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of normal sections of a surface.

Suppose the plane of one of the triangular facets of the polyhedron to
be produced till it cuts the surface. The form of the curve of intersection
\7ill depend on the nature of the surface, and when the size of the triangle
is indefinitely diminished, it will approximate, to the form of a conic section.

For we may suppose a surface of the second order constructed so as to
have a contact of the second order with the given surface at a point within
the angular points of the triangle. The curve of intersection with this surface
will be the conic section to which the other curve of intersection approaches.
This curve will be henceforth called the " Conic of Contact," for want of a better
name.



1)4



TRANSFORMATION OF SURFACES BY BENDING.



To Jind tJie radius of curvature of a normal section
of the surface.

Let ARa be the conic of contact, C its centre, and
CP perpendicular to its plane. rPR a normal section, and
its centre of curvature, then

= 1.^ in the limit, when CR and PR coincide,
^ CP

-s CP'
or calling CP the "sa,gitta," we have this theorem:

"The radius of curvature of a normal section is equal to the square of
the corresponding diameter of the conic of contact divided by eight times the
sagitta."




4. To insciihe a polyhedron in a given surface, all ivhose sides shcdl he
plane quadrilaterals, and all whose solid angles shall he tetraliedral.

Suppose the three systems of curves drawn as described in sect. (1), then
each of the quadrilaterals formed by the intersection of the first and second
systems is divided into two triangles by the third system. If the planes of
these two triangles coincide, they form a plane quadrilateral, and if every such
pair of triangles coincide, the polyhedron will satisfy the required condition.

Let ahc be one of these triangles, and acd the
other, which is to be in the same plane with ahc.
Then if the plane of ahc be produced to meet the
surface in the conic of contact, the curve will pass
through ahc and d. Hence ahcd must be a quad-
rilateral inscribed in the conic of contact.

But since ah and dc belong to the same system of curves, they will be
ultimately parallel when the size of the facets is diminished, and for a similar
reason, ad and ho will be ultimately parallel. Hence ahcd will become a paral-
lelogram, but the sides of a parallelogram inscribed in a conic are parallel to
conjugate diameters.




TRANSFORMATION OF SURFACES BY BENDING. ©5

Therefore the directions of two curves of the first and second system at
their point of intersection must be parallel to two conjugate diameters of the
conic of contact at that point in order that such a polyhedron may be inscribed.

Systems of curves intersecting in this manner will be referred to as "conju-
gate systems."

5. On the elementary conditions of the applicahilitij of two surfaces.

It is evident, that if one surface is capable of being appUed to another by
bending, every point, line, or angle in the first has its corresponding point, line,
or angle in the second.

If the transformation of the surface be eflfected without the extension or
contraction of any part, no line drawn on the surface can experience any change
in its length, and if this condition be fulfilled, there can be no extension or
contraction.

Therefore the condition of bending is, that if any line whatever be drawn
on the first surface, the corresponding curve on the second surface is equal to it
in length. All other conditions of bending may be deduced from this.

6. If two curves on the first surface intersect, the corresponcling curves on the
second surface intersect at the same angle.

On the first surface draw any curve, so as to form a triangle with the
curves already drawn, and let the sides of this triangle be indefinitely dimin-
ished, by making the new curve approach to the intersection of the former
curves. Let the same thing be done on the second surface. We shall then
have two corresponding triangles whose sides are equal each to each, by (5),
and since their sides are indefinitely small, we may regard them as straight
lines. Therefore by Euclid i. 8, the angle of the first triangle formed by the
intersection of the two curves is equal to the corresponding angle of the second.

7. At any given point of the first surface, two directions can he found, which
are conjugate to each other with respect to the conic of contact at that point, and
continue to he conjugate to each other when tJie first surface is transformed into the
second.

For let the first surface be transferred, without changing its form, to a
position such that the given point coincides with the corresponding point of the
second surface, and the normal to the first surface coincides with that of the



96



TRANSFORMATION OF SURFACES BY BENDING.



second at the same point. Then let the first surface be turned about the normal
as an axis till the tangent of any line through the point coincides with the
tangent of the corresponding line in the second surface.

Then by (6) any pair of corresponding lines passing through the point will
have a common tangent, and will therefore coincide in direction at that point.

