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The following calculations shew that the apparent compressibility of the liquid
depends on the compressibility of the vessel, and is independent of the thickness
when the pressures are equal.

A hollow sphere, whose external and internal radii are a^ and a,, is acted
on by external and internal normal pressures h^ and K, it is required to deter-
mine the equilibrium of the elastic solid.

The pressures at any point in the solid are : —

1. A pressure p in the direction of the radius.

2. A pressure q in the perpendicular plane.

These pressures depend on the distance from the centre, which is denoted
by r.

The compressions at any point are -.— in the radial direction, and — in
the tangent plane, the values of these compressions are : —

fr=[h-^^P^''i)*h^ ('")•

T = fe-3fJ(^ + 2,) + l5 (35).

Multiplying the last equation by r, differentiating with respect to r, and
equating the result with that of the first equation, we find

52

THE EQUILIBRITTM OF ELASTIC SOLIDS.

Since the forces whicli act on the particle in the direction of the radius
must balance one another, or

2qdrde +p (rdey =(^p + ^d7^(r + dry 6,

_r dp

therefore ^""-^ = 2 37 ^^^^'

Substituting this value of q -p in the preceding equation, and reducing,

therefore

^ + 2^ = 0.
dr dr

Integrating,
But
and the equation becomes

therefore

p-\-2q = c,.
r dp ,

dp
dr

+ 3^-^-i = 0,

1 c.

Since p = h, when r = a.,, and p = K when r = a,, the value of p at any
distance is found to be

^~ a^-af r' a^-a,'
9- a,'-ai "^^ 7^ <-a/

(37).
.(38).

When r = a„ -y = -^r:^^ - + t ^^ ^^737^3 ^

~ a,' - a/ U 2»i/ a/ - «/ \jx 2wi/ _
When the external and internal pressures are equal

.(39).

h^ = h.,=p = q, and -y-

SV K

.(40),

THE EQUILIBRIUM OF ELASTIC SOLIDS. 53

the change of internal capacity depends entirely on the cubical elasticity of the
vessel, and not on its thickness or linear elasticity.

When the external and internal pressures are inversely as the cubes of the
radii of the surfaces on which they act,

aX = a,%, p = ^ K q= -i^K

when r = r- — ^ '

(41).

V 2 ^^

In this case the change of capacity depends on the linear elasticity alone.

M. Regnault, in his researches on the theory of the steam engine, has
given an account of the experiments which he made in order to determine
with accuracy the compressibility of mercury.

He considers the mathematical formulae very uncertain, because the theories
of molecular forces from which they are deduced are probably far from the
truth ; and even were the equations free from error, there would be much
uncertainty in the ordinary method by measuring the elongation of a rod of
the substance, for it is diflScult to ensure that the material of the rod is the
same as that of the hollow sphere.

He has, .therefore, availed himself of the results of M. Lam6 for a hollow
sphere in three different cases, in the first of which the pressure acts on the
interior and exterior surface at the same time, while in the other two cases
the pressure is applied to the exterior or interior surface alone. Equation (39)
becomes in these cases, —

1. When ^1 = /ij, -^ = — and the compressibility of the enclosed liquid being
/x,, and the apparent diminution of volume S'F,

v-.£-;) «■

2. When /i, = 0,

54 THE EQUILIBRIUM OF ELASTIC SOLIDS.

3. When h,^0,

8V_ h K , 9^\

V a^-a^ \ii ^ m ^ ' V2 J

M. Lamp's equations differ from these only in assuming that fi, = |-m. If
this assumption be correct, then the coefficients /u,, m, and jMj, may be found
from two of these equations ; but since one of these equations may be derived
from the other two, the three coefficients cannot be found when /u, is supposed
independent of m. In Equations (39), the quantities which may be varied at
pleasure are \ and h^, and the quantities which may be deduced from the
apparent compressions are,

'■=G+4)^°<^S-i)=^"

therefore some independent equation between these quantities must be found,
and this cannot be done by means of the sphere alone; some other experiment
must be made on the liquid, or on another portion of the substance of which

The value of /x^, the elasticity of the liquid, may be previously known.

