For example, the parabolic segment ace is equivalent,
since pm varies as */am, to a rod ab whose density varies
as the square root of the distance
from the end a ; so that in this case .
k = k. Hence
This result is useful, because any
small segment of a curve may be
treated approximately as a para-
bolic segment. We observe also
SURFACE AND VOLUME OF HEMISPHERE.
that the area is f ab . cc, a result first obtained by
Archimedes in a different manner.
In a paraboloid, which is the solid got by spinning a
parabola about its axis, and giving a homogeneous strain
to the result, the area of a section parallel to the tangent
plane at a is proportional to mp 2 , and therefore to am ; thus
the solid is equivalent to a uniform triangle ace', its volume
is one-half that of the including cylinder, and the mass-
centre is two-thirds way from a to b.
SURFACE AND VOLUME OF HEMISPHERE.
We go on to consider the area and mass-centre of a
portion of a spherical surface cut out by two parallel planes.
While the circle ab, by revolving about ob, generates a
sphere, the tangent at by the same revolution will generate
a right circular cylinder. Let pq
be a small tangent to the circle,
mn its projection on at, I the pro-
jection on ob of the point of con-
tact r. Then the area traced out
by pq will be equal to pq multiplied
by the length of the path of r (r
being its middle point); that is, it
is 27r.pq.lr. Similarly the area
traced out by mn will be 2-rr . mn . oa. Now
mn : pq =lr : or = lr : oa,
so that these two areas are equal.
Since then an arc of the circle may be indefinitely ap-
proximated to by a circumscribed polygon, it follows that
the area between any two parallel plane sections of the
spherical surface is equal to the area between the same
planes on the circumscribing cylinder whose axis is per-
pendicular to them. And since the small strips of which
these are the sums are respectively not only equal but
equivalent (having the same mass-centres on ob) it follows
also that the mass-centre of the spherical area is midway
between the two planes of section. Thus the mass-centre
of a hemispherical surface is distant half a radius from the
We learn also, in passing, that the whole area of the
sphere is equal to that of the curved surface of the cylinder,
and is therefore = 4?ra 2 . (Archimedes.)
To determine the mass-centre of a hemisphere, we may
regard it as made up of thin con-
centric spherical shells. The mass-
centre of every shell is distant half
its radius from the centre, and the
mass of the shell, if all are of the
same thickness, varies as the square
of the radius. Hence the hemisphere is equivalent to a
straight rod half a radius long, whose density varies as the
squared distance from the centre ; and whose mass-centre
is accordingly distant from the centre f of a radius.
CHAPTER II. SECOND MOMENTS.
IF the density of an area is proportional to the distance
from a line in its plane, being reckoned positive on one
side of the line and negative on the other, the line is
called a neutral axis; the mass-centre of the area, having
that density, is called the pole of
the line in regard to the area; and
the moment of it in regard to the
line is called the second moment of
the uniform area in regard to the
line, or of the line in regard to the
uniform area. The area may be said
to be loaded from the line.
If we consider a small area So. in the neighbourhood
of the point p, and draw pm perpendicular to the given
line, the moment (or, as we may now call it for distinction,
the first moment) of Sa in regard to the line is approxi-
mately mp . 8a, or xSa. If however the density is propor-
tional to the distance from the line, or say the density is
kx, then the mass of So. is approximately kxSa, and its
moment in regard to the line is ka?Sot. Thus the second
moment of the area is what 'ZkofSa indefinitely approxi-
mates to with diminution of the Sa; that is, it is jkx*dct
taken over the whole area. The second moment of any
number of particles in regard to a line is the sum of their
masses, each multiplied by the square of its distance from
The distance of the pole of the line from it is the
moment of the loaded area divided by its mass, or
$ka?da : fkxda, = Jx*dx : fxdx,
which is the ratio of the second moment to the first.
The loaded area may be represented, in the same way
as the first moment in regard to a line, by a solid standing
upon the plane, bounded by straight lines normal to the
plane through the boundary of the area, and by an oblique
plane through the neutral axis. Parts of this solid which
are above the plane are positive, parts below it are nega-
tive. The second moment of the area in regard to the
line is the first moment of this solid in regard to a normal
plane through the line.
If the line pass through the mass-centre of the un-
loaded or uniform area, the loaded area is of zero mass,
there being as much of it on one side
of the line as the other. Let the mass-
centre of one side be p, and of the other
side q; then we have a mass m at p
and a mass m at q. It is easy to
see that the moment of the system
about all lines in the plane having the same direction is
constant, being m . pq . sin 0, where is the angle they
make with pq.
