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Physics

Robb, Alfred Arthur

geometry of motion;
a new view of the theory of
relativity.

OPTICAL GEOMETRY
OF MOTION

A NEW VIEW
OF THE THEORY OF RELATIVITY

ALFRED A. KOBE, M.A., Pn.D.

(ttamfcrftrge :

W. HEFFER AND SONS LTD.
: vSIMPKIN MARSHALL AND Co. LTD,
1911

OPTICAL GEOMETRY
OF MOTION

A NEW VIEW
OF THE THEORY OF RELATIVITY

ALFRED A, ROBB, M.A., Pn.D.

(ftamimfoge :

W. HEFFER AND SONS LTD.

Hotltrott : SIMPKIN MARSHALL AND Co. LTD.

1911

PREFACE.

IN placing before his readers the following brief outline
of a point of view, the writer is well aware that it is far
from complete in many respects. He however believes that,
in the first presentment of a theory, there are considerable
advantages in stating explicitly only its principal features.

To cover np a general standpoint under a mass of detail
is to run the risk of obscuring it altogether. There is always
the danger that the reader may " not be able to see the
wood for the trees " — a danger which is becoming very
real in much modern mathematics.

The substance of the following essay was originally
intended by the writer to form a chapter of a book of
semi-philosophical character upon which he is engaged.

In view, however, of the amount of attention which the
subject of Relativity is at present attracting, it seemed
to him that this portion might prove of sufficient interest
to warrant its separate publication.

From the standpoint of the pure mathematician Geometry
is a branch of formal logic, but there are more aspects of
things than one, and the geometrician has but to look at
the name of his science to be reminded that it had its origin
in a definite physical problem.

That problem in an extended form still retains its
interest.

A. A. ROBB.

Cambridge,

May 13th, 1911.

"Bat deepest of all illusory Appearances, for hiding
Wonder, as for many other ends, are your two grand
fundamental world-enveloping Appearances, SPACE and
TIME. These, as spun and woven for us from before Birth
itself to clothe our celestial ME for dwelling here, and yet
to blind it, — lie all-embracing, as the universal canvas,
or warp and woof, whereby all minor Illusions in this
Phantasm Existence weave and paint themselves. In vaiu,
while here on Earth shall you endeavour to strip them off;
you can, at best, but rend them asunder for moments,
and look through."

CARLYLE, " Sartor Resartus"

OPTICAL GEOMETRY OP MOTION.

Introduction.

foundations of Geometry have been carefully investi-
gated, especially of late years, by many eminent
mathematicians. These investigations have (with the
notable exception of those of Helmholtz) been almost all
directed towards the Logical aspects of the subject, while
tl\e Physical standpoint has received comparatively little
attention.

Speaking of the different " Geometries " which have
been devised, Poincare has gone so far as to say that :
" one Geometry cannot be more true than another ; it can
only be more convenient." In order to support this view it
is pointed out that it is possible to construct a sort of
dictionary by means of which we may pass from theorems
in Euclidian Geometry to corresponding theorems in the
Geometries of Lobatschefskij or Uiemaun.

In reply to this ; it must be remembered that the
language of Geometry has a certain fairly well defined
physical signification which in its essential features must
be preserved if we are to avoid confusion.

As regards the "dictionary," we would venture to add
that it would also be possible to construct one in which the
ordinary uses of the words black and white were inter-
changed, but, in spite of this, the substitution of the word
white for the word black is frequently taken as the very
type of a falsehood.

It is the contention of the writer that the axioms of
Geometry, with a few exceptions, may be regarded as the
formal expression of certain Optical facts. The exceptions
are a few axioms whose basis appears to be Logical rather
than Physical.

It is proposed in the following pages to refer briefly,
in the first place, to certain Optical phenomena which occur
in free space, upon which we might suppose some of the

chief axioms to Iiave their foundations : and then to employ
these to establish on a new basis some of the groundwork
of the theory of " Relativity."

The writer does not propose, in the present paper,
to go into the more minute Logical details of the found-
ations of Geometry ; as it seems to him that these would
tend to obscure the general standpoint which he desires
to emphasize. For this reason he prefers to reserve them
lor a future occasion.

OPTICAL GEOMETRY OF REST.

In the application of ordinary Geometry as distinguished
from Kinematics, we are concerned with systems which
preserve their configuration unchanged. We shall first
consider briefly such systems. Our sense of vision supplies
us with a direct means by which we can tell that a particle
in free space lies in the same straight line as two other
particles. If three particles A, JB, and C do not all lie in
the same straight line, we may also make use of our sense
of vision to determine when a fourth particle D lies in the
same plane as A, B, and C. The test is as follows :

We take a fifth particle E, and then, if D lies in the
plane of A, B, C, we may place E so that it is in the same
straight line as D and one of the three particles, say A,
and is also in the same straight line as the remaining
particles B and C. (In case AD and BC are parallel,
we must interchange either \A and B or A and C in order
to carry out the test,)

(Simple Optical interpretations of like nature may be
devised for various other Geometrical conceptions. As
regards the notions and axioms of congruence these may
be given a very simple interpretation by means of the
properties of plane mirrors, but from a theoretical stand-
point it appears better to regard congruence as based on
the finite propagation of light.

