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other plane object perpendicular to the axis will be distinct, flat and similar
to the object.

When the object is at an infinite distance, the plane of its image is the
principal focal plane, and the point where it cuts the axis is the piincipal
focus. The line joining any point in the object to the corresponding point of
the image cuts the axis at a fixed point called the focal centre. The distance
of the principal focus from the focal centre is called the principal focal length,
or simply the focal length.

There are two principal foci, etc., formed by incident parallel rays passing
in opposite directions through the instrument. If we suppose light always to
pass in the same direction through the instrument, then the focus of incident
rays when the emergent rays are parallel is the Jirst principal focus, and the
focus of emergent rays when the incident rays are parallel is the second
principal focus.

Corresponding to these we have first and second focal centres and focal
lengths.

Now let Q, be the focus of incident rays, P^ the foot of the perpendicular
from ^1 on the axis, Q, the focus of emergent rays, P, the foot of the corre-
sponding perpendicular, F^F^ the first and second principal foci, A^A^ the first and
second focal centres, then

F\F\ _PjQr_FJP,

A^Frp.QrFA.'

lines being positive when measured in the direction of the light. Therefore
the position and magnitude of the image of any object is found by a simple
proportion.



240 ON THE ELEMENTARY THEORY OF OPTICAL INSTRUMENTS.

In one important class of instruments there are no principal foci or focal
centres. A telescope in which parallel rays emerge parallel is an instance. In
such instruments, if m be the angular magnifying power, the linear dimensions

of the image are — of the object, and the distance of the image of the object

from the image of the object-glass is —^ of the distance of the object from

the object-glass. Rules were then laid down for the composition of instruments,
and suggestions for the adaptation of this method to second approximations, and
the method itself was considered with reference to the labours of Cotes, Smith,
Euler, Lagrange, and Gauss on the same subject.



[From the Report of the British Association, 1856.]



XI. On a Method of Drawing the Theoi-etical Forms of Faraday s Lines of
Force without Calculation.



The method applies more particularly to those cases in which the lines
are entirely parallel to one plane, such as the lines of electric currents in a
thin plate, or those round a system of parallel electric currents. In such cases,
if we know the forms of the lines of force in any two cases, we may combine
them by simple addition of the functions on which the equations of the lines
depend. Thus the system of lines in a uniform magnetic field is a series of
parallel straight lines at equal intervals, and that for an infinite straight electric
current perpendicular to the paper is a series of concentric circles whose radii
are in geometric progression. Having drawn these two sets of lines on two
separate sheets of paper, and laid a third piece above, draw a third set of lines
through the intersections of the first and second sets. This will be the system
of lines in a uniform field disturbed by an electric current. The most interesting
cases are those of uniform fields disturbed by a small magnet. If %ve draw a
circle of any diameter with the magnet for centre, and join those points in which
the circle cuts the lines of force, the straight lines so drawn will be parallel and
equidistant; and it is easily shown that they represent the actual lines of
force in a paramagnetic, diamagnetic, or crystallized body, according to the
nature of the original lines, the size of the circle, &c. No one can study
Faraday's researches without wishing to see the forms of the Hnes of force.
This method, therefore, by which they may be easily drawn, is recommended
to the notice of electrical students.



[From the Report of the British Association, 1856.]



XII. On the Unequal Sensibility of the Foramen Centrale to Light of
different Colours.

When observing tlie spectrum formed by looking at a long vertical slit
through a simple prism, I noticed an elongated dark spot running up and down
in the blue, and following the motion of the eye as it moved up and down
the spectrum, but refusing to pass out of the blue into the other colours. It
was plain that the spot belonged both to the eye and to the blue part of the
spectrum. The result to which I have come is, that the appearance is due to
the yellow spot on the retina, commonly called the Foramen Centrale of Soem-
mering. The most convenient method of observing the spot is by presenting
to the eye in not too rapid succession, blue and yellow glasses, or, still better,
allowing blue and yellow papers to revolve slowly before the eye. In this way
the spot is seen in the blue. It fades rapidly, but is renewed every time the
yellow comes in to relieve the effect of the blue. By using a Nicol's prism
along with this apparatus, the brushes of Haidinger are well seen in connexion
with the spot, and the fact of the brushes being the spot analysed by polarized
light becomes evident. If we look steadily at an object behind a series of bright
bars which move in front of it, we shall see a curious bending of the bars as
they come up to the place of the yellow spot. The part which comes over the
spot seems to start in advance of the rest of the bar, and this would seem to
indicate a greater rapidity of sensation at the yellow spot than in the surround-
ing retina. But I find the experiment diflScult, and I hope for better results
from more accurate observers.