PARALLEL AXES. SWING-RADIUS.
The second moment of an area in regard to any line is
equal to the second moment about a parallel line through
the mass-centre, together with the second moment in regard
to the first line of the whole area if collected at the mass-
centre. Let pq be the first line, mn parallel through the
mass-centre g. The second moment about mn is the first
moment of the two solids mnac, mnbd, in regard to a
plane through mn perpendicular to the plane mnab. These
solids and their distances from the plane being both of
opposite signs, the moments are of the same sign. Now
PARALLEL AXES. 17
the sum of these two solids, being of zero mass, has the
same moment about pq as about mn. The second moment
of the area about pq is the first moment of the volume
aefb in regard to a normal plane
through pq ; the plane ef passing
through pq. The difference, there-
fore, between the two second mo-
ments is the moment of cefd in
regard to the normal plane through
pq. Now cefd is equivalent to a
cylinder standing on ab, of height
ce or df. This height is the mo-
ment about pq of a unit of area collected at g. Hence
the moment of cefd in regard to the normal plane through
pq is the first moment about pq of the first moment
of the area supposed to be collected at g, that is, it is the
second moment of the area collected at g.
Otherwise thus: let h be the distance between the two
parallel lines, x the distance of any point from ran, then
h + x is its distance from pq. Hence second moment of
pq = j(x + Tifdi = fz?dot. + 2h fxdit + tfjdz. Now fxdot. is
the fii-st moment of the area in regard to mn, which is
zero because mn passes through the mass-centre. And
fdz is simply the whole area a. Therefore second moment
of pq =fa?doi + A 2 a = second moment of mn + second mo-
ment of whole area supposed to be collected at g.
The second moment of an area in regard to any line,
divided by the area itself, is the square of a length which
is called the swing-radius of the area in regard to the
line, or of the line in regard to the, area.
Thus if k be the swing-radius of pq, and a that of mn,
the second moment of the former is Fa and of the latter
a? a. ; and we have & 2 = a 2 + h*.
CONJUGATE AXES. POLE OF GIVEN AXIS.
If one line pass through the pole of a second in regard
to any area, the second passes through the pole of the first.
Let x be the distance of a point p in the area from the
first line, and y from the second. If the area be loaded
from the second line, the density
at p is proportional to y ; for
shortness suppose it is y. Then
the moment in regard to the first
line is approximately xySx. Thus
the moment in regard to the first
line of the area loaded from the
second is fxydct. We shall call
this the mixed moment of the area in regard to the two
lines. It is clearly the same as the moment in regard to
the second line of the area loaded from the first. If the
first line pass through the pole of the second, this mixed
moment vanishes, and consequently the second line passes
through the pole of the first. Two such lines are called
When a line passes through the mass-centre, the area
loaded from it is of zero mass ; all the lines conjugate to
it are parallel, and the pole is away at an infinite distance.
We may now shew how to find the pole of a given
line, when the swing-radii of all lines through the mass-
centre are known*. Let pq be any line, p'q the parallel
through the mass-centre g, mgn conjugate to p'q and
therefore to pq. Let k be the swing-radius of pq measured
parallel to mn ; that is to say, let & be a line parallel to
mn whose component perpendicular
to pq is the swing-radius of pq.
Let a be the swing-radius of p'q,
measured in the same way; and let
mg = h. Then we know that
k z = a z + h\
Also the distance from pq of its
pole, measured parallel to mn is
the ratio of A; 2 a to hot. Let n be the pole ofpq, then
F a 2 + tf a 2
mn = -T- = i = A + T ;
n n ti
* [It has been pointed out to me that we must know also how to
find the 'conjugate'. Culmann considers the core as the antipolar
reciprocal of the contour with regard to the momental ellipse.]
CORE OF AN AREA. 19
but mg being h, we see that gn = 2 : h, or mg . gn = a 3 .
It follows that if a line move parallel to itself, the nearer
it is to the mass-centre the further away is the pole, and
CORE OF AN AREA.
Suppose any area to have a tight string drawn round
it, so that the string is everywhere either straight or
convex to the outside. Let a line move round touching
the string at the convex parts, coinciding with it when it
is straight, and turning through the necessary angle at
every sharp point. As this line moves, its pole with regard
to the area will trace out a curve. Any
line cutting the area will have its pole
outside this curve ; for it will be nearer
to the mass-centre than the parallel line
whose pole is on the curve. Any line
not meeting the area at all will have its
pole inside the curve. This region, con-
taining the poles of all lines which do not cut the
boundary of the area, is called the core of the area.
