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bending for the lines about which the bending takes place at that instant.
When the bending is such that the position of the lines of bending on the
surface alters at every instant, this is the only condition which is required.
It is therefore called the condition of Instantaneous lines of bending.

17. To find the condition of Permanent lines of bending.

Since q changes with the time, the equation of last article will not be
satisfied for any finite time unless both sides are separately equal to zero. In
that case we have the two conditions



(!)■



d , / ds . ,\ 2ds ^ , ^^
^,log(i^r^^sm<^j + -^,cotc^ = 0,


y


1 ds ^
or - -J- = 0.
r du

|^log(i>r'^,siD<^)+|^cot<^ = 0,'




1 d/ ^
or -, -J-, = 0.
r du





(2).



If the lines of bending satisfy these conditions, a finite amount of bending
may take place without changing the position of the system on the surface.
Such lines are therefore called Permanent lines of bending.

The only case in which the phenomena of bending may be exhibited by
means of the polyhedron with quadrilateral facets is that in which permanent
lines of bending are chosen as the boundaries of the facets. In all other cases
the bending takes place about an instantaneous system of lines which is con-
tinually in motion with respect to the surface, so that the nature of the poly-
hedron would need to be altered at every instant.



14—2



108 TRANSFORMATION OF SURFACES BY BENDING.

We are now able to determine whether any system of lines drawn on a
given surface is a system of instantaneous or permanent lines of bending.

We are also able, by the method of Article (8), to deduce from two con-
secutive forms of a surface, the lines of bending about which the transformation
must have taken place.

If our analytical methods were sufficiently powerful, we might apply our
results to the determination of such systems of lines on any known surface, but
the necessary calculations even in the simplest cases are so compHcated, that,
even if useful results were obtained, they would be out of place in a paper of
this kind, which is intended to afford the means of forming distinct conceptions
rather than to exhibit the results of mathematical labour.

18. On the application of the ordinary unethods of analytical geometry to the
consideration of lines of bending.

It may be interesting to those who may hesitate to accept results derived
from the consideration of a polyhedron, when applied to a curved surface, to
inquire whether the same results may not be obtained by some independent
method.

As the following method involves only those operations which are most
familiar to the analyst, it will be sufficient to give the rough outline, which may
be filled up at pleasure.

The proof of the invariability of the specific curvature may be taken from
any of the memoirs above referred to, and its value in terms of the equation of
the surface will be foimd in the memoir of Gauss.

Let the equation to the surface be put under the form

then the value of the specific curvature is



d\ dh d^



dot? dif dx



~dJz'^ dz^
dx dy\



The definition of conjugate systems of curves may be rendered independent
of the reasoning formerly employed by the following modification.



TRANSFORMATION OF SURFACES BY BENDINO. 109

Let a tangent plane move along any line of the first system, then if the line
of ultimate intersection of this plane with itself be always a tangent to some line
of the second system, the second system is said to be conjugate to the first.

It is easy to show that the first system is also conjugate to the second.

Let the system of curves be projected on the plane of xy, and at the point
(x, y) let a be the angle which a projected curve of the first system makes with
the axis of x, and /8 the angle which the projected curve of the second system
which intersects it at that point makes with the same axis. Then the condition
of the systems being conjugate will be found to be

a and y3 being known as functions of x and y, we may determine the nature
of the curves projected on the plane of xy.

Supposing the surface to touch that plane at the origin, the length and
tangential curvature of the lines on the surface near the point of contact may
be taken the same as those of their projections on the plane, and any change
of form of the surface due to bending will not alter the form of the projected
lines indefinitely near the point of contact. We may therefore consider z as the
only variable altered by bending; but in order to apply our analysis with facility,
we may assume

72

^ = Pg sin' a + PQ- sin' A

d'z

, J = — PQ sin a cos a — PQ~^ sin y3 cos ^,

^ = PQ cos' a + P^-^ cos' /8.

It will be seen that these values satisfy the condition last given. Near the
origin we have



d*z dh d\ I* n- . , / n\



and q=Q'*.