[From the Report of the British Association, 1856.]



XIII. On the TJieory of Compound Colours with reference to Mixtures of Blue

and Yellow Light.

When we mix together blue and yellow paint, we obtain green paint. This
fact is well known to all who have handled colours ; and it is universally
admitted that blue and yellow make green. Red, yellow, and blue, being the
primary colours among painters, green is regarded as a secondary colour, arising
from the mixture of blue and yellow. Newton, however, found that the green
of the spectrum was not the same thing as the mixture of two colours of the
spectrum, for such a mixture could be separated by the prism, while the green
of the specti-um resisted further decomposition. But still it was believed that
yellow and blue would make a green, though not that of the spectrum. As
far as I am aware, the first experiment on the subject is that of M. Plateau,
who, before 1819, made a disc with alternate sectors of prussian blue and gam-
boge, and observed that, when spinning, the resultant tint was not green, but
a neutral gray, inclining sometimes to yellow or blue, but never to green. Prof
J. D, Forbes of Edinburgh made similar experiments in 1849, with the same
result. Prof Helmholtz of Konigsberg, to whom we owe the most complete
investigation on visible colour, has given the true explanation of this phaenomenon.
The result of mixing two coloured powders is not by any means the same as
mixing the beams of light which flow from each separately. In the latter case
we receive all the light which comes either from the one powder or the other.
In the former, much of the light coming from one powder falls on particles of
the other, and we receive only that portion which has escaped absorption by one
or other. Thus the light coming from a mixture of blue and yellow powder,
consists partly of light coming directly from blue particles or yellow particles,
and partly of light acted on by both blue and yellow particles. This latter light
is green, since the blue stops the red, yellow, and orange, and the yellow stops



244 ON THE THEORY OF COMPOUND COLOURS.

the blue and violet I have made experiments on the mixture of blue and
vellow light — by rapid rotation, by con\bined reflexion and transmission, by view-
ing them out of focus, in stripes, at a gre;it distiince, by throwing the colours
of the spectrum on a screen, and by receiving them into the eye directly ; and
I have arranged a portable apparatus by which any one may see the result of
this or any other mLxture of the colours of the spectrum. In all these cases
blue and yellow do not make green. I have also made experiments on the
mixture of coloured powders. Those which I used principally were "mineral
blue" (from copper) and "chrome-yellow." Other blue and yellow pigments gave
curious results, but it was more difficult to make the mixtures, and the greens
were less uniform in tint. The mixtures of these colours were made by weight,
and were painted on discs of paper, which were afterwards treated in the manner
described in my paper " On Colour as perceived by the Eye," in the Transactions
of the Boyal Soi.'icti/ of Edinburgh, Vol. xxi. Part 2. The \'isible effect of the
colour is estimated in terms of the standard-coloured papers : — vermilion (V),
ultramarine (U), and emerald-green (E). The accmucy of the results, and their
sijjnificance, can be best understood by referring to the paper before mentioned.
I shall denote mineral blue by B, and chrome-yellow by Y ; and B, Y, means
a mixture of three parts blue and five parts yellow.

Given Colour. Standard Colours. Coefficient

V. U. E. of brightness.

B, , 100 = 2 36 7 45

B- Y, , 100 = 1 18 17 37

B. Y, , 100 = 4 11 34 49

B, Y, , 100 =9 5 40 54

B, Y. , 100 = 15 1 40 56

B, Y, , 100 = 22 - 2 44 64

B, Y. , 100 = 35-10 51 76

B, Y, , 100 = 64-19 64 109

Y, , 100 = 180 -27 124 277

The columns Y, U, E give the proportions of the standard colours which
are equivalent, to 100 of the given colour; and the sum of V, U, E gives a co-
efficient, which gives a general idea of the brightness. It will be seen that the
tirst admixture of yellow diminishes the brightness of the blue. The negative
vidues of U indicate that a mixture of Y, U, and E cannot be made equivalent
to the given colour. The experiments from which these results were taken had