If the area is a parallelogram, we know that the pole
of ac is at a, on the line joining the middles of ac and bd,
two-thirds way to bd. Similarly the poles of cd, db, ba
are at 7, /3, 8. Now let a line,
coinciding at first with ac, turn
round c until it coincides with
cd. Since ay passes through the
poles of ac and cd, both ac and
cd pass through the pole of ct<y,
which is therefore c. As the line turns round c therefore,
from the position ac to cd, its pole will move along 07
from to 7. As the line turns round d, the pole will
travel from 7 to /3, and so on. Hence 7/3S is the core
of the parallelogram. Or, the core of a parallelogram is
another parallelogram, whose diagonals are the middle
thirds of the median lines.
In the case of a triangle abc, the pole of be is the
mass-centre of the wedge made
by tilting the triangle slightly
about it. Now the mass-centre
of a tetrahedron is half-way be-
tween the middles of opposite
edges; consequently the pole of
be is at a the bisection of ad.
The length gy. = \ga, because
Similarly the poles of ca, ab are at ft, 7, so that g/3 =
gy = \gc. Thus the core of abc is a/37, a similar triangle,
having the same mass-centre, and of one-fourth the linear
dimensions ; which may be called its middle quarter.
To determine the core of a circle, we observe first that
the sum of the second moments of any area about two
lines at right angles through any point is independent of
the direction of the lines, and depends only on the point.
For if x and y be the distances of any small portion 8t
of the area from the two lines, a?Sx + y*Sa = r 2 Sa, where
r is the distance from the point of intersection of the
lines. Thus fx^dot. + jy*da = fr^dx. It follows of course
that the sum of the squares of the swing-radii is con-
stant. The length whose square is equal to this sum may
be called the swing-radius in respect of the point of inter-
section of the two lines.
In the case of the circle, then, the second moment in
regard to any diameter being clearly the same is half the
sum of the second moments in regard to two of them
which are at right angles, or |jV 2 <fo where r is the distance
from the centre. Suppose the circle divided into thin
concentric rings of equal breadth Sr, then the area of one
of them will be 27rrSr, and the second moment in regard
to a diameter is \ \ 27rr 3 dr, = Tra 4 where a is the radius
of the circle. Since the area is Tra 2 , we see that the
squared swing-radius is Ja 2 , or the swing-radius is \a.
It follows that the core is a concentric circle of one-
CORE OF AN AREA. 21
quarter the radius ; which we may call the middle quarter,
as in the case of the triangle.
If we deform an area by parallel projection or homo-
geneous strain, it is easy to see that the core of the area
will be strained into the core of the strained area. Thus
it appears that the core of an ellipse is a similar, similarly
situated, and concentric ellipse, of one-quarter the linear
dimensions; again, as we may say, the middle quarter.
When the core is known, it is easy to construct the
pole of any line, and thence to find its second moment.
Let gx, gy be two conjugate lines through the mass-
centre, ab any other line. To find the pole of ab, take ra
on ag so that ag . gm = square of swing-radius of gx mea-
sured parallel to gy, and n on bg so that bg .gn = square
of swing-radius of gy measured parallel to gx. Then mp,
parallel to gx, is the neutral axis whose pole is a, and np,
parallel to gy, is that whose pole is b. Hence p is the
pole of ab.
All we require, therefore, is to know the poles of two
lines parallel to a pair of conjugate lines through the
If pg meet ab in I, the squared swing-radius of ab,
measured parallel to gl, is Ig . Ip. Hence the second
moment is known.
Let oX, oY be conjugate lines through any point o,
and let on be any other line, the normal ofg to which
makes angles a, /3 with oX, oY. The perpendicular dis-
tance pn of any point p from on is equal to the projection
on og of op, or, what is the same thing, to the sum of the
projections of om and mp on og. Let om = x, mp = y,
then these projections are x cos a, y cos /3. Hence
^m = x cos a + y cos /?.
It follows that the second moment of the area in regard
to on is
/(# cos a + y cos /3) 2 da, = cos 2 a JVcZa + cos 2 /3 /^/ 2 rfa,
since fay da. = because the lines oX, oY are conjugate.