110 TRANSFORMATION OF SURFACES BY BENDING.

Differentiating these values of -y-^ , &c., we shall obtain two values of , ,
and of 1—7—3, which being equated will give two equations of condition.

Now if s' be measured along a curve of the first system, and R be any

function of x and y, then

dE dR dR .

-^j-y = -^j- cos a + -7- sm a,
as dx ay

, dR _ dR ds'
du' ds du '

We may also show that -=-^ = - ,



, ,, , da . da d . (ds' . ,\

and that cos a ;i — sm a ;t- = t- log ( -j—, sm 1 .
cty (j/X cLs \ci/U I



By substituting these values in the equations thus obtained, they are
reduced to the two equations given at the end of (Art. 15). This method of
investigation introduces no difficulty except that of somewhat long equations, and
is therefore satisfactory as supplementary to the geometrical method given at
length.

As an example of the method given in page (2), we may apply it to
the case of the surface whose equation is



(^.) *{rf-j-©'



This surface may be generated by the motion of a straight line whose
equation is of the form



= acosnl — j, 2/ = asinni-f-



t being the variable, by the change of which we pass from one position of the
line to another. This line always passes through the circle

z = 0, ar' + y = a',
and the straight lines z = c, cc=^0,
and z— —c, y = 0,
which may therefore be taken as the directors of the surface.



TRANSFORMATION OF SURFACES BY BENDING. Ill

Taking two consecutive positions of this line, in which the values of t
are t and t + Bt, we may find by the ordinary methods the equation to the
shortest line between them, its length, and the co-ordinates of the point in which
it intersects the first line.



Calling the length 8^,



ac



8C= ,/^ Bin 2tBt,



Ja' + c
and the co-ordinates of the point of intersection are

x = 2a cos' t, y = 2a sin* t, z= —c cos 2t.
The angle 80 between the consecutive lines is

Ja- + c
The distance So- between consecutive shortest lines is

^ 3a'-F-2c*



and the angle S<^ between these latter lines is



sin 2t8t,



'Ja' + c

Hence if we suppose ^, 6, cr, (f), and t to vanish together, we shall have by
integration

(T = ~—, ( 1 — cos 2t),

Ja' + c'

By bending the surface about its generating lines we alter the value of (ft
in any manner without changing 4, 0, or or. For instance, making <^ = 0, all the
generating lines become parallel to the same plane. Let this plane be that of
xy, then ^ is the distance of a generating line from that plane. The projections



o- =



112 TRANSFORMATION OF SURFACES BY BENDING.

of the generating lines on the plane of xy will, by their ultimate intersections,
form a curve, the length of which is measured by a, and the angle which its
tangent makes with the axis of x hj 0, 6 and o- being connected by the equation

^ I 1 - cos 6 ,

which shows the curve to be an epicycloid.

The generating lines of the surface when bent into this form are therefore
tangents to a cylindrical surface on an epicycloidal base, touching that surface
along a curve which is always equally inclined to the plane of the base, the
tangents themselves being drawn parallel to the base.

We may now consider the bending of the surface of revolution

Putting r = Jaf + f, then the equation of the generating line is

r^ + z^ = c^.
This is the well-known hypocycloid of four cusps.

Let s be the length of the curve measured from the cusp in the axis of z,
then,

s = |<jV\
wherefore, r = (|)' c " * 5^.

Let 6 be the angle which the plane of any generating line makes with
that of xz, then s and 6 determine the position of any point on the surface.
The length and breadth of an element of the surface will be Ss and rB$.

Now let the surface be bent in the manner formerly described, so that
becomes 0^, and r, r, when

0^ = 1x0 and r' = -ry
then r' = (f)'c-V"'s'

provided o' = /u,'c.
The equation between r' and s being of the same form as that between
r and ^ shows that the surface when bent is similar to the original surface, its
dimensions being multiphed by fi*.



TRANSFORMATION OF SURFACES BY BENDING. 113

This, however, is true only for one half of the surface when bent. The
other half is precisely symmetrical, but belongs to a surface which is not con-
tinuous with the first.

