus is the curve determined by the intersec-
tion of the two surfaces. When three equations are given, we
have sufficient equations to find the co-ordinates, although there
may be more than one set of values, so that three equations
represent one or more points.
12. The position of a point in space can be defined by
other methods besides the one described in Art. 1.
Another method is the following : an origin 0 is taken, a
fixed line OZ through 0, and a fixed plane XOZ. The
position of a point P is completely determined when its
distance from the fixed point 0, the angle ZOP, and the angle
between the planes XOZ, and POZ are given. These co-
ordinates are called Polar Co-ordinates, and are usually de-
noted by the symbols r, 6 and (/>, and the point is called the
point (r, 6, (f)).
If OX be perpendicular to OZ, and 0 F be perpendicular
to the plane ZOX, we can express the rectangular co-ordinates
of P in terms of its polar co-ordinates.
Draw PX perpendicular to the plane XOY^ and NM
perpendicular to OX, and join ON, Then
x — OM = OX cos (j) — OP sin 6 cos <^ = r sin 6 cos <f),
y = MX= ON s'm ^ = OP sin ^ sin ^ = 7- sin 6 sin <p,
and z = XP = OP cos 6 = r cos 6.
We can also express the polar co-ordinates of any point in
terms of the rectangular. The values are,
r = s/(x'-h f + z'), e = tan-^ VV±j^^ ^^^ ^ = tan"^ ^ .
X
CHAPTER IT.
The Plane.
13. Tojhew that the surface rei:>resented hj the general
equation of the first degree is a i^lane.
The most general equation of the first degree is
Ax+By-\-Cz+D = 0.
If (^V y^y z) and {x^, y^, zj be any two points on the locus,
we have
Ax^ + By^ + C\-\-I) = 0,
^^'^^ Ax^ + %^ + (7^^ + X) = 0.
Multiply these in order by ^'^'-' , and — ^^i— and add:
then we have
A !!^i±^i3 ■;_ ^ ^2 ^1 + ^1 ^2 . r^ '^\ ^1 + ^^ ^. , n = n
This shews [Art. 4] that if the points {x^, y„ z^), {x^, y^, z,) be
on the locus, any other point in the line joining them is also
on the locus; this shews that the locus satisfies Euclid's
definition of a plane.
14. To find the equation ofaj^flane.
Let p be the length of the perpendicular ON from the
origin on the plane, and let /, m, 7i be the direction-cosines of
10
THE PLANE.
the perpendicular. Let P be any point on the plane, and draw
PL perpendicular on XOY^ and LM perpendicular to OX.
Then the projection of OP on ON is equal to the sum
of the projections of OM, ML and LP on ON.
Hence if P be {x, y, z)y we have
Ix -^ my •\- nz = p (i),
the required equation.
By comparing the general equation of the first degree
with (i), we see that the direction-cosines of the normal to the
plane given by the general equation of the first degree are
proportional to A, B, C ; and therefore [Art. 6] are equal to
A B G
V(^' + -S'+C'^)' '^{A' + B'+C')' sJiA'+B'^+Cy
Also the perpendicular from the origin on the plane is
equal to —D
s/{A' + B'+Gy
15, To find where the plane whose equation is
Ax + By+Gz-^D = 0,
meets the axis of x we must put 3/ = 2 = 0 ; hence if the
intercept on the axis of x be a, we have Aa -\- D = 0.
Similarly if the intercepts on the other axes are h and c
we have Bb + B = 0, and Cc ■{• D = 0. Hence the equation
of the plane is
X y z ^
- + ';- + -=i.
a 0 c
This equation can easily be obtained independently.
THE PLANE.
11
16. To find the equation of the plane through three given
points.
Let the three points be (x^, y^, z^), (^^, y^, z^, {x^, y^, z^).
The general equation of a plane is
Ax + By + Cz + D = 0.
If the three given points are on this plane, we have
Axj^ + By^ + C2^ + D = 0,
Ax^ + By^ + Cz^ + D = 0,
and Ax^ + By^ + Cz^ + D = 0.
Eliminating A, B, C, I) from these four equations, we
have for the required equation
X , y , z
X
1 '
1 >
X.
2 '
'2 '
X„
= 0.
2/i.
3/2 >
17. li S=0 and S' = 0 be the equations of two planes,
>S^— X, S' = 0 will be the general equation of a plane through
their intersection. For, since S and S' are both of the first
degree, so also is S — X8' ; and hence >S^ — XS' = 0 represents
a plane. The plane- passes through all points common to
>S^ = 0 and S' = 0; for if the co-ordinates of any point satisfy
>S*= 0 and >S" = 0, those co-ordinates will also satisfy S = \S'.
Hence, since X is arbitrary, 8 — XS' = 0 is the general
equation of a plane through the intersection of the given
planes.
18. To find the conditions that three planes may have a
common line of intersection.
het the equations of the planes be
ax + by-rcz + d = 0 (i),
a'x-\-b'y + cz + d' = 0 (ii),
and a'x + h'y + c'z+d" = 0 (iii).
The equation of any plane through the line of intersection
of (i) and (ii) is of the form
(ax + hy + cz-\-d)+X (ax + Vy + c'z + d') = 0. . .(iv).
12 THE PLANE.
If the three planes have a common line of intersection, we
can, by properly choosing X, make (iv) represent the same
plane as (iii). Hence corresponding coefficients must bo
proportional, so that
a 4- \a h 4- XZ/' c + Xc _d + Xd'
' '' T^' T'^ ZF^ *
a 0 c a
Put each fraction equal to — fi, then we have
a + \a + fia" = 0,
h + W + fMb'' =0,
c + \c + ijlg' =■• 0,
and d + \d' + fjLd'' = 0.
Eliminating X and fj, we have the required conditions,
namely
a , h , c , d I i = 0,
a J h' , c y d!
II 7 n ff -III
a , 0 , c , d
the notation indicatiuG^ that each of the four determinants, ob-
tained by omitting one of the vertical columns, is zero.*
19. We can shew, exactly as in Conies, Art. 26, that if
Ax + B^/ + Cz + D = 0 be the equation of a plane, and w, y, z
be the co-ordinates of any point, then Ax + By + Gz + D
will be positive for all points on one side of the plane, and
negative for all j^oints on the other side.
20. To find the jperpendicular distance of a given point
from a given 23lane.
Let the equation of the given plane be
Ix •\-my + nz=p (i),
and let x\ y\ z be the co-ordinates of the given point P. The
equation
Ix 4- my + ')iz = p (ii)
is the equation of a plane parallel to the given plane.
It will pass through the point (x, y\ z) if
Ix + my' •\-nz =jp>' (iii).
* It is easy to shew that there are only tico independent conditions, as is
geometrically obvious, for if the planes have two points in common they
must have a common line of intersection.
THE PLANE. 13
Now if PL be the perpendicular from P on the plane (i),
and ON, ON' the perpendiculars from the origin on the planes
(i) and (ii) respectively, then will
LP = NN'
=p'-p
= Ix' 4- fny' + nz — p.
Hence the length of the perpendicular from any point on
the plane Ix + my + nz —p = 0 is obtained by substituting the
co-ordinates of the point in the expression lx-\- my -\-nz~-p.
If the equation of the plane be Ax + By + Cz + D = 0, it
may be written
A B G
+ ^ = 0
Avhich is of the same form as (i) ; therefore the length of the
perpendicular from {x , y', z) on the plane is
Ax+By'+Cz+D
*J[A + B' + G'] •
Ex. 1. Find the equation of the plane through (2, 3, - 1) parallel to the /
plane 3x -Ay + lz = Q. Am. Sx - 4?/ + 7z + 13 = 0.
Ex. 2. Find the equation of the plane through the origin and through
the intersection of the two planes 5x-hy + 2z-\-b — 0 and 3a; - 5^/ - 22 - 7 = 0.
Ans. 25ic-23?/ + 22 = 0.
Ex.3. Shew that the three planes 2a; + 5?/ + 3s = 0, x-y + Az = 2, and ^
7y - 52 + 4 = 0 intersect in a straight line.
Ex.4. Shewthatthefoarplanes2x — 3?/ + 22 = 0, CC + ?/- 32 = 4,3.x-2/ + 2=2, /'
and Ix- 5y + 62 — 1 meet in a point.
Ex.5. Shew that the four points (0,-1,-1) (4, 5, 1), (3, 9, 4) and V^
( - 4, 4, 4,) lie on a plane.
Ex. 6. Are the points (4, 1, 2) and (2, 3, - 1) on the same or on opposite ^
sides of the plane 5x-7y-Qz +3 = 0?
Ex. 7. Shew that the two points (1, - 1, 3) and (3, 3, 3) are equidistant ^
from the plane 5x + 2y-7z + 9 = 0, and on opposite sides of it.
Ex. 8. Find the equations of the planes which bisect the angles between
the planes Ax + By + Cz + D = 0, and A'x + B'y + C'z + D'=0. '
Ax + By + Cz + D _ A'x + B'y+C'z + D
u
THE STRAIGHT LINE.
Ex. 9. The locus of a point, whose distances from two given planes are /*
in a constant ratio, is a plane.
Ex. 10. The locus of a point, which moves so that the sum of its distances
from any number of fixed planes is constant, is a plane.
21. The co-ordinates of any point on the line of intersection
of two planes will satisfy the equation of each of the planes.
Hence any two equations of the first degree represent a
straight line. We can find the equations of a straight line in
their simplest form in the following manner.
Let PQ be the straight line, ^^^ its projection on the plane
XOY by lines parallel to OZ. Then the co-ordinates w and y
of any point in PQ are the same as the co-ordinates x and y
of its projection in^q.
Hence if Ix + my = 1 be the equation of pq, the co-ordi-
nates of any point on PQ will satisfy the equation
Ix + my = 1.
Similarly, if the equation of the projection of PQ on the
plane YOZhe ny +pz=l, the co-ordinates of any point on
PQ will satisfy the equation ny+pz = 1. Hence the equations
of the line may be written
lx + my = l, ny+pz— 1,
It should be noticed that the equations of a straight line
contain four independent constants.
The above equations are unsymmetrical and are not so
useful as another form of the equations which we preceed to
find.
THE STRAIGHT LINE.
15
22. Let (a, y5, 7) be any point A on a straight line, and
(x, y, z) any other point P on the hne, at a distance r from
{p, /&, 7) ; and let I, m, n be the direction-cosines of the line.
a
X
Draw through A and P planes parallel to the co-ordinate
planes so as to make a parallelepiped, and let AL, LM, MP
be edges of this parallelopiped parallel to the axes of x, y, z
respectively. Then AL is the projection of AP on the axis
of a?; therefore
X— 0L = Ir, or — i — = r.
I
We have similarly
= r, and = ?'.
Q7i n
Hence the equations of the line are
I m n '
Ex. 1. To find in a symmetrical form the equations of the line of inter-
section of the planes 5x-Ay = l, Sy-5z = 2. ^
x-^ y_ z+"
5^
The equations may be written
Hence the direction-
4 5 3
cosines are proportional to 4, 5, 3. The actual values of the direction-
cosines are therefore f \/2> 2'n/2, i'W2.
Ex. 2. Find in a sjTnmetrical form the equation of the line x-2y=5, ^
Sx+y-7z=0. Am. ^{x-5)=y=z-if,
Ex. 3. Find the direction-cosines of the line whose equations are
12 3
x + y-z + l:=0, 4jc-i-y-2z + 2=0. Ans. 771^ * TnJ* T/u '
Ex. 4. Write down the equation of the straight line through the point ^
(2, 3, 4) which is equally inclined to the axes. Ans. a;-2=:?/-3 = 5-4.
IG
TUE STRAIGHT LINE.
23. To find the equations of a straight line through two
given ])oints.
Let the co-ordinates of the two given points AB be
a\, y^, z^ and x^, y<i,z^\ and let the co-ordinates of any point P
on the line ABhQ x, y, z. Then the ratio of the projections
of ^IPand AB on any axis is equal to AP : AB. Hence
the equations of the line are
^2 ^1 2/2 Vx ^2 ^\
2-i. To find the angle between two straight lines whose
direct ion- cosines are given.
Let I, m, n and l\ m, n be the direction-cosines of the
two lines, and let Q be the ande between them.
Let i^Q be any two points on the first line.
Draw planes through P, Q parallel to the co-ordinate
planes, and let PZ, LM, MQ be edges of the parallelopiped
so formed. Then the projection of PQ on the second line is
equal to the sum of the projections of PL, LM, and MQ on
that line.
Z
^/-^
■>
f-
^K^/"
X
Jj
//^
M
y
1y
Hence PQ, cos Q^PL.V -v LM . m -f- MQ . n\
But PL = l.PQ, LM=m.PQ, and MQ = n.PQ\
THE STRAIGHT LINE. 17
therefore cos 6= W -{• mm + nn\
If the lines are at right angles we have
W + mm' + nn' = 0.
If L, M, N are proportional to the direction-cosines of a
line, the actual direction-cosines will be
L M jsr _
Hence the angle between two lines whose direction-cosines
are proportional to Z, M, N and L', M', N' respectively is
_j LU -F MM' + NN'
^^^ V {U + M' + ]S-') V {L" + M" + IS") '
The condition of perpendicularity is as before
LL' + MM' + NN' = 0.
Ex. 1. Shew that the lines - = ^ = - and - = -^ = - are at right angles.
Ex. 2. Shew that the line 4x=Sy= -z is perpendicular to the line i
3x= -y= -^z.
Ex. 3. Find the angle hetween the lines ~ = ^ = - and - = — -.=-=• v
° 1 1 0 o -4 o
Ans. cos~1tV-
Ex.4. Shew that the lines Sx + 2ij + z-5 = 0 = x + y -2z-S, and ,
8x-4:i/-Az = 0 = 7x + lOy - 8z are at right angles.
Ex. 5. Find the acute angle between the lines whose direction-cosines are ^
V? l,V3and^ i, -^. Ans. 600.
4 ' 4 ' 2 4 ' 4 ' 2 •
Ex. 6. Shew that the straight lines whose direction-cosines are given by y
the equations 2l + 2m-n = 0, and mn + nl + lm = 0 are at right angles.
Eliminating I, we have 2mn- (m + n) {2m-n) = 0, or 2m^-mn-n^ = 0.
Hence, if the direction-cosines of the two lines be l^, m^, tij and l^, W2, Wo, we
«w vn 7 7
have-^^=-i. Similarly -^-^= -i. Hence the condition Ij^l^ + m^m^
+ n^n2 = 0i3 satisfied.
Ex. 7. Find the angle between the two lines whose direction-cosines are /
given by the equations l + m + n = 0, P + m'^-rV^ = 0. Am. 60*^.
Ex. 8. Find the equations of the straight lines which bisect the angles
between the hues - = ^ = - , and - = -^ = - .
I m n V m n
Let P, Q be two points, one on each line, such that OP — OQ=r. Then
the co-ordinates of P are Ir, mr, nr, and of Q are I'r, m'r, n'r; hence the co-
ordinates of the middle point of PQ are ^ (Z -t- V) r,\{m-\- m') r, i (?i + n') r. Since
S. S. G. 2
18 THE STRAIGHT LINE.
the middle point is on the bisector, the required equations are
— Similarly the equations of the bisector of the
l-\-l' m + m' n + n'.
supplementary angle are ^4^, = ^, = ^, .
25. By the preceding Article
cos 6 = ir + rum! + nn ;
therefore sin'* 6 = 1 — {IV + mm + nnY
— {W + mm + nny ;
therefore sin ^ = V { (^^' — '^'w)^ + {nV — n'lf + (Im' — Imf].
20. To find the angle between two planes wJiose equations
are given.
The angle between two planes is clearly equal to the
angle between two lines perpendicular to them. Now we
have seen [Art. 14] that the direction-cosines of the normal
to the plane
Ax-hBy+Cz + D = 0,
are proportional to ^, ^, C. Hence by Article 24 the angle
between the planes whose equations are
Aa) + By+ Cz +D = 0,
A'a) + B'y+az + D'=0,
_i AA'-^BB'+GC
IS cos
V {A' + 5^ + C) V {A'^ + F^ + C") '
Ex. 1. Find the equation of the plane containing the line x + y + z = l,
2x + 3y + 4LZ = 5, and perpendicular to the plane x-y + z = 0.
Ans. x-z + 2 = 0.
Ex. 2. At what angle do the planes x + y + z = i, x-2y-z = 4: cut ? Is the
origin in the acute angle or in the obtuse? Is the point (1,-3, 1) in the
acute angle or in the obtuse ? Ans. cos~^^iy2, acute, obtuse.
Ex. 3. Find the equation of the plane through (1, 4, 3) perpendicular
to the line of intersection of the planes 3a; + 4?/ + 72 + 4 = 0, and x-y + 2z + 3 = 0;
also of the plane through (3, 1, - 1) perpendicular to the line of intersection
of the planes Sx + y -z = 0, 5x-3y + 2z=0.
Ans, 15x + y-'7z + 2 = 0. Ans. x+lly + 14js = 0.
cc *u z
Ex. 4. Shew that the line -= — = - is parallel to the plane
Ix + my +nz+p = 0 if IX + m/uL + ny =0, the axes being rectangular or oblique.
THE STRAIGHT LINE.
19
27. To find the perpendicular distance of a given point
from a given straight line.
Let the equations of the line be
X— a _y — ^ _^ — 7
I m n
Z
a
Let ( f, g, h) be the given point P, and let PQ be the per-
pendicular from P on the line.
Let A be the point (a, /5, 7), and draw through A and P
planes parallel to the co-ordinate planes so as to form a
parallelopiped of which AL, LM, MP are edges parallel to
the axes.
Then AQ is the projection of AP on the given line, and
is equal to the sum of the projections of AL, LM, and MP;
therefore AQ = (/— a) l + (g - ^) m + {h—ry)n.
Hence PQ' = AP' -AQ'
28. To find the condition that two lines may intersect.
Let the equations of the lines be
x — a y — P z —7
and
X — OL
y-fi' _z-r{
I 7n n L m n
If the lines intersect they will lie on a plane ; and, since
the plane passes through (a, /3, 7), we may take for its
equation
\{x-(x)+fjL(j/-^) + v(z-j) = Q .(i).
2—2
20
THE STRAIGHT LINE.
The point (a', ^, y) is on the plane, hence we have
X(oi'-a)-hfM(fi'-^) + v(y'-y) = 0 .(ii).
Also, since the normal to the plane is perpendicular to
both lines, we have
\l + fim +vn =0 (iii),
and \V + fim' + vn' = 0 (iv).
Eliminating X, /j,, v from the equations (ii), (iii) and (iv)
we have the required condition, namely
a - a, ^' - /?, 7' - 7
I J m , n
m
n
= 0.
If this condition be satisfied, by eliminating X, /x, v from
(i), (iv), (iii), we find for the equation of the plane through the
straight lines
m
n
n
= 0.
x-a,y-ff,z-y
I,
I',
If the equations of the lines be a^x + h^y ■\-c^z 4- c^^ = 0,
a^x + h^y 4- c^z + d^ = 0, and a^x + h^y + c^z + d^ = 0, a^x + h^
+ c^z + 5^ = 0, the condition of intersection of the lines is the
condition that the four planes may have a common point,
which is found at once by eliminating x, y, z.
29. To find the shortest distance between two straight
lines whose equations are given.
Let A KB and CLD be the given straight lines, and let
KL be a line which is perpendicular to both. Then KL is
the shortest distance between the given lines, for it is the
projection of the line joining any other two points on the
given lines \
Let the equations of the given lines be
x — a_y — h z — c ^^^ x — a_y — h' z — c'
I
and
m
n
V
m
n
^ We can find KL by the following construction : — draw AE through A
parallel to CD ; let AP be perpendicular to the plane EAB, and let the
plane PAB cut CD in L ; then if LK be drawn parallel to PA it will be the
line required.
THE STKAIGHT LINE.
21
Let the equations of the line on which the shortest
distance lies be
x — OL_y — ^_z^'y
Since the line (i) meets the given lines, we have [Art. 28]
= 0
and
= 0
(i).
^rt
.(ii),
(iii).
a — a, /3 — 5, 7 — c
I y m , n
\ , /Jb , V
a — a, B — b\ 7 — c'
7/ r I
L J m , n
X , ^l y V
Since (i) is perpendicular to the given lines, we have
\l -\- jim +vn =0,
and \l' + fjLm + vn = 0 ;
therefore
7nn —run
nl' — n'l Im! — I'm'
Hence, from (ii) and (iii), we see that (a, y3, 7), which is
an arbitrary point on the shortest distance, is on the two
planes
X — a, y — hy z — c 1 = 0,
I y m , n
ran — m'n, nX — rily Im' — I'm
and
X — a'y y — b'i z — c'
I , m , n
mn — m'riy nl' — n'l, Im' — I'm
= 0.
These planes therefore intersect in the line on which the
shortest distance lies.
We can find the length of the shortest distance from the
fact that it is the projection of the line joining the points
(a, b, c) and (a , h' , c). Now the projection of this line on the
line whose direction-cosines are \, /a, v is
(a — a)\-\-(h — h') fi-\-(c- c) v.
22 THE STRAIGHT LINE.
But as above
\ fl V
mn — m'n nl' — n'l Im! — I'm '
therefore each fraction is equal to
1
^{{mn' - mnf + (nl' - nlf + {Im' - Imf] '
Hence the length of the shortest distance is
(g — a') [mn — mn) + (6 — l>){nl' — n'l) + (c — c){lm' — Tm)
^J[{mn — m'nf + {nV — n'Tf + {Im — VmY]
Ex. 1. Find the perpendicular distance of an angular point of a cube
from a diagonal which does not pass through that angular point.
Ans.
«\/i
Ex. 2. How far is the point (4, 1, 1) from the line of intersection of
/27
x + y-\-z=A, x-2y-z=4L'> Ans. a/tt-
Ex. 3. Shew that the two lines a; - 2 = 2?/ - 6 = 3^, 4x - 11 = 4?/ - 13 = 3z
meet in a point, and that the equation of the plane on which they lie ia
2x-6?/ + 3^ + 14 = 0.
Ex. 4. Find the equation of the plane through the point (a', ^', 7'), and
through the line whose equations are — r— = — = .
I m n
■x-a,y-p, z-y
Ans. a'-a, /3'-^, 7'-7 =0.
i , TO , n
Ex. 5. The shortest distances between the diagonal of a rectangular
parallelopiped and the edges which it does not meet are
he ca ah
where a,b,c are the lengths of the edges.
Ex. 6. Find the shortest distance between the straight lines
|(cc-l)=4(y-2)=2-3, aiB.dy-mx=z=0.
5m - 10
Ans.
V(5w2-16m + 17)'
Ex. 7. Determine the length of the shortest distance between the lines
4x=Sy=-z and 3(ic-l)=-?/-2=-4^ + 2. Find the equations of the
straight line of which the shortest distance forms a part. An^. ■^^.
30. If through any number of points, P,Q, R... lines be
drawn either all through a fixed point, or all parallel to a
fixed lii;^; and if these lines cut a fixed plane in the points
PROJECTIONS. 23
P', Q\R'...; then P', Q\ R'... are called the projections of
P, Q, R... on the plane. If the lines PP', QQ', RR'... are
all perpendicular to the fixed plane, the projection is said to
be orthogonal.
The orthogonal projection of a limited straight line on a
plane is the line joining the projections of its extreniities.
It is easily seen that the projection of a line on a plane
is equal to its length multiplied by the cosine of the angle
between the line and the plane.
31. The orthogonal projection of any plane area on
any other plane is found by multiplying tJie area by the
cosine of the angle between the planes.
Divide the given area into a very great number of
rectangles by two sets of lines parallel and perpendicular to
the line of intersection of the given plane and the plane of
projection. Then, those lines which are parallel to the line
of intersection are unaltered by projection, and those which
are perpendicular are diminished in the ratio 1 : cos 6, where
6 is the angle between the planes. Hence every rectangle,
and therefore the sum of any number of rectangles, is
diminished by projection in the ratio of 1 : cos^. But,
when each of the rectangles is made indefinitely small, their
sum is equal to the given area. Hence any area is diminished
by projection in the ratio 1 : cos 6.
32. If we have more than one plane area, we must
make some convention as to the sign of the projection,
and we have the following definition : the algebraic pro-
jection of any face of a polyhedron on a fixed plane is
found by multiplying its area by the cosine of the angle
between the normal to the fixed plane and the normal
to the face, the normals to the faces being all drawn outwards
or all drawn inwards.
33. Let A be the area of any plane surface ; I, m, n the
direction-cosines of the normal to the plane ; A^, A^, A^ the
projections of A on the co-ordinate planes. Then we have
A^ = l.A, A^^m.A, J^ = n.A.
24
VOLUME OF TETRAHEDRON.
Hence, since P + m^ + n^ = 1,
we have A,' + A^ + A,' = A\
Also the projection of A on any other plane, the direction-
cosines of whose normals are l\ m, n, is ^ cos ^ ; and we
have
A cos 6 = (W + mrn + nn) A
= rA^ + mA^-\-nA^.
Hence to find the projection of any plane area, or of the
sum of any plane areas, on any given plane, we may first
find the projections A^, Ay, A^ on the co-ordinate planes,
and then take the sum of the projections of A^y A^, A^ on
the given plane.
84. To find the volume of a tetrahedron in terms of the
co-ordinates of its angular points.
Let the co-ordinates of the angular points of the tetra-
hedron ABCD be {x^, y^, z,), (^^ y,, ^2), (5, 3/3' ^zl ^^^ (^4^ V^^^a)'
The volume of a tetrahedron is one-third the area of the base
multiplied by the height. Now the equation of the face BCD is
2 '
oc„
= 0.
y y Z y
2/2 » ^2 »
The perpendicular p from A on this is found by sub-
stituting the co-ordinates of A and dividing by the square
root of the sum of the squares of the coefficients of x, y,
and z.
Now the coefficients of x, y, z are
2/2' ^2' 1
»
^2' •^2' -'•
>
^2' 2/2' 1
2/3' ^3' 1
^3, -S'3, 1
^3' 2/3' 1
2/4' ^4' 1
^4 > -^45 J-
^4' 2^4' 1
respectively; and these coefficients are respectively equal
to twice the area of the projection of BCD on the planes
a; = 0, 2/ = 0 and 2^ = 0. Hence the square root of the sum
of the squares of the coefficients of x, y and z is, by the
preceding Article, equal to ^llBCD.
TWO STRAIGHT LINES.
25
Therefore 2p.ABCD= x^, y^, z^, 1
^3 ' 2/3 » -^2 ' ■'■
therefore volume of tetrahedron
1
~ 6
^1'
2/1 >
^1.
1
^2'
2/2 »
^2'
1
^3'
2/3'
^n»
1
^4>
^4'
^4'
1
35. The equations of two straight lines can be found in a
very simple form by a proper choice of axes.
Let 0 be the middle point of CO', the shortest distance
between the two straight lines CD, C D\ Through 0 draw
OA, OB parallel to CD, G'D\ and let OX, OF bisect
the angle AOB. Take OX, OY,OG for axes of co-ordinates ;
then, if AOB be 2at, the equations of OA, OB are 1/ = a; tana
z = 0, and y = — x tan a, ^ = 0.
Hence the equations of the parallel lines CD, G'D' are
y = X tan a, z= c\ and y = — x tan a, z = — c.
When it is not of importance that the axes should be
rectangular, we may take OA, OB, 00 for axes: the equa-
tions of CD, CD' will then be 3/ = 0, ^ = c ; and x = 0, z— — c.
Also CO' may be any straight line which intersects CD and
26 OBLIQUE AXES.
36. Four given planes which have a common line of
intersection cut any straight line in a range of constant cross
ratio.
Let any two lines meet the planes in the points
P, Q, R, S and P\ Q\ R, S' respectively. Let 0, 0' be
any two points on the line of intersection of the given planes,
and let the line of intersection of the two planes OFQRS,
O'P'Q'ES' meet the four given planes m F\ Q", R\ B" respec-
tively. Then, from the pencil whose vertex is 0, we have
[P QBS} = {P"Q"R"S"]\ and, from the pencil whose vertex is 0',
we have [F ' Q" R" S"]=[F Q'R'S']. Hence [P QRS\ = [P'Q'R'S'},
which proves the proposition.
37. Def. Two systems of planes, each of which has
a common line of intersection, are said to be homographic
when every four constituents of the one, and the correspond-
ing four constituents of the other, have equal cross ratios.
An equivalent definition [see Conies, Art. 323] is the
following: — two systems of planes, each of which has a
common line of intersection, are said to be homographic
which are so connected that to each plane of the one system
^.orresponds one plane, and only one, of the other.
Oblique Axes.
38. Some of the preceding investigations apply equally
whether the axes are rectangular or oblique. These may be
easily recognised. We proceed to consider some cases in
which the formulae for oblique and rectangular axes are
different.
39. Let P, Q be two points on a straight line, and
through P, Q draw planes parallel to the co-ordinate planes
so as to form a parallelepiped, and let PL, LK, KQ be
edges parallel to the axes. Then the ratios of PL, LK, KQ
to PQ are called the direction-ratios of the line PQ. It is
clear that the direction of a line is determined by its
direction-ratios.
OBLIQUE AXES. 27
40. To find the angles a line makes luith the axes of
co-ordinates, in terms of its direction-ratios.
Let \, fjb, V be the angles YOZ, ZOX, XOY respectively.
Let I, m, n be the direction-ratios of the line PQ, and let
a, yS, 7 be the angles it makes with the axes. Let PZ, XiT,
KQ be parallel to the axes so that PL = I. PQ, LK = m . PQ,
KQ = n.PQ, as in Art. 39. Then, since the projection of
P^ on the axis of x is equal to the projection of PLKQ,
we have
PQ cos a = PL + LK cos v 4- KQ cos /x ;
therefore cos a =l + m cos v + n cos fjb.
Similarly cosyQ = ?cos^' + m + ncosX,
and cos <y = 1 cos /x + w cos X + n.
41. To find the relation heticeen the direction-ratios of a
line.
Project PL, LK, KQ on PQ, then we have
PL cos a + Z/i cos /3 + iTQ cos 7 = P$ ;
therefore from Art. 40,
l{l-\- m cos V -\-n cos yu,) + m {I cos p-\-m + n cos \)
-f n (Z cos /jb-^-m cos X + n) = 1
or Z** + m'' + n^ + 2mn cos X -f 2?iZ cos /x + 2Z??i cos v=l. . .(i),
which is the required relation.
28 OBLIQUE AXES.
Let the co-ordinates of tlie points P, Q be
^1' I/v ^1 ^^^ ^2' 2/2' ^2-
Then l.PQ^PL = x^-x^, m. PQ = LK=^y^-y^,
and n. P Q= KQ = z^ — z^.
Hence from (i) we have
PQ' = [x, - xf + (y, - y,y + (^, - ^,)^ + 2 (y, - yj (^,-r,)cos \
+ 2 (2^2 - ^J (^2 - ^1) cos /A + 2 {x^ ~ a;J (2/^ - ?/J cos 1/ (ii),
• which gives the distance between two points in terms of their
oblique co-ordinates.
42. To find the angle between two lines whose direction-
ratios are given.
Let I, m, n and Z', m\ n be the direction-ratios of the
lines PQ and PQ' , and let 6 be the angle between them.
Let PL, LKy KQ be parallel to the axes, so that
PL = l.PQ, LK=m.PQ, s^nd KQ = n.PQ.
Project PQ and PLKQ on the line P'Q'; then
PQ cos6 = 1 PQ . cosa +mPQ .cos ^' + nPQ. cos 7',
where a', /8', 7 are the angles the line P'Q' makes with the
axes. Hence, from Art. 40, we have
cos 0=1 (l' + m' cos V -\- n' cos ^i)
4- m {V cos v-\-m +n cos X)
-}- n (If cos jj, -{-m cos X -f ?i')
= Zr + mm + nn + (mn + m'n) cos \+ {nl -\- n'T) cos ^
+ (Zm' -f Im) cos i/.
43. To find the volume of a tetrahedron in terms of three
edges which meet in a point and of the angles they make with
one another.
Take the axes along the three edges, and let a, h, c
be the lengths of the edges, and X, /x, v the angles they make
with one another. Then
Volume = J abc sin v cos 6,
OBLIQUE AXES.
29
where 6 is the angle between OZ and the normal to the
plane XOr.
Let the direction-ratios of the normal to the plane XOY
be I, m, n. Then from Art. 40 we have
I -\-m cos V +n cos /^ = 0,
I cos v + m + n cos X = 0,
I cos fjb-{- m cos \-\-n = cos 0.
Multiply by I, m, n and add, then, from (i) Art 41,
71 cos ^ = 1.
The elimination of I, m, n from the above equations gives
1 , cos V , cos //- , 0 = 0 ;
cos V , 1 , cos \ , 0
cos fi , cos \ , 1 , cos 6
0, 0, cos^, 1
therefore sin'^ v cos'' 6 = 1 , cos i/ , cos fi
cos V , 1 , cos X
cos /i, , cos X , 1
= 1 — cos' X — COS^ yu, — cos'' 1^ + 2 cos X cos /A COS V.
Hence the volume required
= ^ ahc V (1 ~ cos'^ X — cos'^ /a — cos'^ j/ + 2 cos X cos /x cos i/).
TRANSFOR^LA.TION OF CO-ORDINATES.
44. To change the origin of co-ordinates without changing
the direction of the axes.
Let f,g, k be the co-ordinates of the new origin referred
to the original axes. Let P be any point whose co-ordinates
referred to the original axes are x, y, z, and referred to the
new axes x , y\ z. Let PL be parallel to the a5:is of x and
let it meet 70Z in X, and TOZ in L',
30
TRANSFORMATION OF CO-ORDINATES.
Then
therefore
Similarly
and
x — x= LL' =f.
y-y'=9>
z —z=h.
Hence, if in the equation of any surface we write x +/,
y+g, z + h for X, y, z respectively, we obtain the equation
referred to the point (/, g, h) as origin.
45. To change the direction of the axes without changing
the origin, both systems being rectangular.
Let Zj, m^, n^; l^, m^, n^; and l^ m^, n^ be the direction-
cosines of the new axes referred to the old.
JC'
Let P be any point whose co-ordinates in the two systems
are x, y, z and x , y' , z .
Draw PL perpendicular to the plane X! OY' and LM per-
pendicular to OX' \ then OM = x, ML = y', and LP=z',
Since the projection of OP on OX is equal to the sum of
the projections of OM, ML and XP, we have
x=^l^x -\-l^y' + l^z\
Similarly y = m^x +m^y + m, z,
and z^n^x + n^y +n^z^.
TRANSFORMATION OF CO-ORDINATES. 31
These are the formulae required.
Since l^, m^, n^^ ; 7^, m^, n^; and ^3, wig, % are direction-cosines,
we have
// + m; + < = l
C + < + < = !
Also, since 0X\ OY', OZ' are two and two at right
angles, we have
and Z^Zg + mjin^ -{■ nji^ = 0
The six relations between the nine direction-cosines which
we have found above are equivalent to the following :
V + h' + 1: = 1.
< + ^2' + "3' = 1.
»"i^i + '"2^ + ^3^3 = 0, )
Z,??i, -f l^m^ + ^37^3 = 0,
This follows at once from the fact that l^, l^, l^;
in^, m^, m^\ and n^, n^, n^ are the direction-cosines of
OX, OY, OZ referred to the rectangular axes 0X\ OY', OZ',
46. Since
\
IJ^ + m^m^^ -f- n^n^ = 0,
and
^3 + 771,^3 + 71,713 = 0,
we have
l^ TWj n^
m^n^-m^n^ n}^-n}^ l^m^-l^m.'
Hence each fraction is equal to
/f/.__ ..
\2 . /„ 7 .. 7 \-i , n ... 7 ... \1^ — X J- _^iu ^o,\
32
TRANSFORMATION OF CO-ORDINATES.
K^
m^y
'^h
I.
^2'
^2
h^
^3.
^3
Also
= ^i(w.,^3 - '^3^2) + ^, K^3 - ^sO + ^(^2^3 - ^3^0
47. If in Art. 45 the new axes are oblique we still have
the relations
x= 7/ + I J + I/,
y = m^x + W22/' + mg/,
We can deduce the values of x\ y, z in terms of x, y, z'.
the results are
X'
h>
K^
h
=
A,
^3,
X
m^,
*^2'
'"h
^2,
W3,
y
'h^
^2>
^3
^^
^3'
z
.p
c
and two similar equations.
48. The degree of an equation is unaltered hy any trans-
formation of axes.
From the preceding Articles we see that, however the
axes may be changed, the new equation is obtained by sub-
stituting for X, y, z expressions of the form lx + my-\-nz+p.
These expressions are of the first degree, and therefore if
they replace x, y, and z in the equation, the degree of the
equation will not be raised. Neither can the degree of the
equation be lowered; for, if it were, by returning to the
original axes, and therefore to the original equation, the
degree would be raised.
49. We shall conclude this chapter by the solution of
some examples.
(1) A line of constant length has its extremities on two fixed straight lines;
shew that the locus of its middle point is an ellipse.
If we take the axes of co-ordinates as in Art. 35, the equations of the lines
vrill be y = mx, z = C} and y=-7nx, z=-c. Let the co-ordinates of the
EXAMPLES. 33
extremities of the line in any one of its possible positions be x^, t/j, z^ and
^2> 2/25 ^2 ; ^^^ Ist {x, y, z) be the co-ordinates of the middle point of the line.
Then, if 21 be the length of the line, we have
But, since y^=mx-^ and Zy=Cf and y<^— -mx^, Z2= -c, we have
yi-y2=m K + x^) = 2mx,
Zj^-z.2 = 2c, and 2z = z-^ + Z2 — 0.
Hence the locus of the middle point is the ellipse v;hose equations are
2 = 0, l^ = K, + m'x^ + cK
(2) A line moves so as always to intersect three given straight lines,
which are not all parallel to the same plane; find the equation of the
surface generated by the straight line.
Draw through each of the lines planes parallel to the other two ; a
parallelepiped is thus formed of which the given lines are edges. Take the
centre of the parallelepiped for origin, and axes parallel to the edges, then
the equations of the given lines are y = hi z= -c; z = c, x^ -a; and x = a,
y= -b respectively.
Let the equations of the moving line be
x-a __ y -§ _z -y
I m n '
Since this meets each of the given lines we have
b - ^ _ -c-y c-y _ —a-a ^ a- a _ -b- ^
— • , — — , and — ; — = .
m n n I I m
Hence, by multipljnng corresponding members of the three equations, we
see that (a, /3, 7), an arbitrary point on the moving line, is on the surface
whose equation is
(a-x){b-ij) {c-z) + {a + x){b + y) {c + z) = 0,
yz zx xy ^ .
be ca ab
(3) The lines of intersection of corresponding planes of two homographic
systems describe a surface of the second degree.
We may take y—mx, z = c, and y= -mx, z= -c for the equations of the
lines of intersection of the two systems of planes [see Art. 35.]
Let the equations of corresponding planes of the two systems be
y-mx + \(z-c)=0,
and y + mx + \'{z + c)=0.
Since the systems are homographic there is one value of V for every value of
X, and one value of X for every value of X'; hence X, X' must be connected by
a relation of the form
X\' + A\-i-BX + C=0.
S. S. G. 3
34 EXAMPLES ON CHAPTER 11.
Substitute for \ and \', and we have
y^-vi^x^-A (z + c) {y-vix)-B {z-c) {y + mx) + C {z^~c-) = 0.
Hence the line of intersection of corresponding planes describes a surface of
the second degree.
ExAMPLijs ON Chapter II.
1. If P be a fixed poirt on a straight line througli the origin
equally inclined to the three axes of co-ordinates, any plane
through P will intercept lengths on the co-ordinate axes the sum of
whose reciprocals is constant.
2. Shew that the six planes, each passing through one edge
of a tetrahedron and bisecting the opposite edge, meet in a point,
3. Through the middle point of every edge of a tetrahedron
a plane is drawn perpendicular to the opposite edge ; shew that
the six planes so drawn will meet in a point.
4. The equation of the plane through - = -^ = - and which
I 'til ;J¥t
CC 1/ ^ 00 7J i^
is perpendicular to the plane containing? — = - = - and - = ^ = — ,
^ ^ 0)1 Ql C 71 C 111
is X (m — n) + y{n — l)-\-z{l- m) = 0.
5. Shew that the straight lines
X y z X y z
a (3 y' aa bjS cy^
X
7"
11
will lie in one plane, if
- (6 - c) + 7^ (c - a) + - (a - 5) = 0.
a p y
6. Two systems of rectangular axes have the same origin; if
a plane cut them at distances a, b, c, and a, b', c from the origin,
then
1111 11
— t- — -I — = 1- ■ — I .
a 0 c a 0 c
/
EXAMPLES ON CHAPTER II. 35
7. Determine tlie locus of a point which moves so as always \/
to be equally distant from two given straight lines.
8. Through two straight lines given in space two planes are
drawn at ri^ht ansjles to one another ; find the locus of their line
of intersection.
9. A line of constant length has its extremities on two given
straight lines ; find the equation of the surface generated by it,
and shew that any point in the line describes an ellipse.
10. Shew that the two straight lines represented by the
equations ax -\-hy -t cz = 0, yz -{■ zx + xy = 0 will be perpendicular if v
- + T + -==0.
a 0 c
11. Find the plane on which the area of the projection of the
hexagon, formed by six edges of a cube which do not meet a given
diagonal, is a maximum.
12. Prove that the four planes .
r
my + nz = 0, nz + fe = 0, lx + my = 0, lx-{- my + 7iz =p,
form a tetrahedron whose volume is -^ — .
oimn
13. Find the surface generated by a straight line which is
parallel to a fixed plane and meets two given straight lines.
'14. A straight line meets two given straight lines and makes
the same angle with both of them; find the surface which it
generates.
15. Any two finite straight lines are di\Hded in the same
ratio by a straight line ; find the equation of the surface which it
generates.
16. A straight line always parallel to the plane of yz passes
through the curves x^ + y^ = a^, z^O, and cc^ = a;s, 2/ = 0 ; prove
that the equation of the surface generated is
xy = {x'-azY{a^-x%
17. Three straight lines mutually at right angles meet in a
point P, and two of them intersect the axes of x and y respec-
tively, while the third passes through a fixed point (0, 0, c) on the
axis of z. Shew that the equation of the locus of P is
36 EXAMPLES ON CHAPTER II.
1 8. Find the surface generated by a straight line which meets
y = mx, z = c; y = — mx, z = -c ; and y^ + z^z=i c^, x — 0.
19. P, P are points on two fixed non-intersecting straight
lines AB, A'B' such that the rectangle AP, A'P' is constant. Find
the surface generated by the line PP'.
20. Find the condition that
ax~ + hy^ + cz" + 2a yz + Ih'zx + 2c xy — 0
may represent a pair of planes ; and supposing it satisfied, if B be
the angle between the planes, prove that
tan 6 =
2ja'~ + 5"* + c'" — be — ca — ab
a + b + c
21. Find the volume of the tetrahedron formed by planes
whose equations are y + z = 0, z + x = 0, x + y-0, and x + y + z=l.
22. Find the volume of a tetrahedron, having given the
equations of its plane faces.
23. Shew that the sum of the projections of the faces of a
closed polyhedron on any plane is zero.
24. Find the co-ordinates of the centre of the sphere in-
scribed in the tetrahedron formed by the planes whose equations
are x = 0, y = 0, z=0 and x + y + z = I.
25. Find the co-ordinates of the centre of the sphere in-
scribed in the tetrahedron formed by the planes whose equations
are y + z=0, z + x-0, x + y = 0, and x + y + z = a.
CHAPTER III.
Surfaces of the Second Degree.
50. The most general equation of the second degree, viz.
-^ao^ 4- hif + C2;'' + ^fijz + 2^^.r + "ihxy ■{-2ux-^2vy + 2w2 + d = 0,
contains ten constants. But, since we may multiply or divide
the equation by any constant quantity without altering the
relation between x, y, and z which it indicates, there are
really only nine constants which are fixed for any particular
surface, viz. the nine ratios of the ten constants a, h, c, &c. to
one another. A surface of the second degree can therefore
be made to satisfy nine conditions and no more. The nine
conditions which a surface of the second degree can satisfy
must be such that each gives rise to one relation among the
constants, as, for instance, the condition of passing through a
given point. Such conditions as give two or more relations
between the constants must be reckoned as two or more of
the nine.