Let fa? da. = a 2 a, J2/ 2 c?a = 6 2 a,
so that a, b are swing-radii of o Y, oX measured parallel to
oX, o Y respectively. Then if k be the swing-radius of on,
F = a a cos 2 a+S 2 cos 2 /3.
Now if an ellipse be drawn having o for centre and a, b
for conjugate semi-diameters along oX, oY, the tangent to
this ellipse parallel to on will be at a distance p from it,
such that p* = a* cos 8 a + 6 2 cos 2 /? ;
so that k = p.
For let vqu be this tangent, ot=p the perpendicular on
it from the centre, ql parallel
to o Y; and let ol = x, lq = y.
p = ot = ou cos a = ov cos 8.
ol : oa = oa : ou = oa cos a : p,
x _ a cos a _
6 cos 8
I a it
But we know that 2 + fa = 1 ;
,, a 2 cos 2 a + & 2 cos 2 /3
In the case of a hyperbola, it will still be true that
ol : oa = oa : ou,
because, the hyperbola being central projection of a circle,
a'lau is a harmonic range [p. 42]. Hence as before
x _ a cos a y _ b cos 8
a~ p b~ p
and the equation [foot of p. 89]
gives in this case
p 2 = a 2 cos 2 a b 2 cos 2 ft.
Thus, if a pair of parallels be drawn to every line
through o, at distances equal to its swing-radius on either
side of it, these parallels will all touch an ellipse. This
ellipse is called the swing-conic at the point o. The swing-
conic at the mass-centre may be called simply the swing-
conic of the area.
The curve will always be an ellipse in those cases
which occur in practice, because the area being entirely
positive,/# 2 eta, fy 2 da must be also both positive. But if
we consider also ideal areas, parts of which may be nega-
tive, the quantities fx*dz, /yVa may be of different signs,
and then the swing-conic will be a hyperbola.
When the swing-conic at the mass-centre is a hyper-
bola, the second moment of every line touching the con-
jugate hyperbola will be zero. Suppose that fa?dz = a 2
and is positive, fy*da = 6 2 and is negative. Then
k* = a* cos 2 a b 2 cos 2 ft.
The squared swing-radius of a parallel line at distance p is
p 2 + k 2 =p 2 + a 2 cos a. b* cos 2 ft.
This is zero if p 2 = b z cos 2 ft a 2 cos 2 a,
which is the case for tangents of the conjugate hyperbola.
In such a case the swing-radius k is imaginary, since its
square is negative. This conjugate hyperbola is called the
null-conic of the area.
POLES AND POLARS.
A neutral axis and its pole in regard to an area are
related to the null-conic of the area in a very simple way.
The geometrical properties of this relation are most easily
derived from the corresponding theory in the case of the
POLES AND POLARS.
Let the tangents to a circle at p, q meet in t and let ct
meet pq in m. Then we know that cm . ct = ca 2 . The
point is called the pole of pq in
regard to the circle, and pq is
called the polar of . If a line tu
be drawn through t perpendicular
to ct, then tu is the polar of m,
and wi is the pole of tu. So that
a point and a line are pole and
polar in regard to a circle when
the product of their distances
from the centre is equal to the
square of the radius, and when
the line is perpendicular to that
radius of the circle which passes through the point. When
the point is outside the circle, its polar passes through the
points of contact of tangents from it to the circle.
If a point s lies on the polar of t, then t lies on the polar
of s. Draw tn perpendicular to cs; then the triangles
csm, ctn are similar, and cm : cs cn : ct. Therefore
cs . en = cm . ct = ca 2 .
Hence s is the pole of tn, or t lies on the polar of s.
In this case the points s, t are called conjugate points,
and their polars are called conjugate lines in regard to the
It follows that if through any point s we draw a series
of chords pq to meet the circle, the tangents at the extre-
mities of these chords will all meet on a certain straight
line, which is the polar of s. For since each of these
chords passes through s, its pole must lie on the polar
Now this last property of the circle is one which must
belong to all central projections of the circle. For the
projections of the chords pq will be a series of straight
lines passing through the projection of s ; and the tangents
at their extremities will all meet on the projection of tn,
which we know to be a straight line. Hence the property
is true also of the ellipse, parabola, and hyperbola.
In the case of the ellipse, moreover, it remains true
that if ct meets the polar of t in m, and the ellipse in a,
cm . ct = ca*.
