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centre of the triangle, and intersect at right angles the curves of dilatation.

Let the isochromatic lines in Fig. 2 be determined by the equation

<^,{x,y) = I- = (o{q-p)-,

where / is the difference of retardation of the oppositely polarized rays, and
q and p the pressures in the principal axes at any point, z being the thick-
ness of the plate.

Let the lines of equal inclination be determined by the equation

<^2 (^. y) = tan 6,

6 being the angle of inclination of the principal axes ; then the differential
equation of the curves of direction of compression and dilatation (Fig. 4) is

By considering any particle of the plate as a portion of a cylinder whose
axis passes through the centre of curvature of the curve of compression, we find

?-?>=^^ (21).



70 THE EQUILIBRIUM OF EliASTIC SOLIDS.

Let R denote the radius of curvature of the curve of compression at any
point, and let S denote the length of the curve of dilatation at the same
point,

and since {q -p), R and S are known, and since at the surface, where (^^ {x, y) = 0,
j9 = 0, all the data are given for determining the absolute value of p by inte-
gration.

Though this is the best method of finding p and q by graphic construc-
tion, it is much better, when the equations of the curves have been found, that
is, when ^i and <j>^ are known, to resolve the pressures in the direction of the
axes.

The new quantities are p^, p„ and ^3 ; and the equations are

tan^=-^, {p-qY = q.' + (p.-p.y, Pi+P.=P + q-
Pi Pi

It is therefore possible to find the pressures from the curves of equal tint
and equal inclination, in any case in which it may be required. In the mean-
time the curves of Figs. 2, 3, 4 shew the correctness of Sir John Herschell's
ingenious explanation of the phenomena of heated and unannealed glass.



Note A.

As the mathematical laws of compressions and pressures have been very thoroughly
investigated, and as they are demonstrated with great elegance in the very complete and
elaborate memoir of MM. Lamd and Clapeyron, I shall state as briefly as possible their results.

Let a solid be subjected to compressions or pressures of any kind, then, if through any
point in the solid lines be drawn whose lengths, measured from the given point, are pro-
portional to the compression or pressure at the point resolved in the directions in which the
lines are drawn, the extremities of such lines will be in the surface of an ellipsoid, whose
centre is the given point.

The properties of the system of compressions or pressures may be deduced from those
of the ellipsoid.



THE EQUILIBRIUM OF ELASTIC SOLIDS. 71

There are three diameters having perpendicular ordinates, which are called the principal
axes of the ellipsoid.

Similarly, there are always three directions in the compressed particle in which there
is no tangential action, or tendency of the parts to slide on one another. These directions
are called the principal axes of compression or of pressure, and in homogeneous solids they
always coincide with each other.

The compression or pressure in any other direction is equal to the sum of the products
of the compressions or pressures in the principal axes multiplied into the squares of the
cosines of the angles which they respectively make with that direction.



Note B.

The fundamental equations of this paper differ from those of Navier, Poisson, &c., only
in not assuming an invariable ratio between the linear and the cubical elasticity; but since
I have not attempted to deduce them from the laws of molecular action, some other reasons
must be given for adopting them.

The experiments from which the laws are deduced are —

1st. Elastic solids put into motion vibrate isochronously, so that the sound does not
vary with the amplitude of the vibrations.

2nd. Regnault's experiments on hollow spheres shew that both linear and cubic com-
pressions are proportional to the pressures.

3rd. Experiments on the elongation of rods and tubes immersed in water, prove that
the elongation, the decrease of diameter, and the increase of volume, are proportional to the
tension.

4th. In Coulomb's balance of torsion, the angles of torsion are proportional to the
twisting forces.

It would appear from these experiments, that compressions are always proportional to
pressures.

Professor Stokes has expressed this by making one of his coefficients depend on the
cubical elasticity, Avhile the other is deduced from the displacement of shifting produced by
a given tangential force.

M. Cauchy makes one coefficient depend on the linear compression produced by a force
acting in one direction, and the other on the change of volume produced by the same force.

Both of these methods lead to a correct result ; but the coefficients of Stokes seem to
have more of a real signification than those of Cauchy ; I have therefore adopted tiiose of
Stokes, using the symbols m and fi, and the fundamental equations (4) and (5), which define
them.



