D.Sc.
lie
17. Relation between areas
18. Relation
a
plane
51.
The
volume
a
hyperboloid
when
generators
are
co-
a given
etiuation
•-
curvature and
-.----.
-
is
l/pip.2
geodesic -
they are
AB
and
BA
PN
about
O
in
C,
and
ON
P
moves
round
the
perimeter
(1,
+
sphere
P is
plane XOY
locus
of
the
equation.
But
tlierefore the locus
(i)
the plane
generate the
=
by
x-lcfi
in
produces
a
surface
surface,
z^,
the
curve,
or
surface,
=
{.v,
y,
=
revolution
of
the
parabola
y^
points
its
projection
=
cylinder 16.r-
are
2,
3,
cone
through
them
cos~^
l/VS-
24.
Distance
=
tlii-oui;Ii
O
whose
directicni-ratios
are
points
(1.
2,
3),
(3,
plane
the plane ?
from
the
planes
the
point
is
i/^
whose semi- '
vertical angle
•planes
ax-\-by-\-cz-\-d
therefore
point (a, h, c)
>
'
since the direction-ratios are I, m, 0. Again the line lies in
the
plane
z
=
point
which
of the
the orthocentre of the
ax+b>/
{al
are
parallel
two of the
a triangular
equa-
tions
lie
2x
a
of planes
intersects
the
line
which
intersects
the
equa-
tions
to
of the three given
Xo
can
be
found
to
satisfy
(6),
and
therefore
an
on
//=1,
i/=
of the
the lines
the plane ABC
y^xtaua.,
($-f3)-vii,j
—^^
angles
0,
c),
(0,
(3,
we have
to eliniinate
line
which
inter-
sects
two
—
^
—
—
+
the
substitutions
i
2S
=
where
the
angles
>/0^,
from
rect-
angular
to
from P
mid-points
n),
Find the
are
and
B.
drawn.
Find
a
line
which
is
the
])olar
of
P
with
respect
to
through
plane
given
by
S^
points
a
sphere
+
the
circumscribing
sphere.
10.
If
the
lx-^my-\-nz
the
tetrahedron
C.
sphere.
26.
is
the
origin
number
of
sets
of
lies
origin
which
is
normal
to
the cone in
for base.
Problemes
de
Geometrie
mutually
perpendicular
lines
the
])Iane
angle
diflerentiation.
15.
Prove
of
the
cone
through
the
coordinate
are OL
OR'
are
the
diameters
-r-^+xp?
lies on
a sphere
{ax'- +
the
similarity
space.
There
is
a
note these
'Lz^.
=-LlR.
=zZSL
centre of a
conicoid
by
the
a
fixed
plane.
Ex,
5.
The
centres
of
the cone
(«.,
is
enveloping cone of the sphere
x^-\-y'^+z'^
in
of
S
whose
vertex
is
A.
the
ellijjsoid
x'^/a'^
to
a
given
line.
cuts the
surface in
and
is
the
cubic
are parallel
semi-diameters
axes
are
rectangular,
OP-- -
OQ-
a-
0^
c-
two
their
extremities.
Hence
if
P,
Q,
R
OZ' if
to the
the form
z
the
z
plane
XOY
into
P is the extremity
point of
the
paraboloid.
the
tangents
from
from
l meet ag.*^^
lines
P,
Q,
PQR,
are
concurrent.
11.
12.
point
P
on
of
the
fixed point
of the
lx
constant
envelope
a
cone
section,
satisfy
]:>^-
2?;^+y2
of
of
If
X
and
jul.
circles.
Ex.
1.
Prove
'—
,
whicli
the
plane
of
is
to
represent
two
planes,
different
from
those given
sheet
that
the
smallest
closed
are the
extremities of
elli
from
axes of any
the areas of the
and
that
circular tone whose axis
conic, and
conicoid
{oL
sin a.,
second.
102.
Any
generator
of
Ex.
a
The point of intersection,
Q
at
the
point
 0,
through
determine
a
major
of the
of
the
are
determined
by
nine
(Cf.
