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The external magnetic effect will be that of a small magnet whose moment

is jx—i w/sin 6, with its direction along the axis of y, so that the magnetism of

the field would tend to turn it back to the axis of x*.

The existence of these currents will of course alter the distribution of
the electro-tonic functions, and so they will react on themselves. Let the
final result of this action be a system of currents about an axis in the plane
of xy inclined to the axis of x at an angle ^ and producing an external effect
equal to that of a magnet whose moment is FR^.

The magnetic inductive components within the shell are
/i sin ^ — 2/' cos ^ in x,
— 21' sm(f> in. y,
/i cos 6 in 2,

Each of these would produce its own system of currents when the sphere
is in motion, and these would give rise to new distributions of magnetism,
which, when the velocity is uniform, must be the same as the original distri-
bution,

(Ii sin 6 — 21' cos <l>) in x produces 2 t^— r ot {I^ sin 6 — 2 J' cos (f>) in y,

T

( — 2T sin <^) in y produces 2 , m (21' sin ^) in x ;

IiQoad in z produces no currents.

We must therefore have the following equations, since the state of the shell
is the same at every instant,

T

Lam 6- 2r cos <f) = /, sin ^ -^ — — y (o2T sin 6

T

- 2/ sin <^ = -— T oj (/, sin ^- 2r cos <^),

* The expression for p^ indicates a variable electric tension in the shell, so that cuirents might
be collected by wires touching it at the equator and poles.



ON FARADAY 8 LINES OF FORCE. 229

-hence cot <^ = - j w, / = ^ , 5-,^^ /i sin 6.



7-©"



To understand the meanmg of these expressions let us take a particular case.

Let the axis of the revolving shell be vertical, and let the revolution be
from north to west. Let / be the total intensity of the terrestrial magnetism,
and let the dip be d, then Ico3$ is the horizontal component in the direction
of magnetic north.

The result of the rotation is to produce currents in the shell about an

T

axis inclined at a small angle = tan"* ——rco to the south of magnetic west, and

the external effect of these currents is the same as that of a magnet whose
moment is

i , ^"^ i?7cos d.

The moment of the couple due to terrestrial magnetism tending to stop the
rotation is

2i7rk To)



2 24tTrkY + Tq}*



i?Pc08'^,



and the loss of work due to this in unit of time is
24:Trk T(o'



2 247r^?+Pa>'



i?P cos' d.



This loss of work is made up by an evolution of heat in the substance of
the shell, as is proved by a recent experiment of M. Foucault (see Coniptefi
Rendus, XLi. p. 450).



[From the Transacti&M of the Royal Scottish Society of Arts, VoL iv. Part rv.]



IX. Description of a New Form of the Platometer, an Instrument for
measuring the Areas of Plane Figures drawn on Paper*.



1. The measurement of the area of a plane figure on a map or plan is an
operation so frequently occurring in practice, that any method by which it may
be easily and quickly performed is deserving of attention. A very able expo-
sition of the principle of such instruments will be found in the article on
Planimeters in the Reports of the Juries of the Great Exhibition, 1851.

2. In considering the principle of instruments of this kind, it will be most
convenient to suppose the area of the figure measured by an imaginary straight
line, which, by moving parallel to itself, and at the same

time altering in length to suit the form of the area,
accurately sweeps it out.

Let AZ be a fixed vertical line, APQZ the boundary
of the area, and let a variable horizontal line move
parallel to itself firom A to Z, so as to have its extremi-
ties, P, M, in the curve and in the fixed straight line.
Now, suppose the horizontal line (which we shall caU the
generating line) to move from the position PM to QNy
MN being some small quantity, say one inch for distinct-
ness. During this movement, the generating line will
have swept out the narrow strip of the surface, PMNQ,
which exceeds the portion PMNp by the smaU triangle PQp,

But since MN, the breadth of the strip, is one inch, the strip will contain
as many square inches as PM is inches long; so that, when the generating




♦ Bead to the Society, 22nd Jan. 1855.



ON A NEW FORM OF THE PLATOMETER.



231




line descends one inch, it sweeps out a number of square inches equal to the
number of linear inches in its length.