If we now draw the conies of contact belonging to each surface we shall
have two conies with the same centre, and the problem is to determine a pair
of conjugate diameters of the first which coincide with a pair of conjugate
diameters of the second. The analytical solution gives two directions, real,
coincident, or impossible, for the diameters required.

In our investigations we can be concerned only with the case in which these
directions are real.

When the conies intersect in four points, P, Q, R, S, FQES is a parallelo-
gram inscribed in both conies, and the axes CA, CB,
parallel to the sides, are conjugate in both conies.

If the conies do not intersect, describe, through any
point P of the second conic, a conic similar to and con-
centric with the first. If the conies intersect in four
points, we must proceed as before; if they touch in two
points, the diameter through those points and its conju-
gate must be taken. If they intersect in two points only,
then the problem is impossible ; and if they coincide
altogether, the conies are similar and similarly situated,
and the problem is indeterminate.




8. Two surfaces being given as before, one pair of conjugate systems of
curves may be drawn on the first surface, which shall correspond to a pair of
conjugate systems on the second surface.

By article (7) we may find at every point of the first surface two
directions conjugate to one another, corresponding to two conjugate directions on
the second surface. These directions indicate the directions of the two systems
of curves which pass through that point.

Knowing the direction which every curve of each system must have at every
point of its course, the systems of curves may be either drawn by some direct
geometrical method, or constructed from their equations, which may be found by
solving their difierential equations.



TRANSFORMATION OF SURFACES BY BENDING. 97

Two systems of curves being drawn on the first surface, the corresponding
systems may be drawn on the second surface. These systems being conjugate
to each other, fulfil the condition of Art. (4), and may therefore be made the
means of constructing a polyhedron with quadrilateral facets, by the bending of
which the transformation may be effected.

These systems of curves will be referred to as the "first and second systems
of Lines of Bending."

9. General considerations applicable to Lines of Bending.

It has been shewn that when two forms of a surface are given, one of
which may be transformed into the other by bending, the nature of the Hnes
of bending is completely determined. Supposing the problem reduced to its
analyticid expression, the equations of these curves would appear under the
form of double solutions of differential equations of the first order and second
degree, each of which would involve one arbitrary quantity, by the variation of
which we should pass from one curve to another of the same system.

Hence the position of any curve of either system depends on the value
assumed for the arbitrary constant ; to distinguish the systems, let us call one
the first system, and the other the second, and let all quantities relating to
the second system be denoted by accented letters.

Let the arbitrary constants introduced by integration be u for the first
system, and u for the second.

Then the value of lo will determine the position of a curve of the first
system, and that of u a curve of the second system, and therefore u and u will
suffice to determine the point of intersection of these two curves.

Hence we may conceive the position of any point on the surface to be
determined by the values of u and u for the curves of the two systems which
intersect at that point.

By taking into account the equation to the surface, we may suppose x, y,
and 2 the co-ordinates of any point, to be determined as functions of the two
variables u and u. This being done, we shall have materials for calculating
everything connected with the surface, and its lines of bending. But before
entering on such calculations let us examine the principal properties of these lines
which we must take into account.

Suppose a series of values to be given to u and u, and the corresponding
curves to be drawn on the surface.

VOL, I. 13



98 TRANSFORMATION OF SURFACES BY BENDING.

The surface will then be covered with a system of quadrilaterals, the size
of which may be diminished indefinitely by interpolating values of u and u
between those already assumed; and in the limit each quadrilateral may be
regarded as a parallelogram coinciding with a facet of the inscribed polyhedron.

The length, the breadth, and the angle of these parallelograms will vary at
different parts of the surface, and will therefore depend on the values of u
and It.

The curvature of a line drawn on a surface may be investigated by consider-
ing the curvature of two other lines depending on it.

The first is the projection of the line on a tangent plane to the surface at
a given point in the line. The curvature of the projection at the point of
contact may be called the tangential cwvature of the line on the surface. It
has also been called the geodesic curvature, because it is the measure of its
deviation from a geodesic or shortest line on the surface.

The other projection necessary to define the curvature of a line on the
surface is on a plane passing through the tangent to the curve and the normal
to the surface at the point of contact. The curvature of this projection at that
point may be called the normal cw^ature of the line on the surface.

It is easy to shew that this normal curvature is the same as the curvature
of a normal section of the surface passing through a tangent to the curve at
the same point.