72



THE EQUILIBRIUM OF ELASTIC SOLIDS.



Note C.

As the coefficient <w, which determines the optical effect of pressure on a substance,
varies from one substance to another, and is probably a function of the linear elasticity, a
determination of its value in different substances might lead to some explanation of the
action of media on light.



This paper commenced by pointing out the insufficiency of all theories of elastic solids,
in which the equations do not contain two independent constants deduced from experiments.
One of these constants is common to liquids and solids, and is called the modulus of cubical
elasticity. The other is peculiar to solids, and is here called the modulus of linear elasticity.
The equations of Navier, Poisson, and Lam^ and Clapeyron, contain only one coefficient;
and Professor G. G. Stokes of Cambridge, seems to have formed the first theory of elastic
solids which recognised the independence of cubical and linear elasticity, although M. Cauchy
seems to have suggested a modification of the old theories, which made the ratio of linear
to cubical elasticity the same for all substances. Professor Stokes has deduced the theory
of elastic solids from that of the motion of fluids, and his equations are identical with those
of this paper, which are deduced from the two following assumptions.

In an element of an elastic solid, acted on by three pressures at right angles to one
another, as long as the compressions do not pass the limits of perfect elasticity —

1st. The sum of the pressures, in three rectangular axes, is proportional to the sum
of the compressions in those axes.

2nd. The difference of the pressures in two axes at right angles to one another, is
proportional to the difference of the compressions in those axes.



Or, in symbols:



(P. + P..i'J = 3.(^%|4).



(^.-^.)=-(l



(P,



p,)=r,j'y.



(P,-P^ = m



fZz Bx



fi being the modulus of auhical, and m that of linear elasticity.

These equations are found to be very convenient for the solution of problems, some
of which were given in the latter part of the paper.



THE EQUILIBRIUM OF ELASTIC SOLIDS. 73

These particular cases were —

That of an elastic hollow cylinder, the exterior surface of which was fixed, while the
interior was turned through a small angle. The action of a transparent solid thus twisted
on polarized light, was calculated, and the calculation confirmed by experiment.

The second case related to the torsion of cylindric rods, and a method was given by

which m may be found. The quantity E= ^ was found by elongating, or by bending

the rod used to determine m, and fi is found by the equation,

_ Em
^~dm-6E'

The effect of pressure on the surfaces of a hollow sphere or cylinder was calculated,
and the result applied to the determination of the cubical compressibility of liquids and
solids.

An expression was found for the curvature of an elastic plate exposed to pressure on
one side ; and the state of cylinders acted on by centrifugal force and by heat was
determined.

The principle of the superposition of compressions and pressures was applied to the case of
a bent beam, and a formula was given to determine E from the deflection of a beam
supported at both ends and loaded at the middle.

The paper concluded with a conjecture, that as the quantity a (which expresses the
relation of the inequality of pressure in a solid to the doubly-refracting force produced) is
probably a function of m, the determination of these quantities for different substances
might lead to a more complete theory of double refraction, and extend our knowledge of the
laws of optics.



VOL. I. 10



[Extracted from the Cambridge and Dublin Mathematical Journal, Vol. viii. p. 188,

February/, 1854.]



Solutions of Problems.



1. If from a point in the circumference of a vertical circle two heavy particles be suc-
cessively projected along the curve, their initial velocities being equal and either in the same
or in opposite directions, the subsequent motion will be such that a straight line joining
the particles at any instant will touch a circle.

Note. The particles are supposed not to interfere with each other's motion.



The direct analytical proof would involve the properties of elliptic integrals,
but it may be made to depend upon the following geometrical theorems.

(1) If from a point in one of two circles a right line be drawn cutting
the other, the rectangle contained by the segments so formed is double of the
rectangle contained by a line drawn from the point perpendicular to the radical
axis of the two circles, and the line joining their centres.

The radical axis is the line joining the points of intersection of the two
circles. It is always a real hne, whether the points of intersection of the circles
be real or imaginary, and it has the geometrical property — that if from any point
on the radical axis, straight lines be drawn cutting the circles, the rectangle con-
tained by the segments formed by one of the circles is equal to the rectangle
contained by the segments formed by the other.