We shall regard it as an experimental fact that light
takes a finite interval of time in travelling from a particle
A to a particle B and back again.

Let us suppose that we have a particle A which is so
situated with respect to other particles B, C, D, &c., that
a flash of light being sent out simultaneously to B, C, D,
&c., and reflected from these back to A, returns to the latter
simultaneously from all the particles. We shall then define
the stretches AB, A C, AD, &c., as congruent or equal.

We shall define a right angle as follows :

Let D be a particle which lies in the same
straight Hue as two others A and B, and so
that the stretches DA and DB are equal.
Let G be another particle not lying in the
line AB, but such that the stretches CA and
CB are equal. Then we shall define the
angles CD A and CDB as right angles. As
regards other angles we shall define them as
congruent when their trigonometrical ratios
are equal. The method of determining these will be obvious
hereafter.

Since we have ascribed a meaning to the equality of
stretches, we know the meaning of an equilateral triangle.

Suppose now we have five partic.les
D E A, B, C, D, E, of which A, B, and C

lie in a straight line and so that the
stretches B A and BC are equal. Let
the other two particles be so situated
that ADB and BEC are equilateral
triangles, and, further, let them lie in one plane and on the
same side of the Hue AB.

The test for this is that we should be able to place
a sixth particle F so as to lie both in the line AE and
in the Hue CD.

We shall suppose then that observation shows our system
to be such that DBE is also an equilateral triangle.

This excludes the Geometries of Lobatschefskij and
Riemanu, since in these the angle of an equilateral triangle
is either less or more than one-third of two right angles.
We shall find an interpretation of Lobatschefskij 's Geometry
when we come to deal with motion.

It will be shown that a system of three particles diverging
uniformly, with equal relative rapidities, from simultaneous
contact may be regarded as the corners of a Lobatschefskij
triangle.

It thus appears that the fulfilment of our criterion of
a Euclidian system excludes the possibility of our system
of particles diverging from one another in this way —
a possibility which at first sight might appear to lie open.

Since we are not yet in a position to prove this, we
must ignore the seeming possibility until we have shown
that the set of diverging particles has this property, and
it may then be shown (by a process of reductio ad absurdum)
that the seeming possibility has already actually been
excluded.

There is another important restriction which we must
suppose placed on our system of permanent configuration
before we can go on to consider the motion of particles.

Let us consider any three particles A, B, C of our system,
which are not all in the same straight line, and suppose a
flash of light, starting out from one of the particles, say A,
goes thence to B, thence to C, and thence back to A.
Imagine a second flash, starting simultaneously with the
first, and going round in the opposite direction, namely,
from A to G, from C to B, and from B back to A.

Now, from the purely logical standpoint, three possi-
bilities are open :

(1) The first flash may arrive before the second.

(2) The second flash may arrive before the first.

(3) The two flashes may arrive simultaneously.

If, starting from any of the three particles, both flashes
return simultaneously, we regard the system as not rotating
in its own plane.

Thus " absolute rotation " acquires a definite meaning in
our system of Optical Geometry. We shall suppose this
test to be applied to the various sets of three particles which
may be selected from a group of four particles which do not
all lie in one plane. By doing this we ensure the absence
of rotation about any axis of our system of permanent con-
figuration.

Let us now suppose that any selected two of the particles
of the system are at one -half the unit distance apart. As
we have not yet properly defined distance, it would perhaps
be better to say that, having selected a suitable pair of
particles of the system, we define them to be at one-half
the unit distance apart. Having got such a system of
particles, we now proceed to make use of it.

"!NDEX" OF FUNDAMENTAL PAUTICLM AT ANY INSTANT.

We propose to introduce a conception which we shall
call the index of a particle at any instant. This is a number
which we shall define in a certain physical way as associated,
with a particle at an instant. We shall at first define it
only for one of our fundamental particles, leaving the general
definition till later.

Let us take then the particle A, and at a selected instant
suppose a fl?ish of light sent out to B, which is at one-half the
unit distance from it, and reflected from B back to A. We
shall say that the index of A at the instant of departure
is 0, while at the instant of return it is 1. We shall sup-
pose the flash returned once more at B and then back to A,

and shall say that the index of A is then 2. We shall define
the index of A at the nth return as n. We may obviously,
in a similar way, have negative values of the index.

We may interpolate indices to any desired extent by
employing other particles instead of B which are "nearer"
to A than is the latter. Thus, if we employed a particle
B' such that light went to and returned from it to A ten
times during the interval that light went to and returned
from B to A once, we could assign indices to A differing
by •!. Similarly it is theoretically possible to carry this
process out indefinitely and ultimately to treat the index
of -A as a continuum in the usual manner.

It will be seen that the index may be regarded as a
measure of time so long as we confine our attention to the
neighbourhood of the one particle A, but it appears desirable
in view of further developments not to identify index and
time, as the two conceptions seem to ruu to some extent
counter to one another.

DEFINITION OF THE INDEX FOR PARTICLES m GENERAL.

Suppose a flash of light to go out from the particle A to
some other particle P, which may be anywhere in space
and may be " in motion."

Suppose the index of A at the instant of departure be
denoted by Nd and called the index of departure.

Let the flash be reflected back from P to A and let the
index of A at the instant of return be denoted by Nr and
called the index of return.