The linear elasticity m of the vessel may be found by twisting a rod of
the material of which it is made ;

Or, the value of E may be found by the elongation or bending of the

We have here five quantities, which may be determined by experiment.

on sphere.

, audi:

-i^

2
3m

We have here

fiv

(43)

1.

(42)

2.

(31)

3.

(17)

4.
5.

+ — ) by external pressure

Cj = ( j equal pressures.

m by twisting the rod.

/Xj the elasticity of the liquid.

THE EQUILIBRIUM OF ELASTIC SOLIDS.

55

When the elastic sphere is solid, the internal radius a, vanishes, and

fh=p = q, and -y = ^-

When the case becomes that of a spherical cavity in an infinite solid, the
external radius a^ becomes infinite, and

P=K-f{K-K)

r-

= K+i

^i'h-h,)

r

= ^^>i+^^(^>-^^)

1
m

v =

■'-!

(44).

The effect of pressure on the surface of a spherical cavity on any other part
of an elastic solid is therefore inversely proportional to the cube of its distance
from the centre of the cavity.

When one of the surfaces of an elastic hollow sphere has its radius rendered
invariable by the support of an incompressible sphere, whose radius is Oj, then

— = 0, when r = a^,

therefore

2771

q=h

2a^m + 3«//x

3a,V
2a>i + 3a//x

r* 2a^m + 3a//x

IK

W hen r = a,, j-y — lu r-—. ~— ,-

" V -2a>2 + 3a.//i,

K^

1

r* 2a/m + 3a//i

(45).

Case V.

On the equilibrium of an elastic beam of rectangular section uniformly
bent.

By supposing the bent beam to be produced till it returns into itself, we
may treat it as a hollow cylinder.

66 THE EQUILIBRIUM OF ELASTIC SOLIDS.

Let a rectangular elastic beam, whose length is 2irc, be bent into a circular
form, so as to be a section of a hollow cylinder, those parts of the beam which
lie towards the centre of the circle will be longitudinally compressed, while the
opposite parts will be extended.

The expression for the tangential compression is therefore

Br _ r — c
r ~ c '

r

Sr
Comparing this value of — with that of Equation (20),

V=(^-4)<''+-p+«)+^'''

dr

,,. , /I 2\ .,

ion

and by (21), q=p + r

By substituting for q its value, and dividing by r (q- + ^) • the equat:

becomes

dp 2m + 3/x j9 _ 9?n/i. — {m — 3/x) o 9m/x c
dr m + 6fx r~ (m + 6fi) r (m + 6/x) r' *

a linear differential equation, which gives

^ ^ m — 3fir 2m + 3/x

Ci may be found by assumiQg that when r^a^, p = \, and q may be found
from p by equation (21).

As the expressions thus found are long and cumbrous, it is better to use
the following approximations : —

_/_9m^\ y ( )

l^\llcl^ \ (48).

In these expressions a is half the depth of the beam, and y is the distance
of any part of the beam from the neutral surface, which in this case is a cylin-
dric surface, whose radius is c.

These expressions suppose c to be large compared with a, since most sub-
stances break when - exceeds a certain small quantity.

THE EQUILIBRIUM OF ELASTIC SOLIDS. 57

Let b be the breadth of the beam, then the force with which the beam
resists flexure = M

M=lhyq = ^^^-^ = Ef (49),

which is the ordinary expression for the stiffness of a rectangular beam.

The' stiffness of a beam of any section, the form of which is expressed by
an equation between x and y, the axis of x being perpendicular to the plane of
flexure, or the osculating plane of the axis of the beam at any point, is ex-
pressed by

Mc = E{ifdx (50),

M being the moment of the force which bends the beam, and c the radius of
the circle into which it is bent.

Case YI.

At the meeting of the British Association in 1839, Mr James Nasmyth
described his method of making concave specula of silvered glass by bending.

