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TRANSFORMATION OF SURFACES BY BENDING. 83

For, PQ and RS being both perpendicular to Bh, RS may be turned
about Bh till it is parallel to PQ, in which case 8^ becomes = 0.

By repeating this process, we may make all the " shortest lines" parallel to
one another, and then all the generating lines will be parallel to the same
plane.

We have hitherto considered generating lines situated at finite distances from
one another ; but what we have proved will be equally true when their distances
are indefinitely diminished. Then in the limit

du

B0

dO

u,-u,

" du

Str

da-

" du

8(f>

d(f>

Uj — Wi '* du '

All these quantities being functions of u, ^, 0, a- and (f), are functions of u
and of each other; and if the forms of these functions be known, the positions
of all the generating lines may be successively determined, and the equation
to the surface may be found by integrating the equations containing the values
of ^, 0, a- and <j).

When the surface is bent in any manner about the generating lines, C> ^,
and a- remain unaltered, but cf) is changed at every point.

The form of <^ as a function of u will depend on the nature of the
bending ; but since this is perfectly arbitrary, <^ may be any arbitrary function
of u. In this way we may find the form of any surface produced by bending
the given surface along its generating lines.

By making <f) = 0, we make all the generating lines parallel to the same
plane. Let this plane be that of xy, and let the first generating line coincide
with the axis of x, then C will be the height of any other generating line
above the plane of xy, and the angle which its projection on that plane
makes with the axis of x. The ultimate intersections of the projections of the
generating lines on the plane of xy will form a curve, whose length, measured
from the axis of x, will be o-.

84 TRANSFORMATION OF SURFACES BY BENDING.

Since ia this case the quantities C> ^, and cr are represented bj distinct
geometrical quantities, we may simplify the consideration of all surfaces generated
by straight lines by reducing them by bending to the case in which those lines
are parallel to a given plane.

In the class of surfaces in which the generating lines ultimately intersect,

-T- = 0, and ^ constant. If these surfaces be bent so that <j> = 0, the whole of

the generating lines will lie in one plane, and their ultimate intersections will
form a plane curve. The surface is thus reduced to one plane, and therefore
belongs to the class usually described as "developable surfaces." The form of a
developable surface may be defined by means of the three quantities 0, a- and
(f>. The generating lines form by their ultimate intersections a curve of double
curvature to which they are all tangents. This curve has been called the
cuspidal edge. The length of this curve is represented by a, its absolute

curvature at any point by -j- , and its torsion at the same point by ■— .

When the surface is developed, the cuspidal edge becomes a plane curve,
and every part of the surface coincides with the plane. But it does not follow
that every part of the plane is capable of being bent into the original form
of the surface. This may be easily seen by considering the surface when the
position of the cuspidal edge nearly coincides with the plane curve but is not
confounded with it. It is evident that if from any point in space a tangent
can be drawn to the cuspidal edge, a sheet of the surface passes through that
point. Hence the number of sheets which pass through one point is the same
as the number of tangents to the cuspidal edge which pass through that
point ; and since the same is true in the limit, the number of sheets which
coincide at any point of the plane is the same as the number of tangents
which can be drawn from that point to the plane curve. In constructing a
developable surface of paper, we must remove those parts of the sheet from
which no real tangents can be drawn, and provide additional sheets where more
than one tangent can be drawn.

In the case of developable surfaces we see the importance of attending to
the position of the lines of bending; for though all developable surfaces may
be produced from the same plane surface, their distinguishing properties depend
on the form of the plane curve which determines the lines of bending.

TRANSFORMATION OF SURFACES BY BENDING.

85

II.

On the Bending of Surfaces of Revolution.

In the cases previously considered, the bending in one part of the surface
may take place independently of that in any other part. In the case now
before us the bending must be simultaneous over the whole surface, and its
nature must be investigated by a different method.