We shall throughout the present chapter assume that the
equation of the second degree is of the above form, unless it
is otherwise expressed. The left-hand side of the equation
will be sometimes denoted by F{x, y, z).
-^ 51. To find the points where a given straight line cuts
the surface represented by the general equation of the second
degree.
33 THE TANGENT PLANE.
Let the equations of the straight line be
I m n '
To find the points common to this line and the surface,
we have the equation
a (a + IrY + h{^-r mrY + c (7 + nry + 2/ (/S + mr) (7 + nr)
+ 2^ (7 + nr)(a + Ir) + 2/t (a + lr){^ + wr) + 2i^ (a + Zr)
+ 2z; (/3 + mr) + 2w; (7 +wr) + cZ = 0,
or
?'^ (aZ^+ 6m^+ cn^-T 2fmn + 2/7?z?4- 27<Z??i) -\-r\l-^-\-m ;7o +^ 1~
+ i^(«,/3,7) = 0 (i).
Since this is a quadratic equation, any straight line meets
the surface in two points.
Hence all straight lines which lie in any particular plane
meet the surface in two points. So that, all plane sections of
a surface of the second degree are conies.
In what follows surfaces of the second degree will
generally be called conicoids.
— ^ 52. To find the equation of the tangent plane at any
point of a conicoid.
If (a, (3, 7) be a point on F{x, y, -2') = 0, one root of
the equation found in the preceding Article will be zero.
Two roots will be zero if I, m, n satisfy the relation
,dF dF dF ^
^^+^5^+^57=^ w.
The line — -, — = - — — = ~ will in that case be a
I 7)1 n
tangent line to the surface, the point of contact being (a, yS, 7).
If we eliminate I, m, n between the equations of the line,
and the equation (i), we see that all the tangent lines lie in
the plane whose equation is
, . dF , _. dF , . dF _ ....
(.-a)^ + (y-^)^ + (.-T,)^ = 0....(u).
THE POLAR PLANE. 8^
This plane is called the tangent plane at the point (a, jB, 7).
If we write the equation (ii) in full, we obtain
X {ai +'hp + gy-\-n)+y {hoL 4- b/3 + fy + v) + z {gx +fl3-\-cy+w)
. = aa' + h/S^ + C7^ + y/3y -f 2^7X + 2/ia/3 + wa + z;/3 + wj.
Add itoc + v/3 + ivy + c? to both sides, then the right side
becomes F(o^, (3, y), which is zero; we therefore have for the
equation of the tangent plane at (a, ^, y)
X (aa. + h^+gy + u) + y (hx + h/3+/y+v)+z(g2 +fl3 + cy + lu)
+ ua + vl3 + wy-hd = 0. . .(iii).
Ex. 1. Fine! the equation of the tangent plane at the point {x', y', z') on
the surface ax^ + by'^ + cz'^ + d = 0. Ans. ax'x + h])'y + cz'z + d = 0.
Ex. 2. Find the equation of the tangent plane at the point {x', y', z') on
the surface ax'^-\-by^ + 2z = 0. Ans. ax'x + hy'y + z + z' = 0.
^ 53. The condition that the tangent plane at (a, /S, 7)
may pass through a particular point (cc', y\ z) is
X (ax + h/3 + gy+u)+y' {hx+h/3+fy+v)+z' (goL+f^+cy+w)
+ ux -t v/3 + wy + d = 0.
This condition is equivalent to
a{ax +hy +g2' +u) -\-/3 {hx' -\-by' +fi' +v) +y(gx +fy +c/+w)
+ ux + vy + wz + cZ = 0.
From the last equation we see that all the points, the
tangent planes at which pass through the particular point
{x ^ y\ /), lie on a plane, namely on the plane whose equation
is
X (ax + hy' + gz ■\-u)+y {hx + hy -Vfz + v)
+ z {gx ■\-fy + cz + ^^) + ux + vy* + wz' + cZ = 0.
This plane is called the 'polar plane of the point {x , y\ z).
The polar plane of any point P cuts the surface in a conic,
and the line joining P to any point on this conic is a tangent
line. The assemblage of such lines forms a cone, which is
called the tangent cone from P to the conicoid.
The equation of the polar plane of the origin, found by
putting x =y' = z =0 in the above, is
ux-\-vy ■\-wz-\-d = 0.
40 ^ THE POLAR PLANE.
~^=:5r 54. i The condition that the polar plane of {x\ y , z) may
pass through (a, yS, 7) is as above
a {ax + liy +gz+u)+fi {hx + ^'i/' +fz + ?;)
+ 7 {9^' -^fy' + ^•^^ + '^^) + ^^' + ^i/' + '^^•2' + d = 0.
This equation is unaltered if we interchange a and x\
^ and ?/', and 7 and s' ; it therefore follows that if the polar
plane of any point P with respect to a conicoid pass through
a point Q, then will the polar plane of Q pass through P.
— ^ 55. Let R be any point on the line of intersection of the
polar planes of P, Q.
Then, since R is on the polar plane of P and also on the
polar plane of Q, the polar plane of R will pass through P
and through Q, and therefore through the line PQ. Similarly
the polar plane of >S^, any other point on the line of inter-
section, will pass through the line PQ.
Two lines which are such that the polar plane with
respect to a conicoid of any point on the one passes through
the other, are called polar lines, or conjugate lines.
56. If any chord of a conicoid he draivn through a point
0 it ivill be cut harmonically by the surface and the polar
plane of 0.
Take the point 0 for origin, and let the surface be given
by the general equation of the second degree.
Let the equations of any line, which cuts the surface in
P, Q and the polar plane of 0 in P, be
ic _ 2/ _z _
I VI n
To find the points where the line cuts the surface we have,
as in Art. 51, the quadratic equation
r^ {ar + bni^ + cn^ + 2/??m + 2gnl + 2hlm)
+ 2r (ul + vm + wn) -\- d=^0.
Hence tttt + ttts =—-.(ul + vm^ wn).
OP OQ d^
The equation of the polar plane of 0 is
ux + vy + wz + d = 0.
CONDITION OF TANGENCY.
4fl
Hence
11/7 N
TTr = ~ ^ (^^ + ^^^ + ^^) »
therefore
which proves the proposition
J. 1^
OR' '
57. To find the condition that a given plane may touch
a conicoid.
Let the equation of the given plane be
Ix + my + nz +p = 0 (i).
The tangent plane at (x, y, z) is
X (ax + %' + g/ + u) -{- y (hx + by +fz + v)
+ z {gx +fy + cz + w) + ux' + vy + wz + cZ = 0 (ii).
If the planes represented by (i) and (ii) are the same we
have
ax 4- hy + gz -\-u _ hx + hy +fz' -\- v _ gx -\-fy + cz + to
I m n
_ ux + vy' + wz' + cl
Put each fraction equal to — X ; then we have
ax + hy +gz -\-u-\-X I =0,
hx + hy' ■\-fz + V + X, m = 0,
gx ■\-fy' -^-cz ■\-w-\-\ n = 0,
ux + vy -\-wz'+ d + \p = 0.
Also, since [x\ y\ z) is on the given plane,
Ix + my •{• nz^ -h p = 0.
Eliminating x, y , z\ X, we obtain the required condition,
namely
a, h , g , u, Z = 0.
h,
. h,
f> V,
m
9>
f>
c , w,
n
u,
V,
w, d,
P
I,
m ,
n, p,
0
42
TANGENT PLANE.
The determinant when expanded is
AP + Bm' + Ca' + D/ + 2 Fmn + 2 Giil + 2 i7^??i
+ 2 ZJ^p + 2V'mp + 2Wnp = 0,
Avhere A, B, C, &c. are the co-factors of a, h, c, &c. in the
determinant
a, h, g, u
g, f, c, w
u, V , lu, d
We will give special investigations in the two following
cases which are of great importance :
I. Let the equation of the surface be
ax" + hf + cz' +d = 0.
The tangent plane at any point {x\ y\ z') is
ax'x + hi/'y + cz'z + cZ = 0.
Hence, comparing this equation with the given equation
Ix + my -\-nz+p = 0,
ax hy' cz d
we have "-'^ = — = — = - . Each fraction is equal to
I on n p
^l{ax^^r]nr±cz^±d) ^
yc
abed
hence, since ax'^ + by^ + cz'^ + c? = 0,
the required condition of tangency is
abed
II. Let the equation of the surface be
a^' 4- bf + 2z = 0.
The tangent plane at any point (x, y\ z') is
axx + by'y + s + / = 0.
Hence, comparing this equation with the given equation
Ix + my + 7Z^ + 2> = 0,
CENTRE OF A PLANE SECTION. 43
we have -7- = -^ = - = — , Each fraction is equal to
L m n p ^
1(1' m' ^ \'
hence, since ax^ + hy"^ + 2/ = 0,
the required condition of tangency is
72 2
- + -7- + 2n» = 0.
a 0 ^
58. If we find, as in Article 51, the quadratic equation
giving the segments of a chord through {a, /S, y) the roots of
the equation will be equal and opposite, if
,dF dF dF ^
In this case (a, /3, 7) will be the middle point of the chord.
Hence an infinite number of chords of tke conicoid have the
point (a, /3, 7) for their middle point.
If we eliminate I, m, n between the equations of the
chord and (i), we see that all such chords are in the plane
whose equation is
(—«)5^ + (3/-^)^^+ (--7)^ = 0 (n).
Hence (a, ^, 7) is the centre of the conic in which (ii) meets
the surface.
This result should be compared with that obtained in
Art. 52.
Ex, 1. The locus of the centre of all plane sections of a conicoid which
pass through a fixed point is a conicoid.
The equation of the locus is if-x)~ + {g-'y)— + {h-z)-T-=0, where
f, g, h are the co-ordinates of the fixed point.
Ex. 2. The locus of the centre of parallel sections of a conicoid is a
straight line.
44 DIAMETRAL PLANES.
The section whose centre is (a, /3, 7) is parallel to the given plane
lx + m]/ + 7iz=0 if
dF dF dF
da dji dy
I ~ m ~ n '
Hence the locus is the straight line whose equations are
I dx m dy n dz '
The straight lines clearly all pass through the point of intersection of the
dF dF dF ^
planes -3- = 3- = -3-= 0.
dx dy dz
-^ 50. To find the locus of the middle points of a system of
parallel chords of a conicoid.
As in the preceding Article, (1, /9, 7) will be the middle
point of the chord whose direction-cosines are Z, m, n, if
,dF dF dF ^
l-j~+ m Ta + '^^ zr = ^'
di dp dy
Hence the locus of the middle points of all chords whose
direction-cosines are I, m, n is the plane whose equation is
,dF dF dF ^
I -^ — I- m -, \- n —r- = ^«
dx dy dz
Bef. The locus of the middle points of a system of parallel
chords of a conicoid is called the diametral plane.
If the plane be perpendicular to the chords it bisects, it is
called a principal plane.
^^^ 60. To find the equations of the principal planes of a
conicoid.
The diametral plane of the chords whose direction-cosines
are I, m, n is
,dF dF dF ^
dx dy dz
or, writing the equation in full,
I (ax + hy +gz + u) + m (Jix + hy +fz + v)
-f n [gx -Vfy -\- cz ■\- w) — ^ ,
or X {al 4- hm -f- gii) -\- y {hi -}- hm +fn) + z (gl +fm -\- en)
4- id + vm 4- wn = 0.
PRINCIPAL PLANES.
45
If this plane be perpendicular to the chords it bisects,
we have
al + hm+gn _kl+ hm +fii _ gl +fin-\-cn
I in n
Put \ for the common value of these fractions, then
{a — \)l + hm 4- gn — 0,
+ (6 - X) m +fn = 0,
hi
gl -hfm
Eliminating I, m, n we have
a — \, h,
-\-{c-X)n = 0..
= 0,
.(i).
9
f
c — X
or
h, h — X,
9^ f>
\' - (a-^-h + c)\' -h (bc + ca + ah -f - g' ~h')\
- {abc + 2fgh - af - bg' - ch') = 0.
This is a cubic equation for determining X ; and when X is
determined, any two'of the three equations (i) will give the
corresponding values of I, m, n.
Since one root of a cubic is always real, it follows that
there is always one principal plane.
Find the principal planes of the following surfaces :
(i) x^ + y^-z'^ + 2yz + 2zx-2xy = a^.
(ii) 11x2 + 10^2 + 6^2 _ 8yz + 4zx - 12xy = 1.
A71S. (i) x + y + z = 0, x-y = 0, x + y-2z = 0.
Ans. (ii) x + 2y + 2z = 0,2x + y-2z = 0, 2x-2y + z = 0.
61. All parallel plane sections of a conicoid are similar
and similarly situated cmiics.
Change the axes of co-ordinates in such a way that the
plane of xy may be one of the system of parallel planes ; and
let the equation of the surface be the general equation of the
second degree.
Let the equation of any one of the planes be z — k. At
all points of the section of the surface F{Xj y, z) =0, by the
46 PARALLEL SECTIONS ARE SIMILAR.
plane z = h both these relations are satisfied ; we therefore
have
ax" + hif + ch" + y^jlc + 2rjkx + "Ihxy + 2ux + 2vij
■^2Lv7c + d=0 (i).
Now the equation (i) represents a cylinder whose gene-
rating lines are parallel to the axis of z, and which is cut by
the plane ^ = 0 in the curve represented by (i).
Since parallel sections of a cylinder are similar and simi-
larly situated curves, the section of the surface F (x, y, 2) = 0
hy z = k is similar to the conic represented by (i) and 2! = 0;
and all such conies, for different values of k, are clearly
similar and similarly situated : this proves the proposition.
Classification of Conicoids.
62. We proceed to find the nature of the different
surfaces whose equations are of the second degree ; and we will
first shew that we can always change the directions of the
axes of co-ordinates in such a way that the coefficients of yz,
zx, and xy in the transformed equation are all zero.
63. We have seen [Art. 60] that there is at least one
diametral plane which is perpendicular to the chords it
bisects.
Take this plane for the plane ^ = 0 in a new system of co-
ordinates.
The degree of the equation of the surface will not be altered
by the transformation ; hence the equation will be of the form
ax^ + ly^ -f cz^ + 2fyz -\- 2gzx + 2]ixy + 2ux + 2vy + 2wz -\-d = 0.
By supposition the plane z = 0 bisects all chords parallel
to the axis of z ; therefore if {x, y\ z) be any point on the
surface, the point {x, y\ — z) will also be on the surface.
From this we see at once that /= g =w = 0.
2k
Now turn the axes throuojh an ang^le J tan ^ r » then
o o -i a — b
[See Conies, Art. 167] the term involving xy will disappear.
CLASSIFICATION OF CONICOIDS. 47
Hence we have reduced the equation to a form in which the
terms yz, zx, and xy are all absent.
64. When the terms yz, zx, xy are all absent from the
equation of a conicoid, it follows from Art. 60 that the co-ordi-
nate planes are all parallel to principal planes. Hence by
the preceding article, there are always three principal
planes, which are two and two at right angles. This shews
that all the roots of the cubic equation found in Art. 60 are real.
For an algebraical proof of this important theorem see
Todhunter's Theory of Equations.
Q5, We have seen that the general equation of the second
degree can in all cases be reduced to the form
Ax^-\-By''-\-Cz^^2Ux-v2Vy^2Wz + D=0,
I. Let A, Bj (7 be all finite.
We can then write the equation
Hence, by a change of origin, we have
Ax' + By' + Gz'' = D\
If D' be not zero we have
x'^ t/^ ^
A B C
which we can write in the form
.2 2 2
-2 + '-k + % = ^ («X
a 0 c
x" -^ -'
or
:-+!-? =1 (^)'
*> 2 2
X- f Z ^ ( \
4S CLASSIFICATION OF CONICOIDS.
B' B' B'
according as -^ , -yr , -rf- are all positive, two positive and
one negative, or one positive and two negative. [If all three
are negative the surface is clearly imaginary.]
If B' be zero, we have
Ax^-vBxf-\-Gz^=^ (S).
II. Let (7, any one of the three coefficients A^B^ (7, be
zero.
Write the equation in the form
then, if W be not zero, the equation can, by a change of origin,
be reduced to
^ic' + %' + 2TF^ = 0 (e).
If W be zero, we have the form
Ax^'^Bf-^B = ^ (?),
or, if B' be zero, the form
Ax^^Bf^'^ (7;).
III. Let B, G, two of the three coefficients, be zero.
"We then have
A(x + ^]+2Vy+2Wj2+B-~- = 0.
Now take 2F2/+2TF>+D — 7- = ^ ^^^ ^^^® plane y=0, and
the equation reduces to the form
w' = 2ky (d).
If however V= Tr=0, the equation is equivalent to
a)' = k' (t).
66. We now proceed to consider the nature of the
surfaces whose equations are (a), (/5), (t) ; to one of which
forms we have seen that the general equation is reducible.
THE ELIJPSOID. 40
The surface whose equation is
2 2 2
is called an ellipsoid.
Let a, h, c be in descending order of magnitude; then
{w, y, z) being any point on the surface, we have
2 2 _2
a^ a^ a^
and -^+^ + -H:l.
c c c
So that no point on the surface is at a distance from the
origin greater than a, or less than c. The surface is therefore
limited in every direction ; and, since all plane sections of a
conicoid are conies, it follows that all plane sections of an
ellipsoid are ellipses.
The surface is clearly symmetrical about each of the co-
ordinate planes.
If r be the leno-th of a semi-diameter whose direction-
cosines are I, m, n, we have the relation
t' " a^ "^ h' '^ c''
If two of the coefficients are equal, h and c suppose,
the section by the plane x = ^, and therefore [Art. 61]
by any plane parallel to a? = 0, is a circle. Hence the
surface is that formed by the revolution of the ellipse
2 2
-^ + T9 = 1 about the axis of x.
The surface formed by the revolution of an ellipse about
its major axis is called a prolate spheroid ; that formed by
the revolution about the minor axis is called an oblate
spheroid.
li a = b = c the equation of the surface h x^ + y^ + 2^ = a^,
which from Art. 5 represents a sphere.
s. s. G. 4
50 THE HYPERBOLOID OF ONE SHEET.
67. The surface whose equation is
a^ "^ b' e '
is called an Jnjperholoid of one sheet.
The interce^Dts on the axes of oc and y are real, and those
on the axis of z are imaginary.
The surface is clearly symmetrical about each of the co-
ordinate planes.
The sections by the planes x = 0 and y = 0 are hyperbolas,
and that by 2; = 0 is an ellipse.
The section by ^ = A; is also an ellipse, the projection of
X
r
k'
and the section becomes
which on z = 0 is — , + f^ = 1 + 9 »
greater and greater as k becomes greater and greater.
^
A
^^
V
N>---^.^
/
7
/
\/
1 ^^
'A
L
0\
'A
\
V
V^ /
/
^
A
If a — h, the section of the surface by any plane parallel
to 5" = 0 is a circle. Hence the surface is that formed by the
THE HYPERBOLOID OF TWO SHEETS.
51
revolution of the hyperbola -^ — ^ = 1 about its conjugate
Oi c
axis.
The figure shews the nature of the surface.
68. The surface whose equation is
of y^ ^^ — ■]
is called an hyperholoid of two sheets.
The intercepts on the axis of x are real, those on the other
two axes are imacjinarv.
The sections by the planes 2/ = 0 and ^ = 0 are hyper-
bolas.
The section by the plane a^ = 0 is imaginary. The parallel
plane ijc = h does not meet the surface in real points unless
h^ > a^ If F > d^ the section is an ellipse the axes of which
become greater and greater as k becomes greater and greater.
The surface therefore consists of two detached portions as in
the figure.
If & = c, the section by any plane parallel to a? = 0 is
a circle. Hence the surface is that formed by the revolution
x^ if'
of the hyperbola -^ — "^ = 1 about its transverse axis.
\
69. The surface whose equation is Ax^ + B\f + Cz^ = 0 is
a cone.
4—2
52 THE CONE.
A cone is a surface generated by straight lines which
always pass through a fixed point, and which obe}^ some other
law. The lines are called generating lines, and the fixed
point through which they pass is called the vertex of the
cone.
If the vertex of a cone be taken as origin, the equation
of the surface is homogeneous. This follows at once from the
consideration that if (x, y, z) be any point P on the surface,
any other point {kx, ky^ kz) on the line OP is also on the
surface.
Conversely any homogeneous equation represents a cone
whose vertex is the origin of co-ordinates. For, if the values
X, y, z, satisfy a homogeneous equation, so also will kx, ky,
kz, whatever the value of h may be. Hence the line through
the origin and any point on the surface lies wholly on the
surface.
The general equation of a cone of the second degree, or
quadric cone, referred to its vertex as origin is therefore
ax^ + %^ + cz^ + 2fyz + 2gzx + 'Ihxy = 0.
70. If r be the length of the semi-diameter of the
surface ax^ + hy'^ + cz"" = 1, we have the relation
r
Hence the direction-cosines of the lines which meet the
surface at an infinite distance satisfy the relation
ar + hm^ + cn^ = 0.
Such lines are therefore generating lines of the cone
ax^ + hy"^ -f cz"^ = 0.
This cone is called the asymptotic cone of the surface.
71. The equation Ax^ + By^ + 2 Wz = 0 is equivalent to
x^ iP' x^ y^
-j-\-y = 2z, or y — y = 2^, accordlng as the signs of A and B
are alike or different.
THE PARABOLOID.
53
The surface whose equation is
2 2
is called an elliptic paraboloid.
The sections by the planes x = () and y = 0 are parabolas
having a common axis, and whose concavities are in the same
direction.
The section by any plane parallel to z = 0 is an ellipse if
the plane be on the positive side of ^ = 0, and is imaginary if
the plane be on the negative side of z = {). Hence the
surface is entirely on the positive side of the plane ^ = 0, and
extends to an infinite distance.
The surface whose equation is
^ ^_9.
is called an hyperholic paraboloid.
The sections by the planes x = 0 and 2/ = 0 are parabolas
which have a common axis, and whose concavities are in
opposite directions.
The surface is on both sides of the plane s = 0, and
extends to an infinite distance in both directions.
5-i THE PARABOLOID.
The section by the plane 2: = 0 is the two straight lines
of y^
given by the equation y — '^ = 0. The section by any plane
parallel to z = 0 is an hyperbola: on one side of the plane
z = 0 the real axis of the hyperbola is parallel to the axis of
.r, and on the other side the real axis is parallel to the
axis of?/.
The fissure shews the nature of the surface.
7'2. It is important to notice that the elliptic paraboloid
is a limiting form of the ellipsoid, or of the hyperboloid of
two sheets ; and that the hj^perbolic paraboloid is a limiting
form of the hyperboloid of one sheet.
This can be shewn in the following manner.
The equation of the ellipsoid referred to (— a, 0, 0) as
/v.'^ nj^ ^ 2^
origin is -3 + '?^ + -^ = 0. Now suppose that a, h, c all
O/ 0 C Oj
become infinite, while — , - remain finite and equal respec-
a a
y^ z^
tively to I and I' ; then, in the limit, we have ; + 7/ = 2^,
which is the equation of an elliptic paraboloid.
The other cases can be proved in a similar manner.
73. The equation Ao^ + B\j^-{-D = ^ represents a cylinder
[Art. 10], being a hyperbolic cylinder if A and B have dif-
ferent signs, and an elliptic cylinder if A and B have the same
sign. If the signs oi A, B, D are all the same the surface is
imaginary.
The equation Ax^ + Bjf' = 0 represents two intersecting
planes, which are imaginary or real according as the signs of
A and B are alike or different.
The equation x^ = 2hy represents a cylinder whose guiding
curve is a parabola, and which is called a parabolic cylinder.
The equation x^ = lc represents the two parallel planes
x= ± s/lc.
EXAMPLES. 55
Ex. 1. The sum of the squares of the reciprocals of any three diameters
of an ellipsoid which are mutually at right angles is constant.
If r^ be the semi-diameter whose direction-cosines are (Z-^, %, tIj) we
1 Z 111 " 71
have — 5 = -^ + yo- + -7- , and similarly for the other diameters. By addition
1 1 1 111
we have -^ -1- — + — , = -^ + 7^ + - .
Tj^ r^' rg- a^ 0^ c-
Ex. 2. If three fixed points of a straight line are on given planes which
are at right angles to one another, shew that any other point in the line
describes an ellipsoid.
Let A, B, C be the points which are on the co-ordinate planes, and
P {x, y, z) be any other fixed point whose distances from A, B, C are a, h, c.
Then - = L^ = m, and -=n, where I, vi, n are the direction cosines of the
ah c
line. Hence the equation of the locus is -^ -f f- -t- -s=l.
a^ b^ c^
Ex. 3. Find the equation of the cone whose vertex is at the centre of an
ellipsoid and which passes through all the points of intersection of the
ellipsoid and a given plane.
3p lyS ^2
Let the equations of the ellipsoid and of the plane be -^+^2+ ~2~^^ ^^^
lx + my+nz = l. We have only to make the equation of the ellipsoid
homogeneous by means of the equation of the plane : the result is
o '* U
% + j7, + -^={lx + m7j-\-nzy-.
a^ 0- c^
For this equation being homogeneous represents a cone whose vertex is
at the origin ; and it is clear that the plane cuts the cone and the ellipsoid in
the same points.
Ex. 4. Find the general equation of a cone of the second degree referred
to three of its generators as axes of co-ordinates.
The general equation of a quadric cone whose centre is at the origin is
ax"^ + hy^ -t- cz^ + 2fyz + 2gzx + 2hxy = 0,
If the axis of a: be a generating line, then y = 0, z = 0 must satisfy the
equation for all values of x ; this gives a = 0. Similarly, if the axes of y andz
be generating lines, 6 = 0 and c = 0. Hence the most general form of the
equation of a quadric cone referred to three generators as axes is
fyz + gzx + hxy = 0.
Ex. 5. Find the equation of the cone whose vertex is at the centre of a
given ellipsoid, and which goes through all points common to the ellipsoid
and a concentric sphere.
yjj2 ^2 ^2
If the equations of the ellipsoid and sphere be —^ + j-2+~2~^f ^^^
x^ + y^-\- z^=r' respectively ; the equation of the cone will be
56 THE CENTRE.
Ex. 6. Find the equation of the cone whose vertex is the point (a, /3, y)
and whose generating lines pass through the conic — + ^ = 1, 2=0.
Let any generator be —z—=- — -= — -, This meets z=0 where
I m n
or
x = a-l y, and 2/ = ^-^ y. Hence i (a-^7)Vj^ (^-^-^7=1'
--^{an-yl)~ + j-^{^n-ymy=n'^. Substitute for I, m, n from the equations of
1 1
the line, and we have ~T,{az-yx)- + Y^{^z-yy)' — {z-y)^, the required
equation.
74. If the origin be the centre of the surface, it is the
middle point of all chords passing through it; hence if
(x^, 2/j, z^) be any point on the surface, the point (— iCj, — y^, — z^
will also be on the surface.
Hence we have
ax^ + hy^ + cz^ + 2fy^z^ + 2gz^x^ + 2hx^y^ + 2ux^ + 2vy^
+ 2ivz^ + d = 0,
and ax^ + hy^^ + cz^ + 2/1/^^^ + 2^^^a7j + ^hx^y^ — 2ux^ — 2vy^
-2wz^ + d = 0;
therefore ux^ + vy^ + wz^ = 0.
Since this equation holds for all points (aj^, y^, z^) on the
surface, we must have u, v, w all zero.
Hence, when the origin is the centre of a conicoid, the
coefficients of x, y and z are all zero.
75. To find the co-ordinates of the centre of a conicoid.
Let (f , 7], f ) be the centre of the surface ; then if we take
(f , 7], f) for origin, the coefficients of x, y, and z in the trans-
formed equation will all be zero. The transformed equation
will be [Art. 44]
a{x + ^f + h{y + r^Y + c{z+t;Y-^2f{y + 7f){z + ^
+ 2w{z + ^) + d = 0.
THE CENTRE.
57
Hence the equations giving the centre are
a^ + hy + g^ + u = 0,
H + ^V+f^-^ v = 0,
and g^ +fy + c^-\-w = 0,
Therefore
I -V _
■&■
h,
g^
u
a, g.
u
a.
h,
u
h,
f>
V
h, f,
V
h.
b.
V
/.
c,
lU
gy c ,
w
9'
/.
w
-1
a, h,
h, h,
9
f
9> f>
c
The equation of the conicoid when referred to the centre
(?> ^, ?) as origin is
aic' + %' + cz" + 2fijz + 2gzx + 2hxy + d' =- 0 (ii),
where d' = F {^, rj, ^).
Multiply equations (i) in order by f , rj, f and subtract the
sum from F (f , rj, f ) ; then we have
d' = u^-{-V7j + w^+d (iii).
= 0
From (i)
and (ii.
l) we have
a, h, g, u
=
h, h, f, V
g, f, c , w
u, V , IV , d — d'
therefore d'
a, h, g
= ci, h, g,
h, 6, /
h, h, /,
9> f> c
9^ f> c ,
U , V , w
}
u
V
d
\i\).
58 THE CENTRE.
The determinant on the right side of (iv) is called the
discriminant of the function F {x, y, z), and is denoted by the
symbol A.
The determinant on the left side is the discriminant of
the terms in F{x, y, z) which are of the second degree; it is
also the minor of d in the determinant A, and, as in Art. 57,
we shall denote it by D. Equation (iv) may therefore be
written
-f d'D = ^ (v).
76. The equations for finding the centre can also be
obtained from Art. 58 (i); for (^, 77, ^) will be the middle
point of every chord which passes through {^, 7], ^), pro-
vided
dF^cIF_dF_^
d^ dr] d^
It should be noticed that the co-ordinates of the centre
are given by the equations
U V~W~D'
where U, V, W, D have the same meanings as in Art. 57.
77. If, by a change of rectangular axes through the same
origin, ax^ + hy"^ + cz^ + %fyz + 2gzx + 2]ixy
becomes changed into
aV + 6y + cz^ + 2f'yz + 2gzx + 2h'xy ;
then, since x^ •\-y^ -\- z^ is unaltered by the change of axes,
ax"" + hf + c^' + Ifyz + Igzx + "Ihxy - \ {x^ + 7/- + z''). . .(i)
will be chansfed into
o
aV + h'y' + cV + 2/> + 2g'zx + 2Jixy
-X(x'+y' + z') (ii).
The expressions (i) and (ii) will therefore be the product
of linear factors for the same values of X.
INVAKIANTS. 59
The condition that (i) is the product of linear factors is
a — \, h , g 1 = 0,
h , h-\ f
g > f y c-\
that is
X''-X'(a + h + c) + \ {be + ca + ab -f - g-- J^)
-(abc + 2fgh-af'-bg'-cJf) = 0.
The condition that (ii) is the product of linear factors is
similarly
\' - X' (a' + b' + c) + \ (b'c' + c'a' + aV -f - g"" - h'')
- {ab'c + 2fg'h' - a'f - b'g" - cV) = 0.
Since the roots of the above cubic equations in \ are the
same, the coefficients must be equal.
Hence the following expressions are unaltered by any
change of rectangular axes through the same origin, and
are therefore called invariants :
a-\-b + c I,
bc+ca + ab-f'-g^-h'' II,
abc + 2fgh-af'-bg'-ch'' III.
Since the coefficients of the terms of the second degree
are unaltered by a change of origin, the axes being parallel
to their original directions, it follows that the expressions
I, II, and III are unaltered by any change of rectangular
axes.
78. We have seen [Art. 63] that by a proper choice of
rectangular axes ax^ + by^ + c^ -^ 2fyz + 2gzx + 2}ixy can al-
w^ays be reduced to the form ao^ -f ^y^ + 7^^ ; and this re-
duction can be effected without changing the origin, for the
terms of the second degree are not altered by transforming to
any parallel axes.
Now a?'' + 2/^ + ^ is unaltered by a change of rectangular
axes through the same origin. Hence, when the axes are so
changed that
GO THE DISCRIMINATING CUBIC.
ax^ + hy^ + cz^ + 2fyz + 2gzx + ^hxy becomes ax^ + ^y^ + 72:',
ax"" + 6?/' + cz^ + 2/7/^ + 2j^^ + 2]ixy - \ (a;' + 3/' + s'O . . .(i),
will become
ax^ + Py'' + r^z''-\{x^ + y'' + z') (ii).
Both 'these expressions will therefore be the product of
linear factors for the same values of \. The condition that
(i) is the product of linear factors is
= 0.
.(iii).
a — \, h , g
h , h-\, f
9 , f > c-X
But (ii) is the product of linear factors when \ is equal to
a, 13, or 7.
Hence the coefficients a, /3, 7 are the three roots of the
equation (iii).
The equation when expanded is
X' - V (a + 6 + c) + X (a6 + Z^c + ca -f - g^ - It")
- (abc + 2fgh - af - hg' - ch') = 0.
This equation is called the discriminating cubic.
It should be noticed that the equation is the same as that
found in Art. 60.
79. We proceed to shew how to find the nature of a
conicoid whose equation is given.
First write down the equations for finding the centre of
the conicoid ; and from Art. 75 we see that there is a definite
centre at a finite distance, unless the determinant
a, h, g
h h, f
0^ / c
~B
is zero.
CONICOIDS WITH GIVEN EQUATIONS. 61
If D bo not zero, change to parallel axes through the
centre, and the equation becomes
ax^ + hy^ + cz^ + ^fyz + 2gzx + Ihxy + cZ' = 0,
where d' is found as in Art. 75.
Now, keeping the origin fixed, change the axes in such a
manner that the equation is reduced to the form
Then, by Art. 78, a, /3, 7 will be the three roots of the dis-
criminatino; cubic.
[When the discriminating cubic cannot be solved, since its
roots are all real [Art. 64], the number of positive and of
negative roots can be found by Descartes' Rule of Signs.]
Since Dd' = A, the last equation may be written in the
form Bxx' + D/Sf + Dyz' + A = 0.
If the three quantities ^\ ^. ^ are all negative,
the surface is an ellipsoid ; if two of them are negative, the
surface is an hyperholoid of one sheet ; if one is negative, the
surface is an hyperholoid of two sheets ; and if they are all
positive, the surface is an imaginary ellipsoid.
If A = 0, the surface is a co7ie.
Ex. (i). llx''' + 10i/ + (jz^-8ijz + 4:Zx-Uxy + 72x-72y + S6z + 150 = 0.
dF dF dF
The equations for finding the centre are -— = --— = —- = 0, or
ax dy dz
11a;- 6?/ + 22; + 36 = 0,
- Ga; + 10?/-42-36 = 0,
2x- 4?/ + 6z + 18 = 0.
Therefore the centre is ( - 2, 2, - 1).
The equation referred to parallel axes through the centre will therefore be
lla;2 + 10i/2 + 6^2 _ 8^2 + ^zx -12xy-12 = 0. [Art. 75 (iii) .]
The Discriminating Cubic is \-^ - 27\2 + 180\ - 324 = 0 ; the roots of which
are 8, 6, 18. Hence the equation represents the ellipsoid Bx^ + 6y^ + lSz^ = 12
x^ y^ z^ ,
■^+1 + ^=1.
We can find the equations of the axes by using the formulae found in
Art. 60. The direction-cosines of the axes are |, t> l*» l> h "1 5
■1
G2 CONICOIDS WITH GIVEN EQUATIONS.
Ex. (ii). x' + 2y- + 3z--4.xz-Axy + d = 0.
The Discriminating Cubic is X^- 6\2 + 3\-f 14 = 0. All the roots of the
cubic are real ; hence, by Descartes' Eule of Signs, there are two positive
roots and one negative root. The surface is therefore an hyperboloid of
one sheet, an hyperboloid of two sheets, or a cone, according as d is
negative, positive, or zero.
80. Next suppose that D = 0. Then the three planes
[Art. 75 (i)] on which the centre lies will not intersect in a
jDoint at a finite distance from the origin, and we shall have
three cases to consider according as the planes meet in a
point at infinity, or have a common line of intersection, or
are all parallel to one another. These three cases we shall
consider in the following Articles.
It should be observed that when J) = 0 one root of the
discriminating cubic is zero.
81. The conditions that the planes whose equations are
aw + hy + gz + u = 0,
hx + by + fi + V = 0,
and 9^+fy+ cz + w=0,
may be parallel are
aha T h h f
It' h f 9 f c
These conditions may be written
af=gh, hg=-hf, ch=fg (i).
Xow these are the conditions that the terms of the second
degree should be a perfect square ; and when this is the case
it is obvious on inspection.
When the terms of the second degree are a perfect
square, the general equation can be written in the form
fgli[%,-\-'^+j\ +2ux + 2vg + '2w2 + d=0 (ii).
If the plane ux + vy + wz = 0 is parallel to the plane
X y z ^
7^ + ^ + 7 = 0,
f 9 ^^
CONICOIDS WITH GIVEN EQUATIONS. 63
the equation (ii) will represent two 'parallel ])lanes : the con-
ditions for this are
uf= vg =wh (iii).
If the conditions (iii) are not satisfied, the equation (ii) is of
the form Aif +Bx = 0,
which represents a parabolic cylinder whose generating lines
are parallel to y = 0, ^ = 0.
Hence the general equation of the second degree repre-
sents a parabolic cylinder whose generating lines are parallel
to the line
X 11 z
f g h
provided the conditions (i) are satisfied, and that (iii) are not
satisfied.
The latus-rectum of the principal parabolic section can be
found by the same method as that employed in Conies,
Art. 172.
Ex. Find the nature of the conicoid whose equation is
4^2 + 2/- + 4^^ - ^ijz + Qzx - ixy + 2x-4:y + 5z + l = 0.
The equation is
{2x-y + 2z)^ + 2x-4y + 5z+l = 0.
This is equivalent to
{2x-y + 2z + \)^=x {i\ ~2) -y {2\- i) +z {4\- 5) -1.
The planes 2.r-?/ + 22 + X = 0, and x{A\-2)-7j {2\-4)+z{4:\-5)-l = 0,
will be perpendicular, if X = l. Hence the equation of the surface maybe
written {2x-y + 2z + lY = 2x + 2y -z 1,
2x-y + 2z + lY 1 2x + 2y-z~l
or
/2x -y + 2z + lY _
3 • a
Hence, taking 2x-y + 2z+l = 0, and 2x + 2y - z-l = 0 as the planes y = 0
and x = 0 respectively, the equation of the surface will be
2 1
V =— X.
y 3**"
Hence the latus-rectum of 9, principal parabolic section is := .
o
C4
CONICOIDS WITH GIVEN EQUATIONS.
82. Next suppose that the three planes on which the
centre lies are not all parallel, but that they have a common
line of intersection.
If we take any point on the line of centres for origin, the
equation will take the form
ax"" + hf + C2^ + 2/7/^ + 2g2x + 2hxy + cZ' = 0.
Then, keeping the origin fixed, by transformation of axes
the equation will be reduced to the form
ax^-\-Py''-\-d' = 0 (i).
One root of the discriminating cubic is zero, since D = 0 ;
and the roots a, ft 0 are given by the equation
If df = 0, the surface represented by the equation (i) is
Uuo planes, real or imaginary.
If df be not zero, the surface is a cylinder.
The conditions that the three planes
ax + hi/ -\- gz + u = 0, ^
hx + hy +fz + v =0,
gx -\- fy -\- cz -\-w = 0,
may have a common line of intersection, are given by
a, h, g, u
= 0, [Art. is;
h, b, f, V
g, f, c, w
that is,
lf=V=W = D = 0.
Ex. Find the nature of the conicoid whose equation is
S2x^ + y^ + 4:z^ - \^zx - 8xy + 96x - 20?/ - 8^ + 103 = 0.
The equations giving the centre are
B2x - Ay -Sz + 4.8 = 0,
- 4x+ y -10 = 0,
and - 8x +42-4 = 0.
Hence there is a line of centres. Find one point on the line, for example
(0, 10, 1), and change the origin to the point (0, 10, 1) : the equation will
then become 32x2 + 2/^ + 422-16^:0; -8x^ = 1.
CONICOIDS WITH GIVEN EQUATIONS. Go
The Discriminating Cubic is X3-37\^ + 84\ = 0. One root is zero, and the
other two roots are positive ; hence the equation is an elliptic cylinder.
The axis of the cylinder is the line of centres ; and its equations are
x_y-10_z—l
S3. If the planes on ^Yhicll the centre lies meet at a j^oint
at infinity, we proceed as follows.
Since one root of the discriminatinsf cubic is zero, the
equation can always be solved : let the roots be a, ^, 0.
Find the directions of the principal axes of the surface,
by means of the equations of Art. 60; and take axes parallel
to these principal axes. The equation will then become
oix' + 13?/ + 2ux + 2i'ij + 2w'z -\-d = 0,
or, by a change of origin,
ax^+l3/ + 2w'z = 0.
Hence the surface is a paraboloid, the latera recta of its
2w' 2w'
principal parabolic sections being — - and -7^— .
Ex. Find the nature of the surface whose equation is
322_6y^-6^x-7x-5?/ + G2 + 3 = 0.
The Discriminating Cubic is X^- 3\-- 18\ = 0; the roots of which are G,
-3,0.
11-2
The direction-cosines of the prmcij^al axes are -,- , — , —. _- ;
~7q ' ~7q ' "7q ' ^^"^ ~7o ' ~7o ' ^* Hence to find the equation referred to
axes parallel to the principal axes, we must substitute
X y z X y z —2x y
;76 + ;73"^^' ;7B'^73~;/2 ^s^"^^'
for X, y, z respectively. The equation will then become
6x2 - 3 ?/2 - V6a; - 2^3?/ - ^2^ + 3 = 0 ;
or, by changing the origin 6x- — Zy- - fJ2z = 0.
Thus the surface is a hyperbolic paraboloid, the latera recta of the principal
parabolas being -^^2 and 1->J2.
S. S. G. 5
00 CONDITION FOR A CONE.
84. It follows from Art. 75 (ii) and (iv) that when D is
not zero, the necessary and sufficient condition that the
surface represented by the general equation of the second
degree may be a cone is A = 0.
When A = 0 and also D = 0, then will U, V and W be
all zero*: hence [Arts. 81 and 82] the surface must be either
a cylinder or two planes ; and cylinders and planes are
limiting forms of cones. Conversely, when the surface re-
presents a cylinder, or two planes, U, V, W and D are all
zero, and therefore also A = 0.
Hence A = 0 is the necessary and sufficient condition
that the surface represented by the general equation of the
second degree may be a cone.
85. To find the conditions that the surface represented hy
the general equation of the second degree may he a surface of
revolution.
We require the condition that two of the roots of the dis-
criminating cubic may be equal. In that case
ax^ + 6/+ c/ + 2fyz + 2gzx + 2hxy
can be transformed into
ax^ + ay"^ + 7^^
Hence
ax^+hf + cz'+^fyz-^^gzx + 2hxy - \ (^^2/' + ^')...(i),
* This can be proved as follows :
"We have uTJ + vV + loW + dD = ^.,
And, since a determinant vanishes when two of its rows are identical, we
have also
aV+hV+gW+uI) = (i,
hU+bV+fW+vD = 0,
and gU+fV + cW+icD = 0.
Hence when A = 0 and D = 0, unless 77, F, W are all zero, we can eliminate
U, V, W from the first equation and any two of the others : we thus
obtain three determinants which are all zero ; but these determinants are
U, V, and W.
SURFACE OF REVOLUTION. C7
can be transformed into
a£'-\-af^-r^z^-\{x^-Vf-\-z') (ii).
Now if we take X = a, (ii) will be a perfect square.
Hence if the surface is a surface of revolution, we can, by
a proper choice of X, make (i) a perfect square ; and that
square must be
{x si (a -\) + ysl(})-\)^-z ^{c - X)}1
We therefore have
V(c -X) ^{a-X) = h[ (iii).
Hence, iff, g, h he all finite, we have
a_?^ = 6-¥=c-f = X (iv),
the required conditions.
Let h, any one of the three quantities f, g, h, be zero ;
then from (iii) we see that X = a or X = 6, and therefore also
9 = 0 or/=0.
Suppose ^ = 0 and h = 0 ; then X = a, and the condition
for a surface of revolution is
(b-a){c-a) =f (v).
Examples on Chapter HI.