We shall speak of the index of arrival of the flash at P,
meaning thereby the index of the particle P at the instant
of arrival of the light, and shall denote it by Na.

The index of arrival is defined by the following equation :

If we denote — ^- ± by u and — =— — ^ by t, we may

express Na more simply thus

f(*z\

V Uv

It will be found that the index, as thus defined, possesses
remarkable properties.

We shall consider first a system which is permanent
in configuration, and not rotating.

We shall define the distance of any particle P from A
as measured by

In our permanent system this quantity remains fixed for
each particle. If then we select any particle P, and let

/ V 4- N \s

we have NrNd = I ^ ") - **•

This gives

d(NrNd)

Thus the index of arrival in a non-rotating system of
permanent configuration is the arithmetic mean of the indices
of departure and return for light coming from the funda-
mental particle.

It is easy to see that this holds in general for such
a system, and that any particle of it may take the place
of the fundamental particle.

Ln dealing, however, with particles which are in relative
motion, we must fix our attention on one particle and regard
it throughout our investigations as the fundamental particle.

In particular, in the case of a rotating system, since
light takes different intervals in going round a system of
three particles in opposite directions, we should not get
a unique value for the index at a particular instant if we
were to vary our fundamental particle.

It is to be observed that the above result does not
necessarily imply that the instant of arrival is the same
as the instant at which the particle A has the index
N + N

^— — -. In fact, our definition of index is quite com-
patible with the light taking a different time in going from
that which it takes to return.*

UNIFORM MOTION OF PARTICLES.

In the consideration of particles in motion, we refer
them to systems of permaneut configuration which are
non-rotating.

If we send out a flash of light from some particle A
in such a system te a moving particle P, the light will
reach the latter when it has a definite position in the
system. If we imagine a particle B of the system to
occupy as nearly as possible this position, then B is at
a definite distance from the particle A, and this distance

N —N
is —L— — - . Also at the instant when the light reaches P,

which is also the instant at which its position coincides

N + N
with that of B, the particle B has the index — d .

Thus for each value of the index in the system of per-
manent configuration, the particle P occupies some definite
position in the latter. The index of P is. however, quite
different from that of B at the instant when they coincide.

It will be found of great assistance, when the motion
of the particle is confined to one plane, to take the two
co-ordinates x and y as representing the position of the
moving particle with respect to our system of permanent
configuration ; while the third co-ordinate z represents the
index of particles belonging to the system The aggregate
of positions of a moving particle will be represented by
some continuous line ; while the light going from one
particle to another is represented by a straight line making
an angle of 45° with the axis of z.

* This point is philosophically very important, since the theory of relativity,
as usually presented, involves the psychological difficulty of a "time" which is
not unique. Whether or no there may be some deep sense in which this is true,
1 am not prepared to say, nevertheless this difficulty is very likely to stand in the
way of a general acceptance of the theory. Jt rather seems to the writer that the
assumption of a unique time is intimately bound up with the logical principle
of non-contradiction, whereby a thing cannot both be and not be at the same time.
The conception of the index of a particle at an instant appears to avoid this
difficulty.

It is easily seen that this represent-
ation fits in with the result already
obtained that the index of arrival in a
system of permanent configuration which
is not rotating is equal to the arithmetic
mean of the indices of departure and
return. It also fits in with our definition
of the measure of the distance between

N — N
two particles in such a system as — - - -.

This is clear from the diagram.

It is', however, to be noted that distance
defined in this way is absolute distance,
and is always positive.

It is easily seen that all the straight lines through a
point (a, b, c) which make angles of 45° with the axis of
z lie on the cone

Nr-NA

We shall refer to this as the ^standard cone with respect
to the point (a, b, c).

If a particle P be in motion with respect to the system
of' permanent configuration, there is always a tangent to its
path at any instant. If C be any particle of the system
from which the direction of P appears stationary at any
instant, and if a flash of light goes out from C to P and
back, the absolute velocity of P with respect to C is defined

d (N — N )

d at the instant of arrival of the light.

to be

This may also be called the absolute velocity of P with
respect to the system.

A particle in uniform motion in a straight line with
respect to the system will be represented by a straight line
inclined to the axis of z at some angle whose tangent will
represent the absolute velocity with respect to that system.
This angle we shall suppose to be less than 45°, since, if it
were greater than this, a flash of light sent out from C
would never reach the particle P if the instant of departure
of the light were later than the instant at which P leaves
the position of C.

RAPIDITY OF A PARTICLE WITH RESPECT TO A SYSTEM.

We propose now to define what we shall call the rapidity
of a particle with respect to a system of permanent con-
figuration.

If v be the absolute velocity of the particle with respect
to the system, then the inverse hyperbolic tangent of v will
be spoken of as the rapidity.

Thus if o> be the rapidity,

As o> increases from 0 to oo , v increases from 0 to 1.

For small values of o> we have, practically, velocity is
equal to rapidity, but we shall see later that, for large
values, it is the rapidity aud not the velocity which follows
the additive law.

INDKX OF A PARTICLE MOVING WITH UNIFORM VELOCITY
IN A STRAIGHT LINE.