The analytical proof of these propositions is very simple, and may be resorted
to if a geometrical proof does not suggest itself as soon as the requisite figure
is constructed.

If ^, B be the centres of the circles, P the given point in the circle whose
centre is ^, a line drawn from P cuts the first circle in p, the second in Q



SOLUTIONS OF PROBLEMS. 75

and q, and the radical axis in R. If PH be drawn perpendicular to the radical
axis, then

PQ.Pq = 2AB.HP.

CoR. If the line be drawn from P to touch the circle in T, instead of
cutting it in Q and q, then the square of the tangent PT is equal to the
rectangle 2AB . HP.

Similarly, if ph be drawn from p perpendicular to the radical axis

p'P = 2AB.hp.

Hence, if a line be drawn touching one circle in T, and cutting the other
in P and p, then

(PTY : {pT)' :: HP : hp.

(2) If two straight lines touching one circle and cutting another be made
to approach each other indefinitely, the small arcs intercepted by their inter-
sections with the second circle wiU be ultimately proportional to their distances
from the point of contact.

This result may easily be deduced from the properties of the similar
triangles FTP and ppT.

Cor. If particles P, p be constrained to move in the circle A, while
the line Pp joining them continually touches the circle B, then the velocity
of P at any instant is to that of p as PT to pT ; and conversely, if the
velocity of P at any instant be to that of P as PT to pT, then the line
Pp will continue to be a tangent to the circle B.

Now let the plane of the circles be vertical and the radical axis horizontal,
and let gravity act on the particles P, p. The particles were projected from
the same point with the same velocity. Let this velocity be that due to the
depth of the point of projection below the radical axis. Then the square of
the velocity at any other point will be proportional to the perpendicular from
that point on the radical axis ; or, by the corollary to (l), if P and p be at
any time at the extremities of the line PTp, the square of the velocity of P
will be to the square of the velocity of p as PH to ph, that is, as (PTf to
(pTf. Hence, the velocities of P and p are in the proportion of PT to pT,
and therefore, by the corollary to (2), the line joining them will continue a
tangent to the circle B during each instant, and will therefore remain a tangent
during the motion.



76 SOLUTIONS OF PROBLEMS.

The cb'cle A, the radical axis, and one position of the line Pp, are given
by the circumstances of projection of P and p. From these data it is easy to
determine the circle jB by a geometrical construction.

It is evident that the character of the motion will determine the position
of the circle B. If the motion is oscillatory, B will intersect A. If P and p
make complete revolutions in the same direction, B will lie entirely within A,
but if they move in opposite directions, B will lie entirely above the radical axis.

If any number of such particles be projected from the same point at equal
intervals of time with the same direction and velocity, the lines joining successive
particles at any instant will be tangents to the same circle ; and if the time
of a complete revolution, or oscillation, contain n of these intervals, then these
lines will form a polygon of ?i sides, and as this is true at any instant, any
number of such polygons may be formed.

Hence, the following geometrical theorem is true :

"If two circles be such that n lines can be drawn touching one of them
and having their successive intersections, including that of the last and first,
on the circiunference of the other, the construction of such a system of lines
wiU be possible, at whatever point of the first circle we draw the first tangent."



2. A transparent medium is such that the path of a ray of light within it is a given
circle, the index of refraction being a function of the distance from a given point in the
plane of the circle.

Find the form of this function and shew that for light of the same refrangibility —

(1) The path of every ray witJdn the medium is a circle,

(2) All the rays proceeding from any point in the medium will meet accurately in
another point.

(3) If rays diverge from a point without the medium and enter it through a spherical
surface having that point for its centre, they will be made to converge accurately to a point
within the medium.

Lemma I. Let a transparent medium be so constituted, that the refractive
index is the same at the same distance from a fixed point, then the path of
any ray of light within the medium will be in one plane, and the perpen-



SOLUTIONS OF PROBLEMS. 77

dicular from the fixed point on the tangent to the path of the ray at any
point will vary inversely as the refractive index of the medium at that point.