10. General considerations applicable to the inscribed polyhedron.

When two series of lines of bending belonging to the first and second systems
have been described on the surface, we may proceed, as in Art. (l), to describe
a third series of curves so as to pass through all their intersections and form
the diagonals of the quadrilaterals foi-med by the first pair of systems.

Plane triangles may then be constituted within the surface, having these
points of intersection for angles, and the size of the facets of this polyhedron may
be diminished indefinitely by increasing the number of curves in each series.

But by Art. (8) the first and second systems of lines of bending are conju-
gate to each other, and therefore by Art. (4) the polygon just constructed will
have every pair of triangular facets in the same plane, and may therefore be



TRANSFORMATION OF SURFACES BY BENDING. 99

considered as a polyhedron with plane quadrilateral facets all whose solid angles
are formed by four of these facets meeting in a point.

When the number of curves in each system is increased and their distance
diminished indefinitely, the plane facets of the polyhedron will ultimately coincide
with the curved surface, and the polygons formed by the successive edges between
the facets, will coincide with the lines of bending.

These quadrilaterals may then be considered as parallelograms, the length
of which is determined by the portion of a curve of the second system inter-
cepted between two curves of the first, while the breadth is the distance of
two curves of the second system measured along a curve of the first. The
expressions for these quantities will be given when we come to the calculation of
our results along with the other particulars which we only specify at present.

The angle of the sides of these parallelograms will be ultimately the same
as the angle of intersection of the first and second systems, which we may
call <f> ; but if we suppose the dimensions of the facets to be small quantities
of the first order, the angles of the four facets which meet in a point will difier
from the angle of intersection of the curves at that point by small angles of
the first order depending on the tangential curvature of the lines of bending.
The sum of these four angles will differ from four right angles by a small
angle of the second order, the circular measure of which expresses the entire
curvature of the solid angle as in Art. (2).

The angle of inclination of two adjacent facets will depend on the normal
curvature of the lines of bending, and will be that of the projection of two con-
secutive sides of the polygon of one system on a plane perpendicular to a side
of the other system.

11. Explanation of the Notation to be employed in calculation.

Suppose each system of lines of bend-
ing to be determined by an equation con-
taining one arbitrary parameter.

Let this parameter be u for the first
system, and u' for the second.

Let two curves, one from each system,
be selected as curves of reference, and let
their parameters be u^ and u\.




100 TRANSFORMATION OF SURFACES* BY BENDING.

Let ON and OM in the figure represent these two curves.

Let PM be any curve of the first system whose parameter is u, and PN
any curve of the second whose parameter is u, then their intersection P may
be defined as the point (w, u'), and all quantities referring to the point P may
be expressed as functions of u and u.

Let PN, the length of a curve of the second system (u), from N (wj to P
(u), be expressed by s, and PM the length of the curve {u) from {u\) to (u), by
s\ then s and s will be functions of u and u.

Let (w + Sm) be the parameter of the curve QF of the first system consecu-
tive to PM. Then the length of PQ, the part of the curve of the second system
intercepted between the curves (u) and (w + Sw), will be

ds ^
du

Similarly PR may be expressed by

ds\ ,

These values of PQ and PR will be the ultimate values of the length and
breadth of a quadrilateral facet.

The angle between these lines will be ultimately equal to ^, the angle of
intersection of the system ; but when the values of 8w and hu are considered as
finite though small, the angles a, 6, c, d of the facets which form a soHd angle
will depend on the tangential curvature of the two systems of lines.

Let T be the tangential curvature of a curve of the first system at the
given point measured in the direction in which u increases, and let r\ that of the
second system, be measured in the direction in which xC increases.

Then we shall have for the values of the four plane angles which meet at P,

, \ ds ^ , 1 ds^

1 _, 1 c?/ ^ . 1 ds ^

~^ It du It du '

, \ ds rs , \ ds ^
J . I ds' , 1 ds ^



TRANSFORMATION OF SURFACES BY BENDING. 101

These values are correct as far as the first order of small quantities. Those
corrections which depend on the curvature of the surface are of the second order.

