OPTICAL GEOMETRY 
 OF MOTION 
 
 A NEW VIEW 
 OF THE THEORY OF RELATIVITY 
 
 BY 
 
 ALFRED A. ROBB, M.A., PH.D. 
 
 ittamfctfoge : 
 
 W. HEFFER AND SONS LTD, 
 
 : SIMPKIN MARSHALL AND Co. LTD. 
 1911 
 
 Is. net. 
 

OPTICAL GEOMETRY 
 OF MOTION 
 
 A NEW VIEW 
 OF THE THEORY OF RELATIVITY 
 
 BY 
 
 ALFRED A. ROBB, M.A., PH.D. 
 
 (ftamfortoge : 
 
 W. HEFFER AND SONS LTD. 
 : SIMPKIN MARSHALL AND Co. LTD. 
 1911 
 
PREFACE. 
 
 IN placing before his readers tbe following brief outline 
 of a point of view, tbe writer is well aware tbat 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. 
 
 Cambridge, 
 
 May 13/7*, 1911. 
 
 A. A. ROBB. 
 
 330177 
 
" But 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 vain, 
 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 
 the 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 Riernann. 
 
 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 ptanomena which occur 
 in free space, upon which we might suppose some of the 
 
chief axioms to have 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 HEST. 
 
 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 Hue as two other 
 particles. If three particles A, B, 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 I) lies in the 
 same plane as A, J5, 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 Hue as the remaining 
 particles B and C. (In case AD and BC are parallel, 
 AVC 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, J9, &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 line as two others A and B, and so 
 that the stretches DA and DB are equal. 
 Let C be another particle not lying in the 
 line AB, bnt 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 particles 
 A, B, C, D, E, of which A, B, and C 
 lie in a straight line and so that the 
 stretches BA 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 line 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 line 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 Lobatscbefskij 
 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 G 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. 
 
 IJ\ 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 ot our system of permanent'cou- 
 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. 
 
 "INDEX" OF FUNDAMENTAL PARTICLE 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 flash of light sent out to B, which is at one-half the 
 unit distance from it, and reflected from B back to A. W T e 
 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 n ih 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 *1. 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 run to some extent 
 counter to one another. 
 
 DEFINITION OF THR INDEX FOR PARTICLES IN GENERAL. 
 
 Suppose a flash of light to gp 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 N d 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 N r 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 JV a . 
 
 The index of arrival is defined by the following equation: 
 
 d(N,N d ) 
 
 I { 2 r ^ 3 ) 
 
 ^N d 
 
 ~-^ by ^ 
 express JV more simply thus 
 
 N N N + N 
 If we denote ^ by u and : * by t, we may 
 
 du 
 
 
6 
 
 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 
 
 N r -N d 
 2 
 
 In our permanent system this quantity remains fixed for 
 each particle. If then we select any particle "P, and let 
 
 we bare > 
 
 Jf\ S 
 
 .. 
 and 
 
 
 This gives 
 
 d(N r N d ) 
 
 ^ '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 suck 
 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 
 
 <L . 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 permanent configuration which are 
 non-rotating. 
 
 If we send out a flash of light from some particle A 
 in such a system to 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 
 
 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 iu 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 tlie logical principle 
 of non-contradiction, whereby a tiling 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. 
 
8 
 
 N r -Nd 
 
 2 
 
 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 al-'O fits in with our definition 
 of the measure of the distance between 
 
 .. . . -^",-JVa 
 
 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 
 
 (x - a) 2 + (y - b) 2 - (z - c) 2 = 0. 
 
 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 
 iustant, 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 
 
 to be 
 
 *r at the instant of arrival of the light. 
 
 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, 
 
 v = tauh a). 
 
 As o> increases from to GO , v increases from to 1. 
 
 For small values of a> we have, practically, velocity is 
 equal to rapidity, but we shall see later that, for large 
 values, it is the rapidity and not the velocity which follows 
 the additive law. 
 