We shall consider a plane containing the liue of motion
of the particle aud some particle of the system of constant
configuration, and with this particle as origin shall take
axes of x and y. If now we take a three-dimensional set
of rectangular axes Oar, Oy, Qz, and let the z co-ordinate
represent the index of the particles in the system of con-
stant configuration, the moving particle will be represented
by a line whose equations may be taken as

7 -

I m n

where (a?,, yp z^ are the co-ordinates of some point of the
line and I, m, n are its direction cosines.

The standard cone, with respect to any point on this
liue, is

(x - Ir - tf,)2 + (y - mr - yj* -(z- nr -#,)*= 0.

This intersects the axis of z in two points, whose z co-
ordinates are the roots of the equation

z*- 2 (nr + z^z + (nr + #,)'- (Ir + x^f- (mr + y,)2= 0.

Denoting the smaller of these roots by Nd and the larger by
Nr, we have

NrNd = (nr + *,)'- (Ir + ^-(mr+yj*
and Nr + Nd = 2 (nr + zj .

Suppose now a small increase of r to take place, we have
d (NrNd] = 2 {(nr + *,) n - (Ir + x}] I- (mr + yj m} dr
= 2 {(2?22 - 1) r + zji - Xyf — yjn} dr

and d (Nr 4- Nd) = In dr.

d (N N,} 2n* - 1 x 1 + y,m

Thus ~— r = - - r + # - -J— y^- .

V ,F ^

If now we put for r its value - — *, where z is the value

of that co-ordinate for the vertex of the coue, we get
/ _l

Now it is easy to show that if ~z be the ^ co-ordinate
of the point of the line which is nearest to the axis of z then

_ _ (1 — n3) z^ — n (x^

Thus

d(Nr+Nd)
or (if n = cos 7) =^- tan2 7 (z — z).

Also, since — - -- - = z, we have

and accordingly

N.=

/ / d* (NrN
V \ d(Nr + A

- tan* 7)

If we write (z-z] tan 7 = s, and put tan7 = tanha>, where

f.\ ia fV>o TOT\ir1ifxr r\f fVto n^oviljp^ T)Jirticle With I'^snppt' f.n t.n*>

Duration, we have
= V(l — tanhao)) =

If we write (^-^) tan 7 = s, and put tan7 = tanho>, where
o> is the rapidity of the moving particle with respect to the
system of permanent configuration, we have

- .
cosh eo

Thus .ZV = ^ cosh o> — 3 sinh CD.

If we imagine a line drawn parallel to the axis of z through
the point of the line representing the moving particle at the
point where it is nearest to the axis of z, then it is easy

to see that (z — ^)tan7 or s is the distance of the moving
particle from a fixed particle represented by the line parallel
to the axis of z.

GEOMETRICAL CONSTRUCTION.

Let AB be the straight line representing the moving
particle and P the point of it, with respect to which the
standard cone is taken intersecting the axis of z in Nd
and Nr. Let F be the point of AB nearest to the axis of z
and let FG be the common perpendicular. Let a line be
drawn parallel to the axis of z through the point F and
meeting the plane of #, y in R. Let a plane be drawn
through P parallel to the plane of x, y and meeting RF in
H and the axis of z in K. Then the angles PHK and
PHF are both right angles and the angle PFH=y. But
FH= z — ~z and HP = (z — ~z] tan y = s. If then we take a
point M in RH so that HPM = 7, we have

Thus

z —

RECIPROCAL RELATION.

We have seen that in a system of permanent
>B configuration we may take any particle as our
fundamental particle, and the indices defined
with respect to that particle may be employed
to work back from any other particle to the
fundamental one.

Jn defining the index of the fundamental
particle, however, we selected a, perfectly arbi-
trary instant for the zero of index. Thus, as in
the case of potential, it is difference of potential
which is of physical importance ; so in the ease
of index it is difference of index.

We propose now to show that if we have
a particle which moves with constant velocity
with respect to our system of permanent configuration,
and we take the index of the particle with respect to our
fundamental particle, we may use the index so obtained to
work back to the index of the fundamental particle, if the
zero of index has been properly chosen.

If a different zero has been chosen we obtain, not the
index of the fundamental particle itself, but the index plus
a constant. This constant, however, cancels out if we are
dealing with differences of index. In order to show this,
we again make use of our geometrical representation.

Consider then the standard cone taken with respect to
a point- on the z axis, say Nr. Its equation is

This cone meets the line AB representing the moving particle
in two points, the z's of which are given by the equation

2w— 1 2 rt f Ar lx. + my. l + m

or —r—zt~2\Nf+- -#»M

n* n n

n
Multiplying through by , — we may write this equation

in the form

where A and B are constants for the line AB.

Now let z and z" be the roots of this equation. We
have

and

If now Na and Nb be the indices of the moving particle
corresponding to these two points, we have, in accordance
with the result obtained in the previous section,

and

Thus

5 — z -\ -- j- z
n* n*

2n«-l , l-n'

.. x> T « ^

A2n'-

v v~^"

and

2n'-l. , „. 2 (!-«'!_

Suppose now a small increase to take place iu Jfr and
we get

and

Thus ~i-^i r 2w'-l

*J /( *' \

V V2H»-i;

d (Na + N^ 3 ri* cosh2 o>
Thus

Thus we see that if the zero of index of the fundamental
particle be chosen so that # = 0, we get

Thus, if the zero of index be that for which the two particles
are nearest to one another, we may work back from the
moving particle to the fundamental one, and the indices are
connected by a reciprocal relation.