Therefore, if we have a machine with an index of any kind, which, while
the generating line moves one inch downwards, moves forward as many degrees
as the generating line is inches long, and if the generating line be alternately
moved an inch and altered in length, the index will mark
the number of square inches swept over during the whole
operation. By the ordinary method of limits, it may be
shown that, if these changes be made continuous instead
of sudden, the index will still measure the area of the
curve traced by the extremity of the generating line.

3. When the area is bounded by a closed curve, as
ABDC, then to determine the area we must carry the tra-
cing point from some point A of the curve, completely round
the circumference to A again. Then, while the tracing point
moves from A to C, the index will go forward and mea-
sure the number of square inches in ACRP, and, while it
moves from C to D, the index will measure backwards the
square inches in CRPD, so that it will now indicate the

square inches in ACD. Similarly, during the other part of the motion from
D to B, and from B to D, the part DBA will be measured; so that when
the tracing point returns to D, the instrument will have measured the area
ACDB. It is evident that the whole area will appear positive or negative
according as the tracing point is carried round in the direction ACDB or ABDC.

4. We have next to consider the various methods of communicating the
required motion to the index. The first is by means of two discs, the first
having a flat horizontal rough surface, turning on a vertical
axis, OQ, and the second vertical, with its circumference rest-
ing on the flat surface of the first at P, so as to be driven
round by the motion of the first disc. The velocity of the
second disc will depend on OP, the distance of the point of
contact from the centre of the first disc; so that if OP be
made always equal to the generating line, the conditions of the instrument will
be fulfilled.

This is accomplished by causing the index-disc to slip along the radius of




232 ON A NEW FORM OF THE PLATOMETER.

the horizontal disc ; so that in working the instrument, the motion of the index-
disc is compounded of a rolling motion due to the rotation of the first disc,
and a slipping motion due to the variation of the generating line.

5. In the instrument presented by Mr Sang to the Society, the first disc is
replaced by a cone, and the action of the instrument corresponds to a mathe-
matical valuation of the area by the use of oblique co-ordinates. As he has
himself explained it very completely, it will be enough here to say, that the
index-wheel has still a motion of slipping as well as of rolling.

6. Now, suppose a wheel rolling on a surface, and pressing on it with a
weight of a pound; then suppose the coefficient of friction to be |, it will
require a force of 2 oz. at least to produce shpping at all, so that even if the
resistance of the axis, &c., amounted to 1 oz., the rolling would be perfect. But
if the wheel were forcibly pulled sideways, so as to slide along in the direction
of the axis, then, if the friction of the axis, &c., opposed no resistance to the
turning of the wheel, the rotation would still be that due to the forward motion ;
but if there were any resistance, however small, it would produce its effect in
diminishing the amount of rotation.

The case is that of a mass resting on a rough surface, which requires a
great force to produce the shghtest motion; but when some other force acts
on it and keeps it in motion, the very smallest force is sufficient to alter that
motion in direction.

7. This effect of the combination of slipping and rolling has not escaped
the observation of Mr Sang, who has both measured its amount, and shown how
to eliminate its effect. In the improved instrument as constructed by him, I
believe that the greatest error introduced in this way does not equal the ordi-
nary errors of measurement by the old process of triangulation. This accuracy,
however, is a proof of the excellence of the workmanship, and the smoothness
of the action of the instrument; for if any considerable resistance had to be
overcome, it would display itself in the results.

8. Having seen and admired these instruments at the Great Exhibition in
1851, and being convinced that the combination of shpping and roUing was a
drawback on the perfection of the instrument, I began to search for some
arrangement by which the motion should be that of perfect rolling in every



ON A NEW FORM OF TUE PLATOMETER. 233

motion of which the instrument is capable. The forms of the rolUng parts which
I considered were —

1. Two equal spheres.

2. Two spheres, the diameters being as 1 to 2.

3. A cone and cylinder, axes at right angles.

Of these, the first combination only suited my purpose. I devised several modes
of mounting the spheres so as to make the principle available. That which I
adopted is borrowed, as to many details, from the instruments already con-
structed, so that the originality of the device may be reduced to this principle —
The abolition of sUpping by the use of two equal spheres.