 INDUX OF A PARTICLE MOVING WITH UNIFORM VELOCITY 
 IN A STRAIGHT LINE. 
 
 We shall consider a plane containing the Hue of motion 
 of the particle and 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 Oa?, Oy, Oz, 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 
 
 I m n 
 
 where (a?,, y t , 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 
 line, is 
 
 (x - lr x^+(y mr y^)'-- (z nr z^'=. 
 
 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 + #,)* (lr + x^* (mr + y,)* = 0. 
 
 Denoting the smaller of these roots by N d and the larger by 
 N r , we have 
 
 and N r + N d = 2 (nr + zj . 
 
 Suppose now a small increase yf r to take place, we have 
 d(NN d ] = 2 {(nr + z } ) n - (Ir + xJl-far + yJ m] dr 
 = 2 2?i 3 lr + zn-xl m dr 
 
10 
 
 and d (N r + N d ) = 2w dr. 
 
 Thus 4^, = *1 r + *, - i . 
 
 ^* .^_ 'X 
 
 If now we pat for r its value - ! , where z is the value 
 of that co-ordinate for the vertex of the cone, we get 
 
 _ n l-n a ar^ + ^m 
 wV n a w 
 
 Now it is easy to show that if Jz be the z co-ordinate 
 of the point of the line which is nearest to the axis of z then 
 
 i-n* 
 Thus d(N * N > -(*- l 
 
 ~ 
 
 or (if n = cos 7) 
 
 d 
 Also, since r -=z, we have 
 
 and accordingly 
 
 ~ // rf'CJT^, 
 V V *(fr r +V 
 
 If we write (^ 5) tan 7 = 5, and put tan7=tanha>, where 
 cw is the rapidity of the moving particle with respect to the 
 system of permanent configuration, we have 
 
 Thus JV a =jzrcoshfi) 5 sinhw. 
 
 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 
 
11 
 
 to see that [z z] tan 7 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 N d 
 and N r . Let F be the point of AB nearest to the axis of z 
 and let FG be the common perpendicular. Let a Hue 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 a?, 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 PFff=j. But 
 FH=z z and HP = (z ~z) tan 7 = 5. If then we take a 
 point M in RH so that HPM ^^ we have 
 
 Thus 
 
 z s tan 7 = RM- 
 
RECIPROCAL RELATION. 
 
 We have seen that in a system of permanent 
 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. 
 
 In 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 case 
 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 N r . 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 
 
 or 
 
 , 7 
 
 "1 *"" 
 
 r 
 
 l + 
 
 9/ 
 
 Multiplying through by - we may write this equation 
 
 in the form 
 
 where A and B are constants for the line AB. 
 
13 
 
 Now let z and z" be the roots of this equation. We 
 have 
 
 and z'z" = ^ (JV/ 4- B). 
 
 If now N a and N b be the indices of the moving particle 
 corresponding to these two points, we have, in accordance 
 with the result obtained in the previous section, 
 
 AT 
 
 V(l-tan*7) 
 
 and 
 Thus 
 
 A-^F 
 
 and 
 
 /i' 2 ""- 1 
 \/(-tf- 
 
 Suppose now a small increase to take place in N r and 
 we get 
 
and 
 
 14 
 IdN 
 
 + '' J(-4- 
 
 VV2'-1 
 
 and , ^. 
 
 d(N a +Ntf n' ~cosh z o) 
 Thus 
 
 Thus we see that if the zero of index of the fundamental 
 particle be chosen so that ~z = 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 
 h 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, 
 
15 
 
 as before, to represent index in our 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 
 
 *-*i = y-y = z ~\ 
 
 I m n 
 
 and the other by 
 
 I m n 
 
 where (# p y v z^ and (x 2 , y a , z a ) are the co-ordinates of the 
 points which are nearest to the axis of z. 
 