A circular piece of silvered plate-glass was cemented to the opening of an
iron vessel, from which the air was afterwards exhausted. The mirror then
became concave, and the focal distance depended on the pressure of the air.

Buffon proposed to make burning- mirrors in this way, and to produce the
partial vacuum by the combustion of the air in the vessel, which was to be
effected by igniting sulphur in the interior of the vessel by means of a burn-
ing-glass. Although sulphur evidently would not answer for this purpose, phos-
phorus might; but the simplest way of removing the air is by means of the
air-pump. The mirrors which were actually made by Buffon, were bent by
means of a screw acting on the centre of the glass.

To find an expression for the curvature produced in a flat, circular, elastic
plate, by the difference of the hydrostatic pressures which act on each side
of it,—

Let t be the thickness of the plate, which must be small compared with
its diameter.

Let the form of the middle surface of the plate, after the curvature is
produced, be expressed by an equation between r, the distance of any point
from the axis, or normal to the centre of the plate, and x the distance of
the point from the plane in which the middle of the plate originally was, and let

ds=-^{dxY + {dr)\

VOL I. 8

58 THE EQUILIBRIUM OF ELASTIC SOLIDS.

Let A, be the pressure on one side of the plate, and h^ that on the other.

Let p and q be the pressures in the plane of the plate at any point, p
acting in the direction of a tangent to the section of the plate by a plane
passing through the axis, and q acting in the direction perpendicular to that
plane.

By equating the forces which act on any particle in a direction parallel to
the axis, we find

^ drdx , ^ dpdx , ^ d^x ^ ,, j^dr

By making p = when r = in this equation, when integrated,

p-l^l^^ - '^-) ("^-

The forces perpendicular to the axis are

[drV . dpdr , ^ d^r .^ i\dx ^ .

Substituting for p its value, the equation gives

_ (^1 - h^ idr dr dx\ (h^ - h^ /dr ds^d^^ds ^r\ , .
^" t ''[d'sdi'^d^)'^ 2t "^^[didxd^ dxd^)""^ ^'

The equations of elasticity become

dSs (\ 1 \ / ^ h, + h\^p

Differentiating -j- = -^ (""''')' ^^^ ^ ^^ *^^^®

dhr dr dr dSs

dr ~ ds ds ds '

By a comparison of these values of -t— ,

ds

dr\
ds) \9iJ,

, t^rwl 1\/ , ,K + h\,qdrp^ (I l\fdp,dq\

w dr as

THE EQUILIBRIUM OF ELASTIC SOUDS. 59

To obtain an expression for the curvature of the plate at the vertex, let a

be the radius of curvature, then, as an approximation to the equation of the

plate, let

x — — .
2a

By substituting the value of a: in the values of p and q, and in the equa-
tion of elasticity, the approximate value of a is found to be

a =

18m/x, \-\-h^ m- 3/x

. 1 c 1 "T" ' T 7~ ~T~z ; — TT"

.(53).

^i-A, lOm + 51/x A,-^2 lOw + 51/t '

Since the focal distance of the mirror, or -, depends on the difference of

pressures, a telescope on Mr Nasmyth's principle would act as an aneroid baro-
meter, the focal distance varying inversely as the pressure of the atmosphere.

Case VIL

To find the conditions of torsion of a cylinder composed of a great number
of parallel wires bound together without adhering to one another.

Let X be the length of the cylinder, a its radius, r the radius at any point,
hS the angle of torsion, M the force producing torsion, hx the change of length,
and P the longitudinal force. Each of the wires becomes a helix whose radius
is r, its angular rotation Zd, and its length along the axis x-Zx.

Its length is therefore {rZey

— IJ

and the tension is = jE; 1 1 - /[ 1 - - ] V r^ (-]'] .

This force, resolved parallel to the axis, is

60 THE EQUIUBRTCM OF ELASTIC SOUDS.

and since — and r — are small, we may assume

XX

-"-{-l-n?)'} <">■

The force, when resolved in the tangential direction, is approximately

"-■^m'i-m '">

By eliminating — between (54) and (55) we have

X

M: ^^'

^ip.E.^m (56).