The position of any point P on a surface of revolution may be deter-
mined by the distance FV from the vertex, measured
along a generating line, and the angle AVO which
the plane of the generating line makes with a fixed
plane through the axis. Let FV=s and AVO = 6.
Let r be the distance {Pp) of P from the axis ; r
will be a function of s depending on the form of the
generating curve.

Now consider the small rectangular element of the surface at P. Its length
PR = Ss, and its breadth PQ = rhd, where r is a function of s.

If in another surface of revolution r is some other function of s, then the
length and breadth of the new element will be hs and rB\$', and if

r = /xr, and 0' = -0,

rze'=rze,

and the dimensions of the two elements will be the same.

Hence the one element may be applied to the other, and the one surface
may be applied to the other surface, element to element, by bending it. To
effect this, the surface must be divided by cutting it along one of the generating
lines, and the parts opened out, or made to overlap, according as /x is greater
or less than unity.

To find the effect of this transformation on the form of the surface we
must find the equation to the original form of the generating line in terms of
6" and r, then putting / = /ir, the equation between s and r will give the form
of the generating line after bending.

86 TRANSFORMATION OF SURFACES BY BENDING.

When /x is greater than 1 it may happen that for some values of 5, y- is

greater than -. In this case

-j- = fi-j- is greater than 1 ;

a result which indicates that the curve becomes impossible for such values of
s and ft.

The transformation is therefore impossible for the corresponding part of
the surface. If, however, that portion of the original surface be removed, the
remainder may be subjected to the required transformation.

The theory of bending when apphed to the case of surfaces of revolution
presents no geometrical difficulty, and little variety; but when we pass to
the consideration of surfaces of a more general kind, we discover the insufficiency
of the methods hitherto employed, by the vagueness of our ideas with respect
to the nature of bending in such cases. In the former case the bending is
of one kind only, and depends on the variation of one variable ; but the
surfaces we have now to .consider may be bent in an infinite variety of ways,
depending on the variation of three variables, of which we do not yet know the
nature or interdependence.

We have therefore to discover some method sufficiently general to be appli-
cable to every possible case, and yet so definite as to limit each particular case
to one kind of bending easily imderstood.

The method adopted in the following investigations is deduced from the
consideration of the surface as the limit of the inscribed polyhedron, when the
size of the sides is indefinitely diminished, and their number indefinitely increased.

A method is then described by which such a polyhedron may be inscribed
in any surface so that all the sides shall be triangles, and aU the solid angles
composed of six plane angles.

The problem of the bending of such a polyhedron is a question of trigo-
nometry, and equations might be found connecting the angles of the different
edges which meet in each soHd angle of the polyhedron. It will be shewn that

TRANSFORMATION OF SURFACES BY BENDING. 87

the conditions thus obtained would be equivalent to three equations between
the six angles of the edges belonging to each solid angle. Hence three addi-
tional conditions would be necessary to determine the value of every such angle,
and the problem would remain as indefinite as before. But if by any means
we can reduce the number of edges meeting in a point to four, only one con-
dition would be necessary to determine them all, and the problem would be
reduced to the consideration of one kind of bending only.

This may be done by drawing the polyhedron in such a manner that the
planes of adjacent triangles coincide two and two, and form quadrilateral facets,
four of which meet in every solid angle. The bending of such a polyhedron
can take place only in one way, by the increase of the angles of two of the
edges which meet in a point, and the diminution of the angles of the other two.

The condition of such a, polyhedron being inscribed in any surface is then
found, and it is shewn that when two forms of the same surface are given,
a perfectly definite rule may be given by which two corresponding polyhedrons
of this kind may be inscribed, one in each surface.

Since the kind of bending completely defines the nature of the quadrilateral
polyhedron which must be described, the lines formed by the edges of the
quadrilateral may be taken as an indication of the kind of bending performed
on the surface.

These lines are therefore defined as " Lines of Bending."