 Take a point of the first line defined by ?' p 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 
 
 l(z z} m (z z a } 
 
 Thus, for the points of intersection we have 
 
 + - -_ = 
 n ) 
 
 Expanding, we get 
 
 2n'-l . l-n 2 - 
 
16 
 
 Bnt it is easy to show that 
 
 and 
 
 Thus the above may be written 
 
 - 
 
 w" 
 
 or 
 
 //an^l \ 7/2^ -IN 
 
 V \ * / V v ^ 3 / 
 
 9 (1 <M a> \ ( 9*j 3 1 
 
 " I A II I i Z /(, L 
 
 V / ~ J n> \ /* 
 
 ^'SxC'.~f /. 
 
 Bnt this is a quadratic in 
 
 /(2n*-l\ ' 
 V V~^~/ 
 
17 
 
 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 N d and N r , we have 
 
 But if (a?,, y,, z } ) be the point on the first line corresponding 
 to r,, we have 
 
 z z. 
 = r.. 
 
 - 1 l-ri*_ 
 
 2 ^i + 2 ^i 
 
 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. (Jailing this JV a , we have 
 
 2 
 
 or the index of arrival is equal to the arithmetic mean of 
 the indices of departure and return. 
 
 This is the same ns 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 (x - a) 2 + (y - b) 2 + (z- c) 2 - (w-(ty=0, 
 
18 
 
 where w now represents index in the system of permanent 
 configuration. The particles then satisfy the equations 
 
 x 
 
 and 
 
 m n 
 
 where ^ 2 +^ 3 +n 2 +^ 2 = l. 
 
 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 
 oar 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 #, z. 
 
 The equations of such a line may be written 
 
 x - x z tan 7, 
 
 where a? , y are the x and y co-ordinates of the point where 
 the line cuts the plane of a?, y. 
 
 Consider now the intersection of the cone 
 
 with this line. 
 
 We have z 2 
 
 or (1 - tan 2 7) z 1 - 2x tan yz - xj - y* = 0. 
 
19 
 The positive root of this equation in z is 
 
 1 tan 3 7 
 
 Consider now the three parallel lines meeting the plane 
 of #, y in the points 
 
 (0,0), (x\y\ (x",y"), 
 
 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 , 
 
 x tan 7 + yV 2 -H/ 2 f 1 - tan*?)} 
 1 - tan' 7 
 
 (x"-x) tan 7 + V((:c" - a')* -t- (y"-y')' (1 - tan* 7) [ 
 l-tan a 7 
 
 l-tan 3 7 
 
 Next, starting from the origin and going round in the 
 order A- C- B A we get, as before, 
 
 a:" tan 7 + J\x"* + y'"(l -tan 2 7)j 
 
 l-tan 2 7 
 
 -U'"- 
 
 1 tan 3 7 
 -^tan7 + V^ 2 
 
 l-tan 3 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. 
 
 rw 
 
 B D 
 % 
 
 N r +N d N' r +N' d 
 
 2 
 
 A C 
 
 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 and 
 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 the effect is symmetrical about the direction of 
 motion, we may, as before, consider merely the plane of a?, 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;, z, whose equations are 
 
 x = z tan 7, 
 
 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 
 
21 
 
 to a ring of particles moving with it iu the plane of or, y, 
 and suppose the ring to be of such a form that the light 
 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 (c tart 7, 0, c). 
 
 The first cone is 
 
 a;* + y--*:=0. 
 
 The other is 
 
 (x c tan 7)* + y* (z c)*= 0. 
 The common points lie on the surface 
 
 ** 4y- *-{(*-* tan 7)* HY-fc-^'H 
 
 or 
 
 (1 tan*7) = 0. 
 
