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y = BsiD.(nt-mz + a) J

(n'-m'a')A=m^nc'Bl

(n'-m'b')B = 7n'nc'Aj ^ ^'

Phil. Mag. May 1861 [p. 481 of this vol.].



ON PHYSICAL LINES OF FORCE.



511



Multiplying the last two equations together, we find

{)f-m-'cr)(n'-m'h') = m*n'c* (151)

an equation quadratic with respect to 771*, the solution of which is

2n'



a' + h'TJia'-hJ + An'c*



(152).



These values of m" being put in the equations (150) will each give a ratio
of A and B,

A ^ d'-h'T J{(r - b'Y + 4nrc*
B ~ 2nc'

which being substituted in equations (149), will satisfy the original equations
(148). The most general undulation of such a medium is therefore compounded
of two elliptic undulations of different eccentricities travelling with different
velocities and rotating in opposite directions. The results may be more easily
explained in the case in which a = & ; then



m' = — ip — : and A = TB (153).

Let us suppose that the value of A is unity for both vibrations, then we
shall have



X = cos nt



y-



nz \ I nz \ '

— , + cos nt — .

sfcf^cV \ Ja' + n&l

[nt— , \+sm.(nt- , - ^ ]

\ sIce-ncV \ -Ja' + iic-JJ



(154).



The first terms of x and y represent a circular vibration in the negative
direction, and the second term a circular vibration in the positive direction,
the positive having the greatest velocity of propagation. Combining the terms,
we may write

x = 2 cos {nt—jyz) cos qz'] d")

'J



where
and



y = 2 cos (nt —pz) sin qzj
n n



2 Ja^ — nc-
n



+



2Ja'-\-nc'
n



"i-^Ja^



nc-



2^a' +



.(156).



512 ON PHYSICAL LINES OF FORCE.

These are the equations of an undulation consisting of a plane vibration
whose periodic time is — , and wave-length — = X, propagated in the direction

of % with a velocity ~ = v, while the plane of the vibration revolves about the

axis of z in the positive direction so as to complete a revolution when z = — .
Now let us suppose c^ small, then we may write

^=5^<i5=S ('")'

1 T

and remembering that c' = t — v-y, we find

«=i;-^. <-«)•

Here r is the radius of the vortices, an unknown quantity, p is the density
of the luminiferous medium in the body, which is also unknown ; but if we
adopt the theory of Fresnel, and make s the density in space devoid of gross

matter, then

p = si^ (159),

where i is the index of refraction.

On the theory of MacCullagh and Neumann,

p = s (160)

ill all bodies.

/x is the coefficient of magnetic induction, which is unity in empty space
or in air.

y is the velocity of the vortices at their circumference estimated in the
ordinary units. Its value is unknown, but it is proportional to the intensity of
the magnetic force.

Let Z be the magnetic intensity of the field, measured as in the case of
terrestrial magnetism, then the intrinsic energy in air per unit of volume is

1 ^„ 1

where 5 is the density of the magnetic medium in air, which we have reason
to believe the same as that of the luminiferous medium. We therefore put

y=^^ (161),



ON PHYSICAL LINES OF FORCE. 513

X is the wave-length of the undulation in the substance. Now if A be the
wave-length for the same ray in air, and i the index of refraction of that ray in
the body,

>^ = | (162).

Also V, the velocity of light in the substance, is related to F, the velocity of
light in air, by the equation

V
^ = J (163).

Hence if z be the thickness of the substance through which the ray passes, the
angle through which the plane of polarization will be turned will be in degrees,

^ = ^q^ (164);

or, by what we have now calculated,

'='^7-^-y^ i^'^y

In this expression all the quantities are known by experiment except r, the
radius of the vortices in the body, and s, the density of the luminiferous
medium in air.

The experiments of M. Verdet* supply all that is wanted except the deter-
mination of Z in absolute measure ; and this would also be known for all his
experiments, if the value of the galvanometer deflection for a semi-rotation of
the testing bobbin in a known magnetic field, such as that due to terrestrial
magnetism at Paris, were once for all determined.

* Annates de Chimie et de Physique, ser. 3, Vol. xli. p. 370.



VOL. I. 65



[From the London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science.
Vol. XXVII. Fourth Series.]



XXIV. On Reciprocal Figures and Diagrams of Forces.

Reciprocal figures are such that the properties of the first relative to the
second are the same as those of the second relative to the first. Thus inverse
figures and polar reciprocals are instances of two difierent kinds of reciprocity.