1. Determine the nature of the surfaces represented by the
following equations :
(i) X- - 2y^ + 6;2' + 1 2xz + a' = 0. *^^ • <^'^- ^^'^ "
y 7 i / (^0 x^ + y^ + z^ + ixy - 2xz + iyz = 1. »/
(iii) x^ — 2xy — lyz — 2zx = ci?.
(iv) 32x' + 2/' + 4;s^ - 1 ^zx -8xy = l.
(v) Jx+Jy+Jz = 0. ^'^'""
(vi) 2x' + 5y' + z' - 4:xy - 2x- iy - S = 0.
0—2
C8 EXAMPLES ON CHAPTER III.
2. Find the nature of the surfaces represented by the following
equations :
(i) a;' + 2?/^ — 3z^ - 4:yz + Szx — 1 2xt/ + 1=0.
(ii) 2x' + 2i/- 4.Z- - 2yz - 2zx -bxy -2x-2ij ^ z=0.
(iii) 5ic' - ?/" + is" + ^xz + 4:xy + 2x + 4?/ + 6;:; = 8,
(iv) 2x" + 3y^ + Zyz + 2zx + ^xy — 4^/ + 8;2 - 32 = 0.
Find the equations of the axes of (i), and the latera recta of
the principal parabolas of (ii) and of (iii).
3. Shew that the equation
Qi? -V y^ -^ '^ ■\- yz -\- zx ^- xy = \^
represents an ellipsoid the squares of whose semi-axes are 2, 2, \.
Shew also that the equation of its principal axis is x — y-=z.
4. Shew that, if the axes, supposed rectangular, be turned
round the origin in any manner, ir -{-v" ^-vr will be unaltered.
5. Shew that, if three chords of a conicoid have the same
middle point, they all lie in a plane, or intersect in the centre of
the conicoid.
6. Through any point 0 lines are drawn in fixed directions
which meet a given conicoid in points P, F' and Q^ Q' respectively ;
shew that the rectangles OP, OP and OQ, OQ' are in a constant
ratio.
7. If any three rectangular axes through a fixed point 0 cut
a given conicoid in P, P' ; Q, Q' and R, R' ; then will
PP' QQ" RR"
OP'.OP' OQ\OQ'^ OR\OR"'
1 1 1 1
OP.OF^^ OQ. OQ' ^ OR . OR'*
be constant.
CHAPTER IV.
CoNicoiDS Referred to their Axes.
86. In the present chapter we shall investigate some
properties of conicoids, obtained by taking the equations
of the surfaces in the simplest forms to which they can be
reduced.
We shall begin by considering the Sphere.
The Sphere.
87. The equation of the sphere whose centre is {a, h, c)
and radius d is [Art. 5]
The equation of any sphere is therefore of the form
a;2 + 2/2 _^ ^2 _^ 2Ax + 2Bij+2Cz +i)= 0.
Conversely every equation of the above form, that is every
equation in which the coefficients of ^^ y\ and z"^ are equal, and
in which the terms yZj zx, xy do not appear, represents a
sphere.
88. The general equation of a sphere contains four
constants, and therefore a sphere can be made to satisfy /owr
conditions. "We may, for example, find the equation of a
sphere which passes through any four points.
70
THE SPHERE.
If (^j, y^, z^, ^, ?/,, ^,), (x,, 2/3, ^3), (;r,, 7/,, z^ be the four
points the equation of the sphere through them will be.
= 0.
x^ + iy + z\
a; ,
y ^
z ,
1
w,^+y:+z:-.
^i>
^1'
^1'
1
^:+y.'+\\
^2>
2/2'
^2'
1
< + 2// + ^/.
^3,
2/3'
^3>
1
'> , 2 1 2
^4'
2/4.
^4'
1
89. The equation of the tangent plane at any point
■■ {x , y, z) of the sphere whose equation is x^ + ^/^ + ^^ = a^ is
XX -\- yy + zz = a^ [Art. 52, Ex. 1]. This result can be
obtained at once from the fact that the tangent plane at any
point (x\ y, z) on a sphere is perpendicular to the line
joining {x, y\ z) to the centre. This gives for the equation
of the plane
{x - x). x'+{y- y') y' + {z - z) z = 0,
or XX + yy^ + zz = a^.
The polar plane of any point {x, y, /) can be shewn, by
the method of Art. 53, to be
XX + yy + zz = a^.
90. It can be easily shewn, that if >S= 0 be the equation
of a sphere (where S is written for shortness instead of
a? -\- y"^ -{- z^ -\- 2 Ax + 2By + 2Gz + D), and the co-ordinates of
any point be substituted in 8, the result will be equal to the
square of the tangent from that point to the sphere.
Hence, if /S^= 0, and >S' = 0 be the equations of two spheres
(in each of ^vhich the coefficient of x^ is unity), S — S' is the
locus of points, the tangents from which to the two spheres
are equal.
The surface whose equation is >S' — >S" = 0 passes through all
points common to the two spheres >Sf = 0, and >S' = 0 ; for, if
the co-ordinates of any point satisfy the equations 8 = 0 and
8' = 0, they will also satisfy the equation 8 — 8' = 0.
Now 8—8'=0is of the first degree, and therefore represents
a plane. The plane through the points of intersection of two
spheres is called their radical plane.
THE ELLIPSOID. 71
"We have seen that the tangents drawn to two spheres
from any point on their radical plane are equal.
The radical planes of four given spheres meet in a point,
viz. in the point givfen by S^ = S^ = S^ = S^, where /S\ = 0,
S^ = 0, S^ = 0, S^= 0 are the equations of the four spheres,
in each of which the coefBcient of x^ is unity.
This point is called the radical centre of the four spheres,
Ex. 1. Find the equation of the sphere "which has {x^, i/i, z-^) and
{x^, 1/2, z^) for extremities of a diameter.
If {x, y, z) be any point on the sphere, the direction- cosines of the h'nes
joining {x, y, z) to the two given points are proportional to x-x^, y - y■^^,
z — Zj, and x -x^, y-y-2i ^~ ^i-
The condition of perpendicularity of these lines gives the required
equation
{x - x-^ {x - x.^ + (2/ - yi) (y - y.^ + (2 - z^ {z - z^^ = 0.
Ex. 2. The locus of a point, the sum of the squares of whose distances "
from any number of given points is constant, is a sphere.
Ex. 3. A point moves so that the sum of the squares of its distances
from the six faces of a cube is constant; shew that its locus is a sphere.
Ex. 4. A, B are two fixed points, and P moves so that PA=nPB ; shew
that the locus of P is a sphere. Shew also that all such spheres, for different ^
values of n, have a common radical plane.
Ex. 5. The distances of two points from the centre of a sphere are pro-
portional to the distance of each from the polar of the other.
Ex. 6. Shew that the spheres whose equations are
x^ + y^ + z'^ + 2Ax + 2By + 2Cz + D = 0, ^
and <c^ + y^-{-z'' + 2ax + 2by + 2cz~{-d=0,
cut one another at right angles, if
2Aa + 2Bb + 20c - D - d = 0.
91. We proceed to prove some properties of the ellipsoid;
and we shall always suppose the equation of the surface to be
222
\- — A — = 1
unless it is otherwise expressed.
To obtain the properties of the hyperboloids we shall
only have to make the necessary changes in the signs of
h^ and c^.
72 DIRECTOR-SPHERE.
We have already seen [Art. 52] that the equation of the
tangent plane at any point {x, y', z) is
xx' mf , zz .
The length of the perpendicular from the origin on the
tangent plane at the point {x , y' , z) is [Art. 20] given by the
equation
-I '2 '2 '2
1 X y -s ,.'\
-.= "4 + 71 + — (lO-
2^^ a 0 c
Equation (i) is equivalent to Ix -f my + nz = p, where
I _x m _y n _z'
p^d' ' p~P' p~&''
, , aH^ + }fm^ A- c'n' x"" y\z"' ,
therefore o = -^ + 7T + ~^ = ^*
pr ci- b c
Hence the plane whose equation is Ix + my i-nz=p, will
touch the ellipsoid, if
p''=aT + hW + c'n' (iii).
92. To find the locus of the point of intersection of three
tangent planes to an ellipsoid which are mutually at right
angles.
Let the equations of the planes be
l^x + m^y + n^z = J {aH^^ + h^m^ + c'n^),
l,^ X + 77Z, y-{-n,^z = J (aX + Z>X' + ^^W)^
l^x-\- 7/?3 y^Qi^z = ^ (a\^ + h^m^ + &n^).
By squaring both sides of these equations and adding, we
have in virtue of the relations between the direction-cosines
of perpendicular lines
x' + y''-]-z'' = a'' + ¥-\-c\
The required locus is therefore a sphere. This sphere is
called the director-sphere of the ellipsoid.
93. The normal to a surface at any point P is the
straight line through P perpendicular to the tangent plane
at P.
NORMALS. 73
The normal to an ellipsoid at the point {x\ y\ z) is
therefore
x — x'_ y —y _ z — z
X y z
Since /g+^VQ = l, [Art. 91.]
the direction-cosines of the normal are
px py pz
~^' 'W' 'J'
94. If the normal at {x\ y\ z) pass through the par-
ticular point (/, g, h) we have
X y z
^ 'W '^
Put each fraction equal to \ then
. aV , h'g . , cVi
Hence, since
we have
a' "^ 6^ "^ c' '
Since this equation for \ is of the sixth degree, it follows
that there are six points the normals at which pass through a
given point.
Ex. 1. The normal at any point P of an ellipsoid meets a principal ^
plane in G. Shew that the locus of the middle point of PG is an ellipsoid.
Ex. 2. The normal at any point P of an ellipsoid meets the principal ^
planes in G^ G^, G3. Shew that PG^, PG^, PG^ are in a constant ratio.
Ex. 3. The normals to an ellipsoid at the points P, P' meet a principal X
plane in G, G' ; shew that the plane which bisects PP' at right angles bisects ^
GG'.
74- DIAMETRAL PLANES.
Ex. 4. If P, Q be any two points on an ellipsoid, the plane throuph
the centre and the line of intersection of the tangent planes at P, Q, will
bisect FQ.
Ex. 5. P, Q are any two points on an ellipsoid, and planes through the
centre parallel to the tangent planes at P, Q cut the chord PQ in P', Q'. Shew
that PP' = (?(?'.
95. The line whose equations are
x — a__y — ^_z — 7_
I m n- '
meets the surface where
(a + hf {l±mrf {y + nrY _
a' "^" b' "^ 6' '
If (a, p, y) be the middle point of the chord, the two
values of ?• given by the above equation must be equal and
opposite; therefore the coefficient of r is zero, so that we
have
Za 772/3 ny
a 0 c
Hence the middle points of all chords of the ellipsoid
v/hich are parallel to the line
X _ y _z
I m n
are on the plane whose equation is
Ix mv nz
This plane is called the diametral plane of the line
I 111 n '
The diametral plane of lines parallel to the diameter
through the point {x , y\ z) on the surface is
OCX . W' ZZ' /^ /-x
hence the diametral plane of any diameter is parallel to the
taDgent plane at the extremities of that diameter.
CONJUGATE DIAMETERS. 75
The condition that the point {x\ y" , z") should be
on the diametral plane (i) is
r ft I It lit
xx_ y_y_ + ^^-Q
The symmetry of this result shews that if a point Q be on
the diametral plane of OP, then will P be on the diametral
plane of OQ.
^ Let 0^' De, the line of intersection of the diametral
planes of OP, VQ ; then, since the diametral planes of OP,
OQ pass through OR, the diametral plane of OR will pass
through P and through Q, and will therefore be the plane
POQ, so that the plane through any two of the three lines
OP, OQ, OR is diametral to the third.
Three planes are said to be conjugate when each is dia-
metral to the line of intersection of the other two, and three
diameters are said to be conjugate when the plane of any two
is diametral to the third.
96. If {x^, y^, z^, {cc.-,, 7/^, z^) and (x^, y^, z^ be extremities
of conjugate diameters, we have from Art. 95,
^2- + ^2 + ^2 - ^ j
Also, since the points are on the surface,
a' ^ h' ^ & \
a' ^ ¥^ c^ ^ ^
a" "^ h' '^ c^~
7G
CONJUGATE DIAMETERS.
Now from equations (ii) we see that
^1 2/1 ^1 . ^2 3/" ^'^ ' ^
a
h' c' a' h' c' "'^^ a'~h'i'
are direction-cosines of three straight lines, and from equations
(i) we see that the straight lines are two and two at right
angles. Hence, as in Art. 45, we have
ind
2/x +2/2 +2/;
.(iii),
^1^1+ ^22/2 +^.y3=0]
2/1^1+2/2^2+^3^3=01 (iv).
^1^1 + ^2^2 + ^3^3 = 0'
"VVe have also from Art. 46.
= 1, or X
5
a
a
h
b
2/2
b
c
£2
c
1»
X.
2'
CPo
2/1 »
2/2'
2/3 »
= abc
(V).
From (iii) we see that the sum of the squares of the pro-
jections of three conjugate semi-diameters of an ellipsoid on
any one of its axes is constant.
Also, by addition, we have, tJie sum of the squares of three
conjugate diameters of an ellipsoid is constant.
From (v) we see that the volume of the parallelopiped
which has three conjugate semi-diameters of an ellipsoid for
conterminous edges is constant.
In the above the relations (iii) and (iv) were deduced
from (i) and (ii) by geometrical considerations. They
could however be deduced by the ordinary processes of algebra
without any consideration of the geometrical meaning of the
quantities, and hence the results are true for the hyper-
boloids.
CONJUGATE DIAMETERS. V7
97. The two propositions (1) that the sum of the squares
of three conjugate semi-diameters is constant, and (2) that
the parallelepiped which has three conjugate semi-diameters
for conterminous edges is of constant volume, are extremely
important. We append other proofs of these propositions.
Since in any conic the sum of the squares of two conjugate
semi- diameters is constant, and also the parallelogram of
which they are adjacent sides, it follows that in any conicoid
no change is made either in the sum of the squares or in the
volume of the parallelepiped, so long as we keep one of the
three conjugate diameters fixed.
We have therefore only to shew that we can pass from
any system of conjugate diameters to the principal axes of
the surface by a series of changes in each of which we keep
one of the conjugate diameters fixed.
This can be proved as follows : — let OP, OQ, OR, be any
three conjugate semi-diameters, and let the plane Q OB cut a
principal plane in the line 0Q\ and let OR' be in the plane
QOR conjugate to OQ'; then OP, OQ', OR' are three
conjugate semi-diameters.
Again, let the plane POR' meet the principal plane in
which OQ' lies in the line OP", and let OR'^ be conjugate to
OP" and in the plane POR' ; then OP", OQ' and OR" are
semi-conjugate diameters. But, since OR" is conjugate to OP"
and to OQ' , both of which are in a principal plane, it must be
a principal diameter.
Hence, finally, we have only to take the axes of the
section Q'OP" to have the three principal diameters.
98. It is known that any two conjugate diameters of a
conic will both meet the curve in real points when it is an
ellipse ; that 07ie will meet the curve in imaginary points
when it is an hyperbola ; and that both will meet the curve
in imaginary points when it is an imaginary ellipse. Hence,
by transforming as in the preceding Article, we see that
three conjugate diameters of a conicoid will all meet the
surface in real points when it is an ellipsoid ; that one will
meet the surface in imaginary points w^hen it is an hyper-
78 CONJUGATE DIAMETERS.
boloid of one sheet; and that two ^vill meet the surface in
imaginary points when it is an hyperboloid of two sheets.
99. To find the equation of an ellipsoid referred to
three conjugate diameters as axes.
Since the origin is unaltered we substitute for x, y and z
expressions of the form Ix + my + nz in order to obtain the
transformed equation [Art. 47].
The equation of the eUipsoid will therefore be of the form
Ax' + By" + Cz' + ^Fyz -\-2Gzx-\- IHxy = 1.
By supposition the plane ^ = 0 bisects all chords parallel
to the axis of x. Therefore if (^^ y^, z^ be any point on the
surface, (— x^, 3/,, z^ will also be on the surface. Hence
Gz^x^ + B.x(y^ = 0 for all points on the surface : this requires
that G=^H=0.
Similarly, since the plane ?/ = 0 bisects all chords parallel
to the axis of y, we have H = F = 0.
Hence the equation of the surface is
Ax'-j-By'+ Cz'=l,
x^ y^ z' ^
or -7-2 + fa + --, = 1,
a 0 c
where a', U, c' are the lengths of the semi-diameters.
100. We may obtain the relations between conjugate
diameters of central conicoids by the following method : —
The expression
-r^ v^ z^
| + | + '-5 + ^(^' + 2/^+^=)
is transformed, by taking for axes three conjugate diameters
which make angles a, yS, 7 with one another, into the
expression
x' ^/2 ^2
-77^ + J72 + —i + y^ {x^ + y'^ + z^ 4- 2yz cos a + 2zx cos ^+ 2^1/ cos 7).
The two expressions will therefore both split up into
linear factors for the same values of A. Hence the roots of
the cubics
&+^)(J+'^)(?--^'^) = °'
CONJUGATE DIAMETERS.
70
and
a
+ X , X cos 7 , \ cos /3
\ cos 7 , ,-7^ + X ,
X cos a
Xcos/5, X cos a , -t^ + X
= 0
are equal to one another.
Hence, by comparing coefficients in the two equations, we
have
a' + ¥-\-c'' = a^ + h"-\-c' (i),
Jyc" + c'a^ + a'Jf = 6'^c"^ sin^a + c'V sin'y5 +a'^6'^sInV (ii),
and
a6c = tx'Z^'c' ^J{\ - cos^a - cos^yS — cos^ +2 cos acos^cos7)..(iii).
Therefore the sum of the squares of three conjugate
diameters is constant; the sum of the squares of the areas of
the faces of a parallelopiped having three conjugate radii for
conterminous edges is constant ; and the volume of such a
parallelopiped is constant.
Ex. 1. If a parallelopiped be inscribed in an ellipsoid, its edges will be
parallel to conjugate diameters.
Ex. 2. Shew that the sum of the squares of the projections of three
conjugate diameters of a conicoid on any line, or on any plane, is constant.
Ex, 3. The sum of the squares of the distances of a point from the six l^
ends of any three conjugate diameters is constant ; shew that the locus of the
point is a sphere.
Ex. 4. If (Xi7/i^i), {x^y^^,. {x^iJzZi) he extremities of three conjugate -X
diameters of an ellipsoid, the*^equation of the plane through them will be
Ex. 5. Shew that the tangent planes at the extremities of three conju-
gate diameters of an ellipsoid meet on a similar ellipsoid.
Ex. 6. Shew that the locus of the centre of gravity of a triangle whose
angular points are the extremities of three conjugate diameters of an ellipsoid
is a similar ellipsoid.
V
so THE PARABOLOIDS.
The Paraboloids.
101. We have seen that the paraboloids are particular
cases of the central surfaces; properties of the paraboloids
can therefore be deduced from the corresponding properties of
the central surfaces. We will, however, investigate some of
the proj)erties independently.
We shall always suj)pose the equation of the surface
to be
a 0
102. To find the locus of the point of intersection of three
tangent j^lctnes to a j^ci^^ci^oloid which are mutually at right
angles.
Let l^x + m^y + n^z-\-p = 0 be one of the tangent planes;
then, since the plane touches the surface, we have
al^' + hm^ = 2n^ p^. [Art. 57, Ii.]
Hence we may write the equation in the form
IjTijX + ni^n^y + n^^ 2 + h (al^ + hm^) = 0.
We have also
l^n^ X + ???2??2 2/ + 72/ ^ + i {al^ + &»?/) = 0,
and l^n^ x + m^n^ y + n^z + ^ (al^ + hm^) = 0.
Since the planes are at right angles, we have by addition
z+i(a-\-h) = 0;
hence the locus is a plane.
103. The equation of the normal at any point {x, y\ z)
of the paraboloid is
X — X _y —y _z — z
X y —1
a h
PARABOLOIDS. 81 '
The normal at (x, y\ z) will pass through the particular
point (/, g, h), if
f-x' g-y\_fi-z'
a h
Put each fraction equal to X ; then
af , hg , 7 . ^
and substituting in
we have
»2 »2
a b
-f +^j£L^.2(/. + X).
The equation in \ is of the fifth degree; therefore
five normals can be drawn from any point to a paraboloid.
104. The middle points of all chords of the paraboloid
which are parallel to the line
X _ y _z
I m n
are [Art. 59] on the plane whose equation is
,dF dF dF ^
Z-r- + m -7- + ^ 7- =0,
ax ay as
Ix my „
or — + -y^ - ?i = 0.
a 0
Hence all diametral planes are parallel to the axis of the
surface.
It is easy to shew conversely that all planes parallel
to the axis are diametral planes.
A line parallel to the axis of the surface is called a
diameter. Every diameter meets the surface in one point at
a finite distance from the origin ; and this point is called the
extremity of the diameter.
S. S. G. 6
82 PARABOLOIDS.
The two diametral planes whose equations are
Ix my „
a 0
, Ix my , -
and — +— /-?i' = 0,
a 0
are such that each is parallel to the chords bisected by the
other, if
IV mm
- + -T- =0.
a 0
If this condition be satisfied, the planes are called con-
jugate diametral planes.
The condition shews that conjugate diametral planes
meet the plane £^ = 0 in lines which are parallel to conjugate
diameters of the conic
a 0
105. If we move the origin to any point (a, y8, 7) on the
surface, the equation becomes
aba 0
If we take the planes
x = 0, y=0, and — + ^-^=0
•^ ' a b
as co-ordinate planes, and therefore the lines
^ _y _z X _y _z A ^ _y _ ^
for axes, we must [Art. 47] substitute
ax hy ax ^y
ibr .T, y, z respectively.
The transformed equation is
X y ^
a + - Z^ 4- ^
a 0
CONES, ;^3
This is the equation to the surface referred to a point
(a, yS, 7) as origin, two of the co-ordinate planes being parallel
to their original directions, and the third being the tangent
plane at (a, /3, 7).
Ex. 1. Shew that the locus of the centres of parallel sections of a y
paraboloid is a diameter.
Ex. 2. Shew that aU planes parallel to the axis of a paraboloid cut the \/'
surface in parabolas.
Ex. 3. Shew that the latera recta of all parallel parabolLc sections of a y*
paraboloid are equal.
Ex. 4. Shew that the projections, on a plane perpendicular to the axis
of a paraboloid, of all plane sections which are not parallel to the axis, are
similar conies.
Ex, 5. P, Q are any two points on a paraboloid, and the tangent planes
at P, Q intersect in the line US ; shew that the plane through ES and the y^
middle point of PQ is parallel to the axis of the paraboloid.
Ex. G. Shew that two conjugate jjoints on a diameter of a paraboloid
are equidistant from the extremity of that diameter.
Ex. 7. Shew that the sum of the hitera recta of the sections of a
paraboloid, made by any two conjugate diametral planes through a lixegl
point on the surface, is constant.
Cones.
106. The general equation of a cone of the second
degree is
aoi^ + hf + cz^ + Ififz + Igzx 4- llixy = 0.
The tangent plane at any point {x\ y, z) on the
surface is
{x - x) (ax + hj' + fjz) + {y - y) (hx + by +fz)
+ (z- z) {gx -vfij + cz) = 0,
or
X (ax + hy + gz) + y {hx + hy +fz) + z (gx +fy + cz') = 0.
The form of this equation shews that the tangent plane
at any point on a cone passes through its vertex, as is geo-
metrically evident from the fact that the generating line
through any point is one of the tangent lines at that point,
and therefore lies in the tangent plane.
G— 2
84
TANGENT PLANE OF A CONE.
107. To- find the condition that the plane Ix +7)17/ +712=0
may touch the cone whose equation is
ax' + hf + cz' + 2fyz + 2gzx + 2hxi/ = 0.
Comparing the equation of the tangent plane at the point
{x, y\ /), namely
X (ax + hy + gz) + y {lix + hy +fz) 4- z {gx +fy + cz) = 0,
with the given equation, we have
ax_±h/ + gz' ^ /^^' + W +fi' ^ ^-^^ +/y' + ^-' ^
Put each fraction equal to — X, then
ax + Ay' + ^/ + X? =0,
/«:c' -J- 6y' +// + Xm = 0,
and ■ gx -\-fij + cz + X?i = 0.
Also, since [x , y\ z) is on the plane,
Ix + my' + 72/ = 0.
Eliminating x^ y\ /, X, we have the required condition
= 0,
a,
A,
ff^
Z
h,
h.
/,
7?1
fj>
/,
c,
?Z
h
m,
n,
0
or ^r + Bm^ + O/z' + 2Fmn + 2 Gnl+2Hlm = 0,
wdiere Ay B, C, &c. are the minors of a, h, c, &c. in the deter-
minant
a, h, g
h, h, f
108. If through the vertex of a given cone lines be drawn
perpendicular to its tangent planes, these lines generate
another cone tailed the recij^rocal cone.
The line through the origin perpendicular to the plane
X _ y z
7H n '
X
Ix + 7)}y + nz = 0, is y
RECIPROCAL CONE. 85
Hence, from the result of ttie last article, the reciprocal of
the cone
aa? + h-if + cz" + 2fyz + Igzx + 2hxy = 0,
is Ax" + Bf + Cz^ + 2Fijz -{-2GZX + 2Hxij - 0.
Since the minors of A, B, C, &c. in the determinant
A, II, G
H, B, F
G, F, C
are proportional to a, h, c, &c., we see that the relation be-
tween the two cones is a reciprocal one.
As a particular case of the above, the reciprocal of the
cone
x^ ip- z^
aj? + 6/ 4- c/ = 0, is - + f + - = 0.
^ a b c
From this we see at once that a cone and its reciprocal
are co-axial.
109. To find the condition that a cone may have tliree
perpendicular generators.
Let the equation of the cone be
ax" + by"" + cz" + 2fyz + 2gzx + 2hxy = 0 (i).
If the cone have three perpendicular generators, and we
take these for axes of co-ordinates, the equation will [Art. 73,
Ex. 4] take the form
Ayz-\-Bzx-\-Cxy=0 (ii).
Since the sum of the co-efficients of x^, y"^ and z"^ is an in-
variant [Art. 79] and in (ii) the sum is zero ; therefore the
sum must be zero in (i) also. Therefore a wecessarj/ condition
is
a-\-b + c=0 .^. (iii).
If the condition (iii) is satisfied there are an infinite
number of sets of three perpendicular generators. For take
any generator for the axis of x\ then by supposition any
point on the line y = O,z = 0 is on the surface ; therefore the
8G CONE WITH THREE l>ERPENDlCtJLAli GENERATORS.
Co-cfficicnt of x' is zero, so that the transformed equation is of
the form
%'+ c^'+ 2/y^4- 2r;zx-]- 2hxy = 0 (iv);
and since the sum of the co-efficients of x'^, ?/^, z^ is an in-
variant, ^^e have 6 -f c ^ 0.
Now the section of (iv) by the plane £c ^ 0 is the two
straight lines
Z;/+c/ +2/5/^ = 0;
and these arei at right angles^ since 6 + c = 0.
110. If a cone have three perpendicular tangent planes,
the reciprocal cone will have three perpendicular generators.
Hence the necessary and sufficient condition that the
cone
ax^ + hf 4- cz^ + 2ff/z + 2gzx + 2hxy = 0,
may have three perpendicular tangent planes is
Ex. 1, CP, CQ, CR are tliree central radii of an ellipsoid which are
mutually at right angles to one another ; shew that the plane FQli touches
a sphere.
Let the equation of the plane PQR be lx + my + nz=p. The equation of
the cone whose vertex is the origin, and which passes through the intersection
, , ,,. .^x^tPz'^^. ic^ y'^ z'^ (Ix + my+nzV'^
of the plane and the ellipsoid -, + f., + — = 1, is — + jr + .; = '- .
By supposition the cone has three perpendicular generators ; therefore
i 1 1 1
«- 6- 6- i)-
Ex. 2. Any two sets of rectangular axes which meet in a point form six
generators of a cone of the second degree.
Ex. 3. Shew that any two sets of perpendicular i^lanes which meet in
a point all touch a cone of the second degree.
111. To find the equation of the tangent cone from any
jwint to an ellipsoid.
Let the equation of the ellipsoid he
2 2 2
£.+^+1 = 1
TANGENT CONE. 87
Let the co-ordinates of any two points P, Q be x, y, z
and x\ y'\ z" respectively.
The co-ordinates of a point which divides PQ in the ratio
m : n are
Tix + mx" ny + my' nz -f mz"
m + n ' m + n * m + n
If this point be on the ellipsoid, we have
(nx + mx")'^ (ny + mi/'Y inz' -f mz'y , , . «
or n
a b G
„/^^ ?/^ -2^ -\ „ XX yy zz -\
/'2 "2 '"2 \
If the line PQ cut the surface in coincident points, the
7?
above equation, considered as a quadratic in — , must have
equal roots ; the condition for this is
/ '2 '2 '2 \ , //2 "2 ''2 \
Hence, if the point P {x, y\ z) be fixed, the co-ordinates
of any point Q, on any tangent line from P to the ellipsoid,
must satisfy the equation
a' ^ b' ^ e W b' ^ c'
XX yy ^^ ^2
.+f +^-1 =0 (i).
a 0 c J
Hence (i) is the required equation of the tangent cone
from (a?', y\ z') to the ellipsoid.
112. If we suppose the point {x\ y, z) to move to an
infinite distance, the cone will become a cylinder whose
generatiDg lines are parallel to the line from the centre
ot the ellipsoid to the point {x, y, z).
88 ENVELOPING CYLINDER.
Hence, if in the equation of the enveloping cone we put
X = Ir, y = mr, z = ?ir,
and then make r infinitely great, we shall obtain the equation
of the enveloping cylinder whose generating lines are
parallel to
X _ y _z
I m n'
Substituting Ir, mr, nr for x, y , z respectively in the
-equation of the enveloping cone we have
Hence, when r is infinite,
\d' ^ b' ^ c' J W h' cV W 6^ cV
113. The equation of the enveloping cylinder can be
found, independently of the enveloping cone, in the following
manner.
The equations of the straight line which is drawn through
any point {x, y\ z) parallel to
X y _z
I m n *
X — X V — V z — z
are — j— = - — — = = r.
I in n
The straight line will meet the ellipsoid in two points
whose distances from {x, y\ z) are given by the equation
a' b' c
^^+'^+^-i)+^-'r^+iF+c^
„/Z^ m^ it^\ ^
The straight line will therefore touch the surface, if
/x' y" z"" \ (V m^ n\ fix my nzV
EXAMPLES. 89
Hence tlie co-ordinates of any point, whicli is on a
tangent line parallel to
X y _ z
I m n '
satisfy the equation
'x"^ y"^ z^ ^\ fP m^ n'^\ (Ix my nzX^ ^
which is the required equation of the enveloping cylinder.
Ex. (i). To find the condition that the enveloping cone may have three
perpendicular generators.
The equation of the enveloping cone whose vertex is {x', y\ z') is
If this have three perpendicular generators the sum of the coefficients of
x'\ ?/2, and z^ must be equal to zero [Art. 109]. Hence {x', y', z'), the vertex
of the cone, is on the surface
/I 1 iwa;2 y^ t-A-— y^ -
\a' "^ &2 ■*" cV W'^h^'^7^ j ~ a^ "^ 64 + -4-
Ex. (ii). Shew that any two enveloping cones of an ellipsoid intersect in
plane curves.
The equations of the cones whose vertices are [x', y', z') and [x", y'\ z") are
yy +!£_i\2
^ &2 + c2
respectively.
The surface whose equation is
(xx' yy' zz' , y fx'"^ y"'^ 2"2
\ a^ h^ c^ J \a- h^ c^
/xx" yy" zz" ^Y foff"^ !/'2 z''- ,
\ a^ 0^ c^ J \ a- u- c-
passes through their common points, and clearly is two planes.
Ex. (iii). Find the equation of the enveloping cone of the paraboloid
ax^+by^ + 2z = 0.
Ans. (acc2 + hy"^ + 2z) [ax'^ + ly'^ + 2z') = [axx' + Irjif + z+ z'f.
Ex. (iv). Find the locus of a point from which three perpendicular
tangent lines can be drawn to the paraboloid ax- + hy'^ + 2^ = 0.
Ans. ah [x- + y-) + 2 (a + &) 2 = 1.
90
Examples on Chapter IV.
Z' 1. Find the equation of a sphere which cuts four given spheres
orthogonally.
2. Shew that a sphere which cuts the two spheres aS = 0 and
s/^ /S" = 0 at right angles, will cut IS + m8' = 0 at right angles.
3. OP, OQ, OR are three perpendicular lines which meet in
y^ a fixed point 0, and cut a given sphere in the points P, Q, R;
shew that the locus of the foot of the perpendicular from 0 on
the plane FQR is a sphere.
4. Through a point 0 two straight lines are drawn perpen-
dicular to one another and intersecting two given straight lines
at right angles ; shew that the locus of 0 is a conicoid whose
centre is the middle point of the shortest distance between the
given lines.
5. Shew that the cohq Ax'+ By" + Cz^+1Fyz-^2Gzx+2Hxy = 0
wdll have three of its generators coincident with conjugate diameters
v/ of -, +^'+-^=l,if ^ct^ + ^6-' + 6'c^ = 0.
a- h" c^
6. A plane moves so that the sum of the squares of its
distances from n given points is constant; shew that it always
touches an ellipsoid.
7. Tlie normals to a surface of the second degree, at all
points of a plane section parallel to a principal plane, meet two
fixed straight lines, one in each of the other principal planes.
8. Shew that the plane joining the extremities of three
conjugate diameters of an ellipsoid, touches another ellipsoid.
9. Having given any two systems of conjugate semi-diameters
of an ellipsoid, the parallelepiped which has any three for conter-
minous edges is equal to that which has the other three for
conterminous edges.
10. If lines be drawn through the centre of an ellipsoid
parallel to the generating lines of an enveloping cone, the cone so
Ibrmed will intersect the ellipsoid in two planes parallel to the
plane of contact.
EXAMPLES ON CHAPTER IV. 91
11. The enveloping cone from a point P to an ellipsoid has
three generating lines parallel to conjugate diameters of the
ellipsoid ; find the locus of /*.
12. The plane through the three points in which any three
conjugate diameters of a conicoid meet the director-sphere touches
the conicoid.
13. Shew that any two sets of three conjugate diameters
of a conicoid are generators of a cone of the second degree.
14. Shew that any two sets of three conjugate diametral
planes of a conicoid touch a cone of the second degree.
15. Shew that any one of three equal conjugates of an
ellipsoid is on the cone whose equation is
(or + J^ + e) (5 + |I + J) = 3 (»^ +2/' + .').
16. D, E^ /^ and P, Q^ R are the extremities of two sets of
conjugate diameters of an ellipsoid. If jt?, pj, 2^2^ V^ ^^^ ^^^^ P^
pendiculars from the centre and P, Q^ R respectively on
plane DEF, prove that ^
P' + P2 -^Pf = ^P {Pi +P2 +7^3)- ^
17. The sum of the products of the perpendiculars from the
two extremities of each of three conjugate diameters on any
tangent plane to an ellipsoid is equal to twice the square on the
perpendicular from the centre on that tangent plane.
18. The distance r is measured inwards along the normal to
an ellipsoid at any point P, so that pr = 'ni^, where p is the per-
pendicular from the centre on the tangent plane at P ; shew that
the locus of the point so obtained is
2 3 7 " 2 2 2
a X 0 y c z -
(a'--wy {b'-my {c'-m'f
19. Through any point P on an ellipsoid chords PQ, PR, PS
are drawn parallel to the axes ; find the equation of the plane
^P*S', and shew that the locus of E, the point of intersection of
the plane QRS and the normal at P, is another ellipsoid. Shew
also that if the normal at P meet the principal planes in G^, G,^, G^
.1 11 2 111
then will ——V = -rrz^ +
PK PG^ PG^ PG^
92 EXAMPLES ON CHAPTER IV.
20. PK is the perpendicular from any point on its polar
j)lane with respect to a conicoid and this perpendicular meets a
principal plane in G ; shew that, if PK. PG is constant, the locus
of P is a conicoid.
2 2
21. Shew that the cone whose base is the ellipse — , + Vs = !»
^ a' b
2 2
s = 0, and whose vertex is any point of the hyperbola -^ — r^ — ^2
= 1 , 2/ = 0, is a right circular cone.
22. A cone, whose equation referred to its principal axes, is
x^ if
is thrust into an elliptic hole whose equation is —+'—, = 1 ; shew
that when the cone tits the hole its vertex must lie on the ellipsoid
x" y' ofl 1\_,
23. In a cone any system of three conjugate diameters meets
any plane section in the angular points of a triangle self polar
with respect to that section.
24. The enveloping cones which have as vertices two points
on the same diameter of a conicoid intersect in two parallel planes
between whose distances from the centre that of the tangent
plane at the end of the diameter is a mean proportional. What
is the corresponding proposition for a paraboloid?
25. Shew that any two enveloping cones intersect in plane
curves ; and that when the planes are at right angles to one
another, the product of the perpendiculars on one of the planes of
contact from the centre of the ellipsoid and the vertex of the
corresponding cone, is equal to the product of such perpendiculars
on the other plane of contact.
26. If a line through a fixed point 0 be such that its con-
jugate line with respect to a conicoid is perpendicular to it, shew
that the line is a generating line of a quadric cone.
27. The locus of the feet of the perpendiculars let fall from
points on a given diameter of a conicoid on the j^olar planes of
those points is a rectangular hyperbola.
EXAMPLES ON CHAPTER IV. 93
28. Prove that the surfaces
ic' y_' _ 2;s ^ ^ _ ?f ^ ^ _ ^^
<^V^^' <^6/-^' <"V"^~3'
will have a common tangent plane if
2 i
29. Prove that an ellipsoid of semi-axes a, h, c and a concen-
tric sphere of radius , , are so related that an in-
^ Jb'c' + c'a' + d'b'
definite number of octahedrons can be inscribed in the ellipsoid,
and at the same time circumscribed to the sphere, the diagonals of
the octahedrons intersecting at right angles in the centre.
x^ if z^
30. Pind the locus of the centre of sections of — + ^-^ + -„ = ^
a' 6" c
which touch -7^ + , ,., + -7^ = 1.
a 6 ' c
31. Planes are drawn through a given line so as to cut an
ellipsoid; shew that the centres of the sections so formed all lie on
a conic.
32. Pind the locus of the centres of sections of an ellipsoid
by planes which are at a constant distance from the centre.
33. Shew that the plane sections of an ellipsoid which have
their centres on a fixed straight line are parallel to another straight
line, and touch a parabolic cylinder.
34. The locus of the line of intersection of two perpendicular
tangent planes to aaf + ly^ + cz^ = 0 is
a(b + c)x^ + h{G + a)y^ + c{a + h)z^ = 0.
35. The points on a conicoid the normals at which intersect
the normal at a fixed point all lie on a cone of the second degree
whose vertex is the fixed point,
36. Kormals are drawn to a conicoid at points where it is
met by a cone which has the axes of the conicoid for three of its
generating lines; shew that all the normals intersect a fixed
diameter of the conicoid.
94
EXAMPLES ON CHAPTER IV.
37. Shew that the six normals which can be drawn from
any point to an ellipsoid lie on a cone of the second degree, three
of whose generating lines are parallel to the axes of the ellipsoid.
38. Find the equations of the right circular cylinders which
circumscribe an ellipsoid.
39. If a right circular cone has three generating lines
mutually at right angles, the semi-vertical angle is tain~^J'2.
40. If one of the principal axes of a cone which stands
on a given base be always parallel to a given right line, the locus
of the vertex is an equilateral hyperbola or a right line according
as the base is a central conic or a parabola.
41. The axis of the right circular cone, vertex at the origin,
which passes through the three lines, whose direction-cosines are
(/j, m^, nj, (/g, 7??2, n^), {\, w^, n^ is normal to the plane
= 0.
0,
1,
1,
1
X,
K^
h^
^3
2/»
m^.
m^.
71i
n.
n„
42. The equations of the axes of the four cones of revolution
which can be described touching the co-ordinate planes are
a;
siu'a siii'lJ sm'y
o, /?, y being the angles YOZ, ZOX, and XOF respectively.
43. Prove that four right cones may be described, passing
through three given straight lines intersecting in the same point,
and that if 2a, 2/3, 2y be the mutual inclinations of the straight
lines, the equations of the cones referred to the straight lines as
co-ordinate axes will be
sin'a sin'/? sin^y
X
cos^a
X
y
^ sin^a cos^jS cos% .
0, —37- + — ~ + —7-^ = 0,
X
V
sin'yS cos'v _ cos'a
+ -^ - -f - — ~ = 0,
y z X
COS'/? sin^y
y z
= 0.
EXAMPLES ON CHAPTER IV. 95
44. Shew that, if P, Q, B be extremities of three conjugate
diameters of a conicoid, the conic in which the plane PQR cuts
the surface contains an infinite number of sets of three conjugate
extremities, which are at the angular points of maximum triangles
inscribed in the conic PQR.
45. Shew that, if the feet of three of the six normals drawn
from any point to an ellipsoid lie on the plane Ix + my + nz + p = Of
the feet of the other three will be on the plane
ax hy cz \ ^
-J-+ ^ + = 0,
I m n p
the equation of the ellij)soid being ax^ + hy^ + cz' = 1.
46. Prove that the locus of a point with which as a centre of
conical projection, a given conic on a given plane may be projected
into a circle on another given plane, is a plane conic.
47. If C be the centre of a conicoid, and P (Q) denote the
perpendicular from P on tlie polar plane of Q ; then will
P(Q) C{Q)
Q{P)- C{Fy
48. The locus of a point such that the sum of the squares of
its normal distances from a given ellipsoid is constant, is a co-axial
ellipsoid.
49. If a line cut two similar and co- axial ellipsoids in P, P';
Q, Q' ; prove that the tangent plane to the former at P, P\
meet those to the latter at Q or Q' in pairs of parallel lines equi-
distant respectively from Q or Q'.
50. A chord of a quadric is intersected by the normal at a
given point of the surface, the product of the tangents of the
angles subtended at the point by the two segments of the chord
being invariable. Prove that, 0 being the given point and P, P*
the intersections of the normal with two such chords in perpendi-
cular normal planes, the sum of the reciprocals of OP, 0P\ is
invariable.
CHAPTER V.
Plane Sections of Conicoids.
114. We have seen [Art. 51] that all plane sections of a
conicoid are conies, and also [Art. 61] that all parallel
sections are similar conies. Since ellipses, parabolas, and
hyperbolas are orthogonally projected into ellipses, parabolas,
and hyperbolas respectively, we can find whether the curve
of intersection of a conicoid and a plane is an ellipse,
parabola, or hyperbola, by finding the equation of the pro-
jection of the section on one of the co-ordinate planes.
For example, to find the nature of plane sections of a
paraboloid.
The plane Ix + my + ?zj + p = 0 cuts the paraboloid
ax^ + hif -f 2j = 0, in a curve through which the cylinder
a {my + 7?^ + i^)' + hlY + 2Z'^ = 0
passes. The plane x — 0, which is perpendicular to the
generating lines of the cylinder, cuts it in the conic whose
equations are a; = 0, a {my -{■ nz -{■ pf + hFif' -f Wz = 0 ; and
this conic is the projection of the section on the plane ^=0.
If n = 0, the projection will be a parabola; but, if n be not
zero, the projection will be an ellipse or hyperbola accord-
ing as aif {aiif + hl^) - a^m^r^ is positive or negative, or aWn^
positive or negative, that is, according as the surface is an
elliptic or hyperbolic paraboloid.
AREA OF CENTRAL SECTION. 97
Hence all sections of a paraboloid which are parallel
to the axis of the surface are parabolas ; all other sections of
an elliptic paraboloid are ellipses, and of a hyperbolic
paraboloid are hyperbolas.
Ex. 1. Find the condition that the section of aa;^ + Z)?/^ + C2'^ = 1 by the
plane lx + my + nz+p = 0 may be a parabola.
Ans. - + -T- + - = 0.
a b c
Ex. 2. Shew that any tangent plane to the asymptotic cone of a conicoid
meets the conicoid in two parallel straight lines.
115. To find the axes and area of any central plane
section of an ellipsoid.
