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 postulates V, L, R, were still valid it would turn out that the two observers would find their time units in agreement. But, in view of M, the choice of S as fixed would undoubtedly seem perfectly arbitrary to both observers; and the content of the modified postulate R would be essentially different from that of the postulate as we have employed it. Hence, if we accept R as it stands — or, indeed, even a certain part of it, as we have shown above — we must conclude that the time units in the two systems are not in agreement, in fact, that their ratio is that stated in the theorems above.

§ 8. Relation Between the Units of Length. — Let us consider three systems of reference S, $$S_1$$ and $$S_2$$ related in the same manner as in the preceding section except that now the two lines $$l_1$$ and $$l_2$$ coincide. We suppose that $$S_1$$ is moving in the direction indicated by the arrow at A and that $$S_2$$ is moving in the direction indicated by the arrow at B.

We suppose that observers at A and B again measure the velocity of light as it emanates from S, this time in the direction of the line of motion. Each will carry out his observations on the supposition that his system is at rest, for from M it follows that he cannot detect the motion of his system. The observer at A measures the time $$t_1$$ of passage of a ray of light from A to C and return to A, the length of AC being d when the measurement is made with a unit belonging to $$S_1$$. Likewise, the observer at B measures the time $$t_2$$ of passage of a ray of light from B to D and return to B, the length of BD being d when measured with a unit belonging to $$S_2$$.

Just as in the preceding case it may be shown that the two observers must obtain the same estimate for the velocity of light. But the estimate of the observer at A is $$2d/t_1$$ while that of the observer at B is $$2d/t_2$$. Hence

$$t_{1}=t_{2}$$;

that is, the number of units of time required for the passage of the ray at A and of the ray at B is the same, the former being measured on $$S_1$$ and the latter on $$S_3$$. Moreover, the measure of length is the same in the two cases. But the units of time, as we saw in the preceding section, do not have the same magnitude. Hence the units of length of the two systems along their line of motion do not have the same magnitude; and the ratio of units of length is the same as the ratio of units of time.

Combining this result with theorem III, its corollary, and theorem IV. we have the following three results:

V. If two systems of reference $$S_1$$ and $$S_2$$ move with a relative

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 velocity v and β is defined as the ratio of v to the velocity of light estimated in the manner indicated in the first part of § 7, then to an observer on $$S_1$$ the unit of length of $$S_1$$ along the line of relative motion appears to be in the ratio $$\sqrt{1-\beta^{2}}:1$$ to that of $$S_2$$ while to an observer on $$S_3$$ the unit of length of $$S_2$$ along the line of relative motion appears to be in the ratio $$\sqrt{1-\beta^{2}}:1$$ to that of $$S_1$$ (MVLA).

. Theorem V. may be stated as depending on (MVLR') instead of on {MVLA).

VI. If two systems of reference $$S_1$$ and $$S_2$$ move with a relative velocity v and if β is the ratio of v to the velocity of light, then to an observer on $$S_1$$ the unit of length of $$S_1$$ along the line of relative motion appears to be in the ratio $$\sqrt{1-\beta^{2}}:1$$ to that of $$S_2$$ while to an observer on $$S_2$$ the unit of length of $$S_2$$ along the line of relative motion appears to be in the ratio $$\sqrt{1-\beta^{2}}:1$$ to that of $$S_1$$ (MVLR).

We might make an analysis of these results similar to that which we gave for the corresponding results in the preceding section. But it would be largely a repetition. It is sufficient to point out that the remarkable conclusions as to units of length in the two systems rest on just those assumptions which led to the strange results as to the units of time.