9. The instrument (Fig. 1) is mounted on a frame, which rolls on the two
connected wheels, MM, and is thus constrained to travel up and down the
paper, moving parallel to itself

CH is a horizontal axis, passing through two supports attached to the
frame, and carrying the wheel K and the hemisphere LAP. The wheel K rolls
on the plane on which the instrument travels, and communicates its motion to
the hemisphere, which therefore revolves about the axis AH with a velocity
proportional to that with which the instrument moves backwards or forwards.

FCO is a framework (better seen in the other figures) capable of revolving
about a vertical axis, Cc, being joined at C and c to the frame of the instru-
ment. The parts CF and CO are at right angles to each other and horizontal.
The part CO carries with it a ring, SOS, which turns about a vertical axis Oo.
This ring supports the index-.sphere Bh by the extremities of its axis Ss, just
as the meridian circle carries a terrestrial globe. By this arrangement, it will
be seen that the axis of the sphere is kept always horizontal, while its centre
moves so as to be always at a constant distance from that of the hemisphere.
This distance must be adjusted so that the spheres may always remain in con-
tact, and the pressure at the point of contact may be regulated by means of
springs or compresses at and o acting in the direction OC, oc. In this way
the rotation of the hemisphere is made to drive the index-sphere.

10. Now, let us consider the working of the instrument. Suppose the arm
CE placed so as to coincide with CD, then 0, the centre of the index-sphere
will be in the prolongation of the axis HA. Suppose also that, when in this
position, the equator hB of the index-sphere is in contact with the pole A of
the hemisphere. Now, let the arch be turned into the position CE as in the



234 ON A NEW FORM OF THE PLATOMETER.

figure, then the rest of the framework will be turned through an equal angle,
and the index-sphere will roll on the hemisphere till it come into the position
represented in the figure. Then, if there be no slipping, the arc AP = BP, and
the angle ACF = BOP.

Next, let the instrument be moved backwards or forwards, so as to turn
the wheel Kk and the hemisphere LI, then the index-sphere will be turned
about its axis Ss by the action of the hemisphere, but the ratio of their veloci-
ties will depend on their relative positions. If we draw PQ, PR, perpendiculars
from the point of contact on the two axes, then the angular motion of the
index-sphere will be to that of the hemisphere, as PQ is to PR; that is, as
PQ is to QC, by the equal triangles POQ, PQC ; that is, as ED is to DC,
by the similar triangles CQP, CDE.

Therefore the ratio of the angular velocities is as ED to DC, but since
DC is constant, this ratio varies as ED. We have now only to contrive some
way of making ED act as the generating line, and the machine is complete
(see art. 2).

11. The arm CF is moved in the following manner: — Tt is a rectangular
metal beam, fixed to the frame of the instrument, and parallel to the axis AH.
cEe is a little carriage which rolls along it, having two rollers on one side and
one on the other, which is pressed against the beam by a spring. This carriage
carries a vertical pin, E, turning in its socket, and having a collar above,
through which the arm CF works smoothly. The tracing point G is attached
to the carriage by a jointed frame eGe, which is so arranged that the point
may not bear too heavily on the paper.

12. When the machine is in action, the tracing point is placed on a point
in the boundary of the figure, and made to move round it always in one
direction till it arrives at the same point again. The up-and-down motion of
the tracing point moves the whole instrument over the paper, turns the wheel
K, the hemisphere LI, and the index-sphere Bh ; while the lateral motion of
the tracing point moves the carriage E on the beam Tt, and so works the arm
CF and the framework CO; and so changes the relative velocities of the two
spheres, as has been explained,

13. In this way the instrument works by a perfect rolling motion, in what-
ever direction the tracing point is moved; but since the accuracy of the result
depends on the equality of the arcs AP and BP, and since the smallest error