X 24 \ a?/

When P = 0, M depends on the sixth power of the radius and the cube
of the angle of torsion, when the cylinder is composed of separate filaments.

Since the force of torsion for a homogeneous cylinder depends on the
fourth power of the radius and the first power of the angle of torsion, the
torsion of a wire having a fibrous texture will depend on both these laws.

The parts of the force of torsion which depend on these two laws may be
found by experiment, and thus the difference of the elasticities in the direction
of the axis and in the perpendicular directions may be determined.

A calculation of the force of torsion, on this supposition, may be found in
Young's Mathematical Principles of Natural Philosophy; and it \s introduced
here to account for the variations from the law of Case II., which may be
observed in a twisted rod.

Case VIII.

It is well known that grindstones and fly-wheels are often broken by the
centrifugal force produced by their rapid rotation. I have therefore calculated
the strains and pressure acting on an elastic cylinder revolving round its axis,
and acted on by the centrifugal force alone.

THE EQUILIBBIUM OF ELASTIC SOLIDa.

61

The equation of the equilibrium of a particle [see Equation (21)], becomes

dp Air'k ,

where q and p are the tangential and radial pressures, k is the weight in
pounds of a cubic inch of the substance, g is twice the height in inches that
a body falls in a second, t is the time of revolution of the cylinder in seconds.

By substituting the value of q and ^ in Equations (19), (20), and neglect-
ing 0,

-(i-3^)(«|-?-g)-M^S-f-^.^)

which gives

1 TT^k

2gt^\

1 , Tj'k

2+^K + ^«

(-"?)

TT'k

2gf

^=-V + 2^»(-2 + f)^ + c.

(57).

If the radii of the surfaces of the hollow cylinder be a, and cu„ and the
pressures actmg on them h^ and h^, then the values of c^ and c, are

(58).

-f^'-(« - .')S(^-S.

When o, = 0, as in the case of a solid cylinder, c, = 0, and

« = *'+0 {2('^ + «.') + |(3'^-«,')} (59).

When A, = 0, and r^a^,

^ = ^U-2) (60).

When q exceeds the tenacity of the substance in pounds per square inch,
the cylinder will give way; and by making q equal to the number of pounds
which a square inch of the substance will support, the velocity may be found
at which the bursting of the cylinder will take place.

g2 THE EQUILIBRIUM OP ELASTIC SOLIDS.

Since I=ho>(q-p) = '^ (^-2\br', a transparent revolving cylinder, when

polarized light is transmitted parallel to the axis, will exhibit rings whose
diameters are as the square roots of an arithmetical progression, and brushes
parallel and perpendicular to the plane of polarization.

Case IX.

A hollow cylinder or tube is surrounded by a medium of a constant
temperature while a liquid of a different temperature is made to flow through
it. The exterior and interior surfaces are thus kept each at a constant tem-
perature till the transference of heat through the cylinder becomes uniform.

Let V be the temperature at any point, then when this quantity has

reached its limit,

rdv _

v = Ci\ogr + Ci (61).

Let the temperatures at the surfaces be 0^ and 0^, and the radii of the
surfaces a, and a^, then

^ 0^-0^ loga,0^-logaA

^'""logaj-loga/ '~ loga^-loga^
Let the coeflBcient of linear dilatation of the substance be c,, then the
proportional dilatation at any point will be expressed by c,v, and the equations
of elasticity (18), (19), (20), become

r \,9/x 3m/ ^ ^ ^' m
The equation of equHibrivuu is

2-P+r'^ (21),

and since the tube is supposed to be of a considerable length

-J— =c^ a constant quantity.

CL2C

THE EQUILIBRIUM OF ELASTIC SOLIDS. 63

From these equations we find ttat

9/x 3m
and hence v = c^\ogr + Cz, p may be found in terms of r.

Hence ? = (|l + 4) " ^.«' •«§ '- ^. ^ + <'• + (|l + ^) ''.^-

Since I—hco (q —p) = ho)i— + - — ) CjCg — 260)05 -^ ,

the rings seen in this case will differ from those described in Case III. only
by the addition of a constant quantity.