§ 9. Simultaneity of Events Happening at Different Places. — Let us now assume two systems of reference S and $$S'$$ moving with a uniform relative velocity v. Let an observer on $$S'$$ undertake to adjust two clocks at different places so that they shall simultaneously indicate the same time. We will suppose that he does this in the following very natural manner: Two stations A and B are chosen in the line of relative motion of S and $$S'$$ and at a distance d apart. The point C midway between these two stations is found by measurement. The observer is himself stationed at C and has assistants at A and B. A single light signal is flashed from C to A and to B, and as soon as the light ray reaches each station the clock there is set at an hour agreed upon beforehand. The observer on $$S'$$ now concludes that his two clocks, the one at A and the other at B, are simultaneously marking the same hour; for, in his opinion (since he supposes his system to be at rest), the light has taken exactly the same time to travel from C to A as to travel from C to B.

Now let us suppose that an observer on the system S has watched the work of regulating these clocks on $$S'$$. The distances CA and CB appear to him to be

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instead of ½d. Moreover, since the velocity of light is independent of the velocity of the source, it appears to him that the light ray proceeding from C to A has approached A at the velocity c + v, where c is the velocity of light, while the light ray going from C to B has approached B at the velocity c — v. Thus to him it appears that the light has taken longer to go from C to B than from C to A by the amount

But since $$\beta=v/c$$ the last expression is readily found to be equal to

Therefore, to an observer on S the clocks on $$S'$$ appear to mark different times; and the difference is that given by the last expression above.

Thus we have the following conclusion:

VII. ''Let two systems of reference S and S' have a uniform relative velocity v. Let an observer on S' place two clocks at a distance d apart in the line of relative motion of S and S' and adjust them so that they appear to him to mark simultaneously the same time. Then to an observer on S the clock on S' which is forward in point of motion appears to be behind in point of time by the amount''

where c is the velocity of light and $$\beta=v/c$$ (MVLR).

It should be emphasized that the clocks on $$S'$$ are in agreement in the only sense in which they can be in agreement for an observer on that system who supposes (as he naturally will) that his own system is at rest — notwithstanding the fact that to an observer on the other system there appears to be an irreconcilable disagreement depending for its amount directly on the distance apart of the two clocks.

According to the result of the last theorem the notion of simultaneity of events happening at different places is indefinite in meaning until some convention is adopted as to how simultaneity is to be determined. In other words, there is no such thing as the absolute simultaneity of events happening at different places.

§ 10. Transformation of Space and Time Coordinates. — It is now an easy matter to derive the Einstein formulae for the transformation of space

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 and time coordinates. Let two systems of reference S and $$S'$$ have the relative velocity v in the line l. Let systems of rectangular coordinates be attached to the systems of reference S and $$S'$$ in such way that the x-axis of each system is in the line l, and let the y-axis and the z-axis of one system be parallel to the y-axis and the z-axis respectively of the other system. Let the origins of the two systems coincide at the time t = 0. Furthermore, for the sake of distinction, denote the coordinates on S by x, y, z, t and those on $$S'$$ by x', y', z', t'. We require to find the value of the latter coordinates in terms of the former.

From postulate L it follows at once that y' = y and z' = z. Let an observer on S consider a point which at the time t = 0 appears to him to be at distance x from the $$y'z'$$-plane; at time t = t it will appear to him to be at the distance x - vt from the $$y'z'$$-plane. Now, by an observer on $$S'$$ this distance is denoted by $$x'$$. Then from theorem, VI. we have

Now consider a point at the distance x from the yz-plane at time t = t in units of system S. From theorem VII. it follows that to an observer on S the clock on $$S'$$ at the same distance x from the yz-plane will appear behind by the amount

where c is the velocity of light. That is, in units of S this clock would register the time

Hence, by means of theorem IV., we have at once the result

Solving the two equations involving $$x'$$ and $$t'$$ and collecting results, we have

where $$\beta=v/c$$ and c is the velocity of light.

In the same way we may obtain the equations which express t, x, y, z

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 in terms of t', x', y', z'. But these can be found more easily by solving equations (A) for t', x', y', z'. Thus we have

These two sets of equations (A) and ($$A_1$$) are identical in form except for the sign of v. This symmetry in the transformations constitutes one of their chief points of interest.