The kind of reciprocity which we have here to do with has reference to
figures consisting of straight lines joining a system of points, and forming
closed rectilinear figures; and it consists in the directions of all lines in the
one figure having a constant relation to those of the lines in the other figure
which correspond to them.

In plane figures, corresponding lines may be either parallel, perpendicular,
or at any constant angle. Lines meeting in a point in one figure form a
closed polygon in the other.

In figures in space, the lines in one figure are perpendicular to planes in
the other, and the planes corresponding to lines which meet in a point form
a closed polyhedron.

The conditions of reciprocity may be considered from a purely geometrical
point of view; but their chief importance arises from the fact that either of
the figures being considered as a system of points acted on by forces along
the lines of connexion, the other figure is a diagram of forces, in which these
forces are represented in plane figures by lines, and in solid figures by the
areas of planes.

The properties of the "triangle" and "polygon" of forces have been long
known, and the "diagram" of forces has been used in the case of the funicular
polygon; but I am not aware of any more general statement of the method



ON RECIPROCAL FIGURES AND DIAGRAMS OF FORCES. 515

of drawing diagrams of forces before Professor Rankine applied it to frames,
roofs, &c. in his Applied Mechanics, p. 137, &c. The "polyhedron of forces,"
or the equilibrium of forces perpendicular and proportional to the areas of the
faces of a polyhedron, has, I believe, been enunciated independently at various
times; but the application to a "frame" is given by Professor Rankine in the
Philosophical Magazine, February, 1864.

I propose to treat the question geometrically, as reciprocal figures are
subject to certain conditions besides those belonging to diagrams of forces.



On Reciprocal Plane Figures.

Definition.— Tyfo plane figures are reciprocal when they consist of an equal
number of lines, so that corresponding lines in the two figures are parallel,
and corresponding lines which converge to a point in one figure form a closed
polygon in the other.

Note. — If corresponding lines in the two figures, instead of being parallel
are at right angles or any other angle, they may be made parallel by turning
one of the figures round in its own plane.

Since every polygon in one figure has three or more sides, every point in
the other figure must have three or more lines converging to it; and since
every line in the one figure has two and only two extremities to which lines
converge, every line in the other figure must belong to two, and only two
closed polygons. The simplest plane figure fulfilling these conditions is that
formed by the six lines which join four points in pairs. The reciprocal figure
consists of six lines parallel respectively to these, the points in the one figure
corresponding to triangles in the other.

General Relation between the Numbers of Points, Lines, and Polygons in

Reciprocal Figures.

The effect of drawing a line, one of whose extremities is a point connected
with the system of lines already drawn, is either to introduce one new point
into the system, or to complete one new polygon, or to divide a polygon into
two parts, according as it is drawn to an isolated point, or a point already
connected with the system. Hence the sum of points and polygons in the

65—2



516 ON RECIPROCAL FIGURES AND DIAGRAMS OF FORCES.

system is increased by one for every new line. But the simplest figure consists
of four points, four polygons, and six lines. Hence the sum of the points and
polygons must always exceed the number of lines by two.

JSfote. — This is the same relation which connects the numbers of summits,
faces, and edges of polyhedra.

Conditions of indeterminateness and impossibility in drawing reciprocal Diagrams.

Taking any line parallel to one of the lines of the figure for a base,
every new point is to be determined by the intersection of two new lines.
Calling s the number of points or summits, e the number of lines or edges,
and / the number of polygons or faces, the assumption of the first line deter-
mines two points, and the remaining s — 2 points are determined by 2 (s — 2)

lines. Hence if

e = 2s-3,

every point may be determined. If e be less, the form of the figure will be
in some respects indeterminate; and if e be greater, the construction of the
figure will be impossible, unless certain conditions among the directions of the
lines are fulfilled.

These are the conditions of drawing any diagram in which the directions
of the lines are arbitrarily given; but when one diagram is already drawn in
which e is greater than 2s — S, the directions of the lines will not be altogether
arbitrary, but will be subject to e-(2s-3) conditions.

Now if e, s', f be the values of e, s, and / in the reciprocal diagram

e = s+/-2, e' = s'+/'-2.
Hence if s =/, e = 2z - 2 ; and there will be one condition connecting the
directions of the lines of the original diagram, and this condition will ensure
the possibility of constructing the reciprocal diagram. If
s >/, e > 2s - 2, and e' < 2s' - 2 ;
so that the construction of the reciprocal diagram will be possible, but inde-
terminate to the extent of s —f variables.