In particular, if the two particles be such that at any
instant they are in contact, we may select that instant as
the one at which both particles have the index zero.

ARITHMETIC MEAN THEOREM.

We propose now to show that if we have a second
particle moving in the same direction as the first and
having the same velocity, and if we send out a flash of
light from the one particle to the other and back again
to the first, then the index of arrival is equal to the
arithmetic mean of the indices of departure and return.

Let us take our fundamental particle as lying in the
plane of the two moving particles and we shall then be able,

as before, to represent index in oar system of permanent
configuration by a z axis perpendicular to those of x and y,
which give the positions in the plane.

Let the one moving particle be represented by the line

— z.

and the other by

where (#p y,, #,) and (x^ y2, ^2) are the co-ordinates of the
points which are nearest to the axis of z.

Take a point of the first line defined by 1\, so that its
co-ordinates are

The standard cone with respect to this point is

Taking the points of intersection of this cone with the second
line, we have

Thus, for the points of intersection we have

Expanding, we get

\> z

{^
^-^-^--

But it is easy to show that

and lxl+m

Thus the above may be written

{£ _ ) *
^-^l-^l--^j

°r

y I ~

n< ri' **

//2n' — l\ /(2n*—l

V \~^~> V \^~

{^
arf- art - Jr, - -

m
- y, - mrv - - *t

But this is a quadratic in

•4 -t J

-z + ~^-r- z9

I* /M* 2

/(^n'-l\ '

V v~^?~>'

and the two values of this are the indices of the second
moving particle at the instants of departure and return of
light going from it to the first moving particle. If we call
the two roots of this equation Nd and Nr, we have

2ri'- I
n

But if (x^y^ z}) be the point on the first line corresponding
to rp we have

z — ~z.

Thus

/,'2ns-l\

A-sr)

The expression on the right is however the index of the first
particle at the instant of arrival of the light from the second
particle, (.ailing this Na, we have

or the index of arrival is equal to the arithmetic mean of
the indices of departure and return.

This is the same as the result which we obtained for the
case of two particles in our original system of permanent
configuration, and we now find that it also holds for the
case of two particles moving in the same direction and with
the same velocity with respect to that system.

If the fundamental particle does not lie in the plane
containing the lines of motion of the particles, we can no
longer represent index by the third dimension, but the
demonstration is quite analogous.

Instead of the standard cone

we take .r-a

where w now represents index in the system of permanent
configuration. The particles then satisfy the equations

__ y — yl __ z — zt _w —
m n k

and

I m n

where l3+m*+n*+k2= 1.

The demonstration then proceeds exactly as before,
except that we have now three co-ordinates representing
ordinary space instead of two.

ABSENCE OF ROTATION IN UNIFORMLY MOVING SYSTEM.

In obtaining our original system of constant configuration
we made use of a test for absence of rotation, by sending a
flash of light in opposite directions round a set of three
particles.

We now propose to show that if we have a set of three
particles which move in the same direction with respect to
our original system, with equal velocities, then the same
test will be satisfied.

We may, as before, represent index in the original
system by means of the z co-ordinate and shall suppose the
motion to be in the x direction.

A particle moving in this direction will be represented
by a line parallel to the plane of x, z.

The equations of such a line may be written

x - XQ = z tan 7,

where #0, y0 are the x and y co-ordinates of the point where
the line cuts the plane of x, y.

Consider now the intersection of the cone

with this line.

We have z3— (

or ( 1 - tan2 7) z* - 2x0 tan 7^ - x* — y* = 0 .

The positive root of this equation in z is

l-tan27

Consider now the three parallel lines meeting the plane
of x, y in the points

and call these lines A, B, and C respectively.

Starting from the origin and going round in the order
A — B— C — A we get, adding the successive increases of z,

x tan 7 + V{*'f + .y'f (1 - ta.nf 7)}
l-tan'7

1 — tan* 7

-as" tan 7 + ij\x"* + y"* (l - tan2 7) j
1 - tan54 7

V|*"+y"(i-tftn"7)j

l-tan37

Next, starting from the origin and going round in the
order A- C- B — A we get, as before,

"2 + y"2(i - tan27) |

1 —tan2 7

l-tan27

1— tan2 7

Thus the result is the same whichever way we go round,
and so we see that flashes of light starting out simul-
taneously and going round in opposite directions would
arrive simultaneously.

20
APPARENT CHANGE OF DIMENSIONS DUE TO MOTION.

B D

\:\/

Nr+Nd N'r+N'd

If we have two particles moving with the same velocity
and in the same direction with respect to a system of
permanent configuration we may represent them by parallel
straight lines.

In order to get our ideas clear we shall suppose the two
particles to move in the direction of the x axis and take the
z axis to represent indices of our fundamental particle.

Let AB and CD represent the two moving particles
and suppose them to pass through the positions P and Q
respectively, where, if there were particles of the permanent
system, such particles would have equal indices. The
distance between two such particles of the permanent
system is the apparent distance between the two moving
particles as viewed from the permanent system. This is,
however, different from the distance between the two

N — N
moving particles as defined by the quantity — r- between

the two.