When no pressures act on the exterior and interior surfaces of the tube
^j = ^„ = 0, and

/2 . J_V^.^ Aoo-r I ^i'^/ log^i-log«2 , a/logct,-a/logaA

/^ 1_\- I a^a^ log g, - log ct , a^ log a, - a/ log a \

^-1,9,. + 3m/ ^^^3^^^S^ r^ a'-a^ + <-a,^ +V'

\9/x 3m/ ' ' \ r" a{-a^ J

...(62).

There will, therefore, be no action on polarized light for the ring whose

r" = 2 „ log - .

Case X.

Sir David Brewster has observed {Edinburgh Transacticms, Vol. viii.), that
when a solid cylinder of glass is suddenly heated at the cylindric siuface a
polarizing force is developed, which is at any point proportional to the square
of the distance from the axis of the cylinder ; that is to say, that the dif-

64 THE EQUILIBBIUM OF ELASTIC SOLIDS.

ference of retardation of the oppositely polari^ied rays of %ht is proportional
to the square of the radius r, or

/= bCj^cor' = h(o {q —p) = hayr -^ ,
Since if a be the radius of the cylinder, ^ = when r^a,

Hence ?=J(3r'-o").

2

By substituting these values of p and q in equations (19) and (20), and
, . d h' dhr T ^ ,

^=|(4 + li)'-' + »" (««)•

c^ being the temperature of the axis of the cylinder, and c, the coefficient of
linear expansion for glass.

Case XI.

Heat is passing uniformly through the sides of a spherical vessel, such as
the ball of a thermometer, it is required to determine the mechanical state of
the sphere. As the methods are nearly the same as in Case IX., it will be
sufficient to give the results, using the same notation.

, dv c,

dr ^' * r

Ci = aM,— ?, c- = -5-2 —,

o, — o, o, — a,

1 /2 .1 \-^ 1 .

When h, = h, = the expression for p becomes

p = /2 ly- r_aXLl _^A.l^ a.'-a» |

^ \9/t* 3m/ '^ ' ''[a/-a/7^ a,-o^r {0,-0,) (o^-o^)] ^ '

From this value of p the other quantities may be found, as in Case IX.,
from the equations of Case IV.

THE EQUILIBRIUM OF ELASTIC SOLIDS. 65

Case XII.

When a long beam is bent into the form of a closed circular ring (as in
Case v.), all the pressures act either parallel or perpendicular to the direction
of the length of the beam, so that if the beam were divided into planks, there
would be no tendency of the planks to slide on one another.

But when the beam does not form a closed circle, the planks into which it
may be supposed to be divided will have a tendency to slide on one another,
and the amount of sliding is determined by the linear elasticity of the sub-
stance. The deflection of the beam thus arises partly from the bending of the
whole beam, and partly from the sHding of the planks ; and since each of these
deflections is small compared with the length of the beam, the total deflection
will be the sum of the deflections due to bending and sliding.

Let

A=Mc = E\xi/'dy (65).

A is the stiffiiess of the beam as found in Case Y., the equation of the
transverse section being expressed in terms of x and y, y being measured from
the neutral surface.

Let a horizontal beam, whose length is 2l, and whose weight is 2w, be
supported at the extremities and loaded at the middle with a weight W.

Let the deflection at any point be expressed by h^, and let this quantity
be small compared with the length of the beam.

At the middle of the beam, 8,y is found by the usual methods to be

% = ^ {-h^w + ^^l'W) (66).

Let

B = — \xdy = — (sectional area) (jo7).

B is the resistance of the beam to the sliding of the planks. The de-
flection of the beam arising from this cause is

% = 2]b(^'+^^ (68).

VOL. I. 9

66 THE EQXnUBRnJM OF ELASTIC SOLIDS.

This quantity is small compared with S^y, when the depth of the beam is
small compared with its length.

The whole deflection ^y = B^ + S^

A3/ = - (^.Z-^iS + ^ {U +^l) (^^)-

Case XIII.