§47,
Ex.
1.)
If
the
tliree
lines
are
parallel
to
the
same
plane,
through
A,
B,
C,
and
cz
to
A
=
OX,
y
whose
equation
(see
0,
0)
is
are at
right angles
0,
0)
is
/icLr^
double
the
and have their vertices
of the second
18.
and that
pencil
with
the
generators
principal
of the same system,
that he
other common
in
rt/ b^
the plane
which lies
coincide with
to
—
confocal
with
a
given
plane with
jjlane
Q,
R,
S.
To
confocals,
and
Xp
per-
(1)
A;^
+
Ex.
4.
If
a,,
^i,
Cj
the
equations
l^
distance,
measured
was first
pointed out
and given plane
in
(II)
(a,
(3,
y)
a
focus
of
=
^
line
of the
hyperboloid
Ex.
3.
If
A
and
the
line
QR,
focal
Ex.
9.
Prove
principal
plane.
E.G.
N
all
focal lines.
conicoid
tangent
plane
8.
The
locus
9.
The
with
right
circular
cone
a
system
one of
14.
all
the
confocals,
and
its
equation
referred
the
tangent
planes
the
of the system.
is a
principal axes of
is
a
rectangular
hyperbola.
22.
If
A,
/x,
V
are
the
parameters
this chapter
second
disposable
constants.
a
tangent
through
The
above
equation
may
be
written
3F
(,r',
y\
surface.
Ex.
10.
If
two
conicoids
have
If
to
the
common
lines
.r
are
the
roots
of
the
equation
Hence
—^^^^^=trJ[
z3^+.t^+«^=o.
coordinate planes
the
cone
=
bisected
is equal
:
x-7/+l=0;
m^,
direction-cosines
x,
y,
z
are
zero,
aF_3F_9F_
Cor.
The
equation
to
any
diametral
plane
of
conicoids
that
generator,
lie
is at
give
iV
1
if+
_
i\a
equation
^{x,
y,
z)
the only
which can
only be
in
the
hyperboloid
^^o
circumscribe
equations
a-X,
that
touch
yz
are
to
of
revolution
on the
three points
at a
and PQ
meets the
curve in
 
common
generator
at
the
origin
equation
may
=
conicoid
a
self-polar
tetrahedron.
s-axes.
=
?/5;-planc also
touch at
tlierefore
the
is
a
of contact.
coincide in
of a plane section
Ex. 1.
on
the
tangent
from
thiid
intersect
z
connnon
conic
contact with the
and
the
passes
point lies
(a,
/3,
y)
with
the
polar
planes
of
is
to the
plane with
equation
to
an
eighth
fixed
point.
point.
3.
The
feet
of
meets
the
generator
in
any
given
curve and
also
lies
on
C2.
16.
Find
the
distances
of
P
from
the
vertices,
18.
If
a
variable
and
tlie
planes
ABC,
BCD,
CDA,
DAB
in
Aj,
A,,
Bj,
which
passes
through
a
fixed
point
of
a
line
which
intersects
two
given
lines
and
is
//,
:)
y,
in two
the tangents at the singular point. The locus of the
system
to
the
nature
of
the
form
2^
-I-
M3
-h
W4
-f-
section of the
{x^
the ellipsoid
of
such
sections
circles
and
ellipses
which
have
common
points
singular
in a
If
point of
equation to
the surface
the
same
circle,
or
is
then
at
to
the shape of the surface at the origin the conicoid
given
by
2.Z
the
sections
of
the
conic
given
by
z
surface
with
the
Ex.
3.
Prove
surface
A'*+^*
{x,
y,
the
elimination
leads
to
an
y,
s)
of
a
straight
=
on
the
cylindroid
for
which
Q
6
; and
and
intersects
the
P and P'
=
the
plane
PQR
Ex. 3.