Let the equation of the ellipsoid be
2 2 2
— \-~ A — = 1
a 0 c
and let the equation of the plane be
Ix + my ■\-nz = 0 (i).
Every semi-diameter of the surface whose length is r is a
generating line of the cone whose equation is [p. 55, Ex. 5]
x"
&-')-"■&-?) -'G'-J)-" »
This cone will, for all values of r, be cut by the plane in two
straight lines which lie along equal diameters of the section ;
and, when r is equal to either semi-axis of the section, these
equal diameters will coincide. That is, the plane (i) will
touch the cone (ii) when r is equal to either semi-axis of
the section of the ellipsoid by the plane. The condition
of tangency gives
"^1 _ 1^ 1 1 ~ ^
ahc ahc
.(iv),
'^{dH'+bW+c'n') p
where r^, r^ are the semi-axes of the section, and p is the
perpendicular on the parallel tangent plane.
S. s. G. "^
98 PLANE SECTIONS.
From (iv) wc see that the area of the section is equal to
Trabc
116. To find the area of any plane section of an ellipsoid.
Take for co-ordinate planes three conjugate planes of
which ^ = 0 is parallel to the given plane; then the equations
of the surface and of the given plane will be respectively
of the forms
a^ \i^ z^
-2 + -7T2+ -2 = lj and<2r = /j.
a 0 c
The cylinder whose equation is
2 '2 72
a"^b"^c"' '
passes through the curve of intersection of the surface and the
plane ; and the area of the section of this cylinder by ^ = k is
Tra'b' sin v (l 72
V being the angle XOY. The area of the section of the
ellipsoid hy z = 0 is irab' sin v.
Hence, if A be the required area, and AQhe the area of
the parallel central section, we have
Now the tangent plane at (0, 0, c) is z = c ; therefore the
perpendicular distances of the given plane and of the parallel
tangent plane from the centre are in the ratio of k : c.
Hence A = A,(l-Q (i),
where p and^p^ are the perpendicular distances of the given
plane and of the parallel tangent plane from the centre.
This gives the relation between the area of any section
and of the parallel central section ; and w^e have found
in Art. 115, the area of any central section.
PLANE SECTIONS. 99
Hence the area of the sectiou of the ellipsoid whose
equation, referred to its principal axes, is
x"^ y^ z^ _.
a b c
made by the plane whose equation is
lx-\- my -\- nz =p,
irabc / P^ \
'^ V {aH' + Jfm' + cV) V ~ aT+h'm' + enV '
^"""^ ^0 " V(aT + 6WT?/?) '^^'■*' •'•^^-''
and p^'^ = a'r + b'm' + cV [Ai't. 91].
Ex. 1. To find the area of the section of a paraboloid by any plane.
Let the equation of the paraboloid be ax^ + by^ + 2z — 0, and let the equa-
tion of the section be lx + my + nz+p = 0. The projection of the section on
the plane 2 = 0 is the conic
2
ax'^ + by^ — {Lx + my+p) = 0,
The area of the projection is
n^Jab\a &
and. therefore [Art. 31] the area of the section is
TT U2 m2 ^ )
n^Jab^a, b
X^ 7/^ Z^
Ex. 2. To find the area of the section of the cone — [- 4- + — = 0 by the
a b c
plane lx + jny + nz - p.
x^ y^' z-
The area of the section of — r + 77- + 7 = 1 t)y the given plane is
ak bk ck ^ '^
IT s/»bck^ ( ^ p"^
1-
yj ( kal- + kbm^ + kcnP) \ kal^ + kbruP + kcn^) '
If we put A; = 0 the surface becomes the cone. The required area is therefore
irp^ sjabc
Ex. 3. If central plane sections of an ellipsoid be of constant area, their
planes touch a cone of the second degree.
7—2
100
PLA^s^E SECTIONS.
/
Let the area be - , - , and let the equation of one of the planes be
a (
Then we have
Ix+my + nz — O.
irahc irabc
or
j,H^- + b'^m^ + c^n'^=d?;
This shews that the plane lx + my + nz = 0 always touches the cone
y'
x" y z'' _
4 — ^ — 4- =0.
117. We can find, by the method of Art. 11.5, the area
of a central plane section of the surface whose equation is
ax^ + hy^ + cz^ + 2/3/s + "Igzx + thxy = 1.
For the semi-diameters of length r are generating lines of
the cone whose equation is
[a - i) ^^{l- p) f + (c - i) ^^ + 2/yz + ^zx + tlixy = 0.
When r is equal to either semi-axis of the section of the
surface by the plane
Ix + ffiy + ws = 0,
the plane will be a tangent plane of the cone. The condition
of tangency gives, for the determination of the semi-axes, the
equation
a 5
h.
9
f'
t/ •
h.
»-?•
/>
m
g<
/.
1
n
I,
m,
71,
0
= 0.
This result may also be obtained by finding the maxi-
mum value of x^ + y'^ -{- z^ = r^ , subject to the conditions
ax^ + hy"^ + cz^ + 2fyz + 2gzx + iJixy = 1, and Ix 4- my -\-nz = 0.
AXES OF CENTRAL SECTIONS.
101
118. To find the directions of the axes of any central
section of a conicoid.
Let the equation of the surface be
ax^ + hy^ + cz^ + ^fyz + ^gzx + llixy = 1,
and let the equation of the plane be
Ix + Tfiy + n^ = 0.
Then, if P be any point on an axis of the section, the line
joining P to the centre of the section will be perpendicular
to the polar line of P in the plane of the section.
Hence, if P be (f, 77, f), and if the direction-cosines of
the polar line be X, yb, v, we have
Xf + ^7; + ^?=0 (i).
Also, since the polar line is on both the planes
x{a^-\-h7^+gi;)-Vyili^+h^-^f^) + z{gi-\-fri + cQ = l,
and Ix + my + nz = 0,
it is perpendicular to the normals to those planes ; hence
X(a^ + hrj+gO + H'Qi^-^h-^f^) + Hg^+fv + c^) = 0...(u),
and Xl + jJLin -\-vn = 0 (iii).
Eliminating \, jjl, v from the equations (i), (ii), (iii), we
have
= 0.
a^-\-hrj+g^, h^ + hrj+fi, gS + fy + cl^
I, m, n
Hence the required axes are the lines in which the given
plane cuts the cone whose equation is
X, y, z
ax + hy + gz, hx -\-hy -hfz, gx +fy + cz
I, m, n
= 0.
119. To find the angle between the asymptotes of any
plane section of a conicoid.
Let 6 be the angle between the asymptotes of the plane
section, and let the semi-axes of the section be a, /3.
102 CONDITION FOR RECTANGULAR HYPERBOLA.
Then
tan -^ -
= 7'
-^
tan'^ =
—
4a'/3'
This gives the required angle, since we have found, in the
preceding articles, the axes of any plane section.
Ex. 1. Find the angle between the asymptotes of the section of
ax^ + hy" + C2- = 1 by the plane Ix-rmy + nz = 0.
The semi-axes are the roots of the equation
a h :, c - -,
r- ?•" r^
therefore tan^ d= ~ , , ^ ^,
('•1' + r^-? { i2 (6 + c) + m2 (c + a) + n2 (a + 6) } 2
Ex. 2. To find the condition that the section of the conicoid
ax2 + 1?/2 + cz"^ 4- 2fyz + 2(72x + 2hxy= 1
by the plane lx-\-my + nz — Q may be a rectangular hyperbola.
The square of the reciprocal of the semi-diameter whose direction-cosineg
are \, (x, v is given by
- = a\- + 6^2 ^ c^2 ^ 2//X;' + 2gv\ -f 2;jX/i.
Take any three j^erpendicular diameters ; then we have by addition
111
-^ + — + — = « + & + c.
ri2 r^- r^-
Now, if r^, rg be the lengths of any two perpendicular semi-diameters of a
rectangular hyperbola, r^ + r^ = 0.
Hence for any semi-diameter of the conicoid which is perpendicular to
the plane of a section which is a rectangular hyperbola, we have
1 ,
r-
The required condition is therefore
al^ + hrrh^ + cn^ + 2fmn + 2gnl + 2hlni = a-{-h + c = {a + h + c) {l" + m- + n^).
Ex. 3. Shew that the two lines given by the equations ax'^ + by^ + cz^ = Of
Ix + my + nz = 0 will be at right angles, if
P{b + c)+m^c + a)+n'{a + h) = 0.
The lines are the asymptotes of the section of the conicoid ax^ + hy^ + cz- = l
by the plane Ix + my -f nz = 0.
CIRCULAR SECTIONS. 103
120. If two conicoids have one plane section in common
all their^ other points of intersection lie on another plane.
Let the equations of the common plane section be
ax^ + %^ -t- 2hxj/ + 2ux + 2vy + c = 0, 0 = 0.
The most general equations of two conicoids which pass
through this conic are
ax^ + by"^ + 2hxT/ + 2ux + 2vy + c + z (Jx + my + nz +^) = 0,
and
ax^ + hy^ + 2hxy + 2ux + 2vy + c -{■ z {I'x + my 4- nz + ^') = 0.
It is clear that all points which are on both surfaces, and
for which z is not zero, are on the plane given by the
equation
Ix 4- my + nz +p = Tx + vi'y + nz H- p' ;
this proves the proposition.
Circular Sections.
121. To find the circular sections of an ellipsoid.
Since parallel sections are similar, we need only consider
the sections through the centre.
Now all the semi-diameters of the ellipsoid which are of
length r are generating lines of the cone whose equation is
SC rV ' "^ W rV ' \c' r
If there be a circular section of radius r, an infinite
number of generating lines of the cone will lie on the plane
of the section ; hence the cone must be two planes. This
will only be the case when r is equal to a, or 6, or c.
If r = a, the two planes pass through the axis of x, their
equation being
^i?-y+^'fi-a-)=° «•
104? CIRCULAR SECTIONS.
The equations of the other pairs of planes are respectively
^'&-y+-ii-y=o (")'
Of these three pairs of planes, two are imaginary. For,
11 11
if a, h, c be in order of maofnitude, yr, 5 and -o 5 have
the same sign, and therefore the planes (i) are imaginary ;
for a similar reason the planes (iii) are imaginary. Hence,
the only real central circular sections of an ellipsoid pass
through the mean axis, and their equations are
/^-S'^A'-a <'■)■
X
V
Since all parallel sections are similar, there are two
systems of planes which cut the ellipsoid in circles, namely
planes parallel to those given by the equation (iv).
If 5 = c the two planes which give circular sections are
coincident.
122. If the surface be an hyperboloid of one sheet, we
must change the sign of c^ in the equations of the last
Article. In this case the planes which give the real circular
sections are those given by equations (i), a being supposed to
be greater than h.
If the surface be an hyperboloid of two sheets, we must
change the signs of If and c\ In this case the planes which
give the real circular sections are those given by equation
(ii), h being supposed to be numerically greater than c.
123. If a series of planes be drawn parallel to either
of the central circular sections of an ellipsoid, these planes
will cut the surface in circles which become smaller and
smaller as the planes are drawn farther and farther from
the centre ; and, when the plane is drawn so as to touch the
ellipsoid, the circle will be indefinitely small.
CIRCULAR SECTIONS. 105
Def. The point of contact of a tangent plane whicli cuts
a surface in a point-circle is called an umhilic.
124. Any two circular sections of opposite systems are on
a sphere.
The circular sections of the ellipsoid are parallel to the
planes whose equations are
Hence ^ ^g- 1,) +.^(^1-1) +:P = 0,
are the equations of the planes of any two circular sections of
opposite systems.
The equation
is, for all values of X, the equation of a conicoid which passes
through the two circular sections ; and, if X = 1, the equation
represents a sphere ; which proves the proposition.
125. We can find the circular sections of the paraboloid
a^ h '"^'
by writing the equation in the form
l(.= +y + ,^_2a.)+y(l-^)-;;=0.^
It is clear that the two planes given by the equation
cut the paraboloid where they cut the sphere whose equation
is x^ -\- y^ ■\- z^ — laz = 0 ;
106
CIRCULAR SECTIONS.
and, since the planes must cut the sphere in circles, they will
cut the paraboloid in circles.
We can shew in a similar manner that the planes given
by the equation
'1 1\ ^
x" - -
,a bJ 0
will give circular sections of the paraboloid.
Of the two pairs of planes given by the equations
«•e-D-^»."<'='•(^')-^»■
one will be real, if a and h are of the same sign ; but both
pairs of planes will be imaginary if a and h are of different
signs, so that there are no circular sections of a hyperbolic
paraboloid.*
Ex. 1. Shew that the conicoid whose equation is
has the same cyclic planes for all values of X.
Ex. 2. Shew that no two parallel circular sections of a conicoid, which
is not a surface of revolution, are on a sphere.
Ex. 3. Find the circular sections of the conicoid whose equation is
ax^' + lif + cz^ + 2/?/z + Icjzx + Ihxij r= 1.
All semi-diameters which are of length r are generating lines of the cone
whose equation is
(«--^)^'+ (^-72)^'+ ^c-^')s2 + 2/^z + 2^za: + 2/iX7/ = 0...(i).
If therefore r is the radius of a circular section, the cone must be two
planes. The condition for this is
^ A.
1
a — -,
r
•2 '
h,
^-^'
/»
9 1=0.
/
.(ii).
c-
Tf we substitute in (i) any one of the roots of the equation (ii), we shall
obtain the equation of the corresponding planes of circular section.
Ex. 4. Find the real circular sections of the following surfaces
(i) 4x'' + 2i/ + z^ + Syz + zx = l,
(ii) 2x2 + 5t/2-322 + 4xi/ = l.
* This is not strictly true: a section through any generating line by a
plane parallel to the axis of the surface is a circle of infinite radius.
EXAMPLES. 107
Ans. (i) planes parallel to
(ii) planes parallel to
(cc + 2?/)2-4z^ = 0.
Ex. 5. Find the conditions that the plane
lx + my + nz=0,
may cut the conicoid
ax^ + by^ + cz- + 2fyz + 2gzx + 21ixy - 1
in a circle.
As in Ex. 3, the equation
must, for some value of y, be two planes of which the given plane is one.
The equation must therefore be the same as
By comparing the coefficients of yz, zx, xy we have
and two similar equations.
Hence the required conditions are
h'n? + cvi? - 2fmn _ cl^ + an^ - 2gnl _ am^ + 11? - 2hlm
126. We will conclude this chapter by the solution of
two examples.
Ex. 1. With a fixed point 0 on a conicoid as vertex, and plane sections of
the conicoid for bases, cones are described; shew that the cones are cut by any
plane parallel to the tangent plane at 0 in a system of similar conies.
(Chasles.)
The equation of a conicoid, referred to three conjugate diameters as axes,
is of the form
X^ ifi 2^
Hence the equation, referred to parallel axes through the extremity of one of
the diameters, will be
^ y^ z^ 2z^_
a^ b^ c'-^ c
This we will take for the equation of the surface, the common vertex of the
cones being the origin. Let lx + my + nz = l be the equation of any j)lane
section ; then the corresponding cone will be
3.2 y2 ^2 2z
— + f;, + -, + — (Zx + WW + nz) = 0.
o-i b^ c' c ^ ''
lOS EXAMPLES ON CHAPTER V.
The section of this cone by the plane z = k ia clearly similar to the conic
a^ b-
"which proves the ]proposition.
Ex. 2. With a fixed point 0 on a conicoid for vertex, and a plane section
of the conicoid for base, a cone is described ; sheiv (i) that if the cone have
three perpendicular generating lines, the plane base loill meet the normal at O
in a fixed point ; and (ii) that if the normal at O be an axis of the cone, the
plane base will meet the tangent plane at 0 in a fixed straight line.
The most general equation of a conicoid, when the origin is on the
suiface and the plane 2 = 0 is the tangent plane at the origin, is
ax^ + 6?/- + c^^ + '^fyz + Igzx + 2hxy + 2^; = 0.
The equation of the cone whose vertex is the origin, and which passes
through the jpoints of intersection of the conicoid and the plane
lx-\-my-\-nz = l
is ax^ + li/ + cz^ + Ifyz + 2gzx + 2hxy -\-2z{lx + my + nz) = 0.
Now the condition that the cone may have three perpendicular generating
lines is
a + h + c + 2n=0 [Art. 109].
This shews that the intercept on the axis of z is constant ; which proves
(i). The conditions that the axis of z may be an axis of the cone are
[See Art. fiO] g + l = 0, and/+m = 0. Hence the plane meets the axes of x
and y in fixed points; which proves (ii).
Examples on Chapter V,
1. Shew that the area of the section of an ellipsoid, by
a plane which passes through, the extremities of three conjugate
diameters, is in a constant ratio to the area of the parallel central
section.
2. Given the sum of the squares of the axes of a plane
central section of a conicoid, find the cone generated by a normal
to its plane.
3. Shev/ that a plane which cuts off a constant volume from
a cone envelopes a conicoid of which the cone is the asymptotic
cone.
EXAMPLES ON CHAPTER V. 109
4. Shew tliat the axes of plane sections of the conicoid
x" y' z' ,
a 0 c
which pass through the line
X y s
I I 7)i~ n
lie on the cone whose equation is
x^\y zj\h' c'J y\z x,
-'-^Ki-a4G-;)&-»--
5. If through a given point {x^, y^, z^ lines be drawn each
of which is an axis of some plane section of ax^ + hy' + cz^ = 1,
buch lines describe the cone
' x-x^ 'y-y^ ^ ' z-z^
6. If the area of the section of
2 2
4- + - = 2a;
0 c
be constant and equal to a^, the locus of the centre is
7. If a conic section, whose plane is perpendicular to a gene-
rator of a cone, be a circle; the corresponding projection of the
reciprocal cone is a parabola.
8. Shew that the principal semi-axes of the normal section
of the cylinder which envelopes h^c^x^ + c^a^y^ + a^¥z' - d^b^c^y and
whose generating lines are parallel to
X _y _z
are the values of r given by
lib n A
+ T^ + -^ i = 0.
a'-r' b'-r^ c'-r'
110 EXAMPLES ON CHAPTER V,
9. Shew that the section of
2^' _ ^' - ?^
P ~ ? ~ "a
hy the ])hine Ix + my + w;:; = 0 is a rectangular hyperbola, if
10. Shew that all plane sections of
2 2
X y
— = 3
a 6
v/hich are rectangular hyperbolas, and which pass through the
point (a, /3, y), touch the cone
(«-")' ('J-PY A^-yY 0
a 0 a — 0
11. Find the locus of the vertices of all parabolic sections
of a paraboloid, whose planes are at the same distance from its
axis.
1 2. Shew that, if the plane Ix + my + nz = p cut the surface
ax' + hy^ + ca;^ = 1 in a parabola, the co-ordinates of the vertex
of the parabola satisfy the equation
ax
T
(1 2\,L^(1_2),«(1 l) = o.
\6 c) Til \c OjJ n \a bj
13. The area of the section of (ahcfyh\xyzy = 1 by the plane
which pastes through the extremities of its principal axes is
27r , /a + h + c
3^3
. /a ■¥ b + G\
14. A cone is described with vertex {/, g, h) and base the
section of the surface ax^ + by^ + cz^ = 1 made by the plane x=0 ;
shew that the equation of the plane in which this cone again meets
the surface is
X {af + Iff + c7i^ - 1) = 1f{nfx + Igy ^ chz - 1).
EXAMP].ES ON CHAPTER V. Ill
15. Shew tliat the foci of all parabolic sections of
c2
y
a 0
lie on the surface
2/^ z^\ (if _ 'z\ ah fif z^^
a b
a bj 4: \a^ b^J
16. Circles are described on a series of parallel chords of a
fixed circle whose planes are inclined at a constant angle to the
plane of the fixed circle.
Shew that they trace out an ellipsoid, the square on whose
mean axis is an arithmetic mean between the squares on the other
two axes.
17. Shew that if the squares of the axes of an ellipsoid
are in arithmetical progression the umbilici lie on the central
circular sections ; if they are in harmonic progression the circular
sections are at right angles ; if they are in geometrical progression
the tangent planes at the umbilici touch the sphere through the
central circular sections.
18. Points on an ellipsoid such that the product of their
distances from the two central circular sections is constant lie on
the intersection of the ellipsoid with a sphere.
19. If the diameter of the sphere which passes through two
circular sections of an ellipsoid be equal to its mean diameter, the
distances of the planes from the centre are in a constant ratio.
20. A sphere of constant radius cuts an ellipsoid in plane
curves ; find the surface generated by their line of intersection.
21. The hyperboloid x^ + if — '^ tan^ o. = a? is built uj) of thin
circular discs of cardboard, strung by their centres on a straight
wire. Prove that, if the wire be turned about the origin into the
direction (?, m, w), the planes of the discs being kept parallel
to their original direction, the equation of the surface will be
(^ix — Izf + {ny — mzf = n^ (z^ tan^ a + a^).
22. If a series of parallel plane sections of an ellipsoid be
taken, and on any sections as base a right cylinder be erected,
shew that the other plane section, in which it meets the ellijjsoid,
will meet the pla.ne of the base in a straight line whose locus will
be a diametral plane of the ellipsoid.
112 EXAMPLES ON CHAPTER V.
23. Any number of similar and similarly situated conies,
which are on a plane, are the stereographic projections of plane
sections of some conicoid.
24. The tangent plane at an umbilicus meets any enveloping
cone in a conic of which the umbilicus is a focus and the inter-
section of the plane of contact and the tangent plane a directrix.
25. The quadric ax^ + hy^ ^-cz^ =\ is turned about its centre
until it touches a'x' + h'if ■¥ c'z^ = \ along a plane section. Find
the equation to this plane section referred to the axes of either
of the quadrics, and shew that its area is
a + h-\- c — a —h' — g'
abc - ab'c'
CHAPTER VI.
Generating Lines of Conicoids.
127. In cones and cylinders we have met with examples
of curved surfaces on which straight lines can be drawn
which will coincide with the surface throughout their entire
length.
We shall in the present chapter shew that hyperboloids
of one sheet, and hyperbolic paraboloids, can be generated
by the motion of a straight line ; and we shall investigate
properties of those surfaces connected with the straight lines
which lie upon them.
Def. a surface through every point of which a straight
line can be drawn so as to lie entirely on the surface, is
called a ruled surface; and the straight lines which lie upon
it are called generating lines.
A ruled surface on which consecutive generating lines
intersect, is called a developable surface.
A ruled surface on which consecutive generating lines do
not intersect, is called a skew surface.
128, To find where the straight line, whose equations are
I m n '
meets the surface whose equation is F {x, y, z) = 0, we must
substitute a + /r, y8 + mr, and 7 + 72r for x, y, z respectively,
and we obtain the equation F {a + Ir, ^ + 7nr, 7 + nr) = 0.
s. s. G. 8
114 GENERATING LINES.
If the surface is of the k^^ degree, the equation for finding
r is of the k^^ degree ; hence any straight line meets a surface
of the k^'-^ degree in k points.
If, however, for any particular straight line, all the co-
efficients in the equation for r are zero, that equation will be
satisfied for all values of r ; and therefore every point on that
straight line will be on the surface. Since there are ^ + 1
terms in the equation of the k^^ degree, it follows that
k -h 1 conditions must be satisfied in order that a straight line
may lie entirely on a surface of the ^•'^ degree.
Now the general equations of a straight line contain four
independent constants, and therefore a straight line can be
made to satisfy four conditions, and no more.
It follows therefore, that, if the degree of a surface be
higher than the third, no straight line will, in general, lie
altogether on the surface. For special forms of the equations
of the fourth, or higher orders, we may however have
generating lines ; for example, the line whose equations are
y = mx and z = ni^ will, for all values of ???, lie entirely on the
surface whose equation is zx^ = y^.
If the equation of a surface be of the third degree, the
number of conditions to be satisfied is equal to the number
of constants in the general equations of a straight line.
Hence the conditions can be satisfied, and there will be a
finite number of solutions. The actual number of straight
lines (real or imaginary) wdiioh lie on any cubic surface is 27.
[See Cambridge and Dublin Math. Journal, Vol. IV.]
The number of conditions to be satisfied, in order that a
straight line may lie entirely on a conicoid, is three. Since
the number of conditions is less than the number of constants
in the general equations of a straight line, the conditions can
be satisfied in an infinite number of ways, so that there are
an infinite number of generating lines on a conicoid; these
generating lines may however all be imaginary, as is
obviously the case when the surface is an ellipsoid.
129. A generating line on any surface touches the
surface at any point 0 of its length, for it passes through a
GENERATING LINES OF CONICOIDS. 115
point of the surface indefinitely near to 0; hence the tangent
plane to any surface at a point through which a generating
line passes will contain that generating line.
130. The section of a conicoid by the tangent plane at
any point through which a generating line passes, will be a
conic of which the generator forms a part ; the conic must
therefore be two straight lines.
Hence, through any point on a generating line of a
conicoid another generating line passes, and they are both in
the tangent plane at the point.
The two generating lines in which the tangent plane to a
conicoid intersects the surface are coincident when the conicoid
is a cone or a cylinder.
131. Since any plane section of a conicoid is a conic, any
plane which passes through a generating line of a conicoid
will cut the surface in another generating line ; and both
generating lines are in the tangent plane at their point of
intersection. Hence, ani/ plane through a generating line of
a conicoid touches the surface, its point of contact being the
point of intersection of the tw^o generating lines which lie
upon it.
132. To find ivhich of the conicoids are rided surfaces.
If a conicoid have one generating line upon it, and we
draw a plane through that generating line and any point
P of the surface, this plane will cut the surface in another
generating line, which must pass through P.
Hence, if there be a single generating line on a conicoid,
there will be one, and therefore by Art. 13 0^ two generating
lines, through every point on the surface.
We can therefore at once determine whether a conicoid
is or is not a ruled surface, by finding the nature of the inter-
section of the surface by the tangent plane at any particular
point.
The equation of the tangent plane at the point (a, 0, 0) of
x^ v^ z^
the conicoid -5+'7-2+-i = l is x — a\ this meets the surface
a^~ h""- & ' g_2
116 GENERATING LINES OF CONICOIDS.
in straight lines whose projection on the plane x = 0 are
7/' JZ
given by the equation ± yr, + -2 = 0. These lines are clearly
0 c
real when the surface is an hyperboloid of one sheet, and
imaginary when the surface is an ellipsoid, or an hyperboloid
of two sheets.
Hence the hyperboloid of one sheet is a ruled surface.
The hyperbolic paraboloid is a particular case of the
hyperboloid of one sheet ; hence the hyperbolic paraboloid is
also a ruled surface.
This can be proved at once from the equation of the
paraboloid. For, the tangent plane at the origin is 2; = 0, and
this meets the paraboloid ax^ + bif + 22 = 0 in the straight
lines given by the equations ax^ + bif =0, z = 0 ; the lines
are clearly real when a and b have different signs, and are
imaginary when a and 6 have the same sign.
Hence an hyperboloid of one sheet (including an hyper-
bolic paraboloid as a particular case) is the only ruled conicoid
in addition to a cone, a cylinder, and a pair of planes.
133. To shew that there are two systems of generating
lines on an hyperboloid of one sheet.
Since any plane meets any straight line, the tangent
plane at any point P on an hyperboloid of one sheet will
meet all the generating lines of the surface, and the points
of intersection will be on the surface. But the tangent
plane cuts the surface in the two generating lines through
P; hence every generating line of the hyperboloid must
intersect one or other of the two generators PA, PB which
pass through any point P on the surface.
Now no two of the generating lines which meet the same
generator can themselves intersect, for otherwise there would
be three generating lines in a plane, which is impossible,
since every plane section is a conic.
Hence there are two systems of generating lines, which
are such that all the members of one system intersect PB,
but do not themselves intersect ; and all the members of the
GENERATING LINES OF CONICOIDS. 117
other system intersect PA, but do not themselves intersect.
Since the position of P is arbitrary it follows that every
member of one of the two systems of generating lines meets
every member of the other system.
134. If a straight line intersect a conicoid in ^/^reg points,
it will entirely coincide with the surface ; and hence, to have
a generating line of a conicoid given, is equivalent to having
three points given.
To have three non-intersecting generating lines given is
therefore equivalent to having nine points given, so that
[Art. 50] three non-intersecting generators are sufficient to
determine the conicoid on which they lie.
If a Ime meet three non -intersecting lines, it will meet
the conicoid of which they are generators in three points,
namely in the three points in which it intersects the three
lines ; and hence it must itself be a generator of the surface.
Hence, the straight lines which intersect three fixed non-
intersecting straight lines are generators of the same system
of a conicoid, and the three fixed lines are generators of the
opposite system of the same conicoid. [See Art. 49, Ex. 2]
135. Since any line which meets three non-intersecting
straight lines is a generating line of the conicoid on which
they lie, it follows that the only lines which meet the three
lines and which also meet a fourth given straight line are
the generators of the surface, of the system opposite to that
defined by the given lines, which pass through the points
where the conicoid is met by the fourth given straight line.
But the fourth straio-ht line will meet the conicoid in two
points only, unless it be itself a generator of the surface.
Hence two straight lines, and two only, will, in general,
meet each of four given non-intersecting straight lines ; but if
the four given straight lines are all generators of the same
system of a conicoid, then an infinite number of straight
lines will meet the four, which will all be generators of the
opposite system of the same conicoid.
Ex. 1. Two planes are drawn, one through each of two intersecting
generating lines of a conicoid ; shew that the planes meet the surface in two
other intersecting generating lines.
lis GENERATING LINES OF CONICOIDS.
Ex. 2. Shew that the plane through the centre of a conicoid and any
generating line, will cut the surface in a parallel generating line, and will
touch the asymptotic cone.
Ex. 3. A conicoid is described to pass through two non-intersecting given
lines and to touch a given plane. Shew that the locus of the point of contact
is a straight line.
Let the given lines meet the given plane in the points A , B respectively.
Then, the given plane will cut the surface in two generating lines, one
of which will intersect both the given lines; hence, since the points of
intersection must be A and B, the point of contact must be on the
line AB.
Ex. 4. The lines through the angular points of a tetrahedron perpen-
dicular to the opposite faces are generators of the same system of a
conicoid.
Let A A', BB\ CC, DD' be the four perpendiculars, and let a, /S, y, S be
the orthocentres of the faces opposite to A, B, C, D respectively. Then, it is
easy to prove that the lines through a, /3, 7, 5 parallel respectively to
AA', BB', CC\ DD' will meet all the four perpendiculars. Since the four
perpendiculars are met by more than two straight lines, they are generators
of the same system of a conicoid; and the four parallel lines through
a, j3, 7, 5 are generators of the opposite system of the same conicoid.
Ex. 5. If a rectilineal quadrilateral ABCD be traced on a conicoid, the
centre of the surface is on the straight line which passes through the middle
points of the diagonals AC, BD.
The planes BAD, BCD are the tangent planes at ^, C respectively, and
BD is their line of intersection ; hence the centre of the conicoid is on the
plane through BD and the middle point of AC. Similarly the centre is on the
plane through AG and the middle point of BD.
Ex. 6. If a rectilineal hexagon be traced on a conicoid, the three lines
joining opposite vertices will meet in a point, and the three lines of inter-
section of the tangent planes at opposite vertices lie in a plane. [Dandelin.]
Let ABCDEF be the hexagon. Intersecting generators of a conicoid are
of different systems; therefore AB, CD, EF are of one system, and BC, DE,
FA of the opposite system ; so that opposite sides of the hexagon are of
different systems, and therefore will intersect. Each of the diagonals
AD, BE, CF is the line of intersection of two of the planes through pairs of
opposite sides; therefore AD, BE, CF meet in a point, namely in the point
of intersection of the three planes through pairs of opposite sides.
Let X be the point of intersection of AB and DE, Y the point of inter-
section of BC and EF, and Z of CD and FA. The tangent planes at A, D,
namely the planes FAB, CDE, intersect in the line XZ ; the tangent planes
at B, E intersect in the line XY; and the tangent planes at C, F intersect in
the line YZ. Hence the three lines of intersection of the tangent planes at
opposite vertices lie in the plane .X YZ.
Ex. 7. Four fixed generators of the same system cut all generators
of the opposite system in a range of constant cross-ratio. [Chasles.]
Let any three generators of the opposite system cut the fixed generators in
ANGLE BETWEEN GENERATORS. 119
the points A,B,C,D', A', B', C, D' and A", B", C", D" respectively. Then,
the four planes through A"B"C"D" and the fixed generators cut all other
straight lines in a range of constant cross-ratio [Art. 36] ; we therefore have
{A'B'G'D'} = {ABCB}.
Ex. 8. The Hnes joining corresponding points of two homographic
systems, on two given straight lines, are generating hnes of a conicoid.
136. To find the angle between the two generating lines
through any point of an hyperholoid.
The section of an hyperholoid of one sheet hy the
tangent plane at any point is similar and similarly situated to
the parallel central section. Hence the generating lines
through any point are parallel to the asymptotes of the
parallel central section. Let the equation of the surface be
and let /, g, h be the co-ordinates of the point P through
which the generating lines pass.
Let a^, fi- be the squares of the axes of the central section
which is parallel to the tangent plane at P, and let 0 be the
angle between the generating lines through P.
e , s
Then tan^=\/-l-,
^ a
and therefore
tan(9 = 2\/'^-2^^2.
Now the sum of the squares of three conjugate semi-
diameters is constant, and also the parallelepiped of which
they are conterminous edges. Hence
a'-¥^'+OP' = a' + b'-c\
and a^p = J — 1 . ahc.
Hence we have
^''^'=^p{a' + V-,?-OF')'
137. We can write the equation of an hyperholoid of one
120 EQUATIONS OF GENERATORS.
sheet in such a way as to shew at once the existence of
generating lines. For, the equation
d' ^ h' c' '
is equivalent to
x' z' f
and it is evident that all points on the line of intersection of
the planes whose equations are
a c \ bj ' a c \\ h
are on the surface; and by giving different values to A, we
obtain a system of straight lines which lie altogether on the
surface. The generating lines of the other system are
similarly given by the equations
a c V h) ^ a G \\ h
We can find in a similar manner the equations of the
generating lines of the paraboloid
2 2
ci" b' ~ "''•
The equations of the generators of one system aro
a 0 a b \
and of the other system
a b "' tt 6 X*
138. The equations of the generating lines which pass
through any point on an hyperboloid of one sheet can bo
obtained in the following manner.
The co-ordinates of any point on the surface can be
expressed in terms of two variables 6 and <^, where
x = a cos 0 sec (f>, y = b sin 6 sec cp, and z = c tan <^.
GEXERATIXG LINES OF AN HYPERBOLOID. 121
This is seen at once if we substitute in the equation of
the hyperboloid.
The two generating lines through the point P are the
lines of intersection of the surface and the tangent plane at
P. Now, the equation of the tangent plane at {9, (f>) is
- cos 6 sec (b + V sin 9 sec (h tan 6 = 1;
a ^ 0 ^ c
hence the tangent plane meets the plane 2: = 0 in the
line whose equations are
- COS ^ + T^sin ^ = cos<f>, z = 0 (i).
a 0
If this line meet the section of the surface by 2^ = 0 in
the points A, B, whose eccentric angles are a, (3 resj^ectively,
we have from (i)
or a = 6 ■{- (f), Q^idi P = 6 — (f> (ii).
Now AP, BP are the generators through P ; hence from
(ij), ^ + 0 is constant for all points on the generator AP, and
6 — ^ is constant for all points on the generator BP.
The direction-cosines of AP are proportional to
a (cos a — cos ^ sec ^), 6 (sin a — sin ^ sec 0), - c tan ^ ;
or proportional to
cos (6 + (f)) cos (f) — cos d J sin (6 + cf)) cos cf) — sin 0
sm 9 sm 9
or to a sin {6 + <j6), —h cos {9 -{- (j)), c;
hence the equations of AP are
X — a cos 9 sec ^ _y — h sin 9 sec cf) _z — c tan (^
a sin (9 + (j>) —b cos {9 -{■ <^) c
Similarly the equations of BP are
X —a cos 9 sec 4* _y — 6 sin ^ sec (f> z — c tan (/>
a sin {9 — (j)) — 6 cos (^ — </>) "~ — c
122 GENERATING LINES OF A PARABOLOID.
Cor. The equations of the generators, through the point
on the principal elliptic section whose eccentric angle is 0,
are
x— acosd _y — hsmd _ z
a sin 0 —h cos 6 ~ c'
These equations may also be obtained as follows :
The line whose equations are
{c— a cos 6 _y — h sin 0 _z _
I 111 n '
will meet the surface, where
(a cos 6 + Iry (b sin 6 + mrf _ r?V _
a' "^ ¥ ^ ~
Hence, in order that the straight line may be a generating
line, we must have
a' "^ b' c' ~ '
, I cos 6 m sin 6 ^
and 1 ? — = 0.
a 0
Whence
I m n
a b c
sin 6 —cos6 ± 1 *
The equations of the generators are therefore
x — a cos 6 _y — b sin 6 z
a sin 6 —b cos d ~ c'
139. To find the equations of the generating lines
through any point of a hyperbolic paraboloid.
Let the equation of the paraboloid be
2 2
d' b' ~ "''
GENERATING LINES OF A PARABOLOID. 123
Let the equations of any line he
I Til n '
The points of intersection of the line and the surface are
given by the equation
a- V
Hence, in order that the straight line may be a generating
line, we must have
-:2--T7 = 0 (1),
1 2 ?72/3
~~b''
a'
n = 0 (ii),
and ^ - C - 27= 0 (iii).
The equation (iii) is satisfied if (ot, j3, 7) be any point on the
surface ; from (i) we have - = ± -7- ; and, substituting^ in (ii),
a 0 o \ /
we obtain
Z _ m _ n
a + b a _ B '
a 0
Hence the equations of the two generating lines through
the point (a, yS, 7) are
x — OL_y — P_z — 'y
"^~Tr"^r^ '^'''^•
-+ r
a 0
It is clear from the above that any generator of the
paraboloid is parallel to one or other of the two planes
- + V- = 0.
a 0
124 GENERATING LINES OF A PARABOLOID.
Ex. 1. Shew that the projections of the generating lines of an hyper-
boloid on its principal planes are tangents to the principal sections.
The tangent plane at any point P on a principal section is perpendicular
to that section. Hence the jirojection on the principal plane of any hne in
the tangent plane at F is the tangent line which is in the principal plane.
This proves the proposition, since the generating lines through P are in the
tangent plane at P.
Ex. 2, Find the locus of the point of intersection of perpendicular
generators of an hyperboloid of one sheet.
If the generating lines at any point P are at right angles, the parallel
central section is a rectangular hyperbola, and therefore the sum of the
squares of its axes is zero. But the sum of the squares of three conjugate
semi-diameters of the hyperboloid is constant and equal to a^ + 6^ _ c-. Hence
OP- = a' + b''^-C'; so that the points are all on a sphere.
This is the result we should obtain by putting tan ^ = qo in the result of
Art. 136. We could also find the locus by using the equations of Art. 138.
Ex. 3, Find the angle between the generating lines at any point of
a hyperbolic paraboloid.
The result is obtained at once from equations (iv), Art. 139. The gene-
rators are at right angles, if
a2_^2 + ^"_P 0, orif27 + a2_52 = 0.
Thus generators which are at right angles meet on the plane z = ^[b' - a-).
Ex. 4. A line moves so as always to intersect three given straight lines
which are all parallel to the same plane : shew that it generates a hyperbolic
paraboloid.
Ex. 5. A line moves so as always to intersect two given straight lines
and to be parallel to a given plane : shew that it generates a hyjoerbolic
paraboloid.
Ex. 6. AB and CD are two finite non -intersecting straight lines; shew
that the lines which divide AB and CD in the same ratio are generators of
one system of a hyperbolic paraboloid, and that the lines which divide AC
and BD in the same ratio are generators of the opposite system of the same
paraboloid.
Examples on Chapter YI.
1. A straight line revolves about a fixed straight line, find
the surface generated. '^/.'^^h ^ -H y\^ -^ U'l" z 0
2. If four non-intersecting straight lines "be given, shew that
the four hyperboloids which can be described, one through each
set of three, all puss through two other straight lines.
EXAMPLES ON CHAPTER TT. 125
3. Find the equation of the conicoid, three of whose generat-
ing lines are x = i), y = a ; y=^0,z^a; z = 0, x^a. Shew that it
is a surface of revolution, and find the eccentricity of its meridian
section.
4. Find all the straiglit lines w^hich can he drawn entirely-
coinciding (i) with the isurface y^ — z^ = 3a^x; and (ii) with the
surface y'^ — z^ = ia^x.
5. Normals are drawn to an h^'perboloid of one sheet at
every point through which the generators are at right angles ;
prove that the point><, in which the normals intersect any one of
the principal planes, lie in an ellipse.
6. Given any tliree lines, and a fourth line touching the
hyperboloid through the three lines, then will each one of the four
lines touch the hyperboloid through the other three lines.
7. A line is drawn through the centre of ax" + by' + cz" = 1
perpendicular to two parallel generators. Shew that such lines
generate the cone
X- ?/ ;:;-
— +;-+- ^ 0.
a 0 c
S. If two generators of an hyperboloid be taken as two of the
axes of co-ordinates shew that the equation of the surface is
of the form
z^ + 2/yz + 2gzx + 2hxy + 2wz = 0.
9. The generators through any point Ii on a ruled quadric
intersect the generators at a fixed point 0 in P and Q. Shew-
that if the ratio OP : OQ is constant, B lies on a plane section of
the quadric which passes through 0.
10. Find the locus of a point on an hyperboloid the genera-
tors through which intercept on two fixed generators portions
wdiose product is constant.
11. If all the generators to an hyperboloid of one sheet be
projected orthogonally on the tangent plane at any point, their
envelope will be an hyperbola.-
12. Find the equation of the locus of the foot of the perpendi-
126 EXAMPLES OX CHAPTER VI.
cular from the point (a, 0, 0) on the different generating lines
of the surface
^i!^.^^i^l
a' b' c'
13. Prove that the product of the sines of the angles that
any generator makes with the planes of the circular sections is
constant.
14. If CP, CD be conjugate semi-diameters of the principal
elliptic section, and generators through F and U meet in T, prove
that TF' ■= CD' + c^ TD' = CP' + c\
1 5. If two generators drawn from 0 intersect the principal
ellipse in points P, P\ at the ends of conjugate diameters, then will
16. The angle between the generating lines through the point
{xyz) of the quadric — V— -r — — \ is cos~^ ^^ — ~ ^ where \^ X^,
are the roots of the equation
a b c A, — A„
+ ^^f — TT + -. = 0.
a{a-\-X) b {b + X) c{g + X)
17. Shew that the shortest distances between jjeneratins^ lines
of the same system drawn at the extremities of diameters of the
principal elliptic section of the hyperboloid, whose equation is
2 2 2
lie on the surfaces whose equations are
cxy ohz
a? ■\-'if a' — b^'
18. Prove that in general through two non-intersecting
straight lines two and only two conicoids of revolution can be
described.
19. The locus of points on (cibcjgli) {xijzf — \ at which the
generators are at right angles is the intersection of the surface
"WT-th the sphere
' «, ^^ g I
li, h, f (x" + 2/^ + ^') -he ^ ca->t ah ~f' ~ ff - A''.
r
EXAMPLES ON CHAPTER VI. 127
20. Having given two generating lines that intersect and two
j)oints on an hyperboloid, shew that the locus of the centre is
another hyperboloid bisecting the straight lines joining the two
points to the intersection of the generators.
21. Shew that the volume of every parallelopiped which
can be placed so that six of its edges lie along six of the generators
of a given hyperboloid of one sheet is the same.
22. A solid hyperboloid has its generators marked on it and
is then drawn in perspective : shew that the points of intersection
of the representatives of consecutive generators of the same system
will lie on an hyperbola.
23. If two points F, Q be taken on the surface
such that the tangent planes at those points are at right angles to
one another, then will the two generating lines through F appear
to be at right angles when seen from Q.
24. If two conicoids have a common generator, two of their
common tangent planes through that generator have the same
point of contact.
25. If AOA', FOB', COC be any three straight lines, the
lines AF, CA' F'C are generators of one system, and A'F\
C'A, FC are generators of the other system, of the same hyper-
boloid.
26. Deduce Pascal's Theorem from Dandelin's Theorem.
[Ex. 6. Art. 135.]
27. If from any point on a hyperbolic paraboloid perpen-
diculars be let fall on all the generators of the surface of the same
system, they will form a cone of the second degree.