Let p be the normal curvature of a curve of the first system, and p that
of a curve of the second, then the inclination I of the plane facets a and 6,
separated by a curve of the second system, will be

p sin ^ du
as far as the first order of small angles, and the inclination V of h and c will be



7/ 1 0^ ^

/ = -7—. — 7 -J- ou
p Bin.<f> du



to the same order of exactness.




12. On the corresponding polygon on the surface of the sphere of reference.

By the method described in Art. (2) we may
find a point on the sphere corresponding to each
facet of the polyhedron.

In the annexed figure, let a, b, c, d be the
points on the sphere corresponding to the four facets
which meet at the solid angle P. Then the area
of the spherical quadrilateral a, h, c, d will be the
measure of the entire curvature of the solid angle P.

This area is measured by the defect of the sum of the exterior angles
from four right angles ; but these exterior angles are equal to the four angles
a, h, c, d, which form the solid angle P, therefore the entire curvature is
measured by

k = 2'rr-{a + h + c-{-d).

Since a, h, c, d are invariable, it is evident, as in Art. (2), that the entire
curvature at P is not altered by bending.

By the last article it appears that when the facets are small the angles b
and d are approximately equal to <j), and a and c to (tt — ^), and since the sides
of the quadrilateral on the sphere are small, we may regard it as approximately
a plane parallelogram whose angle bad = <f).

The sides of this parallelogram will be I and I', the supplements of the
angles of the edges of the polyhedron, and we may therefore express its area
as a plane parallelogram

k = IV sin <f>.



102



TRANSFORMATION OF SURFACES BY BENDING.



By the expression for I and V in the last article, we find

, 1 ds ds\ ^ ,

k = — r-. — 7 J- J-/ ou du
pp sm<^ du du

for the entire curvature of one solid angle.

Since the whole number of solid angles is equal to the whole number of
facets, we may suppose a quarter of each of the facets of which it is composed
to be assigned to each solid angle. The area of these will be the same as that

of one whole facet, namely,

, ds ds' o ^ ,
sm 9 -J- T-> ou ou ;

therefore dividing the expression for k by this quantity, we find for the value

of the specific curvature at P

1
■^ pp sm'<^
which gives the specific curvature in terms of the normal curvatures of the
lines of bending and their angle of intersection.

13. Further reduction of this expression by rmans of the " Conic of Con-
tact" as defined in Art. (3).

Let a and b be the semiaxes of the conic of contact, and h the sagitta
or perpendicular to its plane from the centre to the surface.

Let CP, CQ be semidiameters parallel to the
lines of bending of the first and second systems, and
therefore conjugate to each other.



By (Art. 3),



, CP"

p=^-hr




and p=i-j^;
and the expression for p in Art. (12), becomes

^~{CP.CQsm(t>)''

But CP .CQbukJ) is the area of the parallelogram CPRQ, which is one

quarter of the circumscribed parallelogram, and therefore by a well-known

theorem

CP .CQsm4> = ah,



TRANSFORMATION OF SURFACES BY BENDING. 103

and the expression for p becomes

or if the area of the circumscribing parallelogram be called A,

The principal radii of curvature of the surface are parallel to the axes of
the conic of contact. Let H and i^ denote these radii, then

and therefore substituting in the expression for p,

1

or the specific curvature is the reciprocal of the product of the principal radii
of curvature.

This remarkable expression was introduced by Gauss in the memoir referred
to in a former part of this paper. His method of investigation, though not
80 elementary, is more direct than that here given, and wUl shew how this
result can be obtained without reference to the geometrical methods necessary
to a more extended inquiry into the modes of bending.

14. 0)1 the variation of normal curvature of the lines of bending as we pa^s
from one point of the surface to another.

We have determined the relation between the normal curvatures of the
lines of bending of the two systems at their points of intersection; we have
now to find the variation of normal curvature when we pass from one hne of
the first system to another, along a line of the second.

In analytical language we have to find the value of

du \pj

Referring to the figure in Art. (11), we shall see that this may be done
if we can determine the difierence between the angle of inclination of the
facets a and h, and that of c and d : for the angle I between a and b is

J 1 ds 5. ,
psiJKp du



104 TRANSFORMATION OF SURFACES BY BENDING.

and therefore the difference between the angle of a and b and that of c and d is

~ du ~ du \psm<f> du'j
whence the differential of p with respect to u may be found

We must therefore find U, and this is done by means of the quadrilateral
on the sphere described in Art. (12).