When the values of the compressions at any point have been found, when
two difierent sets of forces act on a solid separately, the compressions, when
the forces act at the same time, may be found by the composition of com-
pressions, because the small compressions are independent of one another.

It appears from Case I., that if a cylinder be twisted as there described,
the compressions will be inversely proportional to the square of the distance
from the centre.

If two cylindric surfaces, whose axes are perpendicular to the plane of an
indefinite elastic plate, be equally twisted in the same direction, the resultant
compression in any direction may be found by adding the compression due to
each resolved in that direction.

The result of this operation may be thus stated geometrically. Let A^ and
A^ (Fig. 1) be the centres of the twisted cylinders. Join ^1^25 and bisect A^A,
in 0. Draw OBC at right angles, and cut off OB^^ and OB^ each equal to OA^.

Then the difference of the retardation of oppositely polarized rays of light
passing perpendicularly through any point of the plane varies directly as the
product of its distances from B^ and B^, and inversely as the square of the
product of its distances from A^ and A^.

The isochromatic lines are represented in the figure.

The retardation is infinite at the points ^1 and A^; it vanishes at B^^
and jBj ; and if the retardation at be taken for unity, the isochromatic curves
2, 4, surround Aj^ and A^; that in which the retardation is unity has two
loops, and passes through 0; the curves ^, ^ are continuous, and have points
of contrary flexure ; the curve ^ has multiple points at Cj and C,, where

THE EQUILIBEIUM OF ELASTIC SOLIDS.

67

.4,(7, = -4,^,, and two loops surrounding B^ and B^', the other curves, for which
/=l4-» -gS-j ^c-» consist each of two ovals surrounding B^ and jB,, and an
exterior portion surrounding all the former curves.

Fig. 1.

I have produced these curves in the jelly of isinglass described in Case I.
They are best seen by using circularly polarised light, as the curves are then
seen without interruption, and their resemblance to the calculated curves is
more apparent. To avoid crowding the curves toward the centre of the figure,
I have taken the values of / for the different curves, not in an arithmetical,
but in a geometrical progression, ascending by powers of 2.

68

THE EQUILrBRTOM OF ELASTIC SOLIDS.

Case XIV.

On the determination of the pressures which act in the interior of trans-
parent solids, from observations of the action of the solid on polarized light.

Sir David Brewster has pointed out the method by which polarized light
might be made to indicate the strains in elastic solids ; and his experiments on
bent glass confirm the theories of the bending of beams.

The phenomena of heated and unannealed glass are of a much more complex
nature, and they cannot be predicted and explained without a knowledge of the
laws of cooling and solidification, combined with those of elastic equilibrium.

In Case X. I have given an example of the inverse problem, in the case
of a cylinder in which the action on light followed a simple law ; and I now
go on to describe the method of determuiing the pressures in a general case,
applying it to the case of a triangle of unannealed plate-glass.

D D

Fig. 3.

The lines of equal intensity of the action on Hght are seen without
interruption, by using circularly polarized light. They are represented in Fig. 2,
where A, BBB, DDD are the neutral points, or points of no action on light,
and CCC, EEE are the points where that action is greatest ; and the intensity

THE EQUILIBRIUM OF ELASTIC SOLIDS. 69

of the action at any other point is determined by its position with respect to
the isochromatic curves.

The direction of the principal axes of pressure at any point is found by
transmitting plane polarized light, and analysing it in the plane perpendicular
to that of polarization. The light is then restored in every part of the triangle,
except in those points at which one of the principal axes is parallel to the
plane of polarization. A dark band formed of all these points is seen, which
shifts its position as the triangle is turned round in its own plane. Fig. 3
represents these curves for every fifteenth degree of inclination. They correspond
to the lines of equal variation of the needle in a magnetic chart.

From these curves others may be found which shall indicate, by their own
direction, the direction of the principal axes at any point. These curves of
direction of compression and dilatation are represented in Fig. 4 ; the curves
whose direction corresponds to that of compression are concave toward the

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