ON A NEW FORM OF THE PLATOMETER. 235

of adjustment would, in the course of time, produce a considerable deviation
from this equality, some contrivance is necessary to secure it. For this purpose
a wheel is fixed on the same axis with the ring SOs, and another of the same
size is fixed to the frame of the instrument, with its centre coinciding with the
vertical axis through C. These wheels are connected by two pieces of watch-
spring, which are arranged so as to apply closely to the edges of the wheels.
The first is firmly attached to the nearer side of the fixed wheel, and to the
farther side of the moveable wheel, and the second to the farther side of the
fixed wheel, and the nearer side of the moveable wheel, crossing beneath the
first steel band. In this way the spheres are maintained in their proper relative
position; but since no instrument can be perfect, the wheels, by preventing
dei-angement, must cause some slight slipping, depending on the errors of work-
manship. This, however, does not ruin the pretensions of the instrument, for it
may be shown that the error introduced by slipping depends on the distance
through which the lateral slipping takes place ; and since in this case it must
be very small compared with its necessarily large amount in the other instru-
ments, the error introduced by it must be diminished in the same proportion.

14. I have shewn how the rotation of the index-sphere is proportional to
the area of the figure traced by the tracing point. This rotation must be
measured by means of a graduated circle attached to the sphere, and read oti"
by means of a vernier. The result, as measured in degrees, may be interpreted
in the following manner : —

Suppose the instrument to be placed with the arm CF coinciding with CD,
the equator Bh of the index-sphere touching the pole A of the hemisphere, and
the index of the vernier at zero : then let these four operations be performed : —

(1) Let the tracing point be moved to the right till DE = DC, and there-
fore DCE, ACP, and F0B = A5\

(2) Let the instrument be rolled upwards till the wheel K has made a
complete revolution, carrying the hemisphere with it ; then, on account of the
equality of the angles SOP, PC A, the index-sphere will also make a complete
revolution.

(3) Let the arm CF be brought back again till F coincides with D.

(4) Let the instrument be rolled back again through a complete revolution
of the wheel K. The index-sphere will not rotate, because the point of contact
is at the pole of the hemisphere.



236 ON A NEW FORM OF THE PLATOMETER.

The tracing point has now traversed the boundary of a rectangle, whose
length is the circumference of the wheel A", and its breadth is equal to CD;
and during this operation, the index-sphere has made a complete revolution,
360" on the sphere, therefore, correspond to an area equal to the rectangle con-
tained by the circumference of the wheel and the distance CD. The size of
the wheel K being known, different values may be given to CD, so as to make
the instrument measure according to any required scale. This may be done,
either by shifting the position of the beam Tt, or by having several sockets
in the carriage E for the pin which directs the arm to work in.

15. If I have been too prolix in describing the action of an instrument
which has never been constructed, it is because I have myself derived great
satisfaction from following out the mechanical consequences of the mathematical
theorem on which the truth of this method depends. Among the other forms
of apparatus by which the action of the two spheres may be rendered available,
is one which might be found practicable in cases to which that here given
would not apply. In this instrument (Fig. 4) the areas are swept out by a
radius- vector of variable length, turning round a fixed point in the plane. The
area is thus swept out with a velocity varying as the angular velocity of the
radius-vector and the square of its length conjointly, and the construction of the
machine is adapted to the case as follows : —

The hemisphere is fixed on the top of a vertical pillar, about which the rest
of the instrument turns. The index-sphere is supported as before by a ring and
framework. This framework turns about the vertical pillar along with the tra-
cing point, but has also a motion in a vertical plane, which is communicated to
it by a curved slide connected with the tracing point, and which, by means of a
prolonged arm, moves the framework as the tracing point is moved to and from
the pillar.

The form of the curved slide is such, that the tangent of the angle of
inclination of the line joining the centres of the spheres with the vertical is
proportional to the square of the distance of the tracing point from the vertical
axis of the instrument. The curve which fulfils this condition is an hyperbola,
one of whose asymptotes is vertical, and passes through the tracing point, and
the other horizontal through the centre of the hemisphere.

The other parts of this instrument are identical with those belonging to
that alreadv described.



VOL. /. PLATE n.




Fig.iFlan



C %K




FigJF'runr EleuaCion



VOL. I. PLATE U.




Ti^.4.



ON A NEW FORM OF THE PLATOMETER. 237

When the tracing point is made to traverse the boundary of a plane figure,
there is a continued rotation of the radius-vector combined with a change of
length. The rotation causes the index-sphere to roll on the fixed hemisphere,
while the length of the radius-vector determines the rate of its motion about its
axis, so that its whole motion measures the area swept out by the radius-vector
during the motion of the tracing point.