ON THE THEORY OF COMPOUND COLOURS. 245

the negative values tran-sferred to the other side of the equation. They were
all made by means of the colour-top, and were verified by repetition at different
times. It may be necessary to remark, in conclusion, with reference to the mode
of registering visible colours in terms of three arbitrary standard colours, that it
proceeds upon that theory of three primary elements in the sensation of colour,
which treats the investigation of the laws of visible colour as a bmnch of human
physiology, incapable of being deduced from the laws of light itself, as set forth
in physical optics. It takes advantage of the methods of optics to study vision
itself; and its appeal is not to physical principles, but to our consciousness of
our own sensations.



[From the Report of ike British Association, 1856.]



XIV. On an Instrument to illxLstrate Poinsdt's Theory of Rotation.

In studying the rotation of a solid body according to Poinsdt's method, we
have to consider the successive positions of the instantaneous axis of rotation
with reference both to directions fixed in space and axes assumed in the moving
body. The paths traced out by the pole of this axis on the invariable plane and
on the central ellipsoid form interesting subjects of mathematical investigation.
But when we attempt to follow with our eye the motion of a rotating body,
we find it difficult to determine through what point of the body the instantaneous
axis passes at any time, — and to determine its path must be still more difficult.
I have endeavoured to render visible the path of the instantaneous axis, and to
vary the circumstances of motion, by means of a top of the same kind as that
used by Mr Elliot, to illustrate precession^'. The body of the instrument is a
hoUow cone of wood, rising from a ring, 7 inches in diameter and 1 inch thick.
An iron axis, 8 inches long, screws into the vertex of the cone. The lower
extremity has a point of hard steel, which rests in an agate cup, and forms the
support of the instrument. An iron nut, three ounces in weight, is made to
screw on the axis, and to be fixed at any point; and in the wooden ring are
screwed four bolts, of three ounces, working horizontally, and four bolts, of one
ounce, working vertically. On the upper part of the axis is placed a disc of
card, on which are drawn four concentric rings. Each ring is divided into four
quadrants, which are coloured red, yellow, green, and blue. The spaces between
the rings are white. When the top is in motion, it is easy to see in which quad-
rant the instantaneous axis is at any moment and the distance between it and
the axis of the instrument; and we observe, — 1st. That the instantaneous axis
travels in a closed curve, and returns to its original position in the body. 2ndly.

* Transactions of the Royal Scottish Society of Arts, 1855.



ON AN INSTRUMENT TO ILLUSTRATE POINSOT S THEORY OF ROTATION. 247

That by working the vertical bolts, we can make the axis of the instrument
the centre of this closed curve. It will then be one of the principal axes of
inertia. 3rdly. That, by working the nut on the axis, we can make the order
of colours either red, yellow, green, blue, or the reverse. When the order of
colours is in the same direction as the rotation, it indicates that the axis of the
instrument is that of greatest moment of inertia. 4thly. That if we screw the
two pairs of opposite horizontal bolts to different distances from the axis, the
path of the instantaneous pole will no longer be equidistant from the axis, but
will describe an ellipse, whose longer axis is in the direction of the mean axis
of the instrument. 5thly. That if we now make one of the two horizontal axes
less and the other greater than the vertical axis, the instantaneous pole will
separate from the axis of the instrument, and the axis will incline more and more
till the spinning can no longer go on, on account of the obliquity. It is easy
to see that, by attending to the laws of motion, we may produce any of the
above effects at pleasure, and illustrate many different propositions by means of
the same instrument.



[From the Transactions of the Royal Society of Edinburgh, Vol. xxi. Part iv.]



XV. On a Dynamical Top, for exhibiting the phenomena of the motion of a
system of invariable form about a fixed point, with some suggestions as to
the Earth's mx)tion.

(Read 20th April, 1857.)