Our method of proof of these formulae is very different from that of Einstein, as a comparison will readily show. The difference is due primarily to our use of postulates V and L instead of the assumptions of Einstein.

In a paper entitled "The Common Sense of Relativity" Campbell has made some interesting remarks concerning these transformations.

§ II. The Addition of Velocities. — For the sake of completeness in the presentation of the fundamental results of relativity and for use in the next section we derive here the formulae for addition of velocities due to Einstein.

Let the velocity of a point in motion be represented in units belonging to $$S'$$ and to $$S$$ by means of the equations

$$\begin{array}{lll} x'=u_{x'}t, & y'=u_{y'}t, & z'=u_{z'}t,\\ x=u_{x}t, & y=u_{y}t, & z=u_{z}t,\end{array}$$

respectively. In the first of these substitute for $$t', x', y', z'$$ their values given by (A), solve for x/t, y/t, z/t and replace these quantities by their equals $$u_{x},u_{y},u_{z}$$ respectively. Thus we have

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 From these results it follows that the law of the parallelogram of velocities is only approximate. This conclusion of the theory of relativity has given rise, in the minds of some persons, to the most serious objections to the entire theory.

Suppose that both the velocities considered above are in the line of relative motion of S and $$S'$$. Then we have

$$u=\frac{v+u}{1+\frac{vu'}{c^{2}}}$$

This equation gives rise to the following theorem:

VIII. If two velocities, each of which is less than c, are combined the resultant velocity is also less than c (MVLR).

To prove this we substitute in the preceding equation for v and $$u'$$ the values

$$v=c-k,\ u'=c-l,$$

where each of the numbers k and l is positive and less than c. Then the equation becomes

$$u=c\frac{2c-k-l}{2c-k-l+\frac{kl}{c}}.$$

The second member is evidently less than c. Hence the theorem.

If, however, either one (or both) of the velocities v and $$u'$$ is equal to c — and hence k or l (or both) is equal to zero — we see at once from the last equation that u = c. Hence, we have the following result:

IX. If a velocity c is compounded with a velocity equal to or less than c, the resultant velocity is c (MVLR).

Remark. — A conclusion of importance is implicitly involved in the results obtained in §§ 7-11. It can probably be seen in the simplest way by reference to the first two equations (A), these being nothing more nor less than an analytic formulation of theorems IV. and VI. If β is in absolute value greater than 1 — whence $$1-\beta^{2}$$ is negative — the transformation of time coordinates from one system to the other gives an imaginary result for the time in one system if the time in the other system is real. Likewise, measurement of length in the direction of motion is imaginary in one system if it is real in the other. Both of these conclusions are absurd and hence the absolute value of β is equal to or less than 1. If it is one, then any length in one system, however short, would be measured in the other as infinite; and a like result holds for time. Hence β is less than 1. But $$\beta=v/c$$, the ratio of the relative velocity of the two systems to that of light. Hence, the velocity of light is a maximum

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38 better comprehension of the highly cultured, whose life seems to be more and more artificial.

But whether or not, in comparison with the average man of any highly cultured community, to a much greater and wider degree uncultured man is himself, and therefore worth studying.

Mr. drew attention to the following extract from Palmer's History of Wrexham.—P. 123.

The gentleman spoken of died in 1883, aged 59. Though an inhabitant of Wrexham for many years, his birthplace was Cefn, near St. Asaph:—

"I have heard the following story of Mr. John Foulkes, sen., that if in his walks a magpie crossed his path, he would make with his stick the sign of a cross upon the ground, and say, 'Devil, I defy thee.'"

Miss communicated the following notes on present Fifth of November observances in the South of England:—

At Hastings I saw placards announcing the grand procession which would pass through the town on the occasion, carrying effigies (if I remember rightly), and winding up with a bonfire.