The areas measured by this instrument may either lie on one side of the
pillar, or they may extend all round it. In either case the action of the
instrument is the same as in the ordinary case. In this form of the instrument
we have the advantages of a fixed stand, and a simple motion of the tracing
point; but there seem to be difficulties in the way of supporting the spheres
and arranging the shde ; and even then the instrument would require a tall
pillar, in order to take in a large area.

16. It will be observed that I have said little or nothing about the prac-
tical details of these instruments. Many useful hints will be found in the large
work on Platometers, by Professor T. Gonnellu, who has given us an account
of the difficulties, as well as the results, of the construction of his most
elaborate instrument. He has also given some very interesting investio-ations
into the errors produced by various irregularities of construction, although, as
far as I am aware, he has not even suspected the error which the sliding of
the index-wheel over the disc must necessarily introduce. With respect to this,
and other points relating to the working of the instrument, the memoir of
Mr Sang, in the Transactions of this Society, is the most complete that I
have met with. It may, however, be as well to state, that at the time when
I devised the improvements here suggested, I had not seen that paper, though
I had seen the instrument standing at rest in the Crystal Palace.

Edinburgh, 30th January, 1855,
Note. — Since the design of the above instrument was submitted to the Society of Arts,
I have met with a description of an instrument combining simplicity of construction with
the power of adaptation to designs of any size, and at the same time more portable than
any other instrument of the kind. Althougli it does not act by perfect rolling, and there-
fore belongs to a different class of instruments from that described in this paper, I think
that its simplicity, and the beauty of the principle on which it acts, render it worth the
attention of engineers and mechanists, whether practical or theoretical. A full account of
this instrument is to be found in Moigno's " Cosmos," 5th year, Vol. viii., Part viii., p. 213,
published 20th February 1856. Description et Theorie du planiniHre polaire, invents par
J. Amsler, de Schaffuuse en Suisse.
Cambridge, 30th April, 1856.



[From the Cambridge Philosophical Society Proceedings, Vol. i. pp. 173 — 175.]



X. 0?i the Elementary TJieory of Optical Instruments.

The object of this communication was to shew how the magnitude and
position of the image of any object seen through an optical instrument could
be ascertained without knowing the construction of the instrument, by means
of data derived from two experiments on the instrument. Optical questions
are generally treated of with respect to the pencils of rays which pass through
the instrument. A pencil is a collection of rays which have passed through one
point, and may again do so, by some optical contrivance. Now if we suppose
all the points of a plane luminous, each will give out a pencil of rays, and
that collection of pencils which passes through the instrument may be treated
as a beam of hght. In a pencil only one ray passes through any point of
space, unless that point be the focus. In a beam an infinite number of rays,
corresponding each to some point in the luminous plane, passes through any
point; and we may, if we choose, treat this collection of rays as a pencil
proceeding from that point. Hence the same beam of light may be decomposed
into pencils in an infinite variety of ways; and yet, since we regard it as the
same collection of rays, we may study its properties as a beam independently
of the particular way in which we conceive it analysed into pencils.

Now in any instrument the incident and emergent beams are composed
of the same light, and therefore every ray in the incident beam has a
corresponding ray in the emergent beam. We do not know their path within
the instrument, but before incidence and after emergence they are straight
lines, and therefore any two points serve to determine the direction of each.

Let us suppose the instrument such that it forms an accurate image of a
plane object in a given position. Then every ray which passes through a given



ON THE ELEMENTARY THEORY OF OPTICAL INSTRUMENTS. 239

point of the object before incidence passes through the corresponding point of
the image after emergence, and this determines one point of the emergent ray.
If at any other distance from the instrument a plane object has an accurate
image, then there will be two other corresponding points given in the incident
and emergent rays. Hence if we know the points in which an incident ray
meets the planes of the two objects, we may find the incident ray by joining
the points of the two images corresponding to them.

It was then shewn, that if the image of a plane object be distinct, flat, and
similar to the object for two different distances of the object, the image of any



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