We may easily prove that when a ray of light passes through a spherical
surface, separating a medium whose refractive index is /x, from another where
it is /Aj, the plane of incidence and refraction passes through the centre of
the sphere, and the perpendiculars on the direction of the ray before and after
refraction are ir the ratio of /i, to fi^. Since this is true of any number of
spherical shells of different refractive powers, it is also true when the index of
refraction varies continuously from one shell to another, and therefore the
proposition is true.

Lemma II. If from any fixed point in the plane of a circle, a perpen-
dicular be drawn to the tangent at any point of the circumference, the rectangle
contained by this perpendicular and the diameter of the circle is equal to the
square of the line joining the point of contact with the fixed point, together
with the rectangle contained by the segments of any chord through the fixed
point.

Let APB be the circle, the fixed point; then
OY.FE=OP' + AO.OB,




Produce PO to Q, and join QR, then the triangles OYP, PQR are similar;
therefore

OY.PR=OP.PQ

= OP' + OP.OQ;
.: OY.PR=OP' + AO.OB.
If we put in this expression AO . OB = a^,

PO = r, OY=p, PR = 2p,
it becomes 2pp = ?'*+■ a*,



78 SOLUTIONS OF PROBLEMS.

To find the law of the index of refraction of the medium, so that a ray
from A may describe the circle APB, /x must be made to vary inversely as p
by Lemma I.

Let AO = r^, and let the refractive index at A=fii; then generally



h'


_c
p


_ 2C7p .
a' + r''




/^1 =


. 2Cp

a' + r:'




a' + r,'



but at A

therefore

The value of /n at any point is therefore independent of p, the radius of

the given circle; so that the same law of refractive index will cause any other

ray to describe another circle, for which the value of a' is the same. The

a^ . . .

value of OB is — , which is also independent of p ; so that every ray which

proceeds from A must pass through B.

Again, if we assume /x^ as the value of /x when r = 0,

ar + r,'



therefore h' — H-o



d' + r'^-'



a result independent of r^. This shews that any point A' may be taken as
the origin of the ray instead of A, and that the path of the ray will still be
circular, and will pass through another point B' on the other side of 0, such that

Next, let CP be a ray from C, a point without the medium, falling at P
on a spherical surface whose centre is C.

Let be the fixed point in the medium as before. Join PO, and produce

to Q till OQ = jyp. Through Q draw a circle touching CP in P, and cutting

CO in A and B ; then PBQ is the path of the ray within the medium.



SOLUTIONS OF PROBLEMS. 79




Since CP touches the circle, we have
CP'^CA. CB,

= {CO-OA){CO-\-OB);

but 0A= -^;

therefore CF' = CQ + CO (oB - ^^

an equation whence OB may be found, B being the point in the medium
through which all rays from C pass.

Note. The possibility of the existence of a medium of this kind possessing
remarkable optical properties, was suggested by the contemplation of the structure
of the crystalline lens in fish; and the method of searching for these properties
was deduced by analogy from Newton's Principia, Lib. L Prop. vii.

It would require a more accurate investigation into the law of the refractive
index of the different coats of the lens to test its agreement with the supposed
medium, which is an optical instrument theoretically perfect for homogeneous
light, and might be made achromatic by proper adaptation of the dispersive
power of each coat.

On the other hand, we find that the law of the index of refraction which
would give a minimum of aberration for a sphere of this kind placed in water,
gives results not discordant with facts, so far as they can be readily ascertained.



[From the Transactions of the Cambridge Philosophical Society, Vol. ix. Part iv.]



IV. On the Transformation of Surfaces by Bending.



Euclid has given two definitions of a surface, which may be taken as
examples of the two methods of investigating their properties.

That in the first book of the Elements is —

"A superficies is that which has only length and breadth."

The superficies difiers from a line in having breadth as well as length,
and the conception of a third dimension is excluded without being expHcitly
introduced.

In the eleventh book, where the definition of a soHd is first formally
given, the definition of the superficies is made to depend on that of the solid —
" That which bounds a soHd is a superficies."

Here the conception of three dimensions in space is employed in forming
a definition more perfect than that belonging to plane Geometry.