The result of this is that if we have a system of particles
which are all moving with the same uniform velocity aud
in the same direction with respect to a system of permanent
configuration then a sphere in the moving system is
apparently an oblate spheroid as viewed from the former.

Since (he effect is symmetrical about the direction of
motion, we may, as before, consider merely the plane of x, y
and make use of the axis of z to represent index in our
system of permanent configuration.

Consider a line in the plane of a.-, #, whose equations are

x = z tan 7,

y-o,

and let this represent a particle moving with a velocity
equal to tan 7 with respect to our permanent system.
Suppose now a flash of light to go out 'from this particle

a ring of particles moving with it in the plane of x, y,
id suppose the ring to be of such a form that the light

and supp< .__

returns from all the particles simultaneously.

Consider the intersection of two standard cones, the one
with respect to the point (0, 0, 0) and the other with
respect to the point (ctan7, 0, c).

The first cone is

^ + y*_^=0.

The other is

The common points lie on the surface

z —

This is the equation of a plane making an angle 7 with
the axis of x.

Again the common points of the two cones must lie on
the surface

x*+y* - *? -f - -4-7- \z - tan 7 x - - ( 1 - taa* 7) V
1 — tan 7 ( )

n' 7)1 =

- - (1

x* 2 tan2 7 z* 2 tan 7 c* ,.

n^y + •" + i rt^ - I-taH^**- I (' - tan •» - °'

This may be written in the form

This is the equation of an elliptic cylinder whose
generators are parallel to the original line, and which
therefore represent particles moving in the same direction
as the original moving particle and with the same velocity.

The light going out simultaneously from the latter to all
particles represented by generators of this cylinder will
return simultaneously to it.

If we put z — 0 we get the apparent form of this ring of
particles as viewed from the permanent system. We get

- ,

This is an ellipse in the plane of #, y, the ratio of whose

axes is *J(l - tan*7):l. If we put c — —. — — . , this

becomes V(l-tan-7)

But the index of the original moving particle corre-
sponding to the point with respect to which the second
standard cone was taken is

c<J(l -tan* 7) = 2.

Also the index of this particle corresponding to the
instant of departure of the light is zero.
Thus for the moving system we have

The demonstration for the case where the particles are
not confined to one plane is quite analogous.

APPARENT CHANGE OF ANGLES BY MOTION.
From the above it appears that lengths, as measured by
— - - ^, suffer no apparent alteration in a direction at right

angles to that of motion, but, as observed from the original
system, they appear to be shortened in the direction of motion.

This involves au apparent change of angle as determined
from the trigonometrical ratios. Thus if 0' be the angle
between the direction of relative motion and a direction
fixed with respect to the moving particle while 6 is the
corresponding angle observed from the moving particle we
have

tan #'= , say,

and

where — = \/(l — tan2

?
where « is the rapidity. Thus

cosh

tanfl'

COS lift) '

THE GEOMETRY OF A UNIFORMLY MOVING SYSTEM OF
PERMANENT CONFIGURATION is EUCLIDIAN.

A test has already been described to show that the
Geometry of our original system of permanent configuration
was Euclidian.

We suppose three particles A, B, and C to be taken
in a straight line and such that A and G were equidistant
from B. Two other particles D and E were then supposed
taken so that ADB and BEG were equilateral triangles
and D and E both in one plane with the particles A, B, C
and on the same side of the line AB.

We supposed then that observation showed the triangle
BDE was also equilateral.

The test that the particles should lie in one plane was
that it should be possible to place a sixth particle F so as
to be in the same straight line as A and E and also in the
same straight line as C and D. We may imagine a number
of circles as shown in the figure with centres A, B, C, D,

and E aud radii equal to a side of one of the equilateral
triangles. We may suppose the whole figure to be pro-
jected orthogonally upon a plane inclined to its own at an
angle whose cosine is *J(\ — tan" 7), aud then all the circles
are projected into similar equal and similarly situated ellipses,
the ratio of whose axes will all be as 1 : *J(l — tan87). If the
plane of these ellipses be taken as that of #, y, and if A\ B\
C\ ./>', E, F be the projections of A, B, 0, D, E, F respect-
ively, and if straight lines be taken through A, B ', (7, &c.,
which are perpendicular to the line of intersection of the
two planes, and all make angles 7 with the axes of z, then
these lines will represent particles which are all in motion
in the same direction with a velocity equal to tan 7. The
Jines through A\ B', aud G lie in one plane, and so the
particles which these lines represent will all lie in the same
straight line. Similarly, for the particles represented by
the lines through A', F , and E' and also for those repre-
sented by the lines through C, F\ D'. Further, since the
ellipses in the plane of a?, y all show the contraction due to
a uniform velocity tan 7, the three triangles in the moving
system, whose corners are represented by the lines through
A', I)', B', those through B ', E', C\ and those through
B', 1)', E\ have their sides all equal. Thus the Geometry
of the moving system is Euclidian.

COMPOSITION OF RAPIDITIES.

We have already seen that if we have a particle which
moves with uniform velocity in a straight line with respect
to the fundamental particle, in such a way that the two
particles are in contact at a certain instant, then if we take
that instant as that at which the index of the fundamental
particle is zero the process by which the index of the
moving particle is obtained is a reciprocal one. Let us
now consider two particles which are both in contact with
the fundamental particle at the same instant.