 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 
 
 c } 
 
 - - (1 tan 2 7) Y 
 
or 
 
 2 .,_. j ~ 2 tan 7 c* , 3 N 
 
 1 tan 2 7 1 tan* 7 1 tan 2 7 
 
 This may be written in the form 
 (x tany z)* 
 
 + 
 
 -tau'Vr Jr(l-tan*7) 
 
 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 = we get the apparent form of this ring of 
 particles as viewed from the permanent system. We get 
 
 o 9 
 
 x n _ 
 
 J^l-tanSOl 2 * Jc"(l-tan' 7 ) = 
 This is an ellipse in the plane gf #, y, the ratio of whose 
 
 But the index of the original moving particle corre- 
 sponding to the point with respect to which the second 
 standard cone was taken is 
 
 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 
 r , 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. 
 
23 
 
 This involves an 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 
 Jiave 
 
 tan 0'=, say, 
 - 
 
 and 
 
 2 
 
 where q - = V( I - tan* 7) = 
 
 q "cosho,' 
 
 where o> is the rapidity. Thus 
 
 ._ tan 0' 
 cosh to ' 
 
 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 G to be taken 
 in a straight line and such that A and C 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, (7, D, 
 
24 
 
 and E and 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 a,n 
 angle whose cosine is >J(l 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 tan 2 7). If the 
 plane of these ellipses be taken as that of x, y, and if A\ B\ 
 C 1 , D', E', F be the projections of A, B, <?, D, E, F respect- 
 ively, and if straight lines be taken through A', B ', C', &c., 
 which are perpendicular to the Hue of intersection of the 
 two planes, and all make angles 7 with the axes of #, then 
 these lines will represent particles which are all in motion 
 in the same direction with a velocity equal to tan 7. The 
 lines through A\ B\ and C lie in oue 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 aud also for those repre- 
 sented by the lines through G', F', D' . Further, since the 
 ellipses in the plane of x, y all show the coutraction due to 
 a uniform velocity tan 7, the three triangles in the moving 
 system, whose coruers are represented by the lines through 
 A, D', B\ those through B ', E\ C', and those through 
 .B', D\ 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 iustant, 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 
 moviug particle is obtained is a reciprocal one. Let us 
 now consider two particles which are hoth 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 #, ?/, 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 
 
 x _ ;/ z 
 T t ~ ro, ~ n, 
 aud the other by 
 
 x z 
 
25 
 
 Take a point (# 8 , ?/ 2 , z^ on the second line and take the 
 standard cone with respect to it. 
 We have 
 
 Thus the cone is 
 
 -,/-(.,- *.)- (y- 5 4=0. 
 
 If (A-,, y,, z^ be a point where this cone meets the other 
 line we have 
 
 _ ~ y_( Ii /v ^i^r ) (' z ' z 1=0 
 
 ?^, 
 
 * 
 
 If ^ and ^" 2 represent indices of the corresponding 
 particles we have 
 
 /(^^\z-N 
 VI < / l ~ " 
 
 Thn 
 
 The two values of AT, given by this equation are the 
 indices of departure and return of light going from particle 
 1 to particle 2. 
 
 Calling these N d and N r we have 
 
 *' 
 
 Thus 
 
26 
 
 7 ' Tl-T -T \ "l'*o 7-H-T 
 
 and d I JK. + NJ = 2 - i-? cW 
 
 Thns 
 
 and 
 
 Thns 
 
 \ l,l,+ m im ,\'- 
 1 ,, i 
 
 V V 
 
 Thus the index of arrival of liqht coming from particle 
 (1) particle (2) way fo 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 
 
 d(N r +N d ) 3 cosh 2 ay 
 
 for the case of a particle moving with constant rapidity o> 
 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 co l and 2 represent respectively 
 the rapidity of 2 with respect to 3 and of 1 with respect to 
 (3) ; while at the same time we write 
 
 
We have 
 
 27 
 
 1 1 
 
 cosh 2 o), cosli 2 <w 
 
 We must now find the value of the quantity 
 
 If X p /Lt p Vj be the direction cosines of a plane through the 
 first line and the axis of z, we have 
 
 Thus \, = , ' 
 
 w. 
 