The surface in its original form is divided by the plane of xy into two
parts which meet in that plane, forming a kind of cuspidal edge of a circular
form which limits the possible value of s and r.

After being bent, the surface still consists of the same two parts, but the
edge in which they meet is no longer of the cuspidal form, but has a finite

angle = 2 cos"^ - , and the two sheets of the surface become parts of two different

surfaces which meet but are not continuous.



NOTE.

As an example of the application of the more general theory of " lines of bending," let us
consider the problem which has been already solved by Professor Jellett.

To determine the conditions under which one portion of a surface may he rendered rigid, while
the remainder is flexible.

Suppose the lines of bending to be traced on the surface, and the corresponding poly-
hedron to be formed, as in (9) and (10), then if the angle of one of the four edges which
meet at any solid angle of the polyhedron be altered by bending, those of the other three
must be also altered. These edges terminate in other solid angles, the forms of which will
also be changed, and therefore the efifect of the alteration of one angle of the polyhedron will
be communicated to every other angle within the system of lines of bending which defines
the form of the polyhedron.

If any portion of the surface remains unaltered it must lie beyond the limits of the
system of lines of bending. We must therefore investigate the conditions of such a system
being bounded.

The boundary of any system of lines on a surface is the curve formed by the ultimate inter-
section of those lines, and therefore at any given point coincides in direction with the curve of
the system which passes through that point. In this case there are two systems of lines of
bending, which are necessarily coincident in extent, and must therefore have the same boundary.
At any point of this boundary therefore the directions of the lines of bending of the first
and second systems are coincident.

But, by (7), these two directions must be "conjugate" to each other, that is, must corre-
spond to conjugate diameters of the "Conic of Contact." Now the only case in which con-
VOL. I. 15



114 TRANSFORMATION OF SURFACES BY BENDING.

jugate diameters of a conic can coincide, is when the conic is an hyperbola, and both diameters
coincide with one of the asymptotes ; therefore the boundary of the system of lines of bending
must be a curve at every point of which the conic of contact is an hyperbola, one of whose
asymptotes lies in the direction of the curve. The radius of " normal curvature " must there-
fore by (3) be infinite at eveiy point of the curve. This is the geometrical property of
what Professor Jellett calls a " Curve of Flexure," so that we may express the result as
follows :

If one portion of a surface be fixed, while the remainder is bent, the boundary of the fixed
portion is a curve of fiexure.

This theorem includes those given at p. (92), relative to a fixed curve on a surface, for in
a surface whose curvatures are of the same sign, there can be no "curves of flexure," and
in a developable surface, they are the rectilinear sections. Although the cuspidal edge, or
arete de rebroussement, satisfies the analytical condition of a curve of flexure, yet, since its
form determines that of the whole surface, it cannot remain fixed while the form of the surface
is changed.

In concavo-convex surfaces, the curves of flexure must either have tangential curvature or
be straight lines. Now if we put <^=0 in the equations of Art. (17), we find that the
lines of bending of both systems have no tangential curvature at the point where they touch
the curve of flexure. They must therefore lie entirely on the convex side of that curve, and
therefore

If a curve of fiexure be fi^ed, the surface on the concave side of the curve is not flexible.

I have not yet been able to determine whether the surface is inflexible on the convex side
of the curve. It certainly is so in some cases which I have been able to work out, but I
have no general proof.

When a surface has one or more rectilinear sections, the portions of the surface between
them may revolve as rigid bodies round those lines as axes in any manner, but no other motion
is possible. The case in which the rectilinear sections form an infinite series has been discussed
in Sect. (I.).



[From the Cambridge and Dublin Mathematical Journal, Vol. ix.



V. On a particular case of the descent of a heavy body in a resisting

medium.

Every one must have observed that when a slip of paper falls through
the air, its motion, though undecided and wavering at first, sometimes becomes
regular. Its general path is not in the vertical direction, but inclined to it
at aji angle which remains nearly constant, and its fluttering appearance will
be found to be due to a rapid rotation round a horizontal axis. The direction
of deviation from the vertical depends on the direction of rotation.