15. To find the values of hi and U\

In the annexed figure let ahcd repre-
sent the small quadrilateral on the surface
of the sphere. The exterior angles a, h,
c, d are equal to those of the four facets
which meet at the point P of the surface,
and the sides represent the angles which
the planes of those facets make with each
other ; so that

ah = l, lc = l\ cd = l + U, da = l' + Br,

and the problem is to determine Bl and hi" in terms of the sides I and V and
the angles a, h, c, d.

On the sides ha, he complete the parallelogram ahcd.

Produce ad to p, so that ap = aS. Join Bp.
Make eq = cd and join dq.
then Bl = cd- ah,
= cq — ch,
= -(qo + oB),

Now qo = qd tan qdo

= cd sin qcd cot qod,
but cd = I nearly, sin qcd = qcd==(e + h-7r) and qod = <f>;
.'. qo^l (c + h- it) cot <f>.



TRANSFORMATION OF SURFACES BY BENDING. 105

Also oS = -—-^ —
Sin bop

= aB (Bap) — — 7
^ ^' 8m<f>

= l'(a+h-7T)J-r.

Substituting the values of a, h, c, d from Art. (11),
Sl= — (qo + 08)

= —I —, ^- cot <i>Su — V — T—, - — r Bu.
r du ^ r du sm0

Finally, substituting the values of I, V, and Bl from Art. (14),

d ( \ ds"\ sj 5 , cot (/) cZs' 1 (i5 5. ^ , 1 ds I ds' ^ ,

du \p sin <p du / p sm <f> du r du p sm <j> du r du

which may be put under the more convenient form

— n ^ = — 1 / 1 ^^'\ 1 ds , p I ds 1
du^ °'^'~du ^ \sin <j> du) r du ^ p' r du sin <^ '

and from the value of Bl' we may similarly obtain

d ,, '\ _ _^ 1 / 1 ^\ ,i^ +^j_^i^ ^
du ^ ^ ^ ' du' ° \sin <f> du) r du' ^ p r du sin (ft '

We may simplify these equations by putting p for the specific curvature of

the surface, and q for the ratio , , which is the only quantity altered by bending.

We have then

p = — / . , . , and q = —,,
^ pp sm=<^' ^ p

whence p' = q — ^^-r , p'^ = t-tj y

^ ^ p sin <f) 9. P s^ Y

and the equations become

d ,. \ d , ( ^Tl'X 1 ds , , 2 ds 1

In this way we may reduce the problem of bending a surface to the
consideration of one variable q, by means of the lines of bending.

VOL. I. 14



d_

du'



106 TRANSFORMATION OF SURFACES BY BENDING.

16. To obtain the conditio of Instantaneous lines of bending.

We have now obtained tlie values of the differential coefficients of q with
respect to each of the variables u, u.
From the equation

we might find an equation which would give certain conditions of lines of
bending. These conditions however would be equivalent to those which we have
already assumed when we drew the systems of lines so as to be conjugate to
each other.

To find the true conditions of bending we must suppose the form of the
surface to vary continuously, so as to depend on some variable t which we
may call the time.

Of the difierent quantities which enter into our equations, none are changed
by the operation of bending except q, so that in differentiating with respect
to t all the rest may be considered constant, q being the only variable.

Differentiating the equations of last article with respect to t, we obtain
d" ,, . 2 ds 1 d ,, .



Whence



c?" ,, . 2 ds' 1 I d ,. .



A^t'^^'^^^ =



{.4 1- 1 si^)-'^ Tu ^, ii'^^H^'o^'^' 1 1 ii^^ 3^.<(">^*)-



and



(log l)



dududt



( d /2ds 1 \ 2 ds 1 d , } ^ d ,, 2 ds 1 1 d ,, .

{M?d^^^'r-di7^^d^^'^'irqdt^^'^^^

two independent values of the same quantity, whence the requiied conditions
may be obtained.



TRANSFORMATION OF SURFACES BY BENDING.



107



Substituting in these equations the values of those quantities which occur
in the original equations, we obtain



I ds ( d , ,



ds
du



sin



*)



+ - , \, cot <!> y



2 ds
r du



\l ds ( d , f ,ds . A 2 ds . ,\



which is the condition which must hold at every instant during the process of



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