In our analytical treatises on geometry a surface is defined by a function
of three independent variables equated to zero. The surface is therefore the
boundary between the portion of space in which the value of the function is
positive, and that in which it is negative; so that we may now define a
surface to be the boundary of any assigned portion of space.

Surfaces are thus considered rather with reference to the figures which they
limit than as having any properties belonging to themselves.

But the conception of a surface which we most readily form is that of
a portion of matter, extended in length and breadth, but of which the thick-



TKANSFORMATION OF SURFACES BY BENDING. 81

ness may be neglected. By excluding the thickness altogether, we arrive at
Euclid's first definition, which we may state thus —

" A surface is a lamina of which the thickness is diminished so as to become
evanescent."

We are thus enabled to consider a surface by itself, without reference to
the portion of space of which it is a boundary. By drawing figures on the
surface, and investigating their properties, we might construct a system of
theorems, which would be true independently of the position of the surface in
space, and which might remain the same even when the form of the solid of
which it is the boundary is changed.

When the properties of a surface with respect to space are changed, while
the relations of lines and figures in the surface itself are unaltered, the surface
may be said to preserve its identity, so that we may consider it, after the
change has taken place, as the same surface.

When a thin material lamina is made to assume a new form it is said
to be hent. In certain cases this process of bending is called development, and
when one surface is bent so as to coincide with another it is said to be
applied to it.

By considering the lamina as deprived of rigidity, elasticity, and other
mechanical properties, and neglecting the thickness, we arrive at a mathemati-
cal definition of this kind of transformation.

" The operation of bending is a continuous change of the form of a surface,
without extension or contraction of any part of it."

The following investigations were undertaken with the hope of obtaining
more definite conceptions of the nature of such transformations by the aid of
those geometrical methods which appear most suitable to each particular case.
The order of arrangement is that in which the different parts of the subject
presented themselves at first for examination, and the methods employed form
parts of the original plan, but much assistance in other matters has been
derived from the works of Gauss*, Liouvillef, Bertrand^, Puiseux§, &c., references
to which will be given in the course of the investigation.

* Disquisitiones generalea circa superficies curvas. Presented to the Royal Society of Gottingen,
8th October, 1827. Commentationes Recentiores, Tom. vi.

t Liouville's Journal, xii. X ^^'^- ^^'^' § ^^"^■

VOL, I. 11



82



TRANSFORMATION OF SURFACES BY BENDING.



On the Bending of Surfaces generated hy the motion of a straight line in space.



If a straight line can be drawn in any surface, we may suppose that
part of the surface which is on one side of the straight line to be fixed,
while the other part is turned about the straight line as an axis.

In this way the surface may be bent about any number of generating lines
as axes successively, till the form of every part of the surface is altered.

The mathematical conditions of this kind of bending may be obtained in
the following manner.

Let the equations of the generating line be expressed so that the constants
involved in them are functions of one independent variable u, by the variation of
which we pass from one position of the line to another.

If in the equations of the generating line Aa, u = u^, then in the equations
of the line Bh we may put u = U2, and from the equations of these lines we
may find by the common methods the equations of the shortest line PQ between
Aa and Bb, and its length, which we may call S^. We may also find the
angle between the directions of ^a and Bb, and let this angle be SO.

In the same way from the equations of
Cc, in which u = u^, we may deduce the equa-
tions of RS, the shortest line between Bb and
Cc, its length 8^5 and the angle hd^ between
the directions of Bb and Cc. We may also
find the value of QR, the distance between
the points at which PQ and RS cut Bb.
Let QR = h(T, and let the angle between the
directions of PQ and RS be S^.

Now suppose the part of tlie surface between the lines Aa and Bb to be
fixed, while the part between Bb and Cc is turned round Bb as an axis. The
line RS wiU then revolve round the point R, remaining perpendicular to Bhy
and Cc will still be at the same distance from Bb, and wiU make the same
angle with it. Hence of the four quantities S4j S^2> ^cr and 8</>, 8^ alone will
be changed by the process of bending. 8<^, however, may be varied in a
perfectly arbitrary manner, and may even be made to vanish.



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