 Similarly, if X 2 , /^ ? , v 2 be the direction cosines of a plane 
 through the second line and the axis of z, we have 
 
 If then H 3 be the angle between these planes, we have 
 
 l.L+m.m. 
 
 MM // 1?? . 2 \ 7/ 1 ^o 2 \ I.l 9 +m.m 9 
 
 Tims A / -- 5T- A /- -y^- cosO ,=- LJ 
 
 V V ^, / V > w a / W i^ 
 
 ,. ,, 
 
 or tauh o>. tauh o> 9 cos Q, = - 1 - 2 ------ '- * . 
 
 n^ 
 
 If we select the angle Q 3 so that Q 3 =0 corresponds to 
 the same side of the axis of z we take the upper sign. 
 
28 
 
 Thus tanh ft) t tanh co >2 cos G^ = 
 
 which gives 
 
 1 
 
 cosh* o> ( cos I 
 
 cosh 3 ft) 3 {1 -tanh to, tanh o> 2 cosh O 3 ( 3 ' 
 Extracting sequare roots, we get finally 
 
 cosh o> 3 = cosh co l cosh co 2 - siuh o> 1 sinh &) 2 cos O 3 . 
 If Q 3 =7r this gives 
 
 cosh o) 3 = cosh <o 1 cosh &) 2 4- sinh tOj siuh &>, 
 
 = COSh (ft), 4- ft),), 
 
 and thus &),= &>, + o> 2 . 
 
 It will be seen that the formula giving coshco 3 in terms 
 of &)j, o) 2 , and Q 3 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 and meeting it in the point C. 
 
 Suppose the plane meets the lines (1) and 
 (2) in A and B respectively and consider the 
 triangle A, B, C. 
 
 We have 
 
 Also 
 
 a sin (7 
 
 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 Q, be this latter angle we have already seen that 
 
 tan A 
 tanO.- -. 
 
 cosh &> 
 
29 
 But a\b tauh &>, : tanh a> . 
 
 tauh a a>, (1 -cos 2 C) 
 
 I h 11 c 1*1 Tl KL "~* - 
 
 1 cosh'" a> 2 {tanh e 2 tauh a>, cos (7 j 2 " 
 
 cosh a>. cosh &> cosh &), 
 
 But cosC = cosa = ^- 3 . 
 
 sinha> 1 smhft) 2 
 
 Thus tan 2 ^, 
 
 i. /cosh ft), cosh &)., cosh ft),\ a 
 tanlr- )1 ' 
 
 .'.,{-C 
 
 sinner^ sinha> 2 
 
 (, i , /cosh a), cosh ft)., cosh &>,\) 8 
 
 cosh o), ! tanh &) . - tanh co, - } J- 
 
 2 I l V smh&jj smlift) 2 
 
 sinl^co, siuh 3 ft) 2 (cosh &> 1 cosh eo 2 cosh o) 3 ) 2 
 cosh o> cosh o> 
 
 1 cosh 2 a), cosh 2 ft) cosh* o> + 2 cosh &) cosh ft) , cosh &), 
 
 1 2 ^ 3 1 2 3 
 
 (cosh o) 2 cosh ft) 3 cosh coj 2 
 
 ,, 1 1 cosh 2 ft) a cosh 2 co 3 +cosh 2 ft) 2 cosh 2 ft) 3 
 
 cos 2 12j (cosh ( 2 cosh &) 3 cosh coj 2 
 
 sinh 2 ft) 2 sinh 3 ft) 3 
 
 (cosh a> 2 cosh ft) 3 - cosh ewj* 
 Thus, extracting square roots, we get 
 
 _ cosh ft) 2 cosh &) 3 - cosh o) t 
 
 COS 12 : : ^ : 
 
 smho) 3 siuh &) 3 
 or cosh CD, = cosh &) 2 cosh o) 3 sinh &) 2 sinh &) 3 cos O r 
 
 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 iu 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 formulas 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. 
 