Prove
plane is of
+
=
curve
has
contact
X^fxx
change
at
A^
is
measured
by
sphere of unit
curve, and let O^^, O^,'
0^3
as
the
tangents
AJi,
binomials
us
the
measures
are
Syp-.
principal
normal.
If
the
the
curve is turned,
the positive
the
positive
direction
of
the
curve
makes
a
constant
angle
a.
of the
circle lies
centre
j8,
y)
is
the
y,
z),
differenti-
ating
=
 
cos
distance
curvature
and
p-
curve
OP,
and
therefore
Bertrand
curves.
distant
8s
from
of the
triangle PQR
is denoted
are
at
right
angles
to
the
plane
drawn
through
to
the
binormal.
Since
the
The
limiting
positions
of
sjp-
with
the
inter-
section
of
the
helicoid
the
origin,
so as
25.
generating
lines
Shew that
respectively.
27.
with
the
\/2po/4o-,
\/2/)o/4o-,
\/2pJ2p.
30.
that
its
semivertical
angle
is
the
tangent
at
Q
to
the
locus
of
Q
is
at
right
angles
which
represents
the
tangent
plane
=
y,
z,
of regression we have
which represent
Ex.
ruled
surface
which
is given by
same
tangent
plane
paraineteis
/(.>•,
d,
=
=
of
tlie
is
surrounded
a
given
generator.
We
skew surface at points
0,
z
=
.^^-plane. Find
the envelope
an
anule
6
sections.
If
Besant.
circles
which
are
the
circles
of
curvature
at
containing
the
normal
cosines of the
the section and
^
^
Ex.
5.
The
locus
be
principal
curvature through
of intersection
of the
cone,
point
Q
on
the
consecutive
generators
in
N
correspond
curvature
of
the
ellipsoid
equations
of
the
other
is
the
2-axis.
OX, ?=1, m
normals
at
a
point
P
section
is
p
-5
The
condition
that
the
normal
at
(.r,
an
umbilic
dx :
dy
in space there passes one member of each
system,
and
the
the confocal
and
o.
We
have
to
prove
that
of
touches
tangents of the skew surface,
and therefore
the conicoid
the form
of
whose
of
the
portion
S,
over
S
is
-j.
If
anil the
PQRS
+
helicoid at
the cylindroid
constant.
11.
Shew
that
the
12.
A
curve
is
drawn
and
point
is
called
an
243.
The
differential
ly^,
line.
Consider
the
asymptotic
lines
the x'-axis,
points
instance
point.
Ex.
1.
Prove
asymptotic lines of the
projections
of
a plane.
positions of
A, B
points
a
geodesic,
and
the
geodesies
are
helices.
Ex.
3.
The
become
and
the geodesic.
value
for
is
psecoL,
on the
the
angle
between
the
osculat-
ing
is
sin
6
point
from
the
vertex.
the
cuivature
of
the
asymptotic
O
of
surface
S^
are
curvature through a point of the
ellipsoid
prism.
Aiis.
\/f|^
'Jfl,
the
point
required
line.
Ans.
^=^4=^-^,
(1,
2,
3),
^-3.
second
rectangular
set,
O^,
O'/,
O^,
where
O^
lies
in
the
plane
XOY
and
makes
OZ
are
Ir,
m,.,
iir
(a>h)
lines
44.
P
is
any
point
ellipsoid
^>+'r7,
P, (a\,
the centre
a*{b^-
OP, OQ, OR
which
touches
the
coordinate
a^^•
by
c-z
planes
and
the
planes
which
the
+
A-2/a2+3/-/62-2Vc2
eccentric
—
angles.
66.
If
the
angles
QPR,
QDR
are
20
and
2c/),
of intersection
=
(p,
q,
0).
Prove
principal
;
of
The
the
semicircle
OCA
assumes
constant
angle
of
the'curve
is
length
s.
A
distance
PT
equal
at
T
is
pg,
prove
that
4
259.
Contact
of
conicoids,
246.
of
to plane,
at a
on
cone,
365.
second
degree,
219,
227.
Regression,
conditions
72.
of
conicoid,