If the positive directions of an axis be toward the right hand and upwards,
and the positive angular direction opposite to the direction of motion of the
hands of a watch, then, if the rotation is in the positive direction, the hori-
zontal part of the mean motion will be positive.

These efiects are commonly attributed to some accidental peculiarity in the
form of the paper, but a few experiments with a rectangular slip of paper
(about two inches long and one broad), will shew that the direction of rotation
is determined, not by the irregularities of the paper, but by the initial circum-
stances of projection, and that the symmetry of the form of the paper greatly
increases the distinctness of the phenomena. We may therefore assume that
if the form of the body were accurately that of a plane rectangle, the same
effects would be produced.

The following investigation is intended as a general explanation of the true
cause of the phenomenon.

I suppose the resistance of the air caused by the motion of the plane to
be in the direction of the normal and to vary as the square of the velocity
estimated in that direction.

Now though this may be taken as a sufficiently near approximation to the
magnitude of the resisting force on the plane taken as a whole, the pressure

15—2



116 DESCENT OF A HEAVY BODY IN A RESISTING MEDIUM.

on any given element of the surface will vary with its position so that the
resultant force will not generally pass through the centre of gravity.

It is found by experiment that the position of the centre of pressure
depends on the tangential part of the motion, that it lies on that side of the
centre of gravity towards which the tangential motion of the plane is directed,
and that its distance from that point increases as the tangential velocity in-
creases.

I am not aware of any mathematical investigation of this effect. The
explanation may be deduced from experiment.

Place a body similar in shape to the sHp of paper obliquely in a current
of some visible fluid. Call the edge where the fluid first meets the plane the
first edge, and the edge where it leaves the plane, the second edge, then we
may observe that

(1) On the anterior side of the plane the velocity of the fluid increases
as it moves along the surface from the first to the second edge, and therefore
by a known law in hydrodynamics, the pressure must diminish from the first
to the second edge.

(2) The motion of the fluid behind the plane is very unsteady, but may
be observed to consist of a series of eddies diminishing in rapidity as they
pass behind the plane from the first to the second edge, and therefore relieving
the posterior pressure most at the first edge.

Both these causes tend to make the total resistance greatest at the first
edge, and therefore to bring the centre of pressure nearest to that edge.

Hence the moment of the resistance about the centre of gravity will always
tend to turn the plane towards a position perpendicular to the direction of the
current, or, in the case of the slip of paper, to the path of the body itself. It
will be shewn that it is this moment that maintains the rotatory motion of
the falling paper.

When the plane has a motion of rotation, the resistance will be modified
on account of the unequal velocities of difierent parts of the surface. The
magnitude of the whole resistance at any instant will not be sensibly altered
if the velocity of any point due to angular motion be small compared with that
due to the motion of the centre of gravity. But there will be an additional
moment of the resistance round the centre of gravity, which will always act in
the direction opposite to that of rotation, and wOl vary directly as the normal
and angular velocities together.



DESCENT OF A HEAVY BODY IN A RESISTING MEDIUM. 117

The part of the moment due to the obliquity of the motion will remain
nearly the same as before.

We are now prepared to give a general explanation of the motion of the
slip of paper after it has become regular.

Let the angular position of the paper be determined by the angle between
the normal to its surface and the axis of x, and let the angular motion be
such that the normal, at first coinciding with the axis of x, passes towards
that of y.

The motion, speaking roughly, is one of descent, that is, in the negative
direction along the axis of y.

The resolved part of the resistance in the vertical direction will always
act upwards, being greatest when the plane of the paper is horizontal, and
vanishing when it is vertical.

When the motion has become regular, the effect of this force during a
whole revolution will be equal and opposite to that of gravity during the same
time.

Since the resisting force increases while the normal is in its first and third
quadrants, and diminishes when it is in its second and fourth, the maxima of
velocity will occur when the normal is in its first and third quadrants, and
the minima when it is in the second and fourth.

The resolved part of the resistance in the horizontal direction will act in
the positive direction along the axis of x in the first and third quadrants, and
in the negative direction during the second and fourth; but since the resistance
increases with the velocity, the whole effect during the first and third quadrants
will be greater than the whole effect during the second and fourth. Hence
the horizontal part of the resistance will act on the whole in the positive
direction, and will therefore cause the general path of the body to incline in
that direction, that is, toward the right.