At Rye I saw similar placards, announcing the intended doings of the "Borough Bonfire Boys", the route to be taken by the procession, and the place determined for the bonfire, in which the effigies would be consumed, and warning all persons against giving anything towards the funds for the bonfires if not solicited by the authorised "Bonfire Boys".

At Folkestone I saw the procession itself, on Monday, the 6th inst. It consisted of carts or waggons (cars they were styled), decorated, and containing tableaux vivants contributed by the different Friendly and other Societies in the town. Thus, the Ancient Order of Druids sent a party


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 which the relative velocity of two systems may approach but can never reach. This may be formulated in the following theorem:

X. The velocity of light is a maximum which the velocity of a material system may approach but can never reach (MVLR).

It should be pointed out that this theorem may also be proved directly from theorem IX, as one can readily show. This fact will be useful in the next section.

§12. Logical Equivalents of the Postulates. — We shall now show that theorem IX. is in a certain sense a logical equivalent of R. From IX. it follows, as we have seen in theorem X., that the velocity of a material body is less than the velocity of light. But the source of light is always a material body; and therefore no light source can have a velocity as great as that of light. Now, the following is a natural hypothesis:

''B. The velocity of the light source cannot add to the velocity of light a greater velocity than that of the source itself. Likewise, the velocity of a system of reference cannot add to the velocity of light a greater velocity than that of the source itself.''

Now, from theorem IX. it follows that if any velocity less than that of light is compounded with that of light the resultant is the velocity of light. Hence, if we assume theorem IX. and postulate B we can conclude as a consequence postulates $$R'$$ and $$R''$$. Hence we have the following result:

XI. ''Postulates (MVLB) and theorem IX. are a logical equivalent of postulates (MVLR).''

That is, in our system of postulates R may be replaced by theorem IX. and postulate B, and the resulting total body of postulates and theorems will be unaltered.

Now theorem IX. was proved by means of formulae (B) alone; and formulae (B) are a direct consequence of formulae (A) above. Hence postulate R may be proved solely from postulate B and formulae (A) if these are assumed to be true. Further, the third and fourth equations in (A) are equivalent to postulate L. We shall show that a special case of the first two formulae (A) is postulate V. Putting x' = 0 we have x/t = v. That is, to an observer on S the system $$S'$$ appears to move with the velocity v. Now the first two equations in ($$A_1$$) may be obtained algebraically by solving the first two in (A). Then in the second equation of ($$A_1$$) put x = 0; thus we have x't' = -v. That is, to an observer on $$S'$$ the system S appears to move with the velocity -v. These two results together constitute our postulate V. Combining the several conclusions thus reached we have the following theorem:

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 XII. Postulates (MB) and formulae (A) are a logical equivalent of postulates (MVLR).

An analysis of the proof of formulae (A) will show that they follow directly from theorems IV., VI., VII. and postulate V. Hence we have the following theorem as a corollary of the preceding:

XIII. ''Postulates (MVB) and theorems IV., VI., VII. are a logical equivalent of postulates (MVLR).''

 * June, 1912.
 * June, 1912.

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Rh of Ancient Britons; the car provided by the Rev. E. Husband's Working Boys' Club represented "Algeria", where Mr. Husband is at present staying; the Mutual Benefit Society's car represented "Labour", as exemplified by a blacksmith at his forge shoeing a live pony. The Butchers' Trade Car (sent, I fear, by one firm only, not by the trade) conveyed a live bullock, with a man with a poleaxe standing by his head. The Fire Brigades also took part in the procession, and so did no less than four fife-and-arum bands. The whole was lighted by torches and Chinese lanterns, and followed a prescribed route through the town, stopping at intervals to collect money, which was given to the Victoria Hospital. I did not get a very good view of it, but I afterwards obtained a printed programme, which gave me the name of the secretary, Mr. C. Buzan, who is employed in a nursery garden at Folkestone, and from whom I learnt the following particulars:—