The three particles in general define a plane which we
shall take as the plane of a?, y, while we shall, as before,
represent the index of the fundamental particle by the z
co-ordinate.

Let the one particle be represented by the line

aud the other by

Take a point (*•„ y2, z2) on the second line and take the
standard cone with respect to it.
We have

x ,= — z,

and

Thus the cone is

If (a?,, y,, jzj be a point where this cone meets the other
line we have

« o

»,"

If ^ and ^V2 represent indices of the corresponding
particles we have

l,l«-\- m,ma

Thns

The two values of JV, given by this equation are the
indices of departure and return of light going from particle
1 to particle 2.

Calling these Nd and Nr we have

V V < < /

Thu

1 _
and J

V\ < n;

-\ 2n*-l

and

Thus

V \

Thus the index of arrival of light coming from particle
(1) to particle (2) may be obtained by the same formula as
that by which the index of either particle is obtained from
the fundamental one.

It is evident that this holds for as many particles as we
please, provided that they are all in contact at the same
instant.

Now we have already seen (see p. 10) that

for the case of a particle moving with constant rapidity CD
with respect to the fundamental particle.

If, then, we refer to the fundamental particle by the
suffix 3 (having already assigned suffixes 1 and 2 to the
moving particles) and taking the three particles as the
corners of a triangle, let w, and «3 represent respectively
the rapidity of 2 with respect to 3 and of 1 with respect to
(3) ; while at the same time we write

1 w." n*

cosh*

( _ l,l, + m,m,
( «,»,

We liave

cosh3o) cosh2&>

coslr<w3 f _ l^ + mj
We must now find the value of the quantity

If Xp /Lt1? vl be the direction cosines of a plane through the
first line and the axis of ^, we have

V-0,

Thus

and

Similarly, if X2, /A?, v2 be the direction cosines of a plane
through the second line and the axis of z, we have

If then O3 be the angle between these planes, we have

l.L+m.m.

Thus ± . -^
*

± tanh w, tauh or cos Q,= -1-2 --- L— * .

If we select the angle O3 so that Q3=0 corresponds to
the same side of the axis of z we take the upper sign.

Thus tanli &). tanh <w, cos ft = -^ -- ' — 2

cosh11 &), coslra>a

cosh3&), jl -tauhft), tauhft)0coslili,j3

31 I 2 3 J

Extracting stquare roots, we get finally

cosh <»3 = cosh wv cosh &>2 — siuh v^ sinh <w2 cos O3.
If Q3=7r this gives

cosh &) = cosh ft). cosh&)_+ sinho). siiihft),

o 11 1

= cosh (a), + ft)2),
and thus &)3= &)j 4- &>2.

It will be seen that the formula giviug cosh eo3 in terms
of ft)p &)3, and O3 is analogous to the well-known formula in
spherical trigonometry, and, in fact, represents the formula
connecting three sides and an angle for the case of a triangle
on a sphere of radius V(~1). If we prefer so to express it,
it is the formula connecting three sides and an angle in
a Lobatschefskij triangle.

We have now to show that similar re-
lations hold in respect to the other particles.

Consider a plane drawn perpendicular to
the axis of z aud meeting it in the point C.

Suppose the plane meets the Hues (1) and
(2) in A and B respectively and consider the
triangle A, B, C.

We have 6Y=O3.

a sinC

Also tan A

b -a cos C '

Now A is the angle at particle (1) as observed from
particle (3), but is not the angle as observed from (1) itself.
If 11, be this latter angle we have already seen that

But a:b = tauh col : tauh o>2.

mi ir^ tanh2 o>. (1 - cos3 C)

Thus tan £L =

cosh* o>2( tan ho>2- tanhcu, cosC}2 *

n cosh co. cosh co— cosh co.

±>ut cosG = cosQ=- — — — -.

sinha>1smhft>2

Thus tan'O,

12 f /cosh o> cosh co— cosh a>,v*'
tanh w, «1 - (-
( \ smhojj smhco2

i8 f, i , /coshew. cosh to— cosh&>,\) 2

cosh1 co. 4 tanh <*>, - tanh <w, -

( J V smhajjSiuho), /J

_ sinh'a), siuh3a)2— (cosh a>t cosh &)2— cosh &)3)3
(cosh o>8 cosh o>3— cosh Wj)"

_ 1 — cosh2 CD, — cosh2 a>2 — cosh* a>3 4- 2 cosh w, cosh co2 cosh <wa
(cosh a>3 cosh a>3— cosh wj2

m, 1 1 — cosh &> — cosh «,+ cosh-ft)9 cosh-w,

Ihus — , — = - — : — 2— = —

cos ii (cosha) coshw — cosh co )

(cosh &)2 cosh CD3 - cosh wj2
Thus, extracting square roots, we get

cosh coa cosh coa — cosh CD,

cos&= ^-r

sinha>2sinhft)3

or cosh ajj = cosh w2 cosh a>3 — sinh a>2 sinh co3 cos G^

By a similar process we may obtain a third formula of
the same type so that we see that the relation between the
three particles is such that we may regard any one of them
as "at rest/' and the remaining two as in motion with
respect to it.