To those who study the progress of exact science, the common spinning-top
is a symbol of the labours and the perplexities of men who had successfully
threaded the mazes of the planetary motions. The mathematicians of the last
age, searching through nature for problems worthy of their analysis, foimd in
this toy of their youth, ample occupation for their highest mathematical powers.

No illustration of astronomical precession can be devised more perfect than
that presented by a properly balanced top, but yet the motion of rotation has
intricacies far exceeding those of the theory of precession.

Accordingly, we find Euler and D'Alembert devoting their talent and their
patience to the estabhshment of the laws of the rotation of solid bodies.
Lagrange has incorporated his own analysis of the problem with his general
treatment of mechanics, and since his time M. Poins6t has brought the subject
under the power of a more searching analysis than that of the calculus, in
which ideas take the place of symbols, and intelligible propositions supersede
equations.

In the practical department of the subject, we must notice the rotatory
machine of Bohnenberger, and the nautical top of Troughton. In the first of
these instruments we have the model of the Gyroscope, by which Foucault has
been able to render visible the effects of the earth's rotation. The beautiful
experiments by which Mr J. EUiot has made the ideas of precession so familiar
to us are performed with a top, similar in some respects to Troughton's, though
not borrowed from his.



ON A DYNAMICAL TOP. 249

The top which I have the honour to spin before the Society, differs from
that of Mr Elliot in having more adjustments, and in being designed to exhibit
far more complicated phenomena.

The arrangement of these adjustments, so as to produce the desired effects,
depends on the mathematical theory of rotation. The method of exhibiting the
motion of the axis of rotation, by means of a coloured disc, is essential to the
success of these adjustments. This optical contrivance for rendering visible the
nature of the rapid motion of the top, and the practical methods of applying
the theory of rotation to such an instrument as the one before us, are the
grounds on which I bring my instrument and experiments before the Society
as my own.

I propose, therefore, in the first place, to give a brief outline of such parts
of the theory of rotation as are necessary for the explanation of the phenomena
of the top.

I shall then describe the instrument with its adjustments, and the effect of
each, the mode of observing of the coloured disc when the top is in motion, and
the use of the top in illustrating the mathematical theory, with the method of
making the different experiments.

Lastly, I shall attempt to explain the nature of a possible variation in the
earth's axis due to its figure. This variation, if it exists, must cause a periodic
inequality in the latitude of every place on the earth's surface, going through its
period in about eleven months. The amount of variation must be very small,
but its character gives it importance, and the necessary observations are already
made, and only require reduction.



On the Tlieory of Rotation.

The theory of the rotation of a rigid system is strictly deduced from the
elementary laws of motion, but the complexity of the motion of the particles of
a body freely rotating renders the subject so intricate, that it has never been
thoroughly understood by any but the most expert mathematicians. Many who
have mastered the lunar theory have come to erroneous conclusions on this sub-
ject ; and even Newton haa chosen to deduce the disturbance of the earth's axis
from his theory of the motion of the nodes of a free orbit, rather than attack
the problem of the rotation of a solid body.



250 ON A DYNAMICAL TOP.

The method by which M. Poinsot has rendered the theory more manageable,
is by the liberal introduction of "appropriate ideas," chiefly of a geometrical
character, most of which had been rendered familiar to mathematicians by the
writings of Monge, but which then first became illustrations of this branch of
dynamics. If any further progress is to be made in simplifying and arranging
the theory, it must be by the method which Poins6t has repeatedly pointed out
as the only one which can lead to a true knowledge of the subject, — that of
proceeding from one distinct idea to another, instead of trusting to symbols and
equations.

An important contribution to our stock of appropriate ideas and methods has
lately been made by Mr R. B. Hayward, in a paper, "On a Direct Method of
estimatmg Velocities, Accelerations, and all similar quantities, with respect to axes,
moveable in any manner in Space." {Trans. Cambridge Phil. Soc. Vol. x. Part i.)
* In this communication I intend to confine myself to that part of the
subject which the top is intended to illustrate, namely, the alteration of the
position of the axis in a body rotating freely about its centre of gravity. I
shall, therefore, deduce the theory as briefly as possible, from two considera-
tions only, — the permanence of the original angular momentum in direction and
magnitude, and the permanence of the original vis viva.