28. If from any point on the surface of an hyperboloid of one
sheet perpendiculars be drawn to all the generators of the same
system, they will form a cone of the third degree.
29. The normals to a conicoid, at all points of a generating
line, lie on a hyperbolic paraboloid.
30. In every rectilinear octagon AFCDEFGH which is on
a conicoid, the eight lines of intersection of the tangent planes at
A,D', A, F; G,F; G, D; E, II -, E,F\ C, F -, C, H are all
generators of another conicoid. Also the lines AD, AF, GF, GD,
HE, lie, CF, EB are all generators of another conicoid.
CHAPTER VII.
Systems of Coxicoids. Tangential Equations.
ReCIPKO CATION.
140. Since the general equation of the second degree
contains nine constants, it follows that a conicoid will pass
through any nine points, and that an infinite number of
conicoid s will pass through eight points.
If S = 0, and S' = 0 represent any two conicoids which
pass through eight given points, then the equation
S+\S' = 0 w^ill be of the second degree, and will therefore
represent a conicoid, and it is clear that the conicoid
>S' + XS' = 0 will pass through all points common to >S' = 0 and
S' = 0. Also, by giving a suitable value to X, the conicoid
/S'+X>S' = 0 can be made to pass through any ninth point;
and therefore will represent any conicoid through the eight
given points.
Since the conicoid S+\S' = 0 not only passes through
the eight given points, but also through all points on the
curve of intersection of S=i) and S' = 0, we see that all
conicoids through eight given points have a common curve of
intersection.
SELF POLAR TETRAHEDRON.
120
141. Four cones will pass through the curve of inter-
section of two conicoids.
Let the equations of any two conicoids be F^ {x, y, z) =0
and F^ {x, y, z) = 0. The equation of any conicoid througli
their curve of intersection is of the form
F^(x,y,z) + \F^{x,y,z)=0.
The above equation will represent a cone, if
dj + Xo^ , Aj + Xh^ , g^ + Xr/., , u^ + Xu,^ \ = 0.
h^ + Xh^, b^ + Xb^, /i + Vr* ^'i + ^^2
9 1 + ^92 y /l + ^/2 ' ^1 + ^^2 , w^ + Xtu^
u^ + Xu^, v^ + Xv^, w^-\-Xw^, d^-\-Xd,^
Since the equation for determining X is of the fourth
degree, four cones, real or imaginary, will pass through the
points of intersection of two conicoids.
142. The vertices of the four cones through the curve of
intersection of tivo conicoids are the angular j^oints of a
tetrahedron which is self -polar with respect to any conicoid
luhich passes through that curve.
Take the vertex 0 of one of the cones for origin, and
let F^ [x, y, z) =0 and F^ {x, y, z) = 0 be the equations of the
two conicoids. Then the equation of the cone will be of the
form F^ (x, y, z) -r XF^ (x, y, z) = 0. But, since the origin
is at the vertex of the cone, its equation will be homo-
geneous. We therefore have
u^ + Xu^
or
'U V
_1 _ _1 _
w^
d^
d.
(i).
Now the equation of the polar plane of 0 with respect to
any conicoid
■^1 (^> 2/> ^) + H'K {^> y> ^) = 0, is
(^1 + /^^a) ^ + (^ + 1^%) 2/ + K + /^^a) z-\-d^ + fjLd^ = 0',
and, from (i), it is clear that this polar plane coincides with
UjX + v^y + lu^z + c?j = 0
for all values of fx.
S. S. G. 9
130 CONICOIDS THROUGH SEVEN GIVEN POINTS.
Hence 0 lias the same polar plane with respect to all
conicoids throuGjh the curve of intersection of the two driven
conicoids.
Now the polar plane of 0 with respect to any one of the
other cones through the curve of intersection will pass
through the vertex of that cone, and hence the vertices of
the other three cones are on the polar plane of 0 with respect
to any conicoid through the curve of intersection of the given
conicoids: this proves the theorem.
143. If S=0 be the equation of any conicoid, and
ayS = 0 the equation of any two planes, then will 8 — Xol/S = 0
be the general equation of a conicoid which passes through
the two conies in which >S^ = 0 is cut by the planes a = 0
and ^ = 0.
If now the plane a = 0 be supposed to move up to and
ultimately coincide with the plane (3 = 0, we obtain the form
S — X/3^ = 0, which represents a system of conicoids, all of
which touch S = 0 where it is met by the plane /3 = 0.
The surfaces >S^ — X-x/S = 0 and S = 0 touch one another at
the two points where they are cut by the line whose equa-
tions are a = 0, /5 = 0. For at either of these points the
surfaces have two common tangent lines, namely the tangent
lines to the sections by the planes a = 0 and yS = 0.
144. All conicoids which pass through seven given j^oints
pass through another fixed point.
Let >Si, =0, S^ = 0, S^=0 be the equations of any three
conicoids through the seven given points.
Then the conicoid whose equation is S^ + \S^ + fiS^ = 0
will clearly pass through all points common to S^ = 0, 8^ = 0
and >S3 = 0 ; and S^ + \S^ + ^mS^ = 0 can be made to coincide
with any conicoid through the seven given points, for the
two arbitrary constants X and fM can be so chosen that
the surface will pass through any two other points. Now
the three conicoids S^ = 0, 8^ = 0, 8^ = 0 have eight common
points, all of which are on 8^ + \8,^ + yLt>S*3 = 0 ; this proves
the theorem.
ThuS; corresponding to any seven given points there is an
EXAMPLES. 131
eighth point associated with them, such that any conicoid
through seven of the points will also pass through the eighth
point ; and it should be remarked that in order that a system
of conicoids may have a common curve of intersection, they
must have eight points in common which are not so associated.
Ex. 1. All conicoids through the curve of intersection of two rectangular
hyperboloids are rectangular hyperboloids.
[A rectangular hyperboloid is one whose asymptotic cone has three per-
pendicular generating lines.]
The asymptotic cone of a conicoid has three generators at right angles
when the sum of the coefficients of x-, y'^ and z^ in the equation of the surface
is zero. Now the sum of the coefficients of x-, y'^ and z^ in S' + \S'=0 will be
zero, if that sum is zero in S and also in S'. This proves the proposition.
Ex. 2. Any two plane sections of a conicoid and the poles of those planes
lie on another conicoid.
Let ax- + by- + cz"^ + d = 0 be the conicoid, and let {x\y\ z') and (x", y", z")
be any two points. The equations of the polar planes of these points will be
axx' + byy' + cz£ + cZ= 0 and axx" + byy" + czz" + d=0.
The conicoid
X (ax2 + hxf- + cz^ + d) - {axx + byy' + czz' + d) (axx" + byy" + czz" + rf) = 0
is the general equation of a conicoid through the two plane sections. The
conicoid will pass through (x', y , 2;') if \ be such that
X (ax'2 + by'"- + cz"- + d) - (ax'2 + bxj"^ + cz'"- + d) {ax'x" + by'y"+ cz'z" + d) = 0,
or if X=ax'x"+by'y" + cz'z" + d.
The symmetry of this result shews that the conicoid will likewise pass
through (x", y", z").
Ex. 3. Through the curve of intersection of a sphere and an ellipsoid four
quadrlc cones can be drawn; and if diameters of the ellipsoid be drawn
parallel to the generators of one of the cones the diameters are all equal.
Also the continued product of the four values of such diameters is equal to the
continued product of the axes of the ellipsoid and of the diameter of the
sphere.
Let the equations of the ellipsoid and of the sphere be
"^^Vl + t^l
a^ b" c-
and (x - a)2 + (y - /3)2 + (2 - 7)2 = r\
The general equation of a conicoid through the curve of intersection is
9—2
132 EXAMPLES.
This conicoid will be a cone, if the co-ordinates of the centre satisfy the
equations
and ^ax-^y-yz + a- + ^- + y^-r^-\ = 0.
Eliminating x, y, z we have
„2„2 5202 g2-v2
If, for any particular value of X, the conicoid given by (i) is a cone, the
equation of the cone, when referred to its vertex, takes the form
and therefore the direction-cosines of any diameter which is parallel to one
of the generating lines of the cone, satisfy the equation
!l. ^ !^ _ _ 1
^•2 + 5a + ga - X *
Hence the square of the semi-diameter is constant and equal to - \.
Hence also the continued product of the squares of the four values of
the semi-diameters is equal to the product of the four roots of the equation (ii) ;
and the product of the roots is easily seen to be a'-b^c-r^.
Ex. 4. The locus of the centres of all conicoids which pass through seven
given points is a cubic surface, lohich passes through the middle point of the
line joining any pair of the seven given points.
Let ^1 = 0, >S'2 = 0, *S'3 = 0 be any three conicoids through the seven given
points ; then the general equation of the conicoids is
The equations for the centre are
dx dx dx *
dS, ^dS^ dS^ ^
dy dy dy
dz dz dz
TANGENTIAL EQUATIONS. 133
Hence the equation of the locus of the centres, for different vahies of \
and ^l, is
dy'
dS.
dS^
dS^
dz '
dz
=0,
dS^ dS^ dS^
dx ' dx ' dx
dS^
dy
d^
dz
which is a cubic surface, since —^ &c. are of the first degree.
dx
Now, to have the centre of a conicoid given, is equivalent to having three
conditions given ; hence a conicoid which has a given centre can be made to
pass through any six points. Hence, if ^, i> be any two of the seven given
points, one conicoid whose centre is the middle point of AB will pass through
A and through the remaining five points ; and a conicoid whose centre is the
middle point of AB, and which goes through A, must also go through B.
Thus the middle point of ^Z> is a point on the locus of centres ; and so also
is the middle point of the line joining any other pair of the given points.
[Messenger of Mathematics, vol. xiii. p. 145, and xiv. p. 97.]
Tangential Equations.
145. If the equation of a plane be Ix + mi/ + nz +1 = 0,
then the position of the plane is determined if I, m, n are
known, and by changing the values of I, m and n the
equation may be made to represent any plane whatever.
The quantities I, m, and n which thus define the position of
a plane are called the co-ordinates of the plane. These co-
ordinates, when their signs are changed, are the reciprocals of
the intercepts on the axes.
If the co-ordinates of a plane be connected by any relation,
the plane will envelope a surface; and the equation which
expresses the relation is called the tangential equation of the
surface.
146. If the tangential equation of a surface be of the n^^
degree, then n tangent planes can be drawn to the surface
through any straight line. For, let the straight line be given
by the equations ax + by + C2 + 1 = 0, ax + h'y -h c'^ -I- 1 = 0 ;
then the co-ordinates of any plane through the line will be
— , -:. and — . If these co-ordinates be sub-
1+X' H-\ 1 + X
134 CENTRE OF CONICOID.
stituted in the given tangential equation, we shall obtain an
equation of the 7^*'' degree for the determination of \, which
proves the proposition.
Def. A surface is said to be of the ?i"' class when 71
tangent planes can be drawn to it through an arbitrar}^
straicfht line.
147. We have shewn in Art. 57 that the plane
Ix + my + nz + 1 = 0
will touch the conicoid whose equation is
ax^+ hy^+ cz^ + 2fi/z + 2gzx + 2hxy + 2ux-\- 2vy + 2wz +cZ = 0,
if Ar + Enx" + O/i' + 2Fmn + 2Gnl + 2Hlin
+ 2Ul+2Vm+2Wn + D = 0,
where A, B, C... are the co-factors of a, h, c... in the dis-
criminant.
Hence the tangential equation of a conicoid is of the
second degree.
Conversely every surface whose tangential equation is of
the second degree is a conicoid.
148. Since the tangential equation of a conicoid is of the
second degree, which in its most general form contains nine
constants, it follows that a conicoid can be made to satisfy
nine conditions and no more ; and in particular a conicoid
can be made to touch nine given planes.
149. To find the Cartesian co-ordinates of the centre of the
conicoid given by the general tangential equation of the second
degree.
The two tangent planes to the conicoid which are parallel
to the plane x=0 are those for which m = n = 0. The values
of I are therefore given by the equation al'^ + 2id + cZ = 0.
Now the centre of the surface is on the plane midway
between these: and hence the centre is on the plane ic = -, .
a
DIRECTOR-SPHERE. 135
Similarly the centre is on the planes y = -j, and -2^ = 7 •
Hence the required co-ordinates are -^ , -^ , -v . [See
Art. 7G.]
150. We may take the equation of the moving plane to
be Ix -f my -\-nz + p = (d\ and the j^lane will envelope a surface
if I, 711, n, p be connected by a homogeneous equation ; for
any homogeneous equation in I, m, n, p would be equivalent
to an equation between the constants - , — , - .
p p p
If we take Ix -\-my + nz +2^ — ^ fo^ the equation of the
plane, we may suppose I, m, n to be the direction-cosines of
the normal to the plane.
151. To find the director-sphere of a conicoid whose
tangential equation is given.
If we eliminate p between the equation of the surface and
the equation Ix + my + nz -^^ p — O, we shall obtain a relation
between the direction-cosines of any tangent plane which
passes through the particular point {x, y, z). The relation
will be
ar -f- hn^ + cn^ + d(lx + my + nzf -f- 2fmn + 2gnl -f 2hhn
— 2 (ul + vm + wn){lx -f my + nz) = 0.
If {x, y, z) be a point on the director-sphere, three per-
pendicular tangent planes will pass through it ; the above
relation must therefore be satisfied by the direction-cosines
of each of three perpendicular planes. Hence, by addition,
we have
a-{-h + G— 2ux — 2vy — 2wz + d(x'^ + y^-\- z') = 0,
which is the required equation of the director-sphere.
152. If S=0 and 8' = Ohe the tangential equations of
any two conicoids which touch eight given planes, then the
equation S -\- \S' = 0 will be of the second degree, and will
therefore be the tangential equation of a conicoid; and it is
clear that the conicoid S + XS' = 0 will touch the common
13G CONICOIDS WHICH TOUCH SEVEN PLANES.
tangent planes of aS^ = 0 and >S^' = 0, for if the co-ordinates of
any plane satisfy the equations S = 0 and S' = 0, they will
also satisfy the equation S -hXS' = 0. Also, by giving a
suitable value to X, the conicoid >Si + \S' = 0 can be made to
touch any ninth plane : it will therefore represent any coni-
coid touching the eight given planes.
153. If >Sfj = 0, S,^ = 0, >Si3 = 0 be the tangential equations
of any three conicoids which touch seven given planes ; then
the conicoid whose tangential equation is S^ -\- XS,^ + /jlS^ = 0
will touch each of the seven given planes, for if the co-
ordinates of any plane satisfy the three equations S^ = 0,
S^ = 0 and S^ = 0, it will also satisfy the equation
>sf^ + xs,^ + fxS^ = 0.
Also, by giving suitable values to X and fi, the conicoid
S^+XS, + fMS^ = 0
can be made to touch any tiuo other planes ; hence
S, + XS^ + fiS.^ = 0
is the most general equation of a conicoid which touches the
seven given planes.
Similarly, if S, = 0, S,= 0, S,= 0 and >S^, = 0 be the
tangential equations of any four conicoids which touch six
given planes, >S\ -f XS,^ + f^S^ + vS^ = 0 will be the general
tangential equation of the conicoids which touch those six
planes.
Ex. 1. The centres of all conicoids which touch eight given planes are on a
straight line.
II S = 0 and S' = Ohe the equations of any two conicoids which touch the
eight given planes, then S + \S' = 0 will be the general equation of a conicoid
touching them. The centre of the conicoid is given by
_u + Xw' _v + \v' _w + \w'
Eliminating X we obtain the equation of the centre locus, namely
dx-u _ dy-v dz-w ^
d'x -u'~ d'y -v'~ d'z - w' '
hence the locus is a straight line.
EXAMPLES.-^ 137
Ex. 2. The centres of all conicoids which touch seven given planes are on
a plane.
li S = 0, S' = 0, S" = 0 be the equations of three conicoids which touch the
seven given planes, then the general equation of a conicoid which touches the
planes will be S + \S' + fxS" = 0.
Ex. 3. The director-spheres of all conicoids lohich have eight common
tangent planes have a common radical plane.
The director-sphere of the conicoid S + \S' = 0 is
a + b + c- 2ux - 2v7j - 2ivz + d (x- + y^ + z-)
+ \{a' + h' + c'- 2iix - Iv'ij - 2w'z + d' {x" + ?/2 + z") J = 0.
Ex. 4. The director-spheres of all conicoids ivhich touch six given planes
are cut orthogonalhj hy the same sphere. [P. Serret's Theorem.]
If Ci = 0, 0.2 = 0, C3 = 0 and C^ = 0 be the equations of any four conicoids
which touch the six planes; then the general equation of the conicoids
will be
Now from Art. 151 we see that the equation of the director-sphere of a
conicoid is linear in a, b, c, &c. It therefore follows that, if 8^ = 0, So=0,
S^ — 0 and S^ = 0 be the equations of the director-spheres of the conicoids
Ci = 0, C'.3 = 0, C.^ = 0 and 64 = 0 respectively, the equation of the du'ector-
sphere of C^ + XCg -t- fiC^ + vC^ = 0
will be Sj^ + \So + fj^S^ + vS^ = 0.
Now from the condition that two spheres may cut orthopjonally [Art. 90,
Ex. 6J, it follows that a sphere can always be formed which will cut four given
spheres orthogonally; and it also follows that the sphere which cuts
orthogonally the four spheres 8^ = 0, So = 0, S^ = 0 and S^ = 0, will cut
orthogonally any sphere whose equation is Si-^-XS^ + fJ^So + uS^^ — O. This
proves the proposition.
Ex. 5. The locus of the centres of conicoids which touch six planes, and
have the sum of the squares of their axes given, is a sphere. [Mention's
Theorem.]
By Ex. 4 all the director-spheres of the conicoids are cut orthogonally by
the same sphere; and the director-spheres have a constant radius. Hence
their centres, which are the centres of the conicoids, are on a sphere con-
centric with this orthogonal sphere.
Keciprocation.
154. If we have any system of points and planes in
space, and we take the polar planes of those points and the
poles of the planes, with respect to a fixed conicoid G, we
obtain another system of planes and points which is called
138 RECIPROCATION.
the polar reciprocal of the former with respect to the
auxiliary conicoid C.
When a point in one S3"stem and a plane in the reciprocal
are pole and polar plane with respect to the auxiliary
conicoid C, we shall say that they correspond to one another.
If in one system we have a surface >S', the planes
which correspond to the different points of S will all touch
some surface >S". Let the planes corresponding to any
number of points P, Q, R... an a plane section of >S^ meet
in T; then T is the pole of the plane PQR with respect to
0, that is the plane FQR corresponds to T. Now, if the
plane PQR move up to and ultimately coincide with the
tangent plane at P, the corresponding tangent planes to S'
will ultimately coincide with one another, and their point of
intersection T will ultimately be on the surface S'. So that
a tangent plane to the surface S corresponds to a point
on the surface S', just as a tangent plane to >S' corresponds
to a point on S. Hence we see that >S* is generated from S'
exactly as S' is from S.
155. To a line L in one system corresponds the line L'
in the reciprocal system which is the polar line of L with
respect to the auxiliary conicoid.
If any line L cut the surface S in any mimber of points
P, Q, R... we shall have tangent planes to S' corresponding
to the points P, Q, R..., and these tangent planes will
all pass through a line, viz. through the polar line of L with
respect to the auxiliary conicoid. Hence, as many tangent
planes to S' can be drawn through a straight line as there
are points on S lying on a straight line. That is to say the
class [Art. 146] of 8' is equal to the degree of S. Reciprocally
the degree of >§' is equal to the class of S.
In particular, if S be a conicoid it is of the second degree
and of the second class ; hence S' is of the second class and of
the second degree, and is therefore also a conicoid.
156. The reciprocal of a point which is common to
two surfaces is a plane which touches both the reciprocal
surfaces.
RECIPROCATION. 18D
If two surfaces have a common curve of intersection,
they have an infinite number of common points ; the
reciprocal surfaces therefore have an infinite number of
common tangent planes. These common tangent planes
form a surface : and, since the line of intersection of any
two consecutive planes is on the surface, it is a ruled
surface, the generating lines being the lines of intersection
of consecutive planes. Any one of the planes contains
two consecutive generating lines, so that two consecutive
generators must intersect ; hence the surface is a developable
surface.
If all the points of the curve lie on a plane, all the
tangent planes to the developable pass through a point ;
the developable must therefore be a cone. Hence the
reciprocal of a plane curve is a cone.
It follows by reciprocation from Art. 144, that all coni-
coids which touch seven fixed planes will touch an associated
eighth plane.
It also follows from Art. 140 that all conicoids which
touch eight given planes have an infinite number of common
tangent planes, provided that the eight given planes do not
form an associated system.
157. The reciprocation is usually taken with respect to
a sphere, and since the nature of the reciprocal surface is in-
dependent of the radius of the sphere, we only require to
know the centre of the sphere, which is called the origin of
reciprocation.
The line joining the centre of a sphere to any point is
perpendicular to the polar plane of the point. Hence, if P, Q
be any two points, the angle between the polar planes of
these points with respect to a sphere is equal to the angle
that PQ subtends at the centre of the sphere.
158. If any conicoid be reciprocated with respect to a ^^
point 0, the points on the reciprocal surface which corre-
spond to the tangent planes through 0 to the original surface '
must be at an infinite distance.
140 RECIPROCATION.
Hence the generating lines of the asymptotic cone of the
reciprocal surface are perpendicular to the tangent planes of
the enveloping cone from 0 to the original surface.
In particular, if the point 0 be on the director-sphere of
the original surface, that is if three of the tangent planes
from 0 be at right angles, the asymptotic cone of the
reciprocal surface will have three generating lines at right
ansfles.
Corresponding to a point at infinity on the original
surface we have a tangent plane through 0 to the reciprocal
surface.
Hence the tangent cone from the origin to the reciprocal
surface has its tangent planes perpendicular to the generating
lines of the asymptotic cone of the original surface.
In particular, if the asymptotic cone of the original surface
have three perpendicular generating lines, three of the tangent
planes from 0 to the reciprocal surface will be at right angles,
so that 0 is a point on the director-sphere of the reciprocal
conicoid.
159. As an example of reciprocation take the theorem : —
" If tv/o of the conicoids which pass through eight given
points are rectangular hyperboloids, they will all be rect-
angular hyperboloids." If this be reciprocated with respect
to any point 0 we obtain the following, " If the director-
spheres of two of the conicoids which touch eight given
j)lanes pass through a point 0, the director-spheres of all the
conicoids will pass through 0." Hence " the director-spheres
of all conicoids which touch eight given planes have a com-
mon radical plane."
As another example of reciprocation take the theorem : —
" A straight line is drawn to cut the faces of a tetrahedron
A BCD which are opposite to the angles A, B, C, D in
a, h, c and d respectively. Shew that the spheres described
on the straight lines Aa, Bb, Cc, and Del as diameters have
a common radical axis."
Let 0 be a point of intersection of the spheres whose
diameters are Aa, Bb and Cc. If we reciprocate with
RECIPROCATION. 141
respect to 0 we shall obtain another tetrahedron whose
faces and angular points correspond respectively to the
angular points and faces of the original tetrahedron. Corre-
sponding to the four points a, b, c, d which are on a straight
line, we shall have four planes with a common line of inter-
section; and, since a, b, c, d are on the faces of the original
tetrahedron, the corresponding planes will pass through the
angular points of the reciprocal tetrahedron ; also since the
angles AOa, BOb, COc are right angles, the three pairs
of planes corresponding respectively to a and A, to b and
B, and to c and C will be at right angles ; this shews that
the line of intersection of the planes correspondmg to a, b, c, d
will meet three of the perpendiculars of the reciprocal
tetrahedron. But we know [Art. 185, Ex. 4], that every line
which meets three of the perpendiculars of a tetrahedron,
meets the remaining perpendicular ; and hence the planes
corresponding to d and D are at right angles, which shews
that the angle dOD is a right angle. Hence 0 is also on
the sphere whose diameter is Dd.
Ex. 1. The reciprocal of a sphere with respect to any point is a conicoid
of revolution.
Ex. 2. Find the reciprocal of ax^+hj" + cz'^ =l\{iih. respect to the sphere
^ a 0 c
Ex. 3. Shew that the reciprocal of a ruled surface is a ruled surface.
Ex. 4. Shew that if two conicoids have one common enveloping cone
they also have another. [The reciprocal of Art. 120.]
Ex. 5. Either of the two surfaces ax' + by^= ±2z is self reciprocal with
respect to the other.
Examples ox Chapter YII.
1. When three conicoids pass through the same conic, the
planes of their other conies of intersection pass through the same
line.
2. Shew that, if the curve of intersection of two conicoids
cross itself, the conicoids will touch at the point of crossing; and
that if the curve of intersection cross itself twice, it will consist
of two conies.
142 EXAMPLES OX CHAPTER VII.
3. Sbew tliat thi-ee paraboloids will pass througli the curve of
iutersection of any two conicoids.
4. Shew that a surface of revolution will go through the
intersection of any two conicoids whose axes are parallel.
5. If a conicoid have double contact with a sphere, the square
of the tangent to the sphere from any point on the conicoid is in
a constant ratio to the product of the distances of that point from
the planes of intersection.
6. Any two conicoids which have a common enveloping cone
intersect in plane curves.
7. Shew that the polar lines of a fixed line, with respect to a
system of conicoids through eight given points, generate an hyper-
boloid of one sheet.
8. Shew that the polar planes of a fixed point, with respect
to a system of conicoids through seven given points, pass through
a fixed point.
9. Shew that the poles of a fixed plane, with respect to a
system of conicoids which touch seven given planes, lie on a fixed
plane.
10. The polar planes of a point with respect to two given
conicoids are at right angles ; shew that the locus of the point is
another conicoid.
11. All conicoids through the intersection of a sphere and
a given conicoid, have their principal planes, and also their cyclic
planes, in fixed directions.
12. If 0 be any point on a conicoid, and lines be drawn
through 0 parallel to equal diameters of the conicoid, these lines
will meet the surface on a sphere whose centre is on the normal
at 0.
13. If 0 be the centre of any conicoid through the intersec-
tion of a sphere and a given conicoid, the line joining 0 to the
centre of the sphere is perpendicular to the polar plane of 0 with
respect to the given conicoid.
14. Shew that, in a system of conicoids which have a common
curve of intersection, the diametral planes of parallel diameters
have a common line of intersection.
EXAMPLES ON CHAPTER VII. 143
15. If a system of conicoids be drawn through the inter-
section of a given conicoid and a sphere whose centre is 0, the
normals to them from 0 form a cone of the second degree, and
their feet are on a curve of the third order which is the locus of
the centres of all the surfaces.
16. If any point on a given diameter of an ellipsoid be
joined to every point of a given plane section of the surface, the
cone so formed will meet the surface in another plane section,
whose envelope will be a hyperbolic cylinder.
17. A cone is described with its vertex at a fixed point, and
one axis parallel to an axis of a given quadric, and the cone cuts
the quadric in plane curves ; shew that these planes envelope a
parabolic cylinder whose directrix-plane passes through the fixed
point.
18. If two spheres be inscribed in any conicoid of revolution,
any common tangent plane of the spheres will cut the conicoid in
a conic having its points of contact for foci.
19. If the line joining the point of intersection of three, out
of six given planes, to the point of intersection of the other three,
be called a diagonal ; shew that the ten spheres described on the
diagonals have the same radical centre, and the same orthogonal
sphere.
20. The circumscribing sphere of a tetrahedron which is self
polar with respect to a conicoid cuts the director-sphere of the
conicoid orthogonally.
CHAPTER VIII.
confocal conicoids. concycltc conicoids.
Foci of Conicoids.
160. Conicoids whose principal sections are confocal
conies are called confocal conicoids.
The general equation of a system of confocal conicoids is
2 2 2
X y z ^
a" + X Z>'' + X c' + X
Suppose a, h, c to be in descending order of magnitude.
If X is positive, the surface is an ellipsoid, and the
principal axes of the surface will increase as X increases, and
their ratio will tend more and more to equality as X is
increased more and more ; so that a sphere of infinite radius
is a limiting form of one of the confocals.
If X is negative and less than c^ the surface is an ellipsoid ;
but the ellipsoid becomes flatter and flatter as X approaches
the value — c^. Hence the elliptic disc whose equations are
_ x^ y' _-\
a - c h — c
is a limiting form of one of the confocals.
If X is between — c^ and — 6''^ the surface is an hyperboloid
of one sheet. When X is very nearly equal to — c^, the
hyperboloid is very nearly coincident with that part of the
plane z = 0 which is exterior to the ellipse —r, r. + y^ — 7 = 1'
^ ^ a —c 0 —c
CONFOCAL CONICOIDS. 145
When X is very nearly equal to — 6^ the hyperboloid is
very nearly coincident with that part of the plane y = 0
which contains the centre and is bounded by the hyperbola
2 2
i^ 1 19. ■*■•
If X is between — ¥ and — a^, the surface is an hyper-
boloid of two sheets. When \ is very nearly equal to — b'\
the hyperboloid is very nearly coincident with that part
of the plane y = 0 which does not contain the centre and is
w' z"
bounded by the hyperbola — ^ — r^ +-^ — r^ = 1.
When \ is between — o? and — (» the surface is imaginary.
The two conies
x^ . f _
^ = 0,- 5 + 77^^ = 1,
a' — c' h' — &
x'' z"
and- 2/ = 0,;j,-^ + ^,-^,= l,
which we have seen are the boundaries of limiting forms
of confocal conicoids, are called focal conies, one being the
focal ellipse, and the other the focal hyperbola.
161. Three conicoids, confocal luith a given central conicoid,
will pass through a given point ; and one of the three is an
ellipsoid, one an hyj^erboloid of one sheet, and one an hyper-
boloid of two sheets.
Let the equation of the given conicoid bo
2 2 2
-^ + r2+-.= l.
a 0 c
Any conicoid confocal to this is
_-'--, f +^=1 (1).
a'-x 6' - X c'-\
This will pass through the particular point (f g, h) if
/' ib' - \) (c^ - X) + ^^ (c' - X) {a' - X)
+ A'(a^-X)(6'-X)-(a'-X)(6'-X)(c'-X) = 0 (ii).
S. S. G. 10
146 CONFOCAL CONICOIDS.
If we substitute for X the values a^ 6% c\ and — oo in
succession, the left side of the equation (ii) will be +, —,+,—;
hence there are three real roots of the equation, namely one
between d^ and If, one between h^ and c\ and one between
c^ and — 00 . When \ is between c^ and — co , all the
coefficients in (i) are positive, and the surface is an ellipsoid ;
when X is between c^ and 6^ one of the coefficients is
negative, and the surface is an hyperboloid of one sheet ; and
when X is between h^ and oj^ two of the coefficients are negative,
and the surface is an hyperboloid of two sheets.
162. One conicoid of a given confocal system will touch
any plane.
Let the equation of the plane be
Ix + my + nz — p.
The plane will touch the conicoid
+ r^^r^ + 3-7-7 = 1, v^-
a' +\ h' + \ c'+X ■' ^
if (a'^ + X) Z^+ (6'^ + X) m"" + (c' + X)?i^ =p^
which gives one, and only one, value of X. Hence one con-
focal will touch the given plane.
163. Two conicoids of a confocal system will touch any
straight line.
Let the straight line be the line of intersection of the
planes Ix + my + 7iz +p = 0, I'x + m'y + n2+p =0.
Any plane through the straight line will be
(I + kV) x+{m + hni) y-\-{n + hi) z + {p-^ kp) = 0.
This plane will touch the conicoid
x^ 1f^ z^
' ^ ' -1,
a' + X 6' + X c' + X
if {a" + X) (Z + kiy + {If + X) {m + kmj
+ {c'-\-\){n-\-knf^{l->-Vkp')\
CONFOCAL CONICOIDS. 147
Now, if the given line be a tangent line of the conicoid, the
two tangent planes through it will coincide. Hence the roots
of the above equation in k must be equal. The condition for
this gives the following equation for finding \
[{aJ" + X) V^ + (If + X) m" + {c^ + X) n'' - p"]
= [{a^ + X) IV + (6* + X) mm + [c" + X) nn'-pp']\
Since the equation is of the second degree, there are two
confocals which touch the given line.
164. Two confocal conicoids cut one another at right
angles at all their common points.
Let the equations of the conicoids be
a;' ^ ¥ ^ & '
x^ f z"
a' + X 6' + X c' + X
and let {xyz) be a common point ; then the co- ordinates
x\ y, z will satisfy both the above equations. Hence, by
subtraction we have
^" y'^ ^" n r^ '
Now the equations of the tangent planes at the common
point {x'y'z) are
XX ?/?/' zz' ^
a 0 c
J xx' . yy zz - ^. ,
and -^ — - + T~-^ + —, — ^ = 1, respectively,
a' + X 6' + X c' + X ' ^ ^
The condition (i) shews that these tangent planes are at
10—2
riojht anofles.
148 CONFOCAL CONICOIDS.
165. If d straight line touch two confocal conicoids, the
tangent planes at the points of contact will he at right angles.
Let {xy'z), {x'lj'z") be the points of contact, and let the
conicoids be
sr} 7/ /
tt"' -f- X 6' + X c' + \
The tangent planes will be at right angles if
XX It 11 ZZ /N /-v
^ -^ - '^ - -, =0...(i).
But, since the line joining the two points is a tangent line to
both conicoids, each point must be in the tangent jplane at
the other. Hence
XX y y ZZ _-
t If lit III
, XX y y z z ^
and ^TTT^ + 7/rv+-^-TT^ = l-
a+\ o+\ c +\
By subtraction we see that the condition (i) is satisfied.
Ex. 1. The difference of the squares of the perpendiculars from the
centre on any two parallel tangent planes to two given confocal conicoids is
constant, [p^- - p^^ = \- \-]
Ex. 2. The locus of the point of intersection of three planes mutually
at right angles, each of which touches one of three given confocals, is a
sphere. [See Art, 92.]
Ex. 3. The locus of the umbilici of a system of confocal ellipsoids is the
focal hyperbola.
[The umbilici are given by
^/(a■HX) V«'-c2' ^ ' x/(c- + X)
= i /^^^ 1
Ex. 4. If two concentric and co-axial conicoids cut one another everywhere
at right angles they must be confocal.
Ex. 5. P, Q are two points, one on each of two confocal conicoids, and
the tangent planes at P, Q meet in the line ES ; shew that, if the plane
through US and the centre bisect the line PQ, the tangent planes at P and Q
must be at right angles to one another.
CONFOCAL CONICOIDS. 149
Ex. 6. Shew that two confocal paraboloids cut everywhere at right angles.
[The general equation of confocal paraboloids is ^ — i- H — ^-^=2z + \.']
166. We have see'n that three coaicoids confocal with a
given conicoid will pass through any point P, the parameters
of the confocals being the three values of X given by the
equation
x'^
a' + X h' + \ c' + X
where cc, y, z are the co-ordinates of P.
If the roots of the above equation be X^, \^, \, it is easy
to shew that
(a' - b') (a' - c') ■'
with similar values for y'^ and z^.
Hence the absolute values of the co-ordinates of any
point can be expressed in terms of the parameters of the
conicoids which meet in that point, and are confocal with a
given conicoid.
167. The parameters of the two confocals through any
point P of a conicoid are equal to the squares of the axes of
the central section of the conicoid which is parallel to the
tangent plane at P ; and the normals at P to the confocals
are parallel to the axes of that section.
Let (x\ y, z) be any point P on the conicoid whose
equation is
a?^ V^ -2^ -.
— \- — A — = 1 •
a 0 c
then, if P be on the confocal whose parameter is X, we have
'2 '2 '2
x y z -,
a'-X H'-X c'-X
and therefore
T^ ni'^ z'^
150" eONFOCAL CONICOIDS.
The Equation of the central section parallel to the tangent
plane at P is
a' + }^^ + c^ ^•
Hence the equation giving the squares of the axes of the
section is
'2 '2 j.'2
d^ r' h' 7-^ c" r^
^^ a? id' - r') ^ b' {(}' - r') "^ c' (c' - r') " ^''''*
Comparing (i) and (ii), we see that the squares of the
axes of the section are the two values of \.
The equations of the diameter which is parallel to the
normal at P to one of the confocals are
X y z
\
X y
d' -\ ¥-\ c'-X
The length of the diameter will be equal to 2^/\ if it be
one of the generating lines of the cone
the condition that this may be the case is
os" /I 1\ y"' (\ 1\ z'^ (\ \\_
(a' - xy w -xJ "^ {¥ - xy w x) "^ (c^ - xy w x)~^''
and it is clear from (i) that this condition is satisfied.
Hence an axis of the central section is parallel to the
normal to one of the confocals through P, and the square of
the length of the semi-axis is equal to the parameter of
that confocal.
COXFOCAL CONICOIDS. 151
Cor. If diameters of a conicoid be drawn parallel to the
normals to a confocal at all points of their curve of inter-
section, such diameters will be of constant leno-th.
168. Two points {x, y, z), (f, 77, f), one on each of two
co-axial conicoids whose equations are
a 0 c a p 7
respectively, are said to correspond when
^ = f, 1=1 and ? = -^
a a 0 p c y
In order that real points on one conicoid may correspond
to real points on the other, the two surfaces must be of the
same nature, and must be similarly placed.
It follows at once from the equations (i), Art. 96, that if
on one of the conicoids three points be taken which are ex-
tremities of conjugate diameters, the three correspondino-
points on the other conicoid will be at extremities of con-
jugate diameters.
169. The distance between two points, one on each of two
confocal ellipsoids, is equal to the distance between the two
corresponding points.
Let (^,, 7/j, z^), (x,, 3/2, z^) be the two points on one
conicoid, and (fj, tj^, Q, (f^, Vo, Q the corresponding points
on the other conicoid.
Then ^' = is ^ = ^, ^ = £;
a a. b /i^ c 7 '
^r^A 5-& 2/2 _ ^2 ^2_?;
a a 0 p c y
We have to prove that
(^. - ?.)'+ (y^-vd'+ {^- rj = (^. - f.)'+ (2/.- v,y + {^- r,)^
•' ef.-i'.)"+(i'.-f''.)"*es-?'.)'
152 COXFOCAL CONICOIDS.
or
wLicli is clearly the case, since the conicoids are confocal, and
2 * d'i ' 2 2 ' 7 2 •" „a •
a p 7 a 0 c
170. The locus of the poles of a given plane with respect
to a system of confocal conicoids is a straight line.
Let the equation of the confocals be
x'^ y"" z^
of —X h^ — \ c^ — X
and let the equation of the given plane be
Ix + my + nz=l.
The equation of the polar plane of the point {x', y\ z) is
XX yy zz
' ^^ +^. — : =1.
d'-X b'-X c'-\
Comparing this equation with the equation of the given
plane, we have
X , y . z
' I I
therefore ^ - a' = ^ - 6' = - - cl
L "ill n
Hence the locus of the poles is the straight line whose
equations are
X — aH _y — h'm _z — c^n
7 m n
This straight line is perpendicular to the given plane, and
it clearly must pass through the point of contact of that con-
focal which touches the plane. Hence the perpendicular
from any point on its polar plane with respect to a conicoid
meets the polar plane in the point where a confocal conicoid
touches it.
CONFOCAL CONICOIDS. 153
171. The axes of the enveloping cone of a conicoid are
the normals to the confocals which pass through its vertex.
Let OP, OQ, OR be the normals at 0 to the three
conicoids which pass through 0 and are confocal with a given
conicoid; and let P, Q, R be on the polar plane of 0 with
respect to the given conicoid.
By the last article, the line OP is the locus of the poles of
the plane QOR with respect to the system of confocals.
Hence, the pole of the plane QOR wdth respect to the given
conicoid is on the line OP \ the pole is also on the plane
PQR, because PQR is the polar plane of 0 and therefore con-
tains the poles of all planes through 0. Therefore the point
P is the pole of the plane QOR with respect to the given
conicoid. Similarly Q and R are the poles of the planes ROP
and POQ respectively. Hence OPQR is a self -polar tetra-
hedron with respect to the original conicoid.
Now let any straight line be drawn through P so as to
cut the given conicoid in the points A, B and the plane QOR
in G. Then [Art. 56] the pencil 0 [APBG] is harmonic; and
OP and OC are at right angles, hence OP bisects the angle
AOB. This shews that OP is an axis of any cone whose
vertex is at 0, and whose base is a plane section of the
conicoid through P. One such cone is the enveloping cone
from 0 to the given conicoid ; hence OP is an axis of the
enveloping cone. We can shew in a similar manner that OQ
and OR are axes of the enveloping cone.
172. To find in its simplest form the equation of the
enveloping cone of a conicoid.
Let the equation of the conicoid be
^+^V^'=i
d'^h'^c'
The equation of any tangent plane is
lx + my-\-nz = Aj(aT + 6'W + cV).
Hence the direction-cosines of the normal to any tangent
plane which passes through the point (x^, y^, z^ satisfy the
154 CONFOCAL CONICOIDS.
equation
a^Z" + Unv 4- c-/i^ — (Jx^ + my^ + nz^'^ = 0.
Hence the equation of the reciprocal of the enveloping cone
whose vertex is {x^, y^, z^ is
aV + hY + cV - {xx^ + yy, + ^^,)^ = 0 (i).
Similarly the equation of the reciprocal of the enveloping
cone of the conicoid
2 2 8
X If z ^ ....
q!" — X ¥ — \ d^ — \
is (a^-X) x'^- {h'- X) y-+ {c'-X)z'- {xx,+ yy,-\- zz,y= 0. . .(iii).
It is clear from Art. 60, that the cones (i) and (iii) are
co-axial for all values of X. Hence, since a cone and its
reciprocal are co-axial, it follows that all cones which have a
common vertex and envelope confocal conicoids are co-axial ;
and, by considering the three confocals which pass through
the vertex, the enveloping cones to which are the tangent
planes, we see that the principal planes of the system of
cones are the tangent planes to the confocals which pass
through their vertex.
The enveloping cones of the three confocals which pass
through (Xq, y^y z^) are planes, and their reciprocals are
straight lines. Hence the three values of \ for which the
left side of (iii) is the product of linear factors (which are
imaginary) are the three parameters \, \, \ of the con-
focals through (x^, 3/0, Zq).
But [Art. 77] the three values of \ for which the left
side of (iii) is the product of linear factors are the three roots
of the discriminating cubic of (i) .
Therefore the roots of the discriminating cubic of (i) are
X, Xg, X3; so that the equation of the reciprocal of the
enveloping cone, when referred to its axes, is
Hence the equation of the enveloping cone is
^ + l! + £! = o.
Xj \ Xg
CONCrCLIC CONICOIDS. 155
Ex. Find the locus of the vertices of the right circular cones lohicU
circumscribe an ellipsoid.
If a cone be right circular, the reciprocal cone will be right circular.
Hence we require the condition that the cone whose equation is
may be right circular.
If Xq, t/q, Z(^ be all finite, the conditions for a surface of revolution are
[Art. 85] a2- V + V=&'-yo' + yo'-c2- V + V,
so that, unless the surface is a sphere, x^ij^Zq must be zero. If Zq=0, the
condition for a surface of revolution gives
Hence the enveloping cone from any point on the focal ellipse
5^?+65^=^'' = '' «•
is right circular.
Similarly, the enveloping cones from points on
a^2 + ^2=1.2/ = 0..... (ii),
or from points on ~ — n + —r- — o = l, a: = 0 (iii),
u- -a^ c'-a-' ^
are right circular.
The conic (ii) is the focal hyperbola, and (iii) is imaginary.
CONCYCLIC COXICOIDS.
173. The reciprocal of the conicoid
a^ + \ ¥ + \ c' -v\
with respect to the sphere x^ + y""' -\- z^ = k^, is
{a^ -\-\) x^ ■{■ {¥ ■^\)f + {c" + \) z" = k\
It is clear that the reciprocal conicoids have the same
cyclic planes for all values of X.
Hence a system of confocal conicoids reciprocates into a
system of concyclic conicoids.
156
FOCI OF CONICOIDS.
174. The following are examples of reciprocal properties
of confocal and concyclic conicoids.
Three confocals pass througli
any point, namely an ellipsoid, an
liyperboloid of one sheet, and an
hyperboloid of two sheets; also the
tangent planes at the point to the
three surfaces are at right angles.