If s<f, the construction of the reciprocal diagram will be impossible unless
(s-f) conditions be fulfilled in the original diagram.



ON RECIPROCAL FIGURES AND DIAGRAMS OF FORCES.



517



Fig. 1.



If any number of the points of the figure are so connected among them-
selves aa to form an equal number of closed polygons, the conditions of
constructing the reciprocal figure must be found by considering these points
separately, and then examining their connexion with the rest.

Let us now consider a few cases of reciprocal figures in detail. The
simplest case is that of the figure formed by the six lines connecting four
points in a plane. If we now draw the six lines con-
necting the centres of the four circles which pass through
three out of the four points, we shall have a reciprocal
figure, the corresponding lines in the two figures being
at right angles.

The reciprocal figure formed in this way is definite
in size and position ; but any figure similar to it and
placed in any position is still reciprocal to the original
figure. If the reciprocal figures are lettered as in fig. 1,
we shall have the relation

AP^BQ^CR
ap bq cr

In figures 2 and IL we have a pair of reciprocal figures in which the
lines are more numerous, but the construction very easy. There are seven
points in each figure corresponding to seven polygons in the other.

Fig. 2. Fig. II.






The four points of triple concourse of lines ABC, BDE, II I L, LJK
correspond to four triangles, ahc, bde, Ml, Ijk.

The three points of quadruple concourse ADFH, CEGK, IFGJ correspond
to three quadrilaterals, adfh, cegk, ifgj.

The five triangles ADB, BBC, GJK, IJL, HIF correspond to five point.s
of triple concourse, adb, ebc, gjk, ijl, hif.



518



ON RECIPROCAL FIGURES AND DIAGRAMS OF FORCES.



The quadrilateral DEGF corresponds to the point of quadruple concourse
degf.

The pentagon ACKLH corresponds to the meeting of the five lines cvcklh.

In drawing the reciprocal of fig. 2, it is best to begin with a point of triple
concourse. The reciprocal triangle of this point being drawn, determines three
lines of the new figure. If the other extremities of any of the lines meeting
in this point are points of triple concourse, we may in the same way deter-
mine more lines, two at a time. In drawing these lines, we have only to
remember that those lines which in the first figure form a polygon, start from
one point in the reciprocal figure. In this way we may proceed as long as
we can always determine all the lines except two of each successive polygon.

The case represented in figs. 3 and III. is an instance of a pair of reci-
procal figures fulfilling the conditions of possibility and determinateness, but

Fig. III.



Fig. 3.





presenting a slight difficulty in drawing by the foregoing rule. Each figure has
here eight points and eight polygons; but after we have drawn the lines s,
n, 0, k r, we cannot proceed with the figure simply by drawing the last two
lines of polygons, because the next polygons to be drawn are quadrilaterals, and
we have only one side of each given. The easiest way to proceed is to produce
ahcd till they form a quadrilateral, then to draw a subsidiary figure similar to
tlmpq, with abed similarly situated, and then to reduce the latter figure to
such a scale and position that a, h, c, d coincide in both figures.



ON RECIPROCAL FIGURES AND DIAGRAMS OF FORCES.



519



In figures 4 and IV. the condition that the number of polygons is equal
to the number of points is not fulfilled. In fig. 4 there are five points and

Fig. IV.




six triangles; in fig. IV. there are six points, two triangles, and three quadri-
laterals. Hence if fig. 4 is given, fig. IV. is indeterminate to the extent of one
variable, besides the elements of scale and position. In fact when we have drawn
ABC and indicated the directions of P, Q, R, we may fix on any point of P
as one of the angles of XYZ and complete the triangle XYZ. The size of
XYZ is therefore indeterminate. Conversely, if fig. IV. is given, fig. 4 cannot
be constructed unless one condition be fulfilled. That condition is that P, Q,
and R meet in a point. When this is fulfilled, it follows by geometry that
the points of concourse of A and X, B and Y, and C and Z He in one straight
line W, which is parallel to w in fig. 4. The condition may also be expressed
by saying that fig. IV. must be a perspective projection of a polyhedi'on whose
quadrilateral faces are planes. The planes of these faces intersect at the concourse
of P, Q, R, and those of the triangular faces intersect in the line W.