30 
 
 They have also been deduced from Minkowski's theory by 
 Sommerfeld. 
 
 It will be observed that rapidities may be as great as we 
 please, but velocities must always be less than a certain 
 finite quantity which is equal to unity in the units which 
 we have selected. 
 
 OTHER FORMULAE. 
 
 Various other formula, analogous to the formulas of 
 spherical trigonometry, may be obtained connecting the 
 parts of a triangle of rapidities. 
 
 Thus we have, for instance, 
 
 SllTiJ =1- 
 
 siuhft> 2 sinho> 3 
 
 _ (cosh*a> 2 - 1) (cosh a &> 3 - 1)- (cosh &> 2 cosh &> 3 - cosh 
 sin H 
 
 siuh 2 a> 2 sinh 2 a> 3 
 
 V(l -cosh 2 &), - cos h 3 &> 2 - cosh 2 ft> s + 2 cosh w, cosh &> 2 cosh &> s } 
 sinh co l sinh &> 2 sinh &> 3 
 
 From the symmetry of the expression on the right it follows 
 that 
 
 sinO, sinO 2 
 
 sinho) 2 siuho) 3 * 
 Again it is easy to deduce the formula 
 
 cos 2, = cos H 2 cos O 3 + sin Ii a sin Q 3 cosh a) p 
 and two others of the same type. 
 
 LOBATSCHEFSKIJ SYSTEM. 
 
 It is interesting to note that a system of particles 
 diverging in all directions with various uniform relative 
 rapidities from simultaneous contact may be regarded as 
 a kind of Lobatschefskij body. Any three such particles, 
 as we have seen, give a Lobatschefskij triangle of rapidities. 
 If we select any one of the particles, the remainder diverge 
 from it in various directions. If we suppose a small Euclidian 
 system of permanent configuration to be associated with the 
 selected particle, to serve as a system of reference, these 
 
31 "' 
 
 directions will be connected by the relations of spherical 
 trigonometry. As is well known, however, spherical trigo- 
 nometry is common both to Euclidian and Lobatschefskijian 
 geometry, so that the whole system of diverging particles 
 may be regarded as a sort of Lobatschefskij body. 
 
 An ordinary Euclidian body may be regarded as a limiting 
 case in which the instant of simultaneous contact is removed 
 to infinity. 
 
 PHYSICAL SIGNIFICANCE OF THE INDEX OF A PARTICLE. 
 
 In order to obtain a clearer physical conception 
 of the index of a particle which is in motion with 
 - 2 respect to our fundamental particle A, let us sup- 
 n pose the latter to be fixed with respect to a plane 
 w mirror at one-half the unit distance in front of it. 
 
 We shall suppose a second particle P which is 
 * initially in contact with A to move parallel to the 
 e surface of the mirror with uniform velocity v. 
 
 Suppose now. we take the instant at which the 
 particles are in contact as that at which both have 
 the index zero, and suppose that at that instant 
 a flash of light goes out from them to the mirror. 
 Then the index of A at the instant of the n th arrival 
 of the light at A is n ; while it is easy to show 
 that the index of P at the instant of the n th arrival of the 
 light at P is also n. In order to show this we imagine 
 a flash of light to go from A to P directly and back again 
 to A. 
 
 Let N d and N r be the indices of departure and return 
 of the light. 
 
 Since the velocity of the particle is supposed constant, 
 we have 
 
 N r~ N * = y 2 
 
 2 
 
 Also 
 
 Thn 
 and 
 
 This gives N a = ^^ V(l - v'} 
 
32 
 
 If now we pnt N a = 1, we get 
 
 Thus 
 
 2 
 
 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 
 particle P in the same interval is v times this or ^ , 
 
 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 fized for 
 each particular line. This is on the assumption that the 
 velocities are constant. 
 

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