When the lines of bending are given, the forms of the quadrilateral facets
are completely determined ; and if we know the angle which any two adjacent
facets make with one another, we may determine the angles of the three edges
which meet it at one of its extremities. From each of these we may find the
angles of three other edges, and so on, so that the form of the polyhedron
after bending will be completely determined when the angle of one edge is given.
The bending is thus made to depend on the change of one variable only.

In this way the angle of any edge may be calculated from that of any
given edge ; but since this may be done in two different ways, by passing
along two different sets of edges, we must have the condition that these results
may be consistent with each other. This condition is satisfied by the method
of inscribing the polyhedron. Another condition will be necessary that tlie
change of the angle of any edge due to a small change of the given angle,
produced by bending, may be the same by both calculations. This is the con-
dition of " Instantaneous Lines of Bending." That tliis condition mav ccntinue

88 TRANSFORMATION OF SURFACES BY BENDING.

to be satisfied during the whole process we must have another, which is the
condition for " Permanent Lines of Bending."

The use of these lines of bending in simplifying the theory of surfaces is
the only part of the present method which is new, although the investigations
connected with them naturally led to the employment of other methods which
had been used by those who have already treated of this subject. A state-
ment of the principal methods and results of these mathematicians will save
repetition, and will indicate the different points of view under which the
subject may present itself.

The first and most complete memoir on the subject is that of M. Gauss,

The method which he employs consists in referring every point of the
surface to a corresponding point of a sphere whose radius is unity. Normals
are drawn at the several points of the surface toward the same side of it,
then lines drawn through the centre of the sphere in the direction of each of
these normals intersect the surface of the sphere in points corresponding to
those points of the original surface at which the normals were drawn.

If any line be drawn on the surface, each of its points will have a
corresponding point on the sphere, so that there will be a corresponding Hne
on the sphere.

If the line on the surface return into itself, so as to enclose a finite area
of the surface, the corresponding curve on the sphere will enclose an area on
the sphere, the extent of which will depend on the form of the surface.

This area on the sphere has been defined by M. Gauss as the measure of
the "entire curvature" of the area on the surface. This mathematical quantity
is of great use in the theory of surfaces, for it is the only quantity connected
with curvature which is capable of being expressed as the sum of all its parts.

The sum of the entire curvatures of any number of areas is the entire
curvature of their sum, and the entire curvature of any area depends on the
form of its boundary only, and is not altered by any change in the form of
the surface within the boundary line.

The curvature of the surface may even be discontinuous, so that we may
speak of the entire curvature of a portion of a polyhedron, and calculate its
amount.

If the dimensions of the closed curve be diminished so that it may be
treated as an element of the surface, the ultimate ratio of the entire curvature

TRANSFORMATION OF SURFACES BY BENDING. 89

to the area of the element on the surface is taken as the measure of the
" specific curvature " at that point of the surface.

The terms "entire" and "specific" curvature when used in this paper are
adopted from M. Gauss, although the use of the sphere and the areas on its
surface formed an essential part of the original design. The use of these terms
will save much explanation, and supersede several very cumbrous expressions.

M. Gauss then proceeds to find several analytical expressions for the measure
of specific curvature at any point of a surface, by the consideration of three
points very near each other.

The co-ordinates adopted are first rectangular, x and y, or x, y and z, being
regarded as independent variables.

Then the points on the surface are referred to two systems of curves drawn
on the surface, and their position is defined by the values of two independent
variables p and q, such that by varying p while q remains constant, we obtain
the different points of a line of the first system, while p constant and q
variable defines a line of the second system.

By means of these variables, points on the surface may be referred to lines
on the surface itself instead of arbitrary co-ordinates, and the measure of cur-
vature may be found in terms of p and q when the surface is known.

In this way it is shewn that the specific curvature at any point is the
reciprocal of the product of the principal radii of curvature at that point, a
result of great interest.

From the condition of bending, that the length of any element of the
curve must not be altered, it is shewn that the specific curvature at any point
is not altered by bending.