That part of the moment of the resistance about the centre of gravity
which depends on the angular velocity will vary in magnitude, but wUl always
act in the negative direction. The other part, which depends on the obliquity
of the plane of the paper to the direction of motion, will be positive in the
first and third quadrants and negative in the second and fourth ; but as its
magnitude increases with the velocity, the positive effect will be greater than
the negative.

When the motion has become regular, the effect of this excess in the



118 DESCENT OF A HEAVY BODY IN A RESISTING MEDIUM.

positive direction will be equal and opposite to the negative effect due to the
angular velocity during a whole revolution.

The motion will then consist of a succession of equal and similar parts
performed in the same manner, each part corresponding to half a revolution of
the paper.

These considerations will serve to explain the lateral motion of the paper,
and the maintenance of the rotatory motion.

Similar reasoning will shew that whatever be the initial motion of the
paper, it cannot remain uniform.

Any accidental oscillations will increase till their amphtude exceeds half a
revolution. The motion will then become one of rotation, and will continually
approximate to that which we have just considered.

It may be also shewn that this motion will be unstable unless it take
place about the longer axis of the rectangle.

If this axis is incHned to the horizon, or if one end of the slip of paper
be different from the other, the path will not be straight, but in the form of
a helix. There will be no other essential difference between this case and that
of the symmetrical arrangement.

Trinity College, April 5, 1853.



[From the Transactions of the Royal Scottish Society of Arts, Vol. iv. Part in]



VI. On the Theory of Colours in relation to Colour-Blindness.
A letter to Dr G. Wilson.

Dear Sir, — As you seemed to think that the results which I have obtained
in the theory of colours might be of service to you, I have endeavoured to
arrange them for you in a more convenient form than that in which I first
obtained them. I must premise, that the first distinct statement of the theory
of colour which I adopt, is to be found in Young's Lectures on Natural Philo-
sophy (p. 345, Kelland's Edition) ; and the most philosophical enquiry into it
which I have seen is that of Helmholtz, which may be found in the Annals of
Philosophy for 1852.

It is well known that a ray of light, from any source, may be divided by
means of a prism into a number of rays of different refranglbility, forming a
series called a spectrum. The intensity of the light is different at different
points of this spectrum ; and the law of intensity for different refrangibilities
differs according to the nature of the incident light. In Sir John F. W.
Herschel's Treatise on Light, diagrams will be found, each of which represents
completely, by means of a curve, the law of the intensity and refranglbility of
a beam of solar light after passing through -various coloured media.

I have mentioned this mode of defining and registering a beam of light,
because it is the perfect expression of what a beam of light is in itself, con-
sidered with respect to all its properties as ascertained by the most refined
instruments. When a beam of light falls on the human eye, certain sensations
are produced, from which the possessor of that organ judges of the colour and
intensity of the light. Now, though every one experiences these sensations, and
though they are the foundation of all the phenomena of sight, yet, on account
of their absolute simplicity, they are incapable of analysis, and can never become
in themselves objects of thought. If we attempt to discover them, we must



120 THE THEORY OF COLOURS IN RELATION TO COLOUR-BLINDNESS.

do SO by artificial means ; and our reasonings on tKem must be guided by some
theory.

The most general form in which the existing theory can be stated is this, —
There are certain sensations, finite in number, but infinitely variable in

degree, which may be excited by the difierent kinds of light. The compound

sensation resulting from all these is the object of consciousness, is a simple act

of vision.

It is easy to see that the numher of these sensations corresponds to what

may be called in mathematical language the number of independent variables, of

which sensible colour is a function.

This will be readily understood by attending to the following cases : —

1. When objects are illuminated by homogeneous yellow light, the only
thing which can be distinguished by the eye is difference of intensity or
brightness.

If we take a horizontal line, and colour it black at one end, with increasing
degrees of intensity of yellow light towards the other, then every visible object
wiU have a brightness corresponding to some point in this line.



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