Three concyclics touch any plane,
namely an ellipsoid, an hyperboloid
of one sheet, and an hyperboloid of
two sheets; also the lines from the
centre to the points of contact of the
plane are at right angles.
Two confocals touch a straight
line, and the tangent planes at the
points of contact are at right angles.
Two concyclics touch a straight
line, and the lines from the centre
to the points of contact are at right
angles.
One conicoid of a confocal system
touches any plane.
The locus of the pole of a given
plane with respect to a system of
confocals is a straight line.
The principal planes of a cone
enveloping a conicoid are the tangent
planes to the confocals through its
vertex.
One conicoid of a concyclic system
passes through any point.
The envelope of the polar plane
of a given point with respect to a
system of concyclics is a straight line.
The axes of a cone whose vertex
is at the centre of a conicoid and base
any plane section, are the lines from
the centre to the points of contact of
the plane with the concyclics which
touch it.
Foci of Conicoids.
175. There are two definitions of a conicoid which corre-
spond to the focus and directrix definition of a conic.
One definition, due to Mac Cullagh, is as follows : — •
A conicoid is the locus of a point which moves so that its
distance from a fixed point, called the focus, is in a constant
ratio to its distance {measured parallel to a fixed plane) from
a fixed straight line called the directrix.
Let the origin be the focus, and the plane ^ = 0 the fixed
plane.
Also let the equations of the directrix be
I m n '
FOCI OF CONICOIDS. 157
Let X , y , z be the co-ordinates of any point P on the locus,
and let a plane through P parallel to -2: = 0 meet the directrix
in M, then M is 1/+^-^ (/ - U), g + '^ [z- h), z'\ .
Now OP^ = e^ . P3P, e being the constant ratio. Hence
the equation of the locus of (w, y , z) is
a;'+2/'+^'=e'
].-/-^^(.-/o}V{,-^-^|(.-A)f'
• •■(!)•
The locus is therefore a conicoid, and is such that sections
parallel to -s^ = 0 are circles.
If the axes be changed in any manner (i) will always be
of the form
(^-a)'+(i/-/3y+(^-7)'-^ = 0,
where A is the sum of two squares, or is the product of two
imaginary factors. We can therefore find the foci of any
given conicoid whose equation is >S^ = 0, from the consideration
that S-\{[x- af + (y- I3f + (^ - 7)'} will be the product
of imaginary linear factors if (a, yS, 7) be a focus, provided a
suitable value be given to X.
176. The other definition of a conicoid, due to Salmon,
is as follows : —
A cojiicoid is the locus of a point the square of whose
distance from a fixed j^oint, called a focus, varies as the pro-
duct of its distances from two fixed planes.
The equation of the locus is clearly of the form
{x-af-\- {y-py-\-{z- 7)'= kXlx+my + nz +p){Vx+m'y + n'z+p').
We can find the foci of any conicoid according to this
definition by the consideration that
S-X{(x-ar + (y- ^y + (z-yy}
will be the product of real linear factors if (or, ^, 7) be a focus,
provided a suitable value be given to \,
158 FOCAL CONICS.
177. To find the foci of the conicoid whose equation is
ax"" + hf + cz^ = 1.
We liavie seen in Articles 175 and 176 that (a, /3, 7) is a
focus when
ax'+h2f + cz'-l-\{{x-ay+(y-l3y-]-{2-yy} (i)
is the product of linear factors.
Hence \ must be equal to a, or 6, or c.
Let X = a, then (i) becomes
(h - a) y'+ (c - a) /+ 2a2x + 2al3y+ 2ayz - a (a'+ fi'+y') -1,
or
2 ah^"^ acy'^
+ 2a2X — aur — ^ 1.
0 — a c — a
Hence, in order that (i) may be the product of linear
factors, we must have a = 0, and
b a c a
Similarly, if = h, we have /3 = 0 and
— -'-J
1_1 ' 1_1
a b c b
and, if X = c, we have 7 = 0, and
l_l"*"l_l~^*
a c b c
There are therefore three conies, one in each principal
plane, on which the foci lie.
FOCAL LINES OF A CONE. 159
178. If the surface be an ellipsoid whose semiaxes are
a, 6, c, the conies on which the foci lie are
^+6-/^=l.^ = 0 (0,
■ ^^+?t:^=1'2/ = o (").
and / 3+— 2 = 1, x = 0 (iii).
b —a c — a ^ ^
Since a, h, c are in descending order of magnitude (i) is an
ellipse, (ii) is an hyperbola, and (iii) is imaginary. These
conies are called the focal conies ; and, as we have seen in
Art. 160, they are the boundaries of limiting forms of confocal
conicoids.
179. The focal conies of the cone ax"^ + hy^ + C2^ = 0 can
be deduced from the above, or found in a similar manner.
The conies become
b a c a
c b a b
2 2
and 2 = 0, —^ + JI— = 0,
a c b G
One of the focal conies of a cone is therefore a pair of real
straight lines which are called the focal lines ; the other focal
conies are pairs of imaginary straight lines, which we may
consider as point-ellipses.
Ex. 1. Two cones whicli have the same focal lines cut one another at
right angles.
Ex. 2. Shew that the enveloping cones from any point to a system of
confocals have the same focal lines. .
Ex. 3. Shew that the focal conies of & paraboloid are two parabolas.
160 EXAMPLES ON CHAPTER VIII.
180. The focal lines of a cone are perpendicular to the
cyclic planes of the reciprocal cone.
The equations of any two reciprocal cones referred to
their axes are
x^ v^ z^
ax"- -I- h\r + c-3^ = 0, and - + f- + - = 0.
^ a b c
The cyclic planes are [Art. 121]
The focal lines are by the last article
X^ z'^ X^ 2^
a b c b
It is therefore clear that the focal lines of one cone are
perpendicular to the cyclic planes of the other.
Examples on Chapter YIII.
1. Three confocal conicoids meet in a point, and a central
plane of each is drawn parallel to its tangent plane at that point.
Prove that, one of the three sections will he an ellipse, one an
hyperbola, and one imaginary,
2. Plane sections of an ellipsoid envelope a confocal j stew
that their centres lie on a surface of the fourth degree.
3. F, Q are two points on a generator of a hyperholoid; P', Q'
the corresponding points on a confocal hyperholoid. Shew that
FQ' is a generator of the latter, and that PQ = P'Q'.
4. Shew that the points on a system of confocals which are
such that the normals are parallel to a given line are on a rect-
angular hyperbola.
5. If three lines at right angles to one another touch a
conicoid, the plane through the points of contact will envelope
a confocal.
EXAMPLES ON CHAPTER VIII. IGl
6. If three of the generating lines of the enveloping cone of
a paraboloid be mutually at right angles, shew that the vertex will
be on a paraboloid, and that the polar plane of the vertex will
always touch another paraboloid.
7. If through a given straight line tangent planes be drawn
to a system of confocals, the corresponding normals generate a
hyperbolic paraboloid.
8. Shew that the locus of the polar of a given line with respect
to a system of confocals is a hyperbolic paraboloid one of whose
asymptotic planes is perpendicular to the given line.
9. Planes are drawn all passing through a fixed straight line
and each touching one of a set of confocal ellipsoids; find the locus
of their points of contact.
10. At a given point 0 the tangent planes to the three coni-
coids which pass through 0, and are confocal with a given conicoid,
are drawn ; shew that tliese tangent planes and the polar plane of
0 form a tetrahedron which is self-conjugate with respect to the
given conicoid.
11. Through a straight line in one of the principal planes
tangent planes are drawn to a series of confocal ellipsoids ; prove
that the points of contact lie on a plane, and that the normals at
these points pass through a fixed point.
If a plane be drawn cutting the three principal planes, and
through each of the lines of section tangent planes be drawn to
the series of conicoid s, prove that the three planes which are the
loci of the points of contact intersect in a straight line which is
])erpendicular to the cutting plane, and passes through the three
fixed points in which the three series of normals intersect.
12. Any tangent plane to a cone makes equal angles with the
planes through the line of contact and the focal lines.
13. If through a tangent at any point of a conicoid two
tangent planes be drawn to a focal conic, these two planes will be
equally inclined to the tangent plane at 0,
14. The focal lines of the enveloping cone of a conicoid are
the generating lines of the confocal hyperboloid of one sheet which
passes through its vertex.
S. S. G. 11
162 EXAMPLES ON CHAPTER VIII.
15. Any section of a cone which is normal at P to a focal
line, has P for one focus.
16. If a section of an ellipsoid be normal to a focal conic at
P, then P will be a focus of the section.
17. The product of the distances of any point P on a focal
conic of an ellipsoid, from two tangent planes to the surface which
are parallel to one another and to the tangent at P to the focal
conic, is constant for all positions of P.
18. From whatever point in space the two focal conies are
viewed they appear to cut at right angles.
Hence shew that the focal conies project into confocals on any
plane.
19. If two confocal surfaces be viewed from any point, their
apparent contours seem to cut at right angles.
20. If two cylinders with parallel generators circumscribe
confocal surfaces their sections by a plane perpendicular to the
generators are confocal conies.
21. The centres of the sections of a series of confocal conicoids
by a given plane lie on a straight line.
22. Shew that those tangent lines to an ellipsoid from an
external point whose length is a maximum or minimum are normals
at their respective points of contact to confocals drawn through
those points : and further, that the locus of these maximum and
minimum lines to a series of ellipsoids confocal with the original
one is a cone of the second degree.
"J3^
23. A straight line meets a quadric in two points P, Q so
that the normals at P and Q intersect : prove that PQ meets any
confocal quadric in points, the normals at which intersect, and
that if PQ pass through a fixed point it lies on a quadric cone.
24. If from any point 0 normals are dra^vn to a system of
confocals (1) these normals form a cone of the second degree, (2)
the tangent planes at the feet of the normals form a developable
of the fourth degree. Consider the case of 0 being in one of the
principal planes.
EXAMPLES ON CHAPTER VIII. 163
25. The envelope of the polar plane of a fixed point with
respect to a system of confocal quadrics is a developable surface.
Prove this, and shew that the developable surface touches the six
tangent planes to any one of the confocals at the points where the
normals to that confocal through the fixed point meet that confocal.
26. Prove that the developable which is the envelope of the
polar planes of a fixed point P with respect to a system of confocal
quadrics, meet Q the polar plane of P with respect to one of the
confocals in a line, whose polar line with respect to the same
confocal is perpendicular to Q ; and that these polar lines generate
the quadric cone six of whose generators are the normals at P to
the three confocals through P, and the three lines through P
parallel to their axes.
27. Prove that if a model of a hyperboloid of one sheet be
constructed of rods representing the generating lines, jointed at the
points of crossing ; then if the model be deformed it will assume
the form of a confocal hyperboloid, and prove that the trajectory
of a point on the model will be orthogonal to the system of confocal
hyperboloids.
28. The two quadrics
2ayz + 2hzx + 2cxy — 1 and 2dyz + 2h'zx + Ic'xy = 1
can be placed so as to be confocal if
ahc a'h'c' a'b'c' a"b"G" ^
a'+b'+c' "^ a"+ b"+ c" ~ ' (a' + b' + c'f "^ {a"+b" + c'y ~ 2^'
29. Two ellipsoids, two hyperboloids of one sheet, and two
hyperboloids of two sheets belong to the same confocal system;
shew that of the 256 straight lines joining a point of intersection
of three surfaces to a point of intersection of the other three, there
are 8 sets of 32 equal lines, the lines of each set agreeing either in
crossing or in not crossing each of the principal planes.
30. A variable conicoid has double contact with each of three
fixed confocals ; shew that it has a fixed director-sphere.
11—2
CHAPTER IX.
QUADRIPLANAR AND TeTRAHEDRAL CO-ORDINATES.
181. In the quadriplanar system of co-ordinates, four
planes, which form a tetrahedron, are taken as planes of
reference, and the co-ordinates of any point are its perpen-
dicular distances from the four planes. The perpendiculars
are considered positive when they are drawn in the same
direction as the perpendiculars from the opposite angular
points of the tetrahedron.
Since the perpendicular distances of a point from
any three planes are sufficient to determine its position,
there must be some relation connecting the four perpen-
diculars on the planes of reference.
Let A, B, C, D be the angular points of the tetrahedron,
and a, h, c, d be the areas of the faces opposite respectively
to A, B, C, I); then, if a, /3, 7, 8 be the co-ordinates of any
point, the relation will be
where V is the volume of the tetrahedron ABCD. This
is evidently true for any point P within the tetrahedron,
since the sum of the tetrahedra BCDP, CDAP, DABP,
ABCP is the tetrahedron ABCD ; and, regard being had to
the signs of the perpendiculars, it can be easily seen to be
universally true.
TETRAHEDKAL CO-ORDINATES. 165
182. The tetrahedral co-ordinates a, /9, 7, B of any point
P are the ratios of the tetrahedra BCDP, GDAP, PABP,
ABCP to the tetrahedron of reference A BCD. The relation
between the co-ordinates is easily seen to be
a-f/3+7+S=l.
It is generally immaterial whether we use quadriplanar or
tetrahedral co-ordinates, but the latter system has some
advantages, and in what follows we shall always suppose the
co-ordinates to be tetrahedral unless the contrary is stated.
We shall also suppose that the equations are homogeneous,
for they can clearly always be made so by means of the relation
a-f/3-f- 7-1-8 = 1. When the equations are homogeneous we
can use instead of the actual co-ordinates any quantities
proportional to them.
183. The co-ordinates of the point which divides the
line joining (a^, /3,, 7^, 8J and (a^, ^^^y^' K) i^ ^^^^ ^^^io ^ '• H'
are easily seen to be
X-fyU, ' X -f yU, ' X-j-//. ' \ + fJU
184. The general equation of the first degree represents a
jylane.
The general equation of the first degree is
loL -\- m^ -\-ny + pS = 0.
We may shew that this represents a plane by the method
of Art. 13.
Since the equation la. + m/S -1- 717 -f pS = 0 contains three
independent constants it is the most general form of the
equation of a plane.
The equation of the plane through the three points
(^1) l^v Iv ^1)' C^2' ft» 7o, ^2)' («3' ^3> 73' ^3) is
= 0.
a ,
/3,
7 .
h
«i'
^x.
7i.
K
«2'
^2.
72'
K
«3'
^3>
73'
K
166 TETRAHEDRAL CO-ORDINATES.
185. To shew that the perpendiculars from the angular
points of the tetraliedron of reference on the _2^Za?ie whose
equation is Iol + m/S+ny +pB = 0 are proportional to I, m, n,p.
Let Z, M, jS^, P be the perpendiculars on the plane from
the angular points A, B, G, D respectively; the perpendicu-
lars being estimated in the same direction. Let the plane
meet the edge AB in K, then at K we have 7 = 0, 8 = 0
and loL + mB — 0 ; therefore — = — -, .
m I
Now L:M::AK:BK.
But AK : AB :: ACDK : ACDB :: /S : 1;
similarly KB : AB :: KBCD : ABCD :: a : 1;
.-. L :2I::AK:-KB::^:-0L :: I : m;
.*. -y = — , and similarly each = — = — .
0 in '' n p
186. The lengths of the perpendiculars on a plane from
the vertices of the tetrahedron of reference may be called the
tangential co-ordinates of the plane; and, from the preceding
article, the equation of the plane whose tangential co-ordinates
are I, m, n, p is It. + mj3 + ny -\-pS = 0.
The co-ordinates of all planes which pass through the
point whose tetrahedral co-ordinates are a^, ^^, y^, B^, are
connected by the relation loc^ -j- m^^ -\- ny^+pS^ = 0. Hence
the tangential equation of a point is of the first degree.
187. The equation of any plane through the intersection
of the two planes whose equations are
la + m/3 + 7iy + j)8 = 0, and Ta + m^ + ny + p^'S = 0,
is {I + Xl') OL -H {m + \m) ^+ {n + \n') y + {p + \p) 8 = 0.
Hence the tangential co-ordinates of any plane through
the line of intersection of the two planes whose co-ordinates
are I, m, n, p and V, m, n, p are proportional io l-\- Xl',
'in -}- Xm', n -h A?i', p + \p.
TETRAHEDRAL CO-ORDINATES. 167
188. To find the ijerpendicular distance of a point from
a plane.
Let the equation of the plane be
loL -{- m/S -\- ny + pS = 0 (i),
and let its equation referred to any three perpendicular
axes be
Ax+Bij-hCz + I> = 0 :...(ii).
We know that the perpendicular distance of any point
from the plane (ii) is proportional to the result obtained by
substituting the co-ordinates of the point in the left-hand
member of the equation. Hence the perpendicular distance
of any point from (i) is proportional to the result obtained
by substituting the co-ordinates in the expression
It. + mfi -\- ny -\- p8.
Hence, if I, m, n, p be equal to the lengths of the perpendiculars
from the angular points of the tetrahedron of reference, the
perpendicular distance of any other point (a , /3', y', 6') will
be W + mP' -f ny +ph'.
189. If a plane be at an infinite distance from the
angular points of the tetrahedron of reference, the perpen-
diculars upon it from those points are all equal.
Hence the equation of the plane at infinity is
OL+^ + y + h = 0.
This result may also be obtained in the following
manner.
Let ki, h(3y Icy, hh be the co-ordinates of any point ; then
the invariable relation gives kx -f kfi -^-ky ^kh — l, or
a + /3-|-7-f-S = 7^. If therefore k become infinitely great, we
have in the limit a-fyS + 7-fS = 0. This is the relation
which is satisfied by finite quantities that are proportional
to the co-ordinates of any infinitely distant point.
190. Let cfj, /5j,7^, 8^ be the co-ordinates of any point P,
and a, /5, 7, 8 the co-ordinates of a point Q. Also let 6^, 0,^, 6^, 6^
168 TETRAHEDRAL CO-ORDINATES.
be respectively the angles between the line PQ and the
perpendiculars from the angular points A, B, C, D of the
fundamental tetrahedron on the opposite faces.
Then, a, 6, c, d being the areas of the faces opposite to
A, B, C, D respectively, we have
a-a^ = ia.PQcos0^, jS - l3^ = ^h.PQ cos 6^,
ry — Yj = ^c.PQ COS 0^, and S — 8^ = J cZ . PQ cos 6^.
The equations of the straight line through P, whose
direction-angles are 0^^, 0^, 6^, 0^, are therefore
a cos ^^ 5 cos 6^ c cos ^3 c? cos 6^ ^
Since the sum of the projections of the four faces of the
tetrahedron on a plane j)erpendicular to PQ is zero, we have
a COB dj^ + b cos 0^ + c cos 0^-\- d cos ^^ = 0,
or, putting I, m, n, p instead of acos^^, hcosO,^, ccos6^,
d cos 6^ respectively,
I + ni + n-{- 2:> = 0.
Ex. 1. Find the conditions that three planes may have a common line of
intersection.
Ex. 2. Find the conditions that two planes may be parallel.
Ex. 3. Find the equation of a plane through a given point parallel to a
given plane.
[Any plane parallel to la + m^ + ny+p8 = 0, is
Z a + m/3 + 717 + i:» 5 + X ( a + ^ + 7 + 5) = 0.
Hence the parallel plane through {a', /3', 7', 5') is
la + m^ + ny + p8 = [W + m^' + W7' +2)5') (a + /3 + 7 + 5).]
Ex. 4. The equations of the four planes each of which passes through a
vertex of the tetrahedron of reference and is parallel to the opposite face are
^ + 7 + 5==0, 7 + 5+a = 0, o + a + p = 0, and a + j3 + 7 = 0.
Ex. 5. Find the condition that four given points may lie on a plane.
Ex. 6. Find the condition that four given planes may meet in a point.
Ex. 7. The equations of the four planes each of which bisects three of
the edges of a tetrahedron are
a = /3 + 7 + 5, /3 = 7+5 + a, 7 = 5 + a + /3, and 5 = a + /3 + 7.
TETRAHEDRAL CO-ORDINATES. 169
Ex. 8. Shew tliat the lines joining the middle points of opposite edges of
a tetrahedron meet in a point.
Ex. 9. Find the equations of the four lines through A, B, C, D respec-
tively parallel to the line whose equations are
la + TOjS + ny +jpd = 0, I'a + ni'^ + n'y +p'5 = 0.
Ex. 10. A plane cuts the edges of a tetrahedron in six points, and
six other points are taken, one on each edge, so that each edge is divided
harmonically : shew that the six planes each of which passes through one of
the six latter points and through the edge opposite to it, will meet in a
point.
Ex. 11. Lines AOa, BOb, COc, DOd through the angular points of a
tetrahedron meet the opposite faces in a, b, c, d. Shew that the four lines of
intersection of the planes BCD, bed; CD A, cda; DAB, dab; and ABC^ abc
lie on a plane.
[If 0 be (a', /3', y', d') the equation of bed is
^ + 2 + i_^=o.
/s'^y^y a' "'
hence the line of intersection of BCD, bed is on the plane
„,+ .37 + 6 + 77 = 0.]
a p y 0
Ex. 12. If two tetrahedra be such that the straight lines joining
corresponding angular points meet in a point, then will the four lines
of intersection of corresponding faces lie on a plane.
191. We shall write the general equation of the second
des^ree in tetrahedral co-ordinates in the form
qa^ -f rjS^ + 57" + th^ + 2ff3y + 2gyoi + 2112/3
+ 2uah + 2v/3S + 2wyS = 0.
The left side of the equation will be denoted by
F{cL, A 7, 8).
192. To find the points where a given straight line cuts
the surface represented hy the general equation of the second
degree in tetrahedral co-ordiimtes.
Let the equations of the straight line be
I 'iu n p
170 TETRAHEDRAL CO-ORDINATES.
To find the points common to this line and the surface,
we have the equation
F {oL^ + Ip, /5, + mp, 7, + np, \ +pp) = 0,
„, ^ _ f,dF dF dF dF\
or F{cc„^„y,,c,) + p[l^ + m^^+n^+p-^J
+ p''F{l,m,7i,2y) = 0.
Since there are two values of p, the surface is a conicoid.
193. To find the equation of a tangent plane at any point
of a conicoid.
If (a^, /^ij 7i J ^i) be a point on the surface, one root of the
equation found in the preceding article will be zero. Two
roots will be zero, if
,dF dF dF dF ^
l-T- + 'i^^:T^ +n-i- -^-p-jK- =0.
doi^ dp^ dy^ db^
The line will in that case be a tangent line to the surface.
Substituting for I, m, n,p from the equations of the straight
line, we obtain the equation of the tangent plane, namely
But, since the equation F(ol, ^, 7, S) = 0 is homogeneous,
dF \ dF dF ^ dF ^
""^d^^^^^d^.-^-^^d^^^^dEr^'
therefore the equation of the tangent plane at the point
(at,, ^„ 7,, S^) is
dF ^dF dF ^dF ^
axj dp^ dy^ d\
104. It can be shewn by the method of Art. 53, that the
equation of the polar plane of any point (a^, ^^, 7^, 8 J is
^dF o^, ^K + S^=o
d'x^ d(3^ dy^ dS^
195. To find the co-ordinates of the centre of the conicoid.
The polar plane of the centre is the plane at infinity,
whose equation isa + ^+7-f8 = 0.
TETRAHEDRAL CO-ORDINATES. 171
Hence, if (a^, /3j, 7,, 8J be the centre of the conicoid,
we must have
doLj^ dfS^ cZyj dh^'
19G. The diametral plane of a system of parallel chords
of the conicoid can be found from Art. 192. The equation
of the plane is
,dF ^ dF dF dF ^
^^ + ^^ + ^^^+^^ = ^-
Since l + m-{-n+p=0 [Art. 190], it follows that all the
diametral planes pass through the centre, that is through the
point for which
dF_^dF^dF^dF
dj. dB d<y d8 '
197. To find the condition that a given plane may touch
the conicoid.
The condition that the plane Iol + m/3 + 717 -f pS = 0 may
touch the conicoid can be found as in Art. 57. The result is
qr + Rm^ + Sn'' + Tp^ + 2Fmn + 2 Gnl
+ 2Hlm + 2Ulp + 2Vmp + 2Wnp = 0,
where Q, R, S &c. are the co-factors of q^ r, s &c. in the dis-
criminant.
198. To find the condition that the surface represented hy
the general equation of the second degree may he a cone.
The polar planes of the angular points of the fundamental
tetrahedron with respect to a cone meet in a point, namely
in the vertex of the cone. The equations of the polar
planes are
qoL -f hp + gy + u8 = 0,
ha+r^ + fry+v8 = 0,
gx +fP + 57 -\-iuh = 0,
and UOL -t- v^ -\-iuy + tB = 0.
172 TETRAHEDRAL CO-ORDINATES.
The required condition is therefore
<?, /^ g. u \=0.
h, r, f, V
U, V, w, t
190. To shew that any tiuo conicoids have a common self-
polar tetrahedron.
We can shew, as in Art. 142, that four cones can pass
through the intersection of any two conicoids, and that the
vertices of the four cones are the angular points of a tetrahe-
dron self-polar with respect to any conicoid through the
curve of intersection of the given conicoids.
The equation of a conicoid, when referred to a self-polar
tetrahedron, takes the form
For, since a = 0 is the polar plane of the point (1, 0, 0, 0),
we have h=g = u = 0 ; and similarly f= v = w = 0.
200. To find the general equation of a conicoid circum-
scribing the tetrahedron of reference.
If we substitute the co-ordinates of the angular points of
the tetrahedron of reference in the general equation of the
second degree, we have the conditions q = r = s = t = 0.
Hence the general equation of a conicoid circumscribing
the tetrahedron of reference is
//^T + gy^ + ^^^/5 + uoih + v/SB + wyS = 0.
201. To find the general equation of a conicoid which
touches the faces of the tetrahedron of reference.
The planes a=0, /3 = 0, 7 = 0 and 8 = 0 will touch the
conicoid given by the general equation of the second degree if
§ = 0, E = 0, >Sf = 0 and r= 0. [Art. 197.]
Hence conicoids which are inscribed in the tetrahedron of
reference are given by the general equation, with the con-
ditions Q = i^ = >Sf = T= 0.
TETRAHEDRAL CO-ORDINATES. 173
Ex. 1. Find the equation of a conicoid which circumscribes the tetra-
hedron of reference, and is such that the tangent planes at the angular points
are parallel to the opposite faces. Ans. ^y + ya + a^+ad + ^8 + y8 = 0.
Ex. 2. Find the equation of the conicoid which touches each of the faces
of the fundamental tetrahedron at its centre of gravity.
Ans. a- + p'^ + y^ + d^ - ^y - ya- a^ - a8 - ^8 ~y8 = 0.
202. To find the equation of the sphere which circum-
scribes the tetrahedron of reference.
The general equation of a circumscribing conicoid is
fPy + gy^ + h-xP + uoih + vph + wyh = 0.
If the conicoid be the circumscribing sphere, the section
by 3 = 0 will be the circle circumscribing the triangle ABC.
Now the triangular co-ordinates of any point in the plane
8 = 0, referred to the triangle ABC, are clearly the same as
the tetrahedral co-ordinates of that point, referred to the
tetrahedron ABCD. Hence, when we put S = 0 in the equa-
tion of the conicoid, we shall obtain an equation of the same
form as the triangular equation of the circle circumscribing
ABC, Hence, comparing the equations
//37-f5r7a+ 7^3/3 = 0,
and BC^Py + CA'y% + AB'a^ = 0,
we obtain m = UA:^ = jB^'
By considering the sections made by the other faces of
the tetrahedron, we obtain the equation of the circumscribing
sphere in the form
BC^y + CA'ycL + ABh/S + AD'28 4- BD'jSB + CD'yS = 0.
203. To find the conditions that the general equation of
the second degree may represent a sphere.
Since the terms of the second degree in the equations of
all spheres, referred to rectangular axes, are the same ; if
>Si = 0 be the equation of any one sphere, the equation of any
other sphere can be written in the form
S + loL + ml3 ■\- nr^ +ph = 0,
or, in the homogeneous form,
yS-h {h + ml3 + ny+p3) (a -f-/3 + 7 -f 6) = 0.
174 TETRAHEDRAL CO-ORDINATES.
If this be the same conicoid as that given by the general
equation of the second degree, ^' = 0 being the equation of
the circumscribing sphere found in Art. 202, we must have,
for some value of X,
\q = l, \r = m, X5 = n, \t=p;
also 2\f=BC' + m + n,
and five similar equations.
r 4- s — 2f
Hence the required conditions are that ^ should
be equal to the five similar expressions.
The conditions for a sphere may also be obtained by
means of the equation found in Art. 192; or in the following
manner.
To find the points, P^, P^ suppose, where the edge BC
meets the conicoid given by the general equation of the
second degree, we must put a = 0, 3 = 0; and we obtain
r/3' + 57' + 2f/3y = 0 ;
we have also jS -\-y = 1 ;
... ^^2_^5(i_^)2 + 2//3(l-/3) = 0,
and, if the roots be jS^^, ^^, we have
^1^2 = r + s- 2/'
BO' '
hence, if the conicoid be a sphere, and if t^, t^, t^, t^ be the
lengths of the tangents from the points A, B, C, D
respectively, we have
r + s — 2/_ s
BC ^ tj' '
By considering the edges CD, CA we have similarly
s + t —2w _q + s — 2g s
CD' ~ CA' ^ ^i^^'
Hence, as above, the required conditions are that — ^^^ ^
should be equal to the similar expressions.
Now /S,/?^-^^'^^^^^-
EXAMPLES ON CHAPTER IX. 175
Examples on Chapter IX.
1. Shew that, if qa" + r^'^ + sy" + ^8^ = 0 be a paraboloid, it will
touch the eight planes a±^±y=t8 = 0.
2. The locus of the pole of a given plane with respect to a
system, of conicoicls which touch eight fixed planes is a straight
line.
3. The polar planes of a given point, with respect to a system
of conicoids which pass through eight given points, all pass through
a straight line.
4. If two pairs of the opposite edges of a tetrahedron are each
to each at right angles to one another, the remaining pair will be
at right angles. Shew also that in this case the middle points of
the six edges lie on a sphere.
5. Shew that an ellipsoid may be described so as to touch each
edge of any tetrahedron in its middle point.
6. If six points are taken one on each edge of a tetrahedron
such that the three lines joining the points on opposite edges meet
in a point, then will a conicoid touch the edges at those points.
7. If two conicoids touch the edges of a tetrahedron, the
twelve points of contact are on another conicoid.
8. If a conicoid touch the edges of a tetrahedron, the lines
joining the angular points of the tetrahedron and of the polar
tetrahedron will meet in a point.
9. Shew that any two conicoids, and the polar reciprocal of
each with respect to the other have a common sell-polar tetrahedron.
10. A series of conicoids Z7, , U„. U,... are such that U _^, and
. , 1' 2' 3 r+1
t7._^ are polar reciprocals with respect to U^ ; shew that U^^^ and
f7._, are also polar reciprocals with respect to U^.
11. The rectangles under opposite edges of a tetrahedron are
the same whichever pair is taken ; prove that the straight lines
joining its corners to the corners of the polar tetrahedron with
respect to the circumscribed sphere will meet in a point.
176 EXAMPLES ON CHAPTER IX.
12. If four of the eight common tangent planes of three
conicoids meet in a point, the other four will also meet in a point.
13. A plane moves so that the sum of the squares of its
distances from two of the angles of a tetrahedron is equal to the
sum of the squares of its distances from the other two ; prove that
its envelope is a hyperbolic paraboloid cutting the faces of the
tetrahedron in hyperbolas each having its asymptotes passing
through two of the angles of the tetrahedron.
14. If ABCD be a tetrahedron, self-conjugate with respect to
a paraboloid, and DA^ DB, DC meet the surface in A^, £^, C\
respectively j shew that
PA
AA
DB,
+
BB,
DC,
J — — ?
GO,
= 1,
15. If a tetrahedron have a self-conjugate sphere, and if its
radius be B, prove that pr^=r^ = 5 ^77; — t- • where s is the sum of the
bit" zo — OS '
squares of the edges of one face, and S the sum of the squares of
all the edges.
16. Shew that the locus of the centres of all conicoids which
circumscribe a quadrilateral is a straight line.
17. The locus of the pole of a fixed plane with respect to the
conicoids which circumscribe a quadrilateral is a straight line.
18. The polar plane of a fixed point with respect to any conicoid
which circumscribes a given quadrilateral passes through a fixed
line.
19. The sides of a twisted quadrilateral touch a conicoid;
shew that the four points of contact lie on a plane.
20. A system of conicoids circumscribes a quadrilateral : shew
(1) that one conicoid of the system will pass through a given point,
(2) that two of the conicoids will touch a given line, (3) that one
conicoid will touch a given plane. Shew also that the conicoids
are cut in involution by any straight line ; also that the pairs of
tangent planes through any line are in involution.
21. If three conicoids have a common self-polar tetrahedron,
the twenty-four tangent planes at their eight common points touch
a conicoid, and the twenty-four points of contact of their eight
common tangent planes lie on another conicoid.
EXAMPLES ON CHAPTER IX. 177
22. Nine conicoids have a common self-polar tetrahedron;
shew that the eight points of intersection of any three, the eight
jjoints of intersection of any other three, and the eight points of
intersection of the remaining three are all on a conicoid.
23. The sphere which circumscribes a tetrahedron self-polar
with respect to a conicoid cuts the director-sphere orthogonally.
24. The feet of the perpendiculars from any point of the
surface - + -t^ -\ 1--^ = 0, on the faces of the fundamental tetra-
a p y 6
hedron lie in a plane, a, h, c, d being proportional to the volumes
of the tetrahedron formed by the centre of the inscribed sphere
and the feet of the perpendiculars from it on any three of the
faces, and the co-ordinates being quadriplanar.
25. The middle points of the twenty-eight lines which join
two and two the centres of the eight spheres inscribed in any tetra-
hedron are on a cubic surface which contains the edges of the tetra-
hedron. Shew also that the feet of the perpendiculars from any
point of the cubic surface on the faces of the tetrahedron lie on a
plane.
26. The six edges of a tetrahedron are tangents to a conicoid.
The plane through the three points of contact of the three edges
which meet in the same vertex meet the face opposite to that
vertex in a straight line : shew that the four such lines are gene-
rators of the same system of an hyperboloid.
27. When a tetrahedron is inscribed in a surface of the second
degree, the tangent planes at its vertices meet the opposite faces in
four lines which are generators of an hyperboloid.
28. The lines which join the vertices of a tetrahedron to the
points of contact of any inscribed conicoid with the opposite faces
are generators of an hyperboloid.
29. The lines which join the angular points of a tetrahedron
to the angular points of the polar tetrahedron are generators of the
same system of a conicoid.
30. Cones are described whose vertices are the vertices of a
tetrahedron and bases the intersection of a conicoid with the oppo-
site faces. The other planes of intersection of the cones and
conicoid are produced to intersect the corresponding faces of the
tetrahedron. Prove that the four lines of intersection are genera-
ting lines, of the same system, of a hyperboloid.
S. S. G. 13
CHAPTER X.
Surfaces m Geneeal.
204. "We shall in the present Chapter discuss some
properties of surfaces of higher degree than the second.
205. Let F(x, y, z) = 0 be the equation of any surface.
To find the points of intersection of the surface and the
straight line whose equations are
X— X _y — y' __z — z' _
I tn n '
we have the equation
F(x + Ir, y + mr, z' + nr) = 0,
or
„, , , „ f,dF dF dF\
If the equation of the surface be of the n^ degree, the
equation (i) will be of the ?^*^ degree. Hence a straight line
will meet a surface of the n^ degree in n points, and any
plane will cut the surface in a curve of the ?i*^ degree.
206. To find the equation of the tangent plane at any
point of a surface.
If {x', y, z) be a point on F (x, y, z) = 0, one root of the
equation for r, found in the preceding article, will be zero.
INFLEXIONAL TANGENTS, 179
Two roots will be zero if I, m, n satisfy the relation
,dF dF dF ^
^d^'^'^d^'^^'^dz'^^'
LVtAy til U LL/J
The line will in that case be a tangent line to the surface ;
and the locus of all the tangent lines is found by eliminating
Z, m, n by means of the equations of the straight line. We
thus obtain the required equation of the tangent plane
, ,, dF , ,s dF , ,. dF -,
If the equation of the surface be z—f{x, y) = 0, it is easy
to deduce from the above, or to shew independently, that the
equation of the tangent plane at (x , y , z) is
, , ,.df , ,. df
207. The two real or imaginary lines whose direction-
cOvsines satisfy both the relations
,dF dF dF ^
I -j-f + m -J-, -\r n -J-, = Oi
ax ay dz
and (l -r-, + 'in-^—,-\-n -Y-,\ F=Qt
\ ax ay dz J
meet the surface in three coincident points.
Hence at any point of a surface two real or imaginary
tangent lines meet the surface in three coincident points.
These are called the inflexional tangents.
208. The tangent plane at any point of a surface will
meet the surface in a curve of the n^^ degree; and, since
every line which is in the tangent plane, and which passes
through its point of contact, meets the surface, and therefore
the curve of intersection, in two points, it follows that the
point of contact is a singular point in the curve of inter-
section.
When the inflexional tangents are imaginary, the point is
a conjugate point on the curve of intersection. When the
inflexional tangents are real, two branches of the curve of
12—2
ISO INDICATRIX.
intersection pass through the point of contact ; and these
branches coincide when the inflexional tangents are coin-
cident.
209. Tlie section of any surface hy a plane 'parallel and
indefinitely near the tangent plane at any point is a conic.
Let any point on a surface be taken for origin, and let the
tangent plane at the point be the plane ^ = 0. Let the
equation of the surface be z —fix, y) ; then, since 2: = 0 is the
tangent plane at the origin, we have
z = ax^ + 2hxy + hy"^
+ higher powers of the variables.
Hence, if we only consider points so near the origin
that we may neglect the third and higher powers of the
co-ordinates, the section of the given surface by the plane
z = k, is the same as the section of the conicoid whose equa-
tion is
z = ax^ + hif + 2hxy,
by the plane z = k', the section is therefore a conic.
The conic in which a surface is cut by a plane parallel and
indefinitely near the tangent plane at any point, is called the
indicatrix at the point ; and points on a surface are said to
be elliptic, j^cl'^cl^oUc, or hyperbolic, according as the in-
dicatrix is an ellipse, parabola, or hyperbola.
210. If, at the point (x, y, 2') on the surface F{x, y, z) = 0,
we have
dF _dF^^dF^
dx dy dz '
every straight line through the point (x\ y\ z) will meet the
surface in two coincident points.
Such a point is called a singular point on the surface.
All straight lines whose direction-cosines satisfy the relation
fid d , dV T^ ^
will meet the surface in three coincident points and are
r
ENVELOPES. 181
called tangent lines. Eliminating I, m, n, by means of the
equations of the line, we obtain the locus of all the tangent
lines, viz. the cone whose equation is
dy'dz ^ ' ^ ^ dz'dx'
+ 2(.-.')(y-y)^, = 0.
When the tangent lines at any point of a surface form a
cone, the point is called a conical point; and when all the
tangent lines lie in one or other of two planes, the point is
called a nodal point.
Ex. 1. Find the equation of the tangent plane at any point of the
2 2 2 2
surface x^ + y'^ + z^ = a^', and shew that the sum of the squares of the inter-
cepts on the axes, made by a tangent plane, is constant.
Ex, 2. Prove that the tetrahedron formed by the co-ordinate planes,
and any tangent plane of the surface xyz = a^, is of constant volume.
Ex. 3. Find the co-ordinates of the conical points on the surface
X7JZ - a {x^ + y'^ + z-)+4:a^ = 0; and shew that the tangent cones at the conical
points are right circular.
[The conical points are (2a, 2a, 2a,) {2a, -2a, -2a,) (-2a, 2a, -2a) and
(-2a, -2a, 2a), The tangent cone at the first point is
x^ + y^-hz^-2yz-2zx-2xi/ = 0.]
Envelopes.
211. To find the locus of the ultimate intersections of a
series of surfaces, whose equations involve one arbitrary
parameter.
Let the equation of one of the surfaces be
F{x, y, z, a) = 0,
where a is the parameter.
182 ENVELOPES.
A consecutive surface is given by the equation
F (x, y, z,a+ Sa) = 0,
or F(x, y, z, a) +-^F(x, y,z,a)ha-\- = 0.
Hence, when ha is made indefinitely small, we have for the
ultimate intersection of the two surfaces the curve given by
the equations
d
F(w, y, z, a) = 0, and ^ F (x, y, z, a) = 0.
The required envelope is found by eliminating a from these
equations.
The curve in which any surface is met by the consecutive
surface is called the characteristic of the envelope. Every
characteristic will meet the next in one or more points, and
the locus of these points is called the edge of regression! or
cuspidal edge of the envelope.
212. To find the equations of the edge of regression of the
envelope.
The equations of the characteristic corresponding to the
surface F (x, y, z,a) =0 are
F {xy y, z, a) = 0 and -y- F (x, y, z, a) = 0.
The equations of the next consecutive characteristic are
therefore
F{x, y, z, a+ Sa) = 0 and -y- F (x, y, z, a + ha) = 0,
rr dF^ , . .dF ^ d'F^ ,
or F+ -^ oa + ,.. = 0, and -r~ + -r^^ ^^+ =0.
da da da
Hence at any point of the edge of regression we must have
dF d^F
F = 0, :^ = 0, and^ = a
da da
The equations of the edge are found by eliminating a from
the above equations.
ENVELOPES. 183
213. The envelope of a system of surfaces, ivhose equation
involves only one parameter, will touch each of the surfaces
along a curve.
Let A, B, G he three consecutive surfaces of the system ;
and let PQ be the curve of intersection of the surfaces A and
B, and P'Q' the curve of intersection of the surfaces B and
G. Then the curves PQ and P'Q' are ultimately on the
envelope. Let R be any point on the curve PQ ; and let
>S^, T be two points, very near the point R, one on the curve
PQ, and the other on P'Q\ Then the plane R8T will in
the limiting position be the tangent plane at R both to the
surface B and to the envelope ; and hence the envelope
touches the surface B, and similarly every other surface of the
system, along a curve.
214. To fiyid the envelope of a series of surfaces ivhose
equations involve two arbitrary pai^ameters.
Let the equation of any surface of the system be
F{cc,y, z, a, h) = 0,
where a, h are the parameters.
A consecutive surface of the system is
F {x, y, z, a-\- Sa, b + Bb) = 0,
or F (x, y, z, a, h) + Sa -J- + Bb -^ + = 0.
Hence, when 8a and Bb are made indefinitely small, we must
have at a point of ultimate intersection
F= 0, and Ba ^ + Bb ^=0,
da db
or, since Ba and Bb are independent,
T7 r\ dF ^ , dF f.
F= 0, -y- = 0, and -tt = 0.
da do
Hence the curve of intersection of F with any surface
consecutive to it goes through the point which satisfies the
184 FAMILIES OF SURFACES.
e(iHation9
i?^= 0,^=0, and^=a
da do
The required envelope is found by eliminating a and h from
the above equations.
215. To shew that the envelope of a series of surfaces,
whose equations involve two arbitrary parameters, touches each
surface of the series.
Let the curves of intersection of the surface F with
consecutive surfaces of the system pass through the point P ;
then P is a point on the envelope. Let F^, F^ be any
two surfaces consecutive to F, and let Q, R be the points on
the envelope which correspond to these surfaces. Then all
surfaces consecutive to F^^ and therefore the surface F, will
pass through Q ; similarly the surface F will pass through R,
Hence, in the limit, the envelope and the surface F have the
three points P, Q, R, which are indefinitely near to one
another, in common ; they therefore have a common tangent
plane. Hence the envelope touches the surface F, and simi-
larly for any other surface.
Ex. 1. Find the envelope of the plane which forms with the co-ordinate
planes a tetrahedron of constant volume. Ans. a-?/2; = constant.
Ex. 2. Find the envelope of a plane such that the sum of the squares of
2 2 2
its intercepts on the axes is constant. Ans. a;^ + t/"3' + 2'5'= constant.
Ex. 3. Find the equations of the edge of regression of the envelope of the
cz
T^l&ne X sin 9 - y cos 6 = ad - cz. Ans. x' + y' = a^, y = xta,n — .
Families of Surfaces.
216. To find the general functional and differential equa-
tions of conical surfaces.