The 5th November was formerly kept in Folkestone with a great deal of rowdyism, squibbing in the streets, breaking windows, and mischief of all kinds, accompanying the usual carrying of effigies, and burning them in a bonfire on the outskirts of the town. Especially was this the case in the older streets, as High Street and Tontine Street. But I could not learn that the fishing population took any special part, or that there was any feud between them and the landsmen on that occasion. Some five or six years ago an attempt was made by the Friendly Societies of the town to remedy the disorder by organising a joint procession on the lines of the celebration at Eastbourne, which should occupy the hobbledehoys by drawing them to its line of march. They retained the effigies and the bonfire, and paid their expenses and remunerated themselves by the collection made on the way. This only partially succeeded in checking disorder, and when, after November 5th, 1890 (as I understand), there was a difficulty about the accounts, this young man Buzan, and some friends, resolved


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By R. D..

Introduction.
a previous paper I have given an analysis of the characteristic postulates on which the theory of relativity depends and have developed in a general way some of the fundamental conclusions of this theory. The plan of treatment adopted makes it possible to arrive at these results by arguments which are not complicated in form; and the notions which enter into them are of simpler character than those usually to be found in memoirs on this subject.

It seems desirable that the general theory should be further worked out along the same lines, even though this should require in part a repetition of some things already in the memoirs. Accordingly, in the present paper I develop the fundamental properties of mass, force and energy, using as a basis the theorems of my previous paper. In the derivation of these results the theory of electricity is not employed. Consequently the conclusions are of wider applicability than when they are proved by means of electrical theory; in particular, they may be applied to the derivation of results in the theory of electricity itself.

In § 1, after giving some definitions, I state the laws of conservation of momentum and energy and electricity and the principle of least action in the form in which I shall have occasion to use them. The law of conservation of electricity is employed in this paper only in the applications of the results concerning mass and energy.

In § 2 the question of the dependence of mass on velocity is treated. First the transverse mass of a moving body is determined by the elegant method of Lewis and Tolman. The relation between transverse mass and longitudinal mass is found by the method of Bumstead, and thus the longitudinal mass of a moving body is obtained.

In § 3 dimensional equations are employed to derive the relations of acceleration and force in two systems of reference.

In § 4 from considerations concerning the mass of a moving body two essential equivalents of postulate R are determined, each of which

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 Rh other communities in South India, but it is not common. This strange form of it is quite unique. Of course, we have another form of it not far from here—in Ireland—where the feasting seems to be more drinking than eating. Everyone entering the room bends over the corpse and drinks a glass of whisky to begin with.

I have said nothing of the religion of the Kullens. The Brahminical religion, the worship of Siva and Vishnu in their endless forms, incarnations and so on, is very largely influenced by the more primitive cult which underlies it throughout the Presidency. The higher seems to have had little or no effect on the lower, but the lower has had considerable influence over the higher.

As we go south we find the lower and, I believe, earlier cult becoming more distinct, and in the extreme south it is most separated. The profound influence of this lower cult, in the south, cannot be overlooked by the most casual observer. But it is not so easy to say, "it is this," or "it is that," more especially when we are estimating a people whose habit of thought is utterly different from our own.

What is commonly (in my opinion, wrongly) called Demonolatry finds its strongest expression in the southernmost district—Tinnevelly, which, by-the-bye, contains more Christians than the whole of India put together. Every image of a goddess is sword in hand, and doing something bloody; eating her children, usually, or a child in one hand as if being eaten, another at her feet about to be eaten. Perhaps it is in Tinnevelly that the goats have the worst time of it. During certain festivals they are sacrificed by the thousand to the deities of the lower cult.