Figs. 5 and V. represent another case of the same kind. In fig. 5 we
have six points and eight triangles ; fig. V. is therefore capable of two degrees
of variability, and is subject to two conditions.



520



ON RECIPROCAL FIGURES AND DIAGRAMS OF FORCES.



The conditions are that the four intersections of corresponding sides of
opposite quadrilaterals in fig. V. shall lie in one straight line, parallel to the

Fig. V. Fig. 5.




line joining the opposite points of fig. 5 which correspond to these quadrilaterals.
There are three such lines marked a?, y, z, and four points of intersection lie on
each line.

"We may express this condition also by saying that fig. V. must be a per-
spective projection of a plane-sided polyhedron, the intersections of opposite
planes being the lines x, y, z.

In fig. 6, let ABODE be a portion of a polygon bounded by other polygons
of which the edges are PQRST, one or more of these edges meeting each angle
of the polygon.

In fig. VI., let ahcde be lines parallel to ABODE and meeting in a point,
and let these be terminated by the lines pqrst parallel to PQRST, one or
more of these lines completing each sector of fig. VI.

In fig. 6 draw Y through the intersections of ^C and PQ, and in fig.
VI, draw y through the intersections of a, p and c, q. Then the figures of
six lines ABOPQY and ahcpqy will be reciprocal, and y will be parallel to Y.
Draw X parallel to x, and through the intersections of TX and OE draw Z,
and in fig. VI. draw z through the intersections of ex and et ; then ODETXZ



ON RECIPROCAL FIGURES AND DIAGRAMS OF FORCES. 521

and cdetxz will be reciprocal, and Z will be parallel to z. Then through the
intersections oi AE and YZ draw W, and through those of ay and ez draw
w; and since ACEYZW and aceyzw are reciprocal, W will be parallel to w.

Fig. 6.

Fig. VI.





By going round the remaining sides of the polygon ABODE in the same
way, we should find by the intersections of lines another point, the line joining
which with the intersection of AE would be parallel to w, and therefore we
should have three points in one line; namely, the intersection of Y and Z,
the point determined by a similar process carried on on the other part of the
circumference of the polygon, and the intersection of A and E -, and we should
find similar conditions for every pair of sides of every polygon.

Now the conditions of the figure 6 being a perspective projection of a
plane-sided polyhedron are exactly the same. For A being the intersection of
the faces AP and AB, and C that of BC and QC, the intersection ylC will
be a point in the intersection of the faces AP and CQ.

Similarly the intersection PQ will be another point in it, so that Y is the
line of intersection of the faces AP and CQ.

In the same way Z is the intersection of ET and CQ, so that the inter-
section of Y and Z is a point in the intersection of AP and ET.

Another such point can be determined by going round the remaining sides
of the polygon; and these two points, together with the intersections of the
lines AE, must aU be in one straight line, namely, the intersection of the faces
AP and ET.

Hence the conditions of the possibility of reciprocity in plane figures are
the same as those of each figure being the perspective projection of a plane-
sided polyhedron. When the number of points is in every part of the figure
equal to or less than the number of polygons, this condition is ftilfilled of
itself. When the number of points exceeds the number of polygons, there will

VOL. I. 66



522 ON RECIPROCAL FIGURES AND DIAGRAMS OF FORCES.

be an impossible case, unless certain conditions are fulfilled so that certain sets
of intersections lie in straight lines.

Application to Statics.

The doctrine of reciprocal figures may be treated in a purely geometrical
manner, but it may be much more clearly understood by considering it as a
method of calculating the forces among a system of points in equilibrium ; for,

If forces represented in magnitude by the lines of a figure be made to act
between the extremities of the corresponding lines of the reciprocal figure, then
the points of the reciprocal figure will all be in equilibrium under the action
of these forces.

For the forces which meet in any point are parallel and proportional to
the sides of a polygon in the other figure.

If the points between which the forces are to act are known, the problem
of determining the relations among the magnitudes of the forces so as to produce
equilibrium wiU be indeterminate, determinate, or impossible, according as the
construction of the reciprocal figure is so.

Reciprocal figures are mechanically reciprocal; that is,- either may be taken
as representing a system of points, and the other as representing the magnitudes
of the forces acting between them.

In figures like 1, 2 and II., 3 and III., in which the equation

e = 2s-2
is true, the forces are determinate in their ratios; so that one being given,
the rest may be found.