The equation of any cone, when referred to its vertex as
origin, is homogeneous ; and is therefore of the form
F
g' !)=<'•
CONICAL SURFACES. 185
Hence the equation of any cone whose vertex is at the
point (a, /3, 7) is of the form
pfx-a y^\Q (i)_
This is the required functional equation.
The tangent plane at any point of a cone passes through
the vertex of the cone. Hence, if the equation F {x, y,z) = 0
represent a cone whose vertex is (a, /S, <y), we have
. .dF , ri\dF , , . dF ^ ....
(^-c')^ + (y-^)^+ (--7)^=0 K
which is the required differential equation.
217. To find the general fimctional and differential equa-
tions of cylindrical surfaces.
A cylinder is the surface generated by a straight line
which is always parallel to a given straight line, and which
obeys some other law.
Let the equations of the fixed straight line be
X _ y _z
I m n'
The equations of any parallel line arc
X— OL y — ^ _z
■&,
I m n
the two constants a and yS being arbitrar^^
Now, in order that the line (i) may generate a surface,
there must be some relation between the constants a and /3.
Let this relation be expressed by the equation a =y(/9); then,
we have from (i)
or F (nx — Izy ny — mz) =0 (ii),
which is the required functional equation.
18G CONOIDAL SURFACES.
The tangent plane at any point of a cylinder is parallel to
the axis of the cylinder. Hence, if the equation F{x, i/,z) = 0
represent a cylinder, whose axis is parallel to the line
aj_ y _ 2
I on n*
,dF dF dF ^
we have l-r-+m-Y- + n-r- = 0,
ax ay az
which is the required differential equation.
218. To find the general functional and differential equa-
tions of conoidal surfaces.
Def. a conoidal surface is a surface generated by the
motion of a straight line which always meets a fixed straight
line, is parallel to a fixed plane, and obeys some other law.
The surface is called a right conoid when the fixed plane is
perpendicular to the fixed line.
Let the fixed straight line be the line of intersection of
the planes
Ix + my + nz + p =0, l'x-\- m'y + n'z -\-p — 0 ;
and let the fixed plane, to which the moving line is to be
parallel, be
"Xx + ^y + vz = 0.
The equations of any line which satisfies the given
conditions are
Ix + my + nz -\- p-\- A (I'x + m'y + nz-\-p) = 0,
and Xx + //,?/ + vz-^B = 0.
In order that the straight line may generate a surface,
there must be some relation between the constants A and B.
Let this relation be expressed by the equation A=f{B);
then we have
Ix + my -\- nz -\- p ^/^ , , \ r\
V r 7 — ^/ =/ O^cc + iiy + vz) (i),
Ix + my -\-nz+p -^ ^ ^^ ^ ^ ^
the required functional equation.
If we take two of the co-ordinate planes through the fixed
straight line, and the third co-ordinate plane parallel to the
DEVELOPABLE SURFACES. 187
fixed i^lane, the above equation reduces to the simple form
^=/(-) (")•
The differential equation of conoidal surfaces which
corresponds to the functional equation (ii), can be readily
shewn to be
dx "^ dy
The differential equation may also be obtained as follows.
The generator through any point is a tangent line to the
surface ; and the condition that
X y 0 '
may be on the plane
dF dF ^
is x-Y' + y -T-= 0.
ax *^ dy
Ex. 1. Shew that xyz = c (x--t/^) represents a conoidal surface.
Ex. 2. Find the equation of the right conoid whose axis is the axis of z,
and whose generators pass through the circle x = a, y^+z^ = b^.
Ans. a-y^ + x"z^=h^x^.
Ex. 3. Find the equation of the right conoid whose axis is the axis of z,
and whose generators pass through the curve given by the equations
x=acosnz^y = aBinnz. Ans. y=xta,nnz.
Ex. 4. Shew that the only conoid of the second degree is a hyperbolic
paraboloid.
219. Cones, cylinders and conoids are special forms of
ruled surfaces. There are two distinct classes of ruled
surfaces, namely those on which consecutive generators inter-
sect, and those on which consecutive generators do not
intersect ; these are called developable and skew surfaces
respectively. We proceed to consider some properties of
developable and skew surfaces.
188 DEVELOPABLE SURFACES.
220. Suppose we have any number of generating lines
of a developable surface, that is any number of straight lines
such that each intersects the next consecutive. Then, the
plane containing the first two lines can be turned about the
second line until it coincides with the plane containing the
second and third lines ; , this plane can then be turned about
the third line until it coincides with the plane through the
third and fourth lines; and so on. In this way the whole
surface can be developed into one plane without tearing.
221. The tangent plane at any point of a ruled surface
must contain the generator through the point [Art. 129]. If
the surface be a skew surface, the tangent plane will be
different at different points of the same generator ; but, if the
surface be a developable surface, the tangent plane will be
the same at all the different points of a given generator, for
the tangent plane is the limiting position of the plane
through the given generator and the next consecutive
generator.
Since any tangent plane to a developable surface touches
the surface at all points of a straight line, it follows from Art.
213, that a developable surface is the envelope of a plane
whose equation contains only one variable parameter.
222. To find the general differential equation of develop-
able surfaces.
The tangent plane at any point of a developable surface
meets the surface in two consecutive generating lines which
are the two inflexional tangents at the point.
Hence, at any point of a developable surface, the two lines
given by the equations
,dF dF dF ^
ax ay dz
and (^ -7-+ m -T- -f- n ^1 i^'^^O,
V dx dy dzj
must coincide.
DEVELOPABLE SUEFACES.
189
The condition that this may be the case is
d'F d'F d'F dF
dx'' '
d'F
dccdy'
d'F
dxdy
d'F
dxdz ' dx
d'F dF
dif ' dydz' dy
d'F d^T dF
-^ ' dz
0
= 0.
,(i).
dxdz ' dydz ' dz^
dF dj^ dF
dx ' dy ^ dz
This is the required differential equation.
The differential equation may also be obtained from the
property, proved in the last Article, that a developable surface
is the envelope of a plane whose equation involves only one
parameter.
For, the general equation of the tangent plane of a
surface at the point (a?, y, z) is
Hence, if the surface is a developable surface, there must
be some relation connectinsr ~ and — ; that is, connecting:
° dx dy °
dz , dz
-r- and -T-
dx dy
we therefore have
Therefore
dx \dy) *
d^z ^ „, /^
dx^ \dy
d'^
and
d^z
dxdy
^ d'j _
dx^ ' dy^ ~
which is equivalent to (i).
= F'
Hence
dxdy*
dz\ d^z
dy) 'df*
/ d^z Y
\dxdyl '
190 DEVELOPABLE SURFACES.
223. We can find the equation of the developable
surface which passes through two given curves, in the follow-
ing manner. The plane through any two consecutive gene-
rating lines of the surface will pass through two consecutive
points on each of the given curves ; hence the tangent plane
to the required developable surface will touch each of the
given curves.
Now the equation of a plane in its most general form
contains three arbitrary constants, and the conditions of
tan gen cy of the two given curves will enable us to express
any two of these constants in terms of the third, and the
equation of the plane will thus be found in a form involving
only one arbitrary parameter. The developable surface is
then obtained as the envelope of the moving plane.
Ex. Find the equation of the developable surface whose generating lines
pass through the two curves
2/2 = 4ax, 2 = 0 and x^ = iaij, z=c;
and shew that its edge of regression is given by the equations
cx^ - 3ayz=0 = cy^ - Sax [c-z).
Let one of the tangent planes of the developable be lx + my + nz+l=0.
The plane touches the first curve, if lx + my + 1 = 0 touches y^-4:ax = 0; that
is, if l = am'^. The plane touches the second curve, if Ix + my + nc + 1 = 0
touches x^ = 4iay; that is, if m {nc + l) = aP. Hence, the equation of the
tangent plane of the developable is found in the form
am"x + viy + {a^m^-l) - + 1 = 0 (i).
The surface is therefore given by the elimination of m between (i), and
2amx + y + 3 =0 (ii).
c
For points on the edge of regression we have also
„ a^mz - ,...,
ax + d =0 (ill).
c ^ '
From (ii) and (iii) we have m= --^; and therefore, from (iii), cx' = 3ayz.
This is the equation of one surface through the edge of regression. "We
obtain another surface through the edge by substituting m = - — in (i) ; the
result is y'^z = x^ {c-z), and at all points common to the surfaces cx'^ = 3ayz,
and y^z =3i?{c-z), Vfe must have cy^ = dax {c~z).
SKEW SURFACES. 191
224). To shew that a conicoid can he drawn which luill
touch any skew surface along a generating line.
Let AB, A'B' , A"B'' be three consecutive generators of
any skew surface. Then, [Art. 134], a conicoid will have
these three lines as generators of one system, and any line
which intersects the three given lines will be a generator of
the opposite system of the same conicoid. Through any
point Q on A E draw the line PQR to intersect the lines
AB and A"B". Then this line passes through three con-
secutive points of the given surface, and is therefore a taugent
line to the surface. Hence the plane through A'B' and PQR
touches both the given surface and the conicoid. Hence the
conicoid touches the given surface at all points of the line
A'B'.
By means of the above theorem many properties of a
ruled conicoid may be shewn to be true of all skew surfaces.
225. To find the lines of striction of any shew surface.
Def. The locus of the point on a generator of a ruled
surface where it is met by the shortest distance between
it and the next consecutive generator, is called the line
of striction of the surface.
If we know the equations of any generating line, we can
at once find the direction of the shortest distance between it
and the next consecutive generator, and this shortest distance
is a tangent line of the surface. Hence, in order to find the
point on the line of striction, which corresponds to any
particular generator, we have only to write down the con-
dition that the normal at a point on the generator may be
perpendicular to the shortest distance between the given
generator and the next consecutive.
Ex. 1. To find the lines of striction of the hyperboloid
x"^ y^ z^ ^
— V- = 1.
a^ h'^ c^
The direction-cosines of a generator, and of the next consecutive
generator, are proportional respectively to
a sin d, -h cos 6, c, and a sin [Q-\-dd)^ -h cos {d + dd), c.
192 SKEW SURFACES.
Hence the directlou-cosines of the shortest distance are proportional to
- be sin 0, ca cos 6, ah.
Now, if (x, y, z) he the point where the shortest distance meets the con-
secutive generators, the normal at {x, y, z) must be perpendicular to the
given generator, and also to the shortest distance. We therefore have
% oj z
-sin^-r cos^-- = 0,
a b c
and -^sin ^- ^cos0 + „ = 0.
a^ b^ c^
Eliminating 6, we get for the lines of striction the intersection of the
surface and the quartic
Ex. 2. To find the lines of striction of the j)araholoid whose equation is
All the generating lines of one system are parallel to the plane
5_'/ = 0 (i).
a b
The shortest distance between two consecutive generators of this system will
therefore be perpendicular to the plane (i). Hence, at a point on the
corresponding line of striction, the normal to the sui'face is parallel to (i).
The equations of the normal at [x, y, z) are
1-3^ -n-v ^-■g
X = y = -1 ,
a^ ~ b^
Hence one line of striction is the intersection of the surface and the plane
■ — f- -^ =0.
Similarly, the line of striction of the generators which are parallel to the
plane - + f = 0 is the parabola in which the plane ^ - -4- = 0 cuts the
a b a"^ b^
surface.
[See a paper by Prof. Larmor, Quarterly Journal of Mathematics,
Vol. XIX. page 381.]
226. To find the general functional and differential equa-
tions of surfaces of revolution.
Let the equations of the axis of revokition be
X — a _y — h _z~ c
I m n
SURFACES OF REVOLUTION.
193
The equations of a section of the surface by a plane
perpendicular to the axis are of the form
and Ix + my + nz =p.
Hence, since there must be some relation between r^ and
p, the required functional equation is
(x - ay + {y- hf + (2- cy ==f{lx + my + nz).
The normal at every point of a surface of revolution
intersects the axis. The equations of the normal at the point
{x\ y'y z') of the surface F{x, y, z) = 0 are
^ — ^' _y — y^ _ z — z
'W dF dF '
dx dy dz
By writing down the condition that the normal may in-
tersect the axis, we see that at every point of the surface,
dF dF_ dF
dx * dy * dz
x^a, y — by z — c
Z, m, n
= 0;
this is the differential equation of surfaces of revolution.
Note. In the above, and also in Articles 216 and 217,
we have obtained the functional equation and the diffe-
rential equation by independent methods. The differential
equation could however in each case be obtained from the
functional equation; this we leave as an exercise for the
student.
For fuller treatment of Families of Surfaces the student
is referred to Salmon's Solid Geometry, Chapter xiii.
S. S. G.
13
194? EXA3IPLES ON CHAPTER X.
Examples on Chapter X.
1. Prove that a surface of tlie fourth degree can be described
to pass through all the edges of a parallelopiped, and that if it
pass through the centre it also passes through the diagonals of the
ligure.
2. Shew that at any point on the axis of z there are two
tangent planes to the surface a^y^ = x^ (c^ — z').
3. Pind the developable surface which passes through a
parabola and the circle described in a perpendicular plane on the
latus rectum as diameter.
4. Find the equation of the developable surface which
contains the two curves
y^ = Aax, z=0; and (y — by = 4:cz, x = 0;
and shew that its cuspidal edge lies on the surface
(ax + by + czf = Zahx {y + h).
5. The developable surface which passes through the two
circles whose equations are x^ + y^ = a^, z = 0, and x^ + z^ = c", y = 0,
passes also through the rectangular hyperbola whose equations are
z^ — ij^ = —z 5 and x = 0.
G. Prove that the surface
/a;' 2/' z\ _ (x^ y\ z^ , ^
has two conical points, and two singular tangent planes.
7. Explain what is meant by a nodal line on a surface, and
find the conditions for such a line on the surface ^ [x, y, z) = 0.
There is a nodal line on the surface z (x^ + y^) + 2axy = 0 ;
find it.
8. Give a general explanation of the form of the surface
z {x^ + y^) = 2kxy. Shew that every tangent plane meets the
surface in an ellipse whose projection on a plane perpendicular to
the nodal line is a circle.
EXAMPLES ON CHAPTER X. 193
9. Examine the general form of the surface
xyz — a^x — Ify — c^z + 2a6c = 0,
and shew that it has a conical point. Shew also that each of the
planes passing through the conical point and a pair of the inter-
sections with the axes touches the surface along a straight line.
10. If a ruled surface be such that at any point of it a straight
line can be drawn lying wholly on the surface and intersecting the
axis of z, then at every point of the surface
2 d^z ^ (Pz s^'^ _f\
dx^ ^ dxdy ^ d]f~
11. Shew that the surface whose equation is determined by
the elimination of $ between the equations
X cos 6 + y sin 6 = a^
x^in.9 — y cos 0 = -(c9 — z),
is a developable surface, and find its edge of regression.
12. What family of surfaces is represented by the equation
z = <^(-p Describe the form of the surface whose equation is
sin~^ -=n tan~^ - . If w = 2, prove that through any point an
infinite number of planes can be drawn, each of which shall cut
the surface in a conic section.
1 3. At a point on the surface (x — y)z^ + ax{z + a) = 0 there
is in general only one generator, but at certain points there are
two, which are at right angles.
14. Any tangent plane to the surface a (x^ + y^) + xyz = 0
meets it again in a conic whose projection on the plane of xy is a
rectangular hyperbola.
15. Shew that tangent planes at points on a generator of the
surface yx^ — a^z = 0 cut cc = 0 in parallel straight lines.
1 6. Prove that the equation x^ + y^ + z^— Zxyz = a^ represents
a surface of revolution, and find the equation of the generating
curve.
17. From any point perpendiculars are drawn to the
generators of the surface z(x^ + y') — 2mxy = 0; shew that the
feet of the perpendiculars lie upon a plane ellipse.
XO -w
196 EXAMPLES ON CHAPTER X.
18. Shew that all the normals to a skew surface, at points on
a generator, lie on a hyperbolic paraboloid whose vertex is at the
point where the generator meets the shortest distance between it
and the next,
19. A generator PQ of the surface xyz — h(x^ + y^) = 0 meets
the axis of z in P. Prove that the tangent plane at Q meets the
surface in a hyperbola passing through P, and that as Q moves
along the generator the tangent at F to the hyperbola generates a
plane.
20. Prove that all tangent planes to an anchor-ring which
pass through the centre of the ring cut the surface in two circles.
Also if a surface be generated by the revolution of any conic
section about an axis in its own plane, prove that a double tangent
plane cuts the surface in two conic sections.
21. Prove that a flexible inextensible surface in the form of
a hyperboloid of revolution of one sheet, cut open along a
generator, may be bent so that the circle in the principal plane
becomes the axis, and the generators the generating lines of a
conoid of uniform pitch inclined to the axis at a constant angle.
22. Prove that every cubic surface has twenty-seven lines
and forty-five triple tangent planes real or imaginary, and that
every cubic surface which has a double line is a ruled surface.
Discuss some properties of the surface whose equation is
y^ + x^z + yzw = 0.
23. Four tangent planes to any skew surface which are
drawn through the same generator have their cross-ratio equal
to that of their four points of contact.
21:. Any plane through a generator of a skew surface is a
tangent plane at some point F and a normal plane at some point
F' ; shew also that there is a point 0 on the generator such that
the rectangle OF, OF' is constant for all planes through it.
25. Shew that the wave-surface, whose equation is
2 2 7,2 2 2 2
ax by c z
x^ + y^ + z^- a^ x'^ + y^ + z^- b^ x" + y^ + z^- c
has four conical points, and four singular tangent planes.
= 0,
\
CHAPTER XI.
Curves.
227. We have already seen that any two equations will
represent a curve. By means of the two equations of the
curve, we can, theoretically at any rate, express the three
co-ordinates of any point as functions of a single variable ; we
may, for example, suppose the three co-ordinates of any point
of a curve expressed as functions of the length of the arc
measured along the curve from some fixed point.
228. To find the equations of the tangent at any 'point of
a curve.
Let X, y, z be the co-ordinates of any point P on the
curve, and let x+hx, y + hy, z ■\- hz be the co-ordinates of an
adjacent point Q. Then, if hs be the length of the arc PQ,
we have, since the arc is ultimately equal to the chord,
hx''-\-hy''-\-hz'=^^s^',
©'-(I)"-©"--
Also, since the direction -cosines of the chord PQ are
proportional to hx. By, Bz, and the tangent coincides with the
ultimate position of the chord, the direction-cosines of the
tangent are equal to
dx dy dz
ds' ds ^ ds*
so that the required equations of the tangent at {x, y, z) are
^-x r}-y_^-z
dx dy dz *
ds ds ds
198 TANGENT TO A CURVE.
If the curve be the curve of intersection of the two surfaces
F{x, y,z) = Q and G {x, y, z) = 0,
the tangent line at any point is the line of intersection of the
tangent planes of the two surfaces at that point. Hence the
equations of the tangent at any point {x, y, z) are
(?-)f+(.-3/)f+(?-^)f=0,
229. To find on a given surface a curve such that the
tangent line at any point makes a maximum angle with a
given plane.
It is clear that the tangent line to such a curve at any
point is in the tangent plane to the surface at that point, and
is perpendicular to the line of intersection of the tangent
plane and the given plane.
Let the equation of the given plane be
Ix + my + nz = 0.
Then the direction-cosines of the line of intersection of the
given plane and the tangent plane at any point (x, y, z) of
the surface F [Xy y, z) =■ 0, are proportional to
dF dF dF ,dF ,dF dF
m —, n-^ , n ^ I -y- i I -i m-^- ,
dz dy dx dz dy dx
The direction-cosines of the tangent to the curve are
dx dy dz
ds ' ds' ds'
Hence we have
dxf dF_ dF\ dyf dF _,dF\
ds\ dz dy) ds\ dx dz J
dzf dF_ dF\_Q
ds \ dy dx] ~ '
the required differential equation.
CURVES. 199
If the given plane be the plane z = 0, the differential
equation of a line of greatest slope will be
dF dy dFdoc _^
dx ds dy ds
Ex. Find the lines of greatest slope to the plane 2; = 0 on the right conoid
whose equation is x = yf (2).
The differential equation of the projection on 2 = 0 of a line of greatest
slope is X dx + ydy = 0.
Hence the projections of the lines of greatest slope on the plane 2 = 0 are
circles.
230. Definitions. If J., B, C be three points on a curve,
the limiting position of the plane ABC, when A, C are
supposed to move up to and ultimately to coincide with B, is
called the oscidating 2:>lane at B.
The circle ABC in its limiting position is called the circle
of curvature at B, the radius of the circle is the radius of
curvature, and its centre the centre of curvature at B.
The normals to a curve at any point are all in the plane
through the point perpendicular to the tangent to the curve :
this plane is called the normal plane at the point.
The normal which is in the osculating plane at any point
of a curve is called the 'principal normal.
The normal which is perpendicular to the osculating plane
is called the hinormal.
The surface which is the envelope of all the normal planes
of a curve is called the polar developable.
The angle between the osculating planes at any two
points P, Q of a curve is called the whole torsion of the arc
PQ. The limiting value of the ratio of the whole torsion to
the arc is called the torsion at a point.
The radius of the circle whose curvature is equal to the
torsion of the curve at any point, is called the radius of torsion
at that point, and is represented by a.
The radius of the sphere which passes through four
consecutive points of a curve is called the radius of spherical
curvature.
Note. In what follows we shall have frequent occasion
200
CURVES.
to employ differential coefficients with respect to the arc ; and
"we shall for shortness write x ^ x", x" &c. instead of
dx d^x
~ds' di'
— o &C.
231. In the annexed figure A, B, C, D, E, F... are sup-
posed to be consecutive points of a curve, and p, q, r... are
the middle points of the chords AB, BG, CD..., Planes are
THE OSCULATING PLANE. 201
drawn through, p, q, r... perpendicular to the chords AByBC,
CD..., and LP, MQP, NRQ... are the lines of intersection of
the planes through p and q, q and r, r and s,.... The lines pi,
qL are in the plane ABC, and perpendicular respectively to
AB and BC ; the lines qM, rM are in the plane BCD, and
perpendicular respectively to BC, CD.
Then, in the limit, when the chords AB, BC, CD..,
become indefinitely small the planes ABC, BCD,... become
osculating planes of the curve; the planes pLP, qMQ,...
become normal planes of the curve ; the points L, M, N be-
come centres of curvature of the curve ; the lines LP, MQP,
NRQ... become generating lines of the polar surface, and are
called polar lines; and the points P, Q, R... become con-
secutive points on the edge of regression of the polar
surface.
All points on the plane pLP are equidistant from A and
B, all points on the plane qMP are equidistant from B and
C, and all points on the plane rMP are equidistant from C
and D ; therefore a sphere with P for centre will pass through
A, B, C, D; hence the edge of regression of the polar surface
is the locus of the centre of spherical curvature.
232. To find the equation of the osculating plane at any
point of a curve.
Let P, Q, R be three consecutive points on the curve such
that PQ = QR = Bs ; and let 5 be the length of the arc
measured from some fixed point up to Q.
Then, if the co-ordinates of Q be x, y, z, those of P, for
which the arc is s — 65, will be, if we neglect powers of hs
above the second,
x" fc, y" z"
X — xhs^-— hs", y — y'Bs + ^ 8s^ z — z'Bs + ~ Bs^;
^ A A
and the co-ordinates of R will be found by changing the sign
of Bs.
The equation of any plane through Q is of the form
X(?-a;)+i/(7;-2/)-t-iV(r-^) = 0.
202 THE PRINCIPAL NORMAL.
If this plane pass through the points P and R, we must
have
and, eliminating L, M, N, we have the required equation of
the osculating plane, namely
i-x,7]-y,^-z =0.
/ f r
X , y , z
It If 1/
X , y , z
233. To find the equations of the principal normal, and
the curvature, at any point of a curve.
Let P, Q, B be three points on a curve such that
PQ = QR = Bs.
Then, if V be the middle point of PP, QF is in the plane
PQR ; and, since the chords PQ and QR only differ by cubes
of ^5, Q F is ultimately perpendicular to FR, and is therefore
the principal normal at Q.
Then, the co-ordinates of P, Q, R being as in the last
Article, the co-ordinates of F are
t" y" z"
^ + ^^s\ y + \^s\ z^'-Bs\
Hence the equations of Q F are
S-x_7}-y ^-z
(i).
X y z
Again, the circle PQR, in its limiting position, is the
circle of curvature. Hence, if p be the radius of curvature,
we have in the limit
^P-QV'
But Q V = ^ (x" + y''' + z"), and FQ = Bs;
.\l, = x'" + y'' + z"\
P
THE BINORMAL. 203
Hence, the direction-cosines of the principal normal, which
from (i) are proportional to x' , y\ z\ are equal to
px\ py" and pz".
The co-ordinates of the centre of curvature are easily seen
to be
x + p'x", y + p'y\ z + pV\
234. To find the direction-cosines of the binormal.
The binormal is perpendicular to the osculating plane.
Hence, if I, m, 7i be the direction-cosines of the binormal, we
have from Art. 232
I _ m _ n
'/ 77 7 7/ '' 77 ' Ti 7 77 7 i •
y z — zy z X — xz xy —y x
But
{y'z" - zy'J -f iz!^' - xz'J -^ {xy" - y'xj
/ '2 1 '2 1 '2\ / "2 1 "2 1 "2\ / / " 1 ' '/ I » "\2
= (a7^ + 2/^-F-s:')(a; "■ -^-y ^ + z ^)-{xx +yy' + zz f
_1
since x*^ -\- y'"^ + z"^ =1,
and therefore xx" + y'y" -f z'z" = 0.
Hence the required direction-cosines are
p{yz -zy ), p(^« -xz ), pCa;^/ -3/^ ).
235. To find the measure of torsion at any point of a
curve.
Let I, m, n be the direction-cosines of the normal to the
osculating plane at P ; and let l + Bl, m+ Sm, n-\-Bn be the
direction-cosines of the normal to the osculating plane at Q,
where PQ = Bs. Then, if St be the angle between the
osculating planes, we have
sin'^ Bt = (mBn - nBmf H- (nBl - IBnf + (IBm - mBlf,
204 MEASURE OF TORSION.
Hence, in the limit, we have
fdT\^ _ / dn dmV / dl j dnV /, dm
\dsj ~ \ ds ds J \ ds dsj \ ds
diy
or, ~ = (mn' - miif + (nV - n'lf + {Im - VmJ (I).
Now 1 = p {ij'z" — zy") ;
, V t t in I r/f\ , f'P / I If I f/\
:.l=p{yz -zy )-\--^^{yz -zy),
and similarly for m and 7i.
Hence mn — m'n = p^ {zx' — x'z") [xy" — y'oc")
— p [zx —xz ) [xy —yx)
= p X
X
y
z
X , y , z'
X , y y z
We can find similar expressions for nV — nl, and for
Im' — I'm ; and substituting in (i), we have
pV
^ , y > ^
X , y , z
itt I/' ft
X , y , z
236. To find the condition that a curve may he a plane
curve.
Let X, y, z be the co-ordinates of any point P on the
curve, expressed in terms of the arc measured from a fixed
point up to P ; and let Q be the point at a distance a
measured along the curve from P. Then the co-ordinates of
q will be
2 3
205
CONDITION FOR A PLANE CURVE.
If all points of the curve are on the fixed plane
Ax + By + C2 + I> = 0,
the equation
2 3
+ a ^^ + C7/ + 1 ^" + ^ z" + ...) + jD = 0,
will be satisfied for all values of a.
The coefficients of all the different powers of a must
therefore be zero. Hence we have
Ax' +BiJ +C/ =0,
Ax' ^-By" -^Cz" =0,
Ax'" + By'"+Cz" = 0.
The elimination oi A^ B, G gives
« , 2/ >
^ , 2/ ,
X
y
= 0,
a relation which, since P is arbitrary, must be satisfied at all
points of the given curve.
From the result of the preceding Article it will be seen
that the above condition simply expresses the fact that the
torsion is zero at all points of a plane curve.
The condition that a curve may be a plane curve may
also be obtained in the following manner.
The direction-cosines of the normal to the osculating
plane are [Art. 234]
p\y z —zy)> p \zx —xz) and p {xy ^y x ).
206 MEASURE OF TORSION.
Since these are constant, we have
p(yV"-^y") + §(2/V'-^y')=o,
f I III I ff/\ , "'P f I ft f "\ r\
p [ZX — XZ ) + -J- {ZX —XZ)=0,
and p {xy" - yx") + -£ {xy" - y'x") = 0.
Multiply these equations in order by x", y", z" and add :
we then have
/' f ' 11/ ' ni\ , n / r >tr i iii\ , ir t r ni i iti\ r\
X [y z —zy ) +y \z x — xz ) + z [xy —yx)=Of
which is the same condition as before.
237. To find the centre and radius of spherical curvature.
The locus of the centre of spherical curvature is the edge
of regression of the polar surface, that is of the envelope of
normal planes of the curve.
The equation of the normal plane at the point (x, y, z) is
{^-x)c; ^{^-y)y' + {X-z)z' = 0 (i).
Hence [Art. 212] the corresponding point on the edge of
regression is the point of intersection of (i), and the
two planes
{^-x)x'^-{j^-y)y"^{X-z)z"
= x" + y"-\-z" = l (ii),
and {^-X)x"' + (rj-y)y"'j^{^-z)z"= 0...(iii),
lit. III, III f\
since XX +y y + z z =0.
238. In the figure to Art. 231, we have
p =pL = qL, p + Sp = qM = 7'il/,
and Sr = LqM=LPM.
If K be the point of intersection of MQP and qKL, we
have to the second order, Mq = Kq, and KP = LP ;
.-. LK = Bp,
and LP =-^ =~ ultimately (i).
MEASURE OF TORSION.
207
Also
dp
.■.R^ = P' + (Z
(ii).
where R is the radius of spherical curvature.
Projecting the sides of the triangle KLP on the axis of
X, we have, if I, m, n be the direction-cosines of the binormal,
^p.px"-\-^l-'^[l-{-U)=0)
d
d%
., „ 1^- ^ 1 // dp dl dl ds
thereiore ultimately - px = -r- • -r- = -r ~r" >
dr dp ds dr
or px" = al' (iii).
Since 1 = p {yz' — zy") [Art. 234] we have from (iii)
px =(Tp{yz -zy ) + o-^ [y ^ -zy ).
Similarly py' = (Tp [zx" — x z") + <^ -f {^'^" — ^'^")>
and pz" = ap {xy'" - yx'") + cr g (x'y" - y'x).
Multiply the last three equations by x\ y" , z" respectively
and add ; then we have, as in Art. 235,
A
X , y
y
X
X
y"\ z'
.(iv).
239. Since, in the figure to Art. 231, M and L are the
feet of the perpendiculars from q on two consecutive tangents
to the curve PQ^R, if we substitute P, p and r for r, p, yjr in
dr d^D
either of the known formulae r -j- ot p + -y-fa foi" the radius
dp ^ dyjr
of curvature of a plane curve, we shall obtain the radius of
curvature of the edge of regression.
208
THE HELIX.
Hence the radius of curvature of the edge of regression is
equal to
^ dR ^ d^p
[For this and the preceding article see a paper by
Dr Routh, Quarterly Journal, Vol. vii.]
240. The following examples will illustrate the use of
the different formulae we have investigated in this chapter.
Ex. 1. To find the curvature and the torsion of a helix,
A helix is a curve traced on a right circular cylinder so as to cut all the
generating lines at the same angle. Its equations are easily seen to be
ir = acos 6, y = asm.6, z = ad i&na.
Hence a;'=-asin^. 6', y'=acosd .e\ s' = atana. d'.
Square and add, then 1 = a^'^ sec^ a.
COS CL COS CL
"We therefore have x" = - cos 6 , v" = - sin 0 , z" = 0 ;
a ^ a
and also
Hence
and
x'" = -2 sin d cos3 a, y'" = - ^^^ cos ^ cos^ a, z'" ^ 0.
cos^a
a"
a
., ,Orp=: — -:5— ;
i^ ' ^ cos^a
- sin B cos a, cos B cos a, sin a
cos B cos^ a, — sin 0 cos^ a. 0
a a
-oSin^cos^a, — ^.cos^cos^a, 0
a- a^
~ --, cos" a sm a ;
a"*
a
sin a cos a
It should be noticed that the principal normals all intersect perpendicularly
the axis of the cylinder. This is seen at once by writing down the equations
of the principal normal at B, namely
x-a cos B _y -a sin 6 _z-ad tan a
cos^ "" ^md ~ 0 •
EXAMPLES. 209
Ex. 2. To find the equations of the principal normal, and of the
osculating plane at any point of the curve given by the equations
x = 4:a cos^ 6, y = 4:a sin^ d, z = dc cos 2d.
We have x'= - 12a cos^ ^ sin 0 . 6',
y'= 12a sin2 0 cos 0 . e',
z'= -6c sin 2^ . 6'.
Square and add, then 1 = Q J {a^ + c^) siii2e . 6'.
Hence ^'=--n-r — aT^os^, 2/'=-77-^ — -sind, z'= --ry-^ — sv ;
•••^'-121^2)^^^^' 2^' -12]^^°^^^^' ^" = ^-
The equations of the principal normal are therefore
x-4:aco&^d _y -4a sin^ 0 _ 2 -3c cos 29
sin 0 " cos 0 ~ 0 '
The equation of the osculating plane is
x-4:a cos^ 6, y~4:a sin^ 0, z-3c cos 2^ ' = 0.
- a cos 0, a sin ^, - c
sin ^, cos ^, 0
Ex. 3. To find to the third order the co-ordinates of any point of a curve
in terms of the arc, when the axes of co-ordinates are the tangent, the principal
normal, and the hinormal at the point from which the arc is measured.
Let OX, OY, OZ be the tangent, principal normal, and binormal at the
point 0 of a curve. Let x, y, z be the co-ordinates of a point at a distance s
from 0, and let - and - be the curvature and torsion of the curve at 0.
P 0-
Then, at the origin, a;' = l, 2/'=^, z' = 0',
also px"=0, py"=l, z"=0.
"We have, at any point of the curve,
x'x"-ty'y"-hz'z"=0.
Differentiating, we have
^, + x'x"'-hyY' + z'z"' = 0 '(i).
P
Also, by differentiating
1
^'2
f:=x"^ + y"^ + z'
we have at any point
S. S. G. 14
\^=x"x"'+y"y"' + z"z"' (ii).
p^ ds
210 EXAMPLES ON CHAPTER XI.
Also we know that
1
x\
y\
^
p'o-
x".
y'\
z'
X'",
y"\
z'
(iii).
From (i), (ii), (iii) we see that at the origin
/3- *^ p- as pa-
Hence, by Maclaurin's Theorem, we have to the third order
s"^ s^ s^ dp _ s^
^^^~ep''^^Yp~ Gild's* ^"6^'
Examples on Chapter XI.
1. Fincl the equation of the surface generated by the principal
normals of a helix.
2. Find the osculating plane at any point of the curve
cc = 6t cos ^ + 6 sin ^, y = asinO + b cos 6, z = c sin 29,
and shew that it is always inclined at the same angle to the axis
of z.
3. Find the equations of the principal normal at any point of
the curve
2 *> o 9 2
X +y = a , az = X - y .
4. A point moves on an ellipsoid so that its direction of
motion always passes through the perpendicular from the centre
of the ellipsoid on the tangent plane at any point ; shew that the
curve traced out by the point is given by the intersection of the
ellipsoid with the surface
^m-n yn-l ^l-m ^ cOUStaut,
I, m, n being inversely proportional to the squares of the semi-
axes of the ellipsoid.
5. A curve is traced on a right cone so as to cut all the
generating lines at the same angle ; shew that its projection on
the plane of the base is an equiangular spiral.
6. Shew that any curve has an infinite number of e volutes
which lie on its polar developable. Shew also that the locus of
the centre of principal curvature is not an e volute.
EXAMPLES ON CHAPTER XI. 211
7. If a circular helix be drawn passing through four con-
secutive points of a curve in space, prove that when the four
2
points ultimately coincide the radius of the helix equals -^ — , and
p-' + a-'
its slope is tan~^ - .
cr
8. Shew that if the osculating plane at every point of a
curve pass through a fixed point, the curve will be plane.
Hence prove that the curves of intersection of the surfaces whose
equations are x" + y'^ + z' = oJ^, and x^ + ')/ + z^ = —- are circles of
radius a.
9. Prove that the helix is the only curve whose radius of
circular curvature and radius of torsion are both constant.
10. A curve is drawn on the cylinder whose equation is
62 2. 22 2i2 f\
X + ay — a 0 = 0,
cutting all the generators at an angle a ; shew that its radius of
curvature at any point is p cosec^ a, where p is the radrus of
curvature of the principal elliptic section through the point.
11. If a curve in space is defined by the equations
x = 2a cos f, y = 2a sin t, z = hf,
prove that the radius of circular curvature is equal to
«v {
a' + W+hH'i'
12. In any curve if B be the radius of spherical curvature,
p the radius of absolute curvature and- the tortuosity at any
point {x, y, z), then
, f/d'x\' /d'yV /cl^zV) ^ R'
13. If the tangent and the normal to the osculating plane at
any point of a curve make angles a, yS with any fixed line in space,
shew that ~ — ^ • -77; = - , where - , — are the curvature and
sm /j a/3 p ' P o-
tortuosity respectively.
14—2
212 EXAMPLES ON CHAPTER XI.
1 4. Find tlie curvature and torsion at any point of the curve
in question 5.
15. Prove that the origin is the centre of absolute curvature
of the curve a:t? + hif + c;s^ = 1, rx^ + r'lf + r^ = 1 at all points,
whose co-ordinates satisfy the equation
a—r . h-r . c—r . _
X* 4- 2/ + T ^ = ^'
b — c c — a a — 0
16. A curve is drawn on a right circular cone always inclined
at the same angle a to the axis ; prove that (T = p tan a.
17. If p, cr be the radii of curvature and torsion at any point
of a curve in space ; p', a similar quantities at the corresponding
point of the locus of the centre of spherical curvature, then
pp = era ,
18. Every portion of a curve is equal and similar to the
corresponding portion of the edge of regression of the polar sur-
face ; prove that the tangent to it makes an angle of 45" with a
lixed plane, and that its projection on that plane is the evolute of
a circle.
19. Shew that if along the tangent to any curve a point.be
taken at a constant distance c from the point of contact of the
tangent to the given curve, and if pj be the radius of curvature in
the osculating plane of the curve traced out by the point, then
w^here p and o- are the radii of curvature and torsion of the given
curve.
20. A circle of radius a is traced on a piece of paper, Avhich
is then folded so as to become a cylinder of radius b; shew that, if
p be the radius of curvature at any point of the curve which the
Ills
circle now becomes, then — 5 — -s + ri cos*-, where s is the distance,
p a 0 a
measured along the arc, of the point from a certain fixed point of
the curve.
CHAPTER XII.
Curvature of Surfaces.
241. We have already seen, in Art. 209, that the section
of any surface, by a plane parallel to and indefinitely near
the tangent plane at any point 0 on the surface, is a conic,
which is called the Indicatrix, and whose centre is on the
normal at 0.
242. Let any section of the surface, drawn through the
normal OV, cut the indicatrix in the diameter QVQ\ and let
p be the radius of curvature at 0 of the section. Then we
have, in the limit, 2p. OV=QV^. Hence, for different
normal sections through 0, the radius of curvature varies as
the square of the diameter of the indicatrix through which
the section passes.
243. Since the sum of the squares of the reciprocals
of any two perpendicular semi-diameters of a conic is
constant, it follows from the last article that the sum of the
reciprocals of the radii of curvature of any two perpendicular
normal sections through a given point of a surface is con-
stant.
244. Since the semi-diameter of a conic has a maximum
and a minimum value, it follows from Art. 242 that the
radius of curvature of a normal section through any point of
a surface has a maximum and a minimum value, the corre-
sponding sections being those which pass through the axes of
the indicatrix.
214 CURVATURE OF SURFACES.
The maximum and minimum radii of curvature are called
the 'principal radii of curvature, and the corresponding
normal sections are called the principal sections.
The locus of the centres of principal curvature at all
points of a given surface is called its surface of centres.
245. If the axes of x and y be taken in the direction of
the axes of the indicatrix the equation of the surface will he,
when the terms of the third and higher orders are neglected,
2z = ax^ + hif.
Let /Oj, p2 be the principal radii of curvature, that is the
radii of curvature of the sections made by the planes y = 0,
^ = 0 respectively; then it is clear that pj = -, and/02 = -, .
Hence the equation of the surface will be
^ ^ y
Pi P2
The semi-diameter of the indicatrix which makes an
angle 0 with the axis of x is given by
2z cos-^ sin'^
Iir = H •
^ Pi P2
If p be the radius of curvature of the corresponding
section, we have r^ = 2pz.
T-r I cos^^ sin^^
Hence - = ! .
P Pi P2
The results of Articles 243, 244 and 245 are due to Euler.
246. When the indicatrix at any point of a surface is an
ellipse, the sign of the radius of curvature is the same for all
sections ; this shews that the concavity of all sections is
turned in the same direction, so that the surface, in the
neighbourhood of the point, is entirely on one side of the
tangent plane. The surface in this case is said to be
Spiclastic at the point.
When the indicatrix is an hyperbola, the sign of the
radius of curvature is sometimes positive and sometimes
meunier's theoeem. 215
negative, shewing that the concavity of some sections is
turned in opposite directions to that of others. The surface
in this case is said to be Anticlastic at the point.
The radius of curvature of a section which passes through
an asymptote of the indicatrix is infinite ; hence the
asymptotes divide the sections whose concavity is turned one
way from those whose concavity is turned the other way.
In the figure of Art. 71, the concavities of the sections
by the planes x = 0 and y = 0 are turned in opposite direc-
tions ; and the normal sections through the two generating
lines at 0 are the sections of zero curvature.
s
When the indicatrix is a parabola, that is to say is two
parallel straight lines, which become ultimately coincident,
one of the principal radii of curvature is infinite ; and, if p^
be the finite radius of principal curvature, the curvature of
any other normal section is given by the lormuia - = .
247. To find the radius of curvoiure of any oblique
section of a surface.
Let any oblique section through the point 0 of a surface
cut the indicatrix in the line JRKR', and let the normal
section through the same tangent line cut the indicatrix
in the line Q VQ' parallel to RKE . Let K, V be the middle
points of MR', QQ' respectively, and let p, p^ be the radii of
curvature of the sections ROR', QOQ' respectively.
Then we have, in the limit,
2p.0K = RK^,
and 2p^.JV=QV\
But OV, and therefore VK, is small compared with QV;
hence RR' and QQ' are ultimately equal. Also
0F=0^ cos 6*,
where 6 is the angle between the planes ROR' and QOQ'.
21G LINES OF CURVATURE.
Hence we have ultimately,
or P — po cos 0.
This is called Meunier's Theorem.
248. From Meunier's Theorem, and the theorem of Art.
245, it follows that if two surfaces touch one another, and
have the same radii of principal curvature at the point of
contact, then all sections through that point have the same
curvature.
249. The following proof of Meunier's Theorem is due to
Dr Besant.
Let OT be any tangent line at the point 0 of a surface,
and let P be a point contiguous to 0 on the normal section
through OT, and Q a point contiguous to 0 on an oblique
section through OT. Then a sphere can be described to
touch OT at 0, and to pass through P and Q; and the
sections of this sphere by the planes TOQ, TOP are
ultimately the circles of curvature at 0 of the sections of
the surface by those planes. Hence, as Meunier's Theorem
is obviously true for a sphere, it is true for the surface.
Ex. 1. Find the principal radii of curvature at the origin of the surface
2z = 6x2 -Sxy- 6y'-. Ans. ^%, - -^\.
Ex. 2. Find the radius of principal curvature at any point of the curve
of intersection of two surfaces.
Let p be the required radius of curvature at any point P. Let the
surfaces intersect at an angle a, and let 6, a- 6 be the angles between the
principal normal of the curve of intersecticn, and the normals to the two
surfaces. Let p^, p^ be the radii of curvature of normal sections of the two
surfaces through the tangent Hne at P. Then, by Meunier's Theorem,
p=Pi cos 6, and p = P2 cos (a - 6).
Hence, eliminating d, we have
sin2a_ 1 1 2 cos a
P' Pi' Pi PiP-2.
LINES OF CURVATURE. 217
250. Def. a line of curvature on any surface is a curve
such that the tangent line to it at any point is a tangent line
to one of the principal sections of the surface at that point.