I thank you all for your kindness in having listened to me. Whether you all believe in the power of folk-lore and the help it promises to give us concerning the earlier part of man's history, you must feel that intimacy with uncultured man is otherwise beneficial and helpful towards


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 furnishes a possible means for the experimental proof or disproof of the theory of relativity. The researches of Bucherer have been thought to afford the requisite experimental confirmation in the first case; this matter is treated in § 5. In § 6 suggestions are given for a new crucial experiment for testing the theory of relativity, this being associated with the second essential equivalent of R in § 4. The writer desires to call especial attention to this proposed experiment.

In § 7 the intimate relation of the mass and the total energy of a body is pointed out and two theoretical means are suggested for determining the velocity of light indirectly, that is, without direct measurement of this velocity. These experiments, if they could be performed with sufficient accuracy, would afford an interesting and striking confirmation of the theory of relativity, provided of course that they turn out according to the predictions of this theory.

A few remarks on the principle of least action are found in § 8, and § 9 is given to some speculative considerations which are intended as brief suggestions of means by which one may represent to himself the conclusions of relativity as natural parts of a consistent view of physical phenomena.

§ 1. .
If m, M and v are respectively the mass, momentum and velocity of a body we shall assume (as in the classical mechanics) that they are connected by a relation of the form

and hence any one of these quantities is determined in terms of the other two (except that mass is not thus determined when velocity and momentum are zero). We shall take mass and velocity to be the two fundamental quantities, so that momentum is defined in terms of them.

Likewise we shall define the kinetic energy E of a moving body by means of the usual relation

Later we shall see that "mass" is variable and is not in general independent of the direction in which it is measured; consequently, we must take for m in the above formulae the mass of the body in the direction of its motion.

We shall take for granted the following laws of conservation of momentum and energy and electricity:

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 $$C_1$$. The sum total of momentum in any isolated system remains unaltered, whatever changes may take place in the system, provided that it is not affected by any forces from without.

$$C_2$$. The sum total of energy in any isolated system remains unaltered, whatever changes may take place in the system, provided that it is not affected by any forces from without.

$$C_3$$. The sum total of electricity in any isolated system remains unaltered, whatever changes may take place in the system, provided that the system as a whole neither receives electricity from nor gives out electricity to bodies not belonging to the system.

The "action" of a moving body in passing from one position to another may be defined as the space integral of the momentum taken over the path of motion. If we denote this action by A we have therefore

Now ds = vdt, so that we have also

If several bodies are involved we have

where the summation is for the various bodies in the system.

We may state the fundamental principle of least action in the following form:

. The free motion of a conservative system between any two given configurations has the property that the action A is a minimum, the admissible values $$\overline{A}$$ of the action with which A is compared being obtained from varied motions in which the total energy has the same constant value as in the actual free motion.

§ 2. .
Suppose that we have two systems of reference $$S_1$$ and $$S_2$$ moving with a relative velocity v. We inquire as to whether, and in what way, the mass of a body as measured on the two systems depends on v. Will a given body have the same measure of mass when that mass is estimated in units of $$S_1$$ and in units of $$S_2$$? And will the mass of a body depend on the direction of its motion by means of which that mass is measured? Our purpose in this section is to answer these two questions.

The two most important directions in which to measure the mass of a body are, first, that perpendicular to the line of relative motion of $$S_1$$ and $$S_2$$ and, secondly, that parallel to this line of motion. For convenience

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 in distinguishing these we shall speak of the "transverse mass" of a body as that with which we have to deal when we are concerned with the motion of the body in a direction perpendicular to the line of relative motion of $$S_1$$ and $$S_2$$; when the motion is parallel to this line we shall speak of the "longitudinal mass" of the body.

Lewis and Tolman determine what they call the "mass of a body in motion," employing for this purpose a very simple and elegant method. This "mass" is what we have just defined as the transverse mass of the body. We employ the excellent method of these authors in deriving the formula for transverse mass.

Suppose that an experimenter A on the system $$S_1$$ constructs a ball $$B_1$$ of some rigid elastic material, with unit volume, and puts it in motion with unit velocity in a direction perpendicular to the