When e>2.s-2, as in figs. 4 and 5, the forces are indeterminate, so that
more than one must be known to determine the rest, or else certain relations
among them must be given, such as those arising firom the elasticity of the
parts of a frame.

When e<2s-2, the determination of the forces is impossible except under
certain conditions. Unless these be fulfilled, as in figs. IV. and V., no forces
along the lines of the figure can keep its points in equilibrium, and the figure,
considered as a frame, may be said to be loose.

When the conditions are fulfilled, the pieces of the frame can support forces,
but in such a way that a small disfigurement of the frame may produce in-



ON RECIPROCAL FIGURES AND DIAGRAMS OF FORCES.



523



finitely great forces in some of the pieces, or may throw the frame into a loose
condition at once.

The conditions, however, of the possibility of determining the ratios of the
forces in a frame are not coextensive with those of finding a figure perfectly
reciprocal to the frame. The condition of determinate forces is

e = 2s - 2 ;
the condition of reciprocal figures is that every line belongs to two polygons

only, and

e = s+f-2.

In fig. 7 we have six points connected by ten lines in such a way that
the forces are all determinate ; but since the line Z is a side of three triangles,
we cannot draw a reciprocal figure, for we should have to draw a straight line
I with three ends.

If we attempt to draw the reciprocal figure as in fig. VII., we shall find
that, in order to represent the reciprocals of all the lines of fig. 7 and fi-x
their relations, we must repeat two of them, as h and e by h' and e, so as
to form a parallelogram. Fig. VII. is then a complete representation of the rela-
tions of the force which would produce equilibrium in fig. 7 ; but it is redundant
by the repetition of h and e, and the two figures are not reciprocal.

Fig. VII.





On Reciprocal Figures in three dimensions.

Definition. — Figures in three dimensions are reciprocal when they can be so
placed that every line in the one figure is perpendicular to a plane face of the
other, and every point of concourse of lines in the one figure is represented by
a closed polyhedron with plane faces.

66—2



524 ON RECIPROCAL FIGURES AND DIAGRAMS OF FORCES.

The simplest case is that of five points in space with their ten connecting
lines, forming ten triangular faces enclosing five tetrahedrons. By joining the five
points which are the centres of the spheres circumscribing these five tetrahedrons,
we have a reciprocal figure of the kind described by Professor Rankine in the
Philosophical Magazine, February 1864; and forces proportional to the areas of
the triangles of one figure, if applied along the corresponding lines of connexion
of the other figure, will keep its points in equilibrium.

In order to have perfect reciprocity between two figures, each figure must
be made up of a number of closed polyhedra having plane faces of separation,
and such that each face belongs to two and only two polyhedra, corresponding
to the extremities of the reciprocal line in the other figure. Every line in the
figure is the intersection of three or more plane faces, because the plane face in
the reciprocal figure is bounded by three or more straight lines.

Let s be the number of points or summits, e the number of lines or edges,
/ the number of faces, and c the number of polyhedra or cells. Then if about
one of the summits in which polyhedra meet, and a edges and -q faces, we
describe a polyhedral cell, it will have ^ faces and cf summits and -q edges,

and we shall have

i7 = <^ + o-^2 ;

s, the number of summits, will be decreased by one and increased by <t\

c, the number of cells, wiU be increased by one ;

/, the number of faces, wiU be increased by <^ ;

e, the number of edges, will be increased by -q;
so that c + c-(s4-/) will be increased by 77+ 1 -(cr + <^- 1), which is zero, or
this quantity is constant. Now in the figure of five points already discussed,
e = 10, c = 5, s = 5, /= 10 ; so that generally

e + c=^s+f,
in figures made up of ceils in the way described.

The condition of a reciprocal figure being indeterminate, determinate, or im-
possible except in particular cases, is

e = 3s -5.

<

This condition is sufficient to determine the possibiHty of finding a system of
forces along the edges which will keep the summits in equilibrium ; but it is



ON RECIPROCAL FIGURES AND DIAGRAMS OF FORCES. 525

manifest that the mechanical problem may be solved, though the reciprocal figure
cannot be constructed owing to the condition of all the sides of a face lying
in a plane not being fulfilled, or owing to a face belonging to more than two
cells. Hence the mechanical interest of reciprocal figures in space rapidly
diminishes with their complexity.

Diagrams of forces in which the forces are represented by lines may be
always constructed in space as well as in a plane, but in general some of the



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