Thus instead of a Euclidian triangle of velocities, we get
a Lobatschefskij triangle of rapidities. For small rapidities,
however, we may identify rapidity and velocity, and the
Lobatschefskij triangle may be treated as a Euclidian one.
It is also seen that rapidities in the same, straight line are
additive.

The formulae which we have obtained agree with those
of Einstein, if we take the " velocity of light " as unity and
express the results in terms of velocities instead of rapidities.

Tlr'V have also heen deduced from I\l inkowski's theory l>v

Sommerfeld,

It will !)«• observed (li;i( rapidities may he n.s «;Ter,l us we

please, Inil velocities must :il\v:i.ys be less than a. certain

Unite <piant ity which is efjna.l to unity in the units which
we have selected.

Various other formula', analogous to the formula) of
spherical I ri-onomet ry, may be obtained connect .iu»- (lie
parts of a, triangle of ra pidit ies.

Thus we have, lor instance,

/cosh oi cosh n> — cosh &>V
> = 1 -

\ sinhfD^sinlKWj /

^- 1) (cosl^ft^- 1) - ieosh o)., cosh r,>, cosh o),)7

siniJt,
sinlun,

_ \/| 1 - cosll'fti, - cosh'V, cosh"o)( i '.! cosh w} cosh «•»>.. cosh r,>J
siuh ft), siidi M^ sinh «>i

From the symmetry of the expression on thiM-i^ht it fallows

that

sin 12, siuil.. _ sinii.(

Mlllln*. ~ MIlllO)^ "

i
A ain it is eas to deduce the formula

, cos Q,= — cos ft, cosH8+ siu i\ siu Q3 cosh a>p

and two others of (he same type.

Lor.ATSCllKKSKU SVsTlM.

It is int crest iui;- to unit' (lint a system of particles
diver^iiiL;- in nil directions \\ith various uniform relative
rapidities Iroiu Mmultaueous t-ontact may he i-e^anh-d as
a kind of Lohatschefskij hody. Any three such particles,
ha\e seen. :'j\e a Lohatschefskij triangle of rapidit it's.
If we select an\ one of the particles, (he remainder di\
from it in various direct ions. 1 f we suppose a small Euclidian
system of permanent couli-iirat ion to he associated with the
selected particle, to serve as a system of ivtereiice, these

tlirootions \vill he eonuootod hy tho rolations of sphorieal

Momotrv. As is well known, however, sphorioal t.|
nomotry is oommon hoth to Kuelidian and Kohalsohol'sk
livomotry, so that tho whole system ol' divor^in:; partteles
may ho regarded MS a sort of 1 ,oh:it sehofskij hodv.

An ordinary Kuelidian hod\ may he iv^nrolt'o! ;^ M liinilin^-
t-:is(^ in \vhioh ll»o inshint of .siiuultjinoons t-onlMiM is mi
tv» infinit.

OF THE Ixi'i-A 01 A

In onlor to ohtain a eloarer physieal eoneoption
of tlu1 index of a parfiele whieh is in motion with
rospoel to our fundamental parlielo .-1. lot us sup
pose tho latter to be fixed with reboot to a plane
mirror at one-hall' t he unit distanee in front ol'il.

\Yo shall suj>poso a seeond partielo /' whieh is
initially in eonlaet with .1 to move parallel to the
surfaeo of the mirror with uniform veloeil\

Suppose now we take tin* instant at wlueh the
partieles are in eontaet as that at w hieh hoth have
the index . ero. and suppose that at that instant
a. Hash of li.",ht :;oos out from them to tho mirror.
Then the index of .1 at tho i nst ant of the //'' arrival
at .1 is i; ; while' it is easv to show

of tlu>

that the index •
li-ht at /' is a
a. Hash of li-Jil
to .-I.

Let A', and
ot th(> li:vhl.

Sinee t he velocity of the partielo is supposed eonstant,
we have

+ M

Also

/' at the instant of t hi' .'."' arrival of the

i ;/. In onlor to show this wo imagine

<>,» trom .1 to /' direel 1\ and ha.-k a",. tin

ho the iiuliees of departure and return

Thus
and

Tins gives

,/l.v \ a

«/i-vr i .v.,1-

32
If DOW we pnt JVa= l, we get

Thiw ''-•"< = jfi^r: •

This is the distance from A of the position of P, when P
has the index unity.

The distance travelled by light in going from A to the
mirror and from the mirror to this position is

But in our system of units the distance travelled by the

v
particle P in the same interval is v times this or — — - — - ,

V \1 27 J

which is the distance from A of the position of P at the
instant when P has the index unity. This proves the result
stated.

Now, if we have any number of systems of permanent
configuration which are moving with respect to one another
with uniform velocities in fixed directions and without
rotation, we may always imagine one particle of each
system such that all such particles are in contact simul-
taneously. The index of these might be supposed to be
given by the mirror method, while the index of any other,
moving in the same direction and with the same velocity,
might be supposed to be determined by the arithmetic mean
theorem.

We may also offer the following suggestion as to index,
which, if permissible, renders its meaning more definitely
physical :—

The number 'of vibrations corresponding to a definite
spectrum line of a particular substance, which are executed
in any interval, is proportional to the difference of index of
the particle emitting the light at the beginning and end
of the interval, the constant of proportion being fixed for
each particular line. This is on the assumption that the
velocities are constant.

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