251. The normals to any surface at consecutive points of
one of its lines of curvature intersect.
Let P be an extremity of an axis of the indicatrix which
corresponds to the point 0 of a surface, then 0, P are
consecutive points on a line of curvature.
Let V be the centre of the indicatrix, then OV will be
the normal to the surface at 0.
The tangent line at P to the indicatrix is perpendicular
to the normal to the surface at P ; it is also perpendicular to
OV; and, since P is an extremity of an axis of the indicatrix,
the tangent line is perpendicular to PV. Hence OF, PV,
and the normal at P are in a plane, and therefore the
normals at 0 and P will intersect.
Conversely, if the normals at P and 0 intersect, the tan-
gent line at P to the indicatrix will be perpendicular to the
plane which contains the normals at 0 and P ; therefore the
tangent line will be perpendicular to PF, and hence PF is
an axis of the indicatrix.
252. To find the differential equations of the lines of
curvature on any surface.
Let F{x, y, z) — 0 be the equation of the surface. Then
the equations of the normal at any point {x, y, z) are
^ — x_r} — y_ ^— -g
'dF^'lF dJ^ '
dx dy dz
The normal at the consecutive point
{x -^dx, y + dy, z + dz) is
^— x — dx 7) — y — dy t, — z — dz
dF ^fdF\ ~ ^ , ^ (dl\ ~ dF /dF^
dx \dxj dy \dy J dz \dz ^
218
LINES OF CURVATURE.
The condition of intersection of the two normals gives
tlie equation
clF
dx '
clF
dz
dF
dz
^m, dr'
\dx J
Uy}' '^{dlj
= 0...(i).
Since (x -\- dx, y + dij, z + dz) is on the surface, we have
also
dF , dF , dF , ^ ,..,
-dx + ^dy + ^dz=0...{n).
The equations (i) and (ii) are the required differential
equations.
253. To find the principal radii of curvature, and the
lines of curvature, on a surface of revolution.
It is clear that the normals to the surface at all points on
a meridian lie in the plane through the axis and that
meridian ; hence normals at consecutive points on a meridian
intersect, so that any meridian is a line of curvature. It is
also clear that the normals to the surface at all points of any
circle whose plane is perpendicular to the axis of the surface,
meet the axis in the same point, and therefore any such
circle is a line of curvature. Hence the lines of curvature
are the meridians, and the circular sections which are per-
pendicular to the axis.
It is easy to see that one of the principal radii at any
point P is the radius of curvature of the generating curve at
P ; and that the other principal radius is the length of the
normal intercepted between P and the axis.
254. The tangent plane to a developable touches the
surface at all points of a generating line. The normals
to the surface at all points of a generating line are therefore
parallel; hence normals at consecutive points intersect, so
that one set of the lines of curvature of a developable are the
LINES OF CURVATURE.
219
generating lines, the corresponding radii of curvature being
infinite.
The other lines of curvature are carves which cut all the
generating lines perpendicularly ; and hence, if the surface
be developed into a plane, the lines of curvature will become
involutes of the curve into which the edge of regression
developes.
In the particular case of the developable being a cone,
the lines of curvature will cut the generating lines at a
constant distance from the vertex, and hence they are the
curves of intersection of the surface and spheres with the
vertex for centre.
Ex. 1. Find the surface of revolution whicli is such that the indicatrix
at any point is a rectangular hyperbola.
The principal radii of curvature must be equal and opposite at any point.
Hence the radius of curvature at any point of the generating curve must be
equal and opposite to the normal: this is a known property of a catenary.
Hence the surface is that formed by the revolution of a catenary about
its axis.
Ex. 2. Shew from the general differential equations of lines of curvature,
that one system of lines of curvature on a cone are the generating lines,
and the other system are the curves of intersection of the surface and con-
centric spheres.
The equations are
dx
dF
dx
dy
dF
dy
dz
dF
dz
fdF^
(^)' "(f)' 'K^)
and
dF
dF
dF
(i),
dx^''^'dbj'^y^~dz'^'=^
jXJu ^^U v4/^
.(ii).
Since the surface is a cone whose vertex is at the origin, we have
dF dF dF ^
' dx
dz
.(iii),
therefore from (ii)
^Ks
)-''(f)
■\-zd
'i)-"-
(iv).
220 LINES OF CURVATUIIE.
Multiply the terms of the columns in (i) by x, y, z respectively, and add ;
then on account of (iii) and (iv), (i) will become
dx , dij , xdx + ydy + zdz ]=0.
dF dF
dx * dy
, fdF\ ^ fdF\
Ileuce either xdx + ydy + zdz = 0 (v),
1(1) <|) £ ^
dF dF dF
^/- (vi).
dx dy dz
From (v) we have ar + 2/^ + 2^ = constant,
shewing that one series of the lines of curvature are the curves of inter-
section of the surface and concentric spheres.
From (vi) we have
dF dF dF
dx _ dy _ dz
I m n *
where /, m, n are constants. Hence, from (iii), we have
Ix + my + nz = 0,
which shews that the other series of lines of curvature are the generating
lines.
Ex. 3. If two surfaces cut one another at a constant angle, and the
curve of intersection be a line of curvature on one of the surfaces, it "will be
a line of curvature on the other.
Let P, Q be any two consecutive points on the curve of intersection, and
let Oab be the line of intersection of the normal planes of the curve at P, Q,
where 0 is in the osculating plane of the arc PQ. If the curve of inter-
section be a line of curvature on one of the surfaces, the normals to that
surface at P, Q must intersect, they will therefore meet the line Oab in the
same point, a suppose.
Let the normals to the other surface at P, Q meet Oab in c, c' respectively.
The triangles OFa, OQa are equal in all respects, for PO = QO, Pa = Qa,
and Oa is common. And, since the surfaces intersect at a constant angle,
the angles aPc and aQc' are equal. Therefore the angle OPc, OQc' are equal.
But the angles POc, QOc' are equal, and PO = QO. Therefore Oc=Oc'.
This proves the proposition.
Ex. 4. If the line of intersection of two surfaces be a line of curvature
on both, the two surfaces cut at a constant angle.
For let P, Q be any two consecutive points on the curve of intersection ;
let the normals to one surface at P, Q meet in a, and the normals to the
other surface meet in b. Then, we have Pa=Qa, Pb = Qb, and ab common
to the two triangles aPb, aQb. Hence the angles aPb and aQb are equal.
dupin's theorem. 221
Ex. 5. If a line of curvature be a plane curve its plane will cut the
surface at a constant angle.
Any line is a line of curvature on a plane (or on a sphere). The theorem
therefore is a particular case of Ex. 4.
255. If three series of surfaces intersect at right angles at
all their common points, the curve of intersection of any tiuo
is a line of curvature on each. (Dupin's Theorem.)
Take for origin a point of intersection of three of the
surfaces, one of each series, and let the three perpendicular
tangent planes be taken for co-ordinate planes. The equa-
tions of the three surfaces will then be
^x+ay" -\-h£' -\-2hyz 4- = 0 (i),
2y + aV -I- 6V -f ^Kzx + = 0 (ii),
2z+a"(c'+Vy'+2h"xy-\- = 0 (iii).
At a consecutive point common to (i) and (ii) we have
^ = 0, y = 0,z = z', where z' is very small ; and the tangent
planes to (i) and (ii) at (0, 0, /) are ultimately
X + hzz + hyz = 0,
y + a!zz-\- h'xz = 0.
The condition that these may be at right angles gives
or, ultimately, h + h' = 0. We have similarly, since the other
surfaces cut at right angles, h' + K' = 0, and h" -\-h = 0.
Hence h = li =h'' = 0, and therefore the axes are tangents to
the lines of curvature on each surface. This being true at all
points of intersection of three surfaces, it follows that all
curves of intersection of two surfaces of different systems are
lines of curvature on each.
We have proved in Art. 164 that confocal conicoids cut
one another at right angles at all their common points.
Hence, one system of the lines of curvature of an ellipsoid
are its curves of intersection with confocal hyperboloids of one
sheet, and the other system of lines of curvature are the
curves of intersection with confocal hyperboloids of two sheets.
222 PRIXCirAL RADII OF CURVATURE.
256. To find the 7)?'i/ici2^a^ radii of curvature at any
'point of a surface.
Let f, 77, ^ be the co-ordinates of the point of intersec-
tion of the normals at two consecutive points (oc, y, z) and
{x -{- dx, y + dy, z + dz) of a surface, and let p be the radius
of curvature at (^, y, z) of the normal section through those
points. Then [Art. 251] p is one of the principal radii of
curvature, and we have
^ — x v — y_ K— ^ P P
dx dy dz y ]}\dx ) \dy) \dz ) ]
^ odF odF ^ ^pclF
K dx "^ K dy K dz
And, since (^, rj, f) is also on the normal at {x + dx,
y j^ (Jy^ z + dz^, we have by differentiating the preceding
equations, considering ^, 77, f, p as constant,
and tw^o similar equations.
Since
^dF\ d'F , d'F , d'F
\dx) dx^ dxdy dxdz "^
and similarly for d i—r- j and d ( ;t -) , the equations may be
written
^ /«r d''F\ , d'F , d'F , dfc dF
0 = - +-T-^ ) dx + -j—j- dy + -j-~r- dz -7- ,
\p ax J dxdy ^ dxdz k dx
^ d'F , fK crF\ , d'F , dKdF
^ = :r-j- dx + [-^ -^—, dy + -7—7- dz -y- ,
dxdy \p dy J '^ dydz k dy
0
d^'F , d'F , [k d'F\ , die dF
dxdz dydz ^ \p dz^ J k dz
We have also
^ dF, dF , ^ dF,
0 = -Y- dx -h -T- dy + -f- dz.
dx dy '^ dz
UMBILICS.
223
Eliminating dx, dy, dz, die we have for the determination
of the principal radii the equation
fc d^
p^ dx"
d'F
dxdy'
d'F
d'F
dxdy '
K d'F
p-^df'
d'F
d'F
dxdz'
d'F
dydz '
fc cFF
p'^ dz''
dF
dz'
dF =0.
dx
dF
dy
dF
dz
0
dx dz ' dy dz '
dF dF
dx^ dy'
257. To find the umhilics of any surface.
At an umbilic the indicatrix is a circle.
Let the equation of the surface be F {x, y, z) = 0, and
let {x, y\ z) be any point on it. The equation of the surface
referred to parallel axes through {x\ y\ z) will be
dF dF dF , f d
d ^^y r _ 0
dy' dzj
dx' dy' dz \ dx
Hence the indicatrix is similar to the section of the
conicoid
d^F 2 d'^F , d'F . ^
d'F
dx''
dy'
dz
yz
dy dz
^ d'F ^ d'F , ^ ...
+ 2 , , , ,zx-\-2-r-T-r-r^I/-\- 1 = 0 ...(i),
dz'dy
dx dy
by the plane
dF dF dF
ax '^ dy dz
0.
.(ii),
and we have already found [Art. 125, Ex. 5] the conditions
that a given section of a conicoid may be circular.
From the result of Art. 256 it is clear that the two values
of - are the squares of the axes of the section of (i) by (ii).
224 PRINCIPAL RADII OF CURVATURE.
258. To find the radii of principal curvature, and the
lines of curvature, of the surface ivhose equation is z =f{oc, y).
Let (f, 7;, f) be one of the centres of principal curvature
at the point (x, y, z), and let p be the corresponding radius
of curvature. Then, the equations of the normal at {x, y, z)
AY ill be
^-■x_7j^y_^— z p
therefore ^— x = —p (^ — z),
and 7}-y = -q(^-z).
Since the normal at (x+dx, y + dy, z + dz) also passes
through (f, 7], f) we have, by differentiating the preceding
equations,
— dx = —dp (f — z) + pdz,
and —dy = — dq (f — z) + qdz ;
that is —dx=p (pdx + qdy) ~ (^—z) (rdx + sdy).. .(i)
and — dy = q {pdx + qdy) — (^— 3) (sdx + tdy) . . . (ii).
Eliminating ^—z from (i) and (ii) we have
(1 + p^) dx +pqdy _ pqdx + (1 + q^) dy _
rdx + sdy sdx + tdy '
therefore (1 +/) 5 -pqr + {(1 +p')t-{l+ q") r] ^
+ {pqt-s(l+q^)}[^J = 0...(iii),
which is the differential equation of the projection of the
lines of curvature on the plane z= 0.
Again, from (i) and (ii) we have, putting k for
Jl+p' + q\
{^+P' + '^)do^+[pq + f)dy = 0,
and (^2 + Tt) ^'^'^ (^ "^ ^'"^ ^) ^^^^'
GAUSS' aiEASURE OF CURVATURE. 225
Heuee (1+ / + ^) (l + 3" + J) - (pg + f) = 0,
or
(rt -s') p'' -h K {t(l +p') -{- r (1 + q') - 22Jqs] p -h k'' = 0...(iv),
which is an equation giving the principal radii of curvature.
259. At an umbilicus the directions of principal curva-
ture are indeterminate ; hence the conditions for an umbilicus
are, from equation (iii) of the last Article,
1 +p^_ l-\-q^ _pq
r t s '
260. Def. The ivhole curvature of any portion of a
surface, bounded by a closed curve, is equal to the area cut
off from a sphere of unit radius by radii which are parallel to
the normals to the surface at all points of the curve.
The average curvature of any portion of a surface is the
ratio of the whole curvature to the area of that portion.
The measure of curvature at any point is the average
curvature of a very small portion which includes the point.
These definitions, which are analogous to the definitions
in plane curves, are due to Gauss.
The curve traced out on the unit sphere as above is
called the horograph of the given portion of the surface.
261. To sheiu that the measure of curvature at any point
of a surface is the reciprocal of the product of the principal
radii of curvature of the surface at that point.
Consider a small portion PQRS of the surface bounded
by lines of curvature ; then PQRS is ultimately a rectangle
whose area is PQ . PS.
Let lines parallel to the normals at P, Q, R, S, drawn
through the centre of a sphere of unit radius, meet the sphere
in p, q, r, s. Then, since the principal planes at any point of a
surface are at right angles, the angles p, q, r, s are right
angles, and therefore pqrs is ultimately a rectangle whose area
is pq.ps. But the angle between the normals atP and Q
S. S. G. 15
220 GEODESIC LINES.
PO
is ultimately — — , and the angle between the normals at P
PS
and S is ultimately — , where p^ , p^ are the principal radii
Pa
pn pa
of curvature at P. Hence pa = — ^ , and ps = — , so that the
area of pqrs is ultimately ^ . Hence the measure of
curvature at P, which by definition is the limiting value of
area pqrs . 1
area PQUS ' p^p^ '
Geodesic Lines.
262. Def. a geodesic line on a surface is such that any
small element AB is the shortest line which can be drawn
on the surface from A to B.
The length of the line joining any two indefinitely near
points will clearly be least when the curvature is least. But
by Meunier's theorem, the curvature of a surface through a
given tangent line is least when the section is a normal
section. Hence at any point of a geodesic line on a surface the
plane of the curve contains the normal to the surface, so that
the principal normal of the curve coincides with the normal
to the surface. We therefore have at any point of a geodesic
line on a surface
d^x d^y d"z
ds^ _ ds^ _ ds^
dF_~ dF_~ dF '
dx dy dz
Curvature of Conicoids.
263. Since all parallel sections of a conicoid are similar,
it follows that the indicatrix at any point P of a conicoid is
similar to the central section which is parallel to the tangent
plane at P. Hence the tangents to the lines of curvature
at any point P are parallel to the axes of that central section.
CURVATURE OF CONICOIDS.
227
Now, by Art. 167, the lines which are parallel to the axes
of the central section are the tangent lines at P to the curves
of intersection of the conicoid with the confocals which go
through P. Hence, as we have already proved in Art. 255,
the lines of curvature of a conicoid are the curves of intersec-
tion with confocal conicoids.
264. We can shew that the lines of curvature on a
conicoid are its curves of intersection with confocals in the
following manner.
At points common to
a 0 c
and
of
+
y
a+X ' b +\
we have, by subtraction,
c + X
= 1
•(i),
•(iiX
X
r
+
a(a + \) h (b + \) c(c + X)
Differentiating (ii) and (iii) we have
xdx ydy . zdz
.(iii).
and
a + X
tA/\A/tA/
+
b+\ c+\
+ T-7:
ydy
+
= 0.
zdz
= 0
(iv),
a (a + X) ' 6(6 + X) c(c+X)
The elimination of a + X, 6 + X, c + X from (iii), (iv),
(v) gives
00
y
z
a'
b '
c
dx,
dyy
dz
dx
1l'
dy
b '
dz
c
= 0
•(vi),
which is the differential equation of the curve of intersection
of (i) and any one of its confocals ; and it is easy to see, by
comparing with (i), Art. 252, that (vi) is the differential
equation of a line of curvature.
15—2
228 LINES OF CURVATURE OF CONICOIDS.
2G5. The radius of curvature of any normal section of a
central conicoid may be found as follows.
The radius of curvature of any central section of a coni-
coid through a point P is, by a well-known formula, equal to
— , where d is the semi-diameter parallel to the tangent at P,
and p is the perpendicular from the centre on the tangent
at P. Hence, by Meunier's Theorem, the radius of curvature
of any normal section of a conicoid through the point P is
equal to — , where p^ is the perpendicular from the centre
on the tangent plane at P, and d is the semi-diameter
parallel to the tangent line at P ; for the cosine of the angle
between the normal section and the central section is — .
P
266. At any point of a line of curvature of a central
conicoid, the rectangle contained hy the diameter parallel to the
tangent at that point and the perpendicular from the centre on
the tangent plane at the point is constant.
Let p be the perpendicular from the centre on the tangent
plane at any point P of a given line of curvature, and let a, j3
be the semi-axes of the central section parallel to the tangent
plane at P. Then, one of the axes, a suppose, is parallel to
the tangent at P to the line of curvature, and the other axis
is of constant length for all points on the line of curvature
[Art. 167, Cor.]. Hence, since jjayS is constant, it follows
that poL is constant throughout the line of curvature.
267. At any point of a geodesic on a central conicoid, the
rectangle contained hy the diameter 'parallel to the tangent at
that p)oint and the perpendicular from the centre on the tangent
plane at the pioint is constant.
The differential equations of a geodesic on the conicoid
aa^ + by"^ + c^^ = 1 are
d^x d^y d^z
ds^ _ ds'^ _ ds^
ax by C2 *
GEODESICS ON CONICOIDS. 229
// // II
or — = 7-= — = A, (i).
ax by cz
We have to prove that 'pr is constant, where
\= ax^ ■\-ly'^ ■\- cz"" (ii),
and -2 = aV + 6y + cV (iii).
'p
Differentiating ax' -}- hy^ -{■cz'^ = 1 twice with respect to s,
we have
ax"^ + hy'^ + cz'^ + axx' + hyy" + czz" = 0 (iv).
From (i) we have
axx" + hyy" + czz" r>^ c r--\ j /• \
a'«' + by + &z^ r' ^ ^ ^ ^
I It , 1 I tl t 'If 6 7
., ^ axx +bijy + cz z r as „ .... ^ r--\
Also X = —T, — -, — r^r^^, 7, — r = -\ — 7- , irom (ii) and (in).
a XX + b'yy + czz 1 dp ^ ' ^ ^
]f ds
TT 1 dr 1 dp ^
Hence - -r -^ t- = ^>
r as p ds
and therefore pr is constant.
Ex. 1. The constant pr is the same for all geodesies which pass through
an umbilic.
This follows from the fact that the central section parallel to the tangent
plane at an umbilic is a circle, and therefore the semi-diameter parallel to
the tangent to any geodesic through an umbilic is of constant length.
Ex. 2. The constant pr has the same value for all geodesies which touch
the same line of curvature.
At the point of contact of the line of curvature and a geodesic which
touches it, both p and r are the same for the line of curvature and for the
geodesic.
Ex. 3. Two geodesies which touch the same line of curvature make equal
angles with the lines of curvature through their point of intersection.
From Ex. 2, the semi-diameters parallel to the tangents to the two
geodesies, at their point of intersection P, are equal to one another, and are
therefore equally inclined to the axes of the central section which is parallel
to the tangent plane at P. But the axes of the central section are parallel to
the tangents to the lines of curvature through P; this proves the proposition.
230 EXAMPLES ON CHAPTER XII.
Ex. 4. Two geodesies which pass through umbilics make equal angles
with the lines of curvature through their point of intersection.
Ex. 5. Any geodesic through an umbilic will pass through the opposite
umbilic.
Ex. 6. The locus of a point which moves so that the sum, or the differ-
ence, of its geodesic distances from two adjacent umbilics is constant, is a
line of curvature.
Ex. 7. All geodesies which join two opposite umbilics are of constant
length.
Ex. 8. The point of intersection of two geodesic tangents to a given line
of curvature, which intersect at right angles, is on a sphere.
Let i\, r^ be the semi-diameters parallel to the tangents to the geodesies
at P, their point of intersection. Then, since the geodesies cut at right angles,
where a and j3 are the semi-axes of the central section parallel to the tangent
plane at P. But, if p be the perpendicular on the tangent plane at P, then
25j'i=:pr2 = constant, from Ex. 2. Hence, since pa^ is constant, and also
a^+^^ + OP^, it follows that OP is constant.
Ex. 9. The point of intersection of two geodesic tangents, one to each of
two given lines of curvature, which cut at right angles, is on a sphere.
Examples on Chapter XII.
1. A surface is formed by the revolution of a parabola about
its directrix ; shew that the principal curvatures at any point are
in a constant ratio.
2. If p, p' be the principal radii of curvature of any point of
an ellipsoid on the line of its intersection with a given concentric
sphere, prove that the expression , will be invariable.
p + p
3. If itj -1- w^ + 1^3 + w ^ = 0 be the equation to a surface
where u^ is a homogeneous function of x, y, z, of the rth degree,
then u^ + u^ + u^ (Ix + my + nz) = 0 will be the general equation of
surfaces of the second order having the same curvature at the
ori":in.
EXAMPLES ON- CHAPTER XII. 231
4. The normal at each point of a principal section of an
ellipsoid is intersected by the normal at a consecutive point not
on the principal section ; shew that the locus of the point of inter-
section is an ellipse having four (real or imaginary) contacts with
the evolute of the principal section.
5. In the surface y cos — x sin - = 0,
the principal radii of curvature at (x, y, z) are =»=
a
C. Shew that the umbilici of the surface
'^\^ . fy\' . /^^' = l
©*Hi)'^(:
lie on a sphere whose centre is the origin and whose radius is
- , abc
equal to —, — ^ .
ao + DC -^ ca
7. The centres of curvature of plane sections of a surface at
any point lie on the surface
{x' + 2/ + z') ("- + ^^ = ;s (a;^ + f).
8. Prove that the line which separates the synclastic from
the anticlastic parts of a surface is a line of curvature, and that
ulong it the inflexional tans^ents coincide.
9. The projections of the lines of curvature of an ellipsoid on
the cyclic planes, by lines parallel to the greatest axis of the
surface, are confocal conies.
10. If one of the lines of curvature on a developable surface
lies on a sphere all the other lines of curvature, other than the
rectilineal ones, lie on concentric spheres.
11. A plane curve is wrapped upon a developable surface.
If p is the radius of curvature of the plane curve at any point, p
the corresponding radius of circular curvature of the curve upon
the surface, R the corresponding princij^al radius of curvature of
the surface, and ^ the angle at which the curve intersects the
sin^ (^ _ 1 I
K p p
generator of the surface.
232
EXAMPLES ON CHAPTER XII.
12. If one system of lines of curvature of a surface are
ciicles, the surface is the envelope of a sphere whose centre moves
on a given curve.
13. If a geodesic line is either a line of curvature or a plane
curve it is both ; but a plane line of curvature is not necessarily
geodesic.
Shew that if one series of the lines of curvature is geodesic
they are all repetitions of the same plane curve.
14. Shew that if the normal to a surface always passes
through a given curve, one set of the lines of curvature are circles;
and that those normals which pass through a given point on the
curve are generating lines of a right cone whose axis is the
tangent at that point. Hence shew that if the normal always
passes through two curves, these curves must be conies in planes
at right angles, the foci of one being the vertices of the other.
15. Find the differential equation of the projection on the
plane xy of each family of lines of curvature of the surface which
is the envelope of a sphere whose centre lies on the parabola
o? + ^ay = 0, ;:; = 0, and which passes through the origin.
16. Shew that the principal curvatures at any point of a
surface are given by the equation
dx
dm
dx
dn
dx
1
+ -,
dl_
dy
dm
dy
dn
dy
+
dl
Tz
dmx
dz
dn
dz
1
+ -
= 0,
P
I, m, n are the direction-cosines of the normal at the
where
poiut.
17. The tangent planes to the surface of centres at the two
points where any normal meets it are at right angles.
IS. Shew that the point for which cc = y = » is an umbilic of
a;"* + 2/"* + z"^ = a"*,
and the radius of curvature there is
a
m — 1
(3)
TO— 2
S'l
EXAMPLES ON CHAPTER XII. £33
19. In a hyperbolic paraboloid, of whicli the principal para-
bolas are equal, the algebraic sum of the distances of all points of
the same line of curvature from two fixed rectilinear generators is
constant.
20. Along the normal at a point P of an ellipsoid is measured
PQ of a length inversely proportional to the perjDendicular from
the centre on the tangent j)lane at P; prove that the locus of Q is
another ellipsoid, and that the envelope of all such ellipsoids is the
"surface of centres," that is the locus of the centres of principal
curvature.
21. Shew that the specific curvature at any point of the
surface xyz = abc varies as the fourth power of the perpendicular
from the origin on the tangent plane at the point, and that at an
umbilicus it is 4 (ahc)~^.
22. If a surface have one principal radius of curvature con-
stant it is the envelope of a sphere of constant radius.
x^ ij^ ^
23. Find the umbilici of the surface v ^ + — = k", and
abc
X II z
shew that at the umbilicus - = ^ = - the directions of the three
abc
lines of curvature are given by the equations
dx chi dy dz dz dx . .
— = -,- , -, - = and — = - — respectively.
abbe c a ^ "^
24. If two geodesies be drawn on an ellipsoid from any point
to two fixed points, the sine of the angle between them varies as
the perpendicular on the tangent plane at the point.
25. Shew that on a surface of revolution, the distance of any
point of a geodesic from the axis varies as the cosecant of the
angle between the geodesic and the meridian.
26. If a geodesic line be drawn on a developable surface and
cut any generating line of the surface at an angle »//■ and at a
distance t from the edge of regression measured along the generator,
prove that
-y- + cot W . t = p,
dij/
234 EXAMPLES ON CHAPTER XII.
where p is the radius of curvature of the edge of regression
at the point where the generator touches it.
27. Shew that the tangent to a geodesic or line of curvature
on a quadric always touches a geodesic or line of curvature
respectively on a confocal quadric.
28. Shew that the reciprocals of the radii of curvature and
torsion of a curve drawn on a developable surface are
sin^ 9 , sin 6 cos 0 da
and 1- -r- ,
p cos a p as
where p is the principal radius of curvature of the surface at the
point, 6 the angle the tangent line to the curve makes with the
generator through the point, and a the angle between the normal
to the surface and the principal normal of the curve.
If a geodesic on a developable surface be a plane curve it must
be one of the generators or else the surface must be a cylinder.
29. If - and - be the curvature and tortuosity at any point of
a <]jeodesic drawn on a surface, and — , — be the principal curvatures
. Pi P2
of the surface at that point, shew that
o" vPi py \P2 p
30. Through a given generator of a hyperboloid of one sheet,
draw a variable plane ; this will touch the surface at some point
A on the generator and will contain the normal to the surface at
another point B. Shew that the sum of the square roots of the
measures of curvature of the surface at A and B is constant for all
planes through this generator.
Hence shew that the same proposition is true for any skew
surface.
31. If tcT be the pitch of the screw by which any generator of
a skew surface twists into its consecutive position, shew that
m^ + pp = 0, where p, p' are the principal radii of curvature at tho
point where the shortest distance between the two consecutive
irenerators meets them.
EXAMPLES ON CHAPTER XII. 235
32. If a geodesic be drawn on an ellipsoid from an umbilicus
to an extremity of the mean axis, prove that its radius of torsion
at the latter point is
r, I . sin 6 cos 6,
where a, h, c are the semi-axes of the ellipsoid arranged in
descending order of magnitude.
33. If from any point on a surface a number of geodesic
lines be drawn in all directions, shew (1) that those which have
the greatest and least torsion bisect the angles between the
principal sections, and (2) that the radius of torsion of any line,
making an angle 6 with a principal section, is given by the
equation
where p^ , p^ are the radii of curvature of the principal sections.
34. Find the equation to the surface which is the locus of the
central circular sections of a series of confocal ellipsoids. Prove
that this surface cuts all the ellipsoids orthogonally, and that the
orthogonal trajectories of the circles, drawn upon the surface, are
lines of curvature upon two hyperboloids confocal with the
ellipsoids.
35. If a cone of revolution circumscribe an ellipsoid, prove
that the plane of contact divides the ellipsoid into two portions
whose total curvatures are 27r(l +sina) and 27r(l — sina), where
2a is the vertical angle of the cone.
36. If any cylinder circumscribes an ellipsoid it divides it into
portions whose integral curvatures are equal.
37. The measure of curvature at any point of the surface
2.22 2
T^ 2 = 1 IS -—, 3V2 >
a c (c + r')
where r is the length of the generator through the point cut off
by the plane z = 0.
38. Prove that, if radii be drawn to a sphere parallel to the
principal normals at every point of a closed curve of continuous
2o() EXAMPLES ON CHAPTER XII.
curvature, the locus of their extremities divides the surface of the
sphere into two equal parts.
Hence shew that the total curvature of a geodesic triangle on any
surface is equal to the excess of its angles over two right angles.
39. Define the radius of geodesic curvature of a curve drawn
upon a surface, and shew that at any point it is equal to R cot <^,
where R is the radius of curvature of the normal section contain-
ing the tangent to the given curve, and ^ is the inclination of the
osculating plane to that section.
40. If a surface roll on a second surface without rotation
about the common normal, and the trace on one surface is a
geodesic, the trace on the other surface is a geodesic.
Hence prove that Gauss's measure of curvature is constant for
all areas enclosed by geodesies.
MISCELLANEOUS EXAMPLES.
1. The inclinations to the horizon of two lines which are at
right angles to one another are a, ^, the lines being on a plane in-
clined to the horizon at an angle 0; shew that sin^^ = sin^a + sin^yS.
2. Shew that the volume of the tetrahedron of which a pair
of opposite edges is formed by lengths r, r' on the straight lines
whose equations are
x-a y -h z- G ^ x-a y -h' z-c'
I m ti I'
m' 7t' '
is \rr a - a', h -h', c-c'
V
n
, 'ill , n
3. A parallelogram of paper is creased along its shorter
diagonal, and the two halves are folded so as to make an angle 6
with each other : find the distance between the extremities of the
longer diagonal, and prove that it is equal to the shorter, if
Q
sin^ - = cot a cot /?, where a and y8 are the angles the sides make
with the shorter diagonal.
4. The ends of a straight line lie on two fixed planes which
are at right angles to one another, and the straight line subtends
a right angle at each of two given points: shew that the locus
of its middle point is a plane.
5. The equations of three straight lines are oj — z = l, a; = 0;
z-x=l^ y = 0; and x — y = l, z — 0; prove that the locus of all
straight lines which intersect the three lines is
x^ + y^ + z^ - 2yz - 2zx - 2xy = 1.
238
MISCELLANEOUS EXAMPLES.
6. Three fixed lines are cut by any other line in the points
A, i?, C, and D is the point on the line ABC such that {ABCD}
is harmonic: shew that the locus of /) is a straight line.
7. A point moves so that its perpendicular distances from
two given lines are in a constant ratio: shew that its locus is an
hyperboloid whose circular sections are perpendicular to the given
lines.
8. A straight line slides upon two fixed straight lines in
such a way that the part intercepted subtends a right angle at
a fixed point : shew that the line generates a conicoid.
9. A sphere touches the six edges of a tetrahedron : shew
that the three lines joining pairs of opposite points of contact
will meet in a point.
10. A straight line moves in such a manner that each of
four fixed points on the line is always on a given plane; shew
that any other fixed point on the line describes a plane ellipse.
11. Any three points jP, Q, JR, and the polar planes of those
points with reference to any conicoid are taken. JPQi, -P-^j are
the perpendiculars from F on the polar planes of Q and H respec-
tively; QF^, QP^ are the perpendiculars from Q on the polar
planes of R and F respectively ; and RF^ , FQ^ are the perpen-
diculars from F on the polar planes of F and Q respectively.
Shew that FQ^ . QF^_ . FF^ = FF^ . QF„ . FQ^.
1 2. Shew that, if the equation
ax^ + 6y- + cz^ + ^fyz 4- "Igzx + Ihxy = 0,
represent two planes, the planes which bisect the angles between
them are given by the equation
X
ax + hy + gZy
1
y ,
hx -I- hy -{-fzy
1
gx +fy + cz
1
ch-fg
13.
of- gh hg - hf
Shew that, if the equation
ace" + hrf + c:^ + yyz + Igzx + Ihxy = 0,
0.
MISCELLANEOUS EXAMPLES.
239
represent two planes, the product of the perpendiculars on the
planes from the point (x, y, z) is
ax^ + by'' + cz^ + ^fyz + ^gzx + Ihxy
(a + b + cf + ^{r- bcY + 4 (/ - caf + 4(A' - ahf '
14. li U= {abcdlmnpqr) {xyzwY = 0 is the equation of a cone,
shew that the co-ordinates of the vertex satisfy the equations
dU
dU
dU
da
db
dl
^^_ -—
: mm^ —
aA
oA
' 8A
da
db
dl
= 0,
where A is the discriminant.
15. Shew that, if the equation
ax^ + by^ + cz^ + ^fyz + ^gzx + ^hxy + ^ux + ^vy + Iwz + cZ .= 0,
represent a paraboloid of revolution, c = b ^a. Shew also that if
c = 6 + a, the equations of the axis of the paraboloid will be
cz + w-0, {ex + u) J a + {cy + v) Jb = 0.
16. Shew that the three principal planes of the surface
ax^ + by~ + cz- + 2fyz + 'igzx + 2hxy = 1
are given by the equations
ax + hy + gz, hx +by -{-fz , gx +fy + cz
Ax + Hy-\- Gz, Hx + By + Fz, Gx + Fy + Cz
, a; ,2/ > ^
where -4, B, C ... are the minors of a, 6, c in the determinant
' a, h, g
K \ f
17. If ^ be any semi-axis of the conicoid
ax^ + b'lf -^c^ + Ifyz + 2gzx + 2Aa:y - 1,
prove that the values of r will be given by
9^ . ¥ , fg
+
+
gh-af-^-^- hf-bg + ^, fg-ch+'^^.
1.
240 MISCELLANEOUS EXASIPLES.
18. The ellipse lfx^ + a-if-a%^ = 0, ^=0 is a plane section
of a cone whose equation, referred to its principal axes, is
/Jyx" + yay"^ + afiz^ = 0.
Shew that the vertex of the cone is on the curve
\ a + /3 + y j (. ySy + ya + ayS
r 27 2 2 \ 2
I a 0 z \
19. Shew that the conicoid ax" + hy^ + cz^ + d=0 is its own
polar reciprocal with respect to any one of the conicoids
± ax' ± hy' =•= cz" =fc cZ = 0.
20. Find the locus of the centre of the sphere w^hich passes
through two circular sections of a conicoid which are of opposite
systems and whose planes are equidistant from the centre.
21. Prove that the foci of sections of an ellipsoid made by
a series of parallel planes lie on an ellipse.
22. Shew that the perpendicular from the centre on the
/y*" J_ /^fi" ^"^ (lie
tangent plane at any point of ~ s = 1 is — ^^ ; , where r
is the length of a generator through the point cut oflf by the plane
of xy.
23. The six lines AB', B'C, Cxi\ A'B, BC\ C'A are six gene-
rators of the hyperboloid ax' + h'if + cz^ = \, and AB', B'C, CA',
are respectively parallel to A'B, BC, C'A ; shew that, if the
parallelo piped of which the six generators are edges be completed,
the corners which are not on the hyperboloid will be on
ax' + hy^ + cz' + 3 = 0.
24. Shew that at any point the rate per unit of length of
X' + ?/ z^
generator at which the normal to the hyperboloid ~ — ~2 = 1
a c
ft
twists round a generator as we move along it is — „ , where r
c + r^
is the distance, measured along the generator, of the point from
the plane of xy.
MISCELLANEOUS EXAMPLES. 241
25. ABGBQ is a twisted polygon all whose angles are right
angles; AB, CD lying on fixed straight lines. Shew that if ^,
B, C, D be any points on their respective lines, the locus of P or
Q is an hyperboloid of one sheet.
26. If I be the latus-rectum of a parabola, and l^, l^, l^ the
latera recta of its orthogonal projections upon a rectangular system
of co-ordinate planes making angles a, /3 and y respectively with
the plane of the original parabola, then
2 cos^a cos'^B cos^v
l^ It li li
27. If the six points on a conicoid, normals at which meet
in a point, are joined in pairs by three lines, prove that whatever
set of joining lines is taken the sum of the squares of the semi-
diameters parallel to them is constant.
28. A conicoid whose centre is D touches the three planes
YOZ, ZOX, XOY in A, B, G respectively : shew that the lines
through A, B, C parallel respectively to OX, OY, OZ, and the
line OD are four generators of an hyperboloid of one sheet.
29. Three perpendicular tangent planes are drawn, one to
each of three confocal conicoids : shew that the normals at the
points of contact of the planes, and the line joining their point
of intersection to the centre of the conicoids are generators of an
hyperboloid of one sheet.
30. If any line through a fixed point 0 meet any number of
fixed planes in the points -4, B, C , and on the line a point X
be taken such that -^-p. = ^^ + ^ = + y— + . . . ; shew that the locus
(JX (J A (JJj (JL/
of X will be a plane.
31. If any line through a fixed point 0 meet any given sur-
face in the points Ay B, C, D..., and X be taken such that
OX^OA^OB^OC'^iW ''"''' ^^^"^ ""'^^ *^^ ^''''''' of Z be a
plane.
S. S. G. 16
242 MISCELLANEOUS EXAMPLES.
32. Two straight lines drawn in fixed directions through any
point 0 meet a given surface in the points A, B, C, D... and
A, B,C , D ...; shew that ^^, ^^, ^^, ^jjr~ is constant.
33. Prove that the pedal of a helix with regard to any point
on its axis is a curve lying on a hyperboloid of one sheet ; and
that, if the pitch of the helix be ^tt, this curve will cut perpen-
dicularly all the generators of one system of the hyperboloid.
34. A curve is di-awn on a sphere of radius a cutting all the
meridians at a constant angle ; shew (i) that the foot of the per-
pendicular from the centre of the sphere upon the osculating plane
is the centre of curvature ; (2) that if p, o- be the radii of curva-
ture and torsion crp' = a^,
35. Prove that the shortest distance of the tangents at tv.^o
points FQ of any curve is ultimately equal to -- — , where p and
(T are the radii of curvature and torsion.
36. Tangent planes to a conicoid are drawn at points along a
line of curvature : shew that the perpendiculars from the centre
on their planes lie on a quadric cone, that the different cones so
formed are confocal, and that the focal lines of the cones are
perpendicular to the circular sections of the conicoid.
37. A curve is drawn making a constant angle a with the
axis of a paraboloid of revolution : prove (i) that its projection
on a plane perpendicular to the axis is the involute of a circle
of radius Zcota, (ii) that its radii of curvature p and tori<ion a-
are given by the equations p^sin^a = o-^ sin^a cos^a = r° — -^^ cot^a,
where r is the distance of the point from the axis, and I is the
semi-latus rectum of the generating parabola.
CAIilBRIDGE : PlilNTIiD DY C. J. CLAY, M,A. AND bOX, AT THE UNIVERSITY PRESS.
By the same Author.
AN
ELEMENTAEY TEEATISE
ON
CONIC SECTIONS.
Fourth Edition. Crowu 8vo. 7s. 6d.
The Academy says : — " The best elementary work on these curves which
has come under our notice. A student who has mastered its contents is in a
good position for attacking scholarship papers at the universities. ... There is
an ample store of exercises, and many useful examples are worked out in a
very suggestive manner."
The Journal of Education says : — "We can hardly recall any mathematical
text-book which in neatness, lucidity, and judgment displayed, alike in choice
of subjects and of the methods of working, can compare with this..., We have
no hesitation in recommending it as the book to be put in the hands of the
beginner."
Nature says: — "A thoroughly excellent elementary treatise. For a long
time we have been exercised in mind when asked to recommend a book on
Conies. To all its predecessors, with their varying shades of goodness and
badness, we had some objection or other to urge. Mr Smith has just met our
want ; his book is right up to the time, and is admirably adapted for the pre-
paration of pupils for college scholarships ; for students at the University it
is a fitting introduction to that as yet unapproached work, Salmon's treatise
on these curves. The text is excellent, full in alternative proofs, suggestive
in its methods ; the numerous worked-out exercises in addition to those col-
lected at the close of the several chapters, render the reader independent
of any other work,"
The Glasgow Herald says: — "This is a valuable contribution to mathe-
matical literature. The arrangement will be generally admitted as judicious.
He commences with investigations of the more elementary properties of the
ellipse, parabola, and hyperbola, as the best preliminary to the consideration
of the general equations of the second degree... Abundant examples, many of
them with complete solutions, accompany each chapter, and add greatly to
the value of the book."
MACMILLAN AND CO., LONDON.
By the same Author.
ELEMENTARY ALGEBEA.
Globe 8vo. 4s. GJ.
In this work the Author has endeavoured to explain the principles of
Algebra in as simple a manner as possible for the benefit of beginners,
bestowing great care upon the explanations and proofs of the fundamental
operations and rules.
The Athenceum says : — "This Elementary Algebra treats the subject up to
the binomial theorem for a positive integral exponent, and so far as it goes
deserves the highest commendation. Mr Smith has avoided the danger
which, as the preface shows, besets writers of treatises like the one before us
— that of 'paying too little attention to the groundwork of their subject.'
All through the volume the reasoning underlying the processes of algebra is
kept prominently in view, and thus a real interest is infused into the subject,
while the educational value of the study is immensely increased. This
valuable characteristic of the book is observable as much in the earUest as in
the most advanced chapters, and we doubt not that beginners will appreciate
it... The examjjles, which are very numerous, are a notable feature of the book,
and, so far as we have investigated them, are singularly well selected and
arranged, and the solution of them on the students' part, after careful
perusal of the chapters to which they are appended, cannot fail to be greatly
'for the benefit of beginners.' "
The Schoolmaster says: — "The examples are numerous, well selected, and
carefully arranged. The volume has many good features in its pages, and
beginners will find the subject thoroughly placed before them, and the road
through the science rendered easy to no small degree."
The School Guardian says : — "The examples and exercises are skilfully
constructed and grouped... It extends as far as the simpler cases of the
binomial theorem, and, no matter at what page it may be opened, it will be
found a model of accurate and strict method."
Nature says: — "It is a pleasure to come across an algebra-book which has
manifestly not been written in order merely to prepare students to pass an
examination. Not that we think Mr Smith's book unsuitable for this purpose ;
indeed, with its carefully-worked examples, graduated sets of exercises, and
regularly-recurring miscellaneous examination papers, it compares favourably
with the most approved 'grinders' books... He shows to great advantage
as a teacher, his style of exposition being most lucid: the average student
ought to find the book easy and pleasant reading. The second set of exercises
on the Binomial Theorem is worth specially noting."
The Educational Times says:— "Mr Charles Smith, Tutor of Sidney
Sussex College, Cambridge, whose Elementary Treatise on Conic Sections is so
well known to most students of Mathematics, has done us very good service
in publishing an Elementary Algebra. There is a logical clearness about his
expositions and the order of his chapters for which both schoolboys and
schoolmasters should be, and will be, very grateful. His treatment of the
Theory of Indices, for instance, though really a very simple matter, is admir-
able for the way in which it sets forth the difficulties of the subject, and then
solves them."
ALGEBRA FOR SCHOOLS AND COLLEGES.
Globe 8vo. In the Press.
MACMILLAN AND CO., LONDON.
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