The rest of the memoir is occupied with the consideration of particular
modes of describing the two systems of lines. One case is when the lines of.
the first system are geodesic, or "shortest" lines having their origin in a point,
and the second system is drawn so as to cut off equal lengths from the curv^es
of the first system.

The angle which the tangent at the origin of a line of the first system
makes with a fixed line is taken as one of the co-ordinates, and the distance
of the point measured along that line as the other.

It is shewn that the two systems intersect at right angles, and a simple
expression is found for the specific curvature at any point.

M. Liouville (Journal, Tom. xii.) has adopted a different mode of simpli-

VOL. I. 22

90 TRANSFORMATION OF SURFACES BY BENDING.

tying the problem. He has shewn that on every surface it is possible to find
two systems of curves intersecting at right angles, such that the length and
breadth of every element into which the surface is thus divided shall be equal,
and that an infinite number of such systems may be found. By means of these
curves he has found a much simpler expression for the specific curvature than
that given by M. Gauss.

He has also given, in a note to his edition of Monge, a method of testing
two given surfaces in order to determine whether they are applicable to one
another. He first draws on both surfaces lines of equal specific curvature, and
determines the distance between two corresponding consecutive lines of curvature
in both surfaces.

If by assuming the origin properly these distances can be made equal for
every part of the surface, the two surfaces can be applied to each other. He
has developed the theorem analytically, of which this is only the geometrical
interpretation.

When the lines of equal specific curvature are equidistant throughout their
whole length, as in the case of surfaces of revolution, the surfaces may be
applied to one another in an infinite variety of ways.

When the specific curvature at every point of the surface is positive and
equal to a^, the surface may be applied to a sphere of radius a, and when the
specific curvature is negative = —a" it may be applied to the surface of revo-
lution which cuts at right angles all the spheres of radius a, and whose centres
are in a straight line.

M. Bertrand has given in the Xlllth Vol. of Liouville's Journal a very
simple and elegant proof of the theorem of M. Gauss about the product of

He supposes one extremity of an inextensible thread to be fixed at a point
in a surface, and a closed curve to be described on the surface by the other
extremity, the thread being stretched all the while. It is evident that the
length of such a curve cannot be altered by bending the surface. He then
calculates the length of this curve, considering the length of the thread small,
and finds that it depends on the product of the principal radii of curvature
of the surface at the fixed point. His memoir is followed by a note of
M. Diguet, who deduces the same result from a consideration of the area of
the same curve ; and by an independent memoir of M. Puiseux, who seems to
give the same proof at somewhat greater length.

TRANSFORMATION OF SURFACES BY BENDING. 91

Note. Since this paper was written, I have seen the Rev. Professor Jellett's Memoir, On
the Properties of Inextensible Surfaces. It is to be found in the Transactions of the Royal Irish

Professor Jellett has obtained a system of three partial differential equations which express
the conditions to which the displacements of a continuous inextensible membrane are subject.
From these he has deduced the two theorems of Gauss, relating to the invariability of the product
of the radii of curvature at any point, and of the " entire curvature" of a finite portion of the
surface.

He has then applied his method to the consideration of cases in which the flexibihty of the
surface is limited by certain conditions, and he has obtained the following results : —

If the displacements of an inextensible surface he all parallel to the same plane, the mrface
moves as a rigid body.

Or, more generally,

If the movement of an inextensible surface, parallel to any one line, be that of a rigid body, the
entire movement is that of a rigid body.

The following theorems relate to the case in which a curve traced on the surface is rendered
rigid :—

// any curve be traced upon an inextensible surface whose principal radii of curvature are finite
and of the same sign, and if this curve he rendered immoveable, the entire surface will become
immoveable also.

In a developable surface composed of an inextensible membrane, any one of its rectilinear
sections may be fixed without destroying the fiexibility of the membrane.