•"' The mathematical difiSculties of the theory of rotation arise chiefly from
the want of geometrical illustrations and sensible images, by which we might
fix the results of analysis in our minds.

It is easy to understand the motion of a body revolving about a fixed axle.
Every point in the body describes a circle about the axis, and returns to its
original position after each complete revolution. But if the axle itself be in
motion, the paths of the different points of the body will no longer be circular
or re-entrant. Even the velocity of rotation about the axis requires a careful
definition, and the proposition that, in all motion about a fixed point, there is
always one Hne of particles forming an instantaneous axis, is usually given in
the form of a very repulsive mass of calculation. Most of these difficulties may
be got rid of by devoting a little attention to the mechanics and geometry of
the problem before entering on the discussion of the equations.

Mr Hayward, in his paper already referred to, has made great use of the
mechanical conception of Angular Momentum.

* 7th May, 1857. The paragraphs marked thus have been rewritten since the paper was read.



ON A DYNAMICAL TOP. 251

Definition. — Jlie Angular Momentum of a particle about an axis is mea-
sured by the product of the mass of the particle, its velocity resolved in the normal
plane, and the perpendicular from the axis on the direction of motion.

^' The angular momentum of any system about an axis is the algebraical
sum of the angular momenta of its parts.

As the rate of change of the linear momentum of a particle measures the
moving force which acts on it, so the rate of change of angular momentum
measures the moment of that force about an axis.

All actions between the parts of a system, being pairs of equal and opposite
forces, produce equal and opposite changes in the angular momentum of those
parts. Hence the whole angular momentum of the system is not aflfected by
these actions and re-actions.

* When a system of invariable form revolves about an axis, the angular
velocity of every part is the same, and the angular momentum about the axis is
the product of the angular velocity and the moment of inertia about that axis.

* It is only in particular cases, however, that the whole angular momentum
can be estimated in this way. In general, the axis of angular momentum differs
from the axis of rotation, so that there will be a residual angular momentum
about an axis perpendicular to that of rotation, imless that axis has one of three
positions, called the principal axes of the body.

By referring everything to these three axes, the theory is greatly simplified.
The moment of inertia about one of these axes is greater than that about any
other axis through the same point, and that about one of the others is a mini-
mum. These two are at right angles, and the third axis is perpendicular to
their plane, and is called the mean axis.

* Let A, B, C be the moments of inertia about the principal axes through
the centre of gravity, taken in order of magnitude, and let Wj oj., cd^ be the
angular velocities about them, then the angular momenta wHl be Ao)„ Bco.
and Cwj .

Angular momenta may be compounded like forces or velocities, by the
law of the "parallelogram," and since these three are at right angles to each
other, their resultant is

JA^:^JTB%JTC^' = H (1),

and this must be constant, both in magnitude and direction in space, since no
external forces act on the body.



252 ON A DYNAMICAL TOP.

We shall call this axis of angular momentum the invariable axis. It is

perpendicular to what has been called the invariable plane. Poins6t calls it

the axis of the couple of impulsion. The direction-cosines of this axis in the

body are,

, A(o, B(o. Ca)o

« = ^, m = -^, ^ = ^-

Since I, m and n vary during the motion, we need some additional
condition to determine the relation between them. We find this in the property
of the vis viva of a system of invariable form in which there is no friction.
The vis viva of such a system must be constant. We express this in the
equation

Aoj,' + B(o,'+C(o,'=V (2).

Substituting the values of Wi, w^, Wj in terms of I, m, n,



Let -i=a\ -T, = h\ ^=c\



= e'



A ' B ' C~ ' W
and this equation becomes

a'Z' + 6W + cV = e» (3),

and the equation to the cone, described by the invariable axis within the
body, is

(a'-e')x' + {h'-e')y'-\-{c'-e')z' = (4).

The intersections of this cone with planes perpendicular to the principal
axes are found by putting x, y, or z, constant in this equation. By giving
e various values, all the different paths of the pole of the invariable axis,
corresponding to different initial circumstances, may be traced.

*In the figiu-es, I have supposed a' = 100, 6'= 107, and c" = 110. The
first figure represents a section of the various cones by a plane perpendicular
to the axis of x, which is that of greatest moment of inertia. These sections



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