But in the case of the hyperbola, when the polar meets
both branches of the curve, this statement requires inter-
pretation. In that case, as the figure shews, ct does not
meet the hyperbola itself; but if it meets the conjugate
hyperbola in a, we shall still have
cm : ac = ca : ct, or cm . ct = ca 2 .
Hence in order to apply the theorem in this case, we must
regard the distance from the centre to the original hyper-
bola, measured along ct, as being the square root of ca 2 ,
or caV( !) This V( 1) however is not the operation
of turning through a right angle ; for if we turn ca through
a right angle, we no longer get a distance measured along
ct. It must be treated for the present simply as a means
of simplifying the statement of certain propositions. To
determine the points where ct meets the hyperbola, we
must measure off on either side of c the distance ca \/( 1).
Let a 1} a/ be the points so found ; we say that these
points are invisible ; but still, since ca* = ca 2 , it follows
that cm . ct = ca 2 .
In this way we shall take the squared semi-axes of the
hyperbola to be a 2 , b 2 ; and the squared semi-axes of the
conjugate hyperbola will be a 2 , 6 2 . It will be seen at
once that by this consideration many formulae relating to
the hyperbola become reconciled with the corresponding
formulae for the ellipse. For example, in both curves the
APPLICATION TO THE NULL-CONIC. 27
squared distance of the foci from the centre is equal to the
difference of the squares of the semi-axes.
The pole of pq in regard to the conjugate hyper-
bola is a point t' such that cm . ct' ca?. Thus t, t'
are opposite points in regard to the centre c. The
conjugate hyperbola has its squared semi-axes equal in
magnitude but opposite in sign to those of the original
hyperbola. Thus when two conies have their axes equal
in magnitude and opposite in sign, the poles of the same
line in regard to them are opposite points in regard to
their common centre.
APPLICATION TO THE NULL-CONIC.
We may shew very easily that the pole of any .line
in regard to an area is the opposite point of its pole in
regard to the swing-conic at the mass-centre. Let pq be
the line, c the mass-centre, cam conjugate to a line through
c parallel to pq. We have seen
that the swing-radius of this
parallel line is equal to the
perpendicular from c on the
tangent at a to the swing-conic ;
for two lines conjugate in re-
gard to the area are conjugate
diameters of the swing-conic at
their intersection, so that the j>
tangent at a will be parallel to
pq. Hence the swing-radius of the parallel diameter, mea-
sured parallel to ca, will be ca itself. Therefore the pole
of pq in regard to the area is at a point t such that
me . ct = ca*.
Now its pole in regard to the swing-conic is a point t' such
that cm . ct' = ca 2 . Therefore tc = ct', or t, t' are opposite
points in regard to c.
If therefore the swing-conic is a hyperbola, the pole of
a line in regard to the area is simply its geometric pole in
regard to the null-conic. But also when the swing-conic
is an ellipse, there is a certain convenience in saying the
same thing. For since cm . ct = cd*, if we write
ca, 2 = ca 2 , or ca t = ca \/( 1), we shall have cm . ct = ca*.
Hence if we construct a conic having the same centre and
direction of axes as the swing-conic, but having for squared
semi-axes a 2 and 6 2 instead of a 2 and 6 2 , so that every
diameter of it is equal to the corresponding diameter of
the swing-conic multiplied by \/( 1), then the pole of
any line in regard to the area is its geometric pole in
regard to this conic. The conic of course is altogether
invisible ; yet it gives rise to a visible and perfectly
definite system of poles and polars, so that we can find the
pole of any given line or the polar of any given point in
regard to it. The only difference is that whereas in regard
to an ellipse the pole and polar are always on the same
side of the centre, and in regard to a hyperbola they are
sometimes on the same and sometimes on opposite sides,
according as the polar does not or does cut both branches ;
in regard to this invisible conic the pole and polar are
always on opposite sides of the centre. If we agree to
take the notion of a conic as involving not only the points
and tangents of the curve but also the system of poles and
polars to which it gives rise, we may say that there always
is a null-conic, but that when the area is all positive its
points and tangents are invisible; and that the pole of
any line in regard to the area is its geometric pole in
regard to the null-conic.
EQUIVALENT TRIAD OF PARTICLES.
We shall now prove that for purposes of calculating
second moments, an area may be replaced by tbree particles.
Take any point x in the plane of the area; let y be a
point on its neutral axis (or
polar, as we may now call
it), and let z be the pole of
xy. Then xyz is a triangle
of which every vertex is the