In convexo-concave surfaces, there are two directions passing through every point of the
surface, such that the curvature of a normal section taken in these directions vanishes. We
may therefore conceive the entire surface to be crossed by two series of curves, such that
a tangent drawn to either of them at any point shall coincide with one of these direc-
tions. These curves Professor Jellett has denominated Curves of Flexure, from the following
properties : —

Any curve of fiexure may he fi^ed without destroying the fiexibility of the surface.

If an arc of a curve traced upon an inextensible surface be rendered fixed or rigid, the entire of
the quadrilateral, formed by drauring the two curves of fiexure through each extremity of the curve,
become fixed or rigid also.

Professor Jellett has also investigated the properties of partially inextensible surfaces, and
of thin material laminae whose extensibility is small, and in a note he has demonstrated the
following theorem : —

If a closed oval surface he perfectly inextensible, it is also perfectly rigid.

A demonstration of one of Professor Jellett's theorems will be found at the end of this paper.

J. C. M.
Aug. 30, 1851

92 TRANSFORMATION OF SURFACES BY BENDING.

On the properties of a Surface considered as the limit of the inscribed

Polyhedron.

1. To inscribe a polyhedron in a given surface, aU whose sides shall he
triangles, and all whose solid angles shall he hexahedral.

On the given surface describe a series of curves
according to any assumed law. Describe a. second series
intersecting these in any manner, so as to divide the
whole surface into quadrilaterals. Lastly, describe a
third series (the dotted lines in the figure), so as to
pass through all the intersections of the first and second
series, forming the diagonals of the quadrilaterals.

The surface is now covered with a network of curvilinear triangles. The
plane triangles which have the same angular points will form a polyhedron
fulfilling the required conditions. By increasing the number of the curves in
each series, and diminishing their distance, we may make the polyhedron
approximate to the surface without limit. At the same time the polygons
formed by the edges of the polyhedron will approximate to the three systems
of intersecting curves.

2. To find the measure of the ''entire curvature" of a solid angle of the
'polyhedron, and of a finite portion of its surface.

From the centre of a sphere whose radius is unity draw perpendiculars to
the planes of the six sides forming the solid angle. These lines will meet the
surface in six points on the same side of the centre, which being joined by
arcs of great circles will form a hexagon on the surface of the sphere.

The area of this hexagon represents the entire curvature of the solid angle.

It is plain by spherical geometry that the angles of this hexagon are the
supplements of the six plane angles which form the solid angle, and that the
arcs forming the sides are the supplements of those subtended by the angles
of the six edges formed by adjacent sides.

The area of the hexagon is equal to the excess of the sum of its angles
above eight right angles, or to the defect of the sum of the six plane angles
from four right angles, which is the same thing. Since these angles are

TRANSFORMATION OF SURFACES BY BENDING. 93

invariable, the bending of the polyhedron cannot alter the measure of curvature
of each of its solid angles.

If perpendiculars be drawn to the sides of the polyhedron which contain
other solid angles, additional points on the sphere will be found, and if these
be joined by arcs of great circles, a network of hexagons will be formed on
the sphere, each of which corresponds to a solid angle of the polyhedron and
represents its " entire curvature."

The entire curvature of any assigned portion of the polyhedron is the sum
of the entire curvatures of the solid angles it contains. It is therefore repre-
sented by a polygon on the sphere, which is composed of all the hexagons
corresponding to its solid angles.

If a polygon composed of the edges of the polyhedron be taken as the
boundary of the assigned portion, the sum of its exterior angles will be the
same as the sum of the exterior angles of the polygon on the sphere ; but
the area of a spherical polygon is equal to the defect of the sum of its
exterior angles from four right angles, and this is the measure of entire curva-
ture.

Therefore the entire curvature of the portion of the polyhedron enclosed
by the polygon is equal to the defect of the sum of its exterior angles from
four right angles.

Since the entire curvature of each solid angle is unaltered by bending,
that of a finite portion of the surface must be also invariable.

3. On the " Conic of Contact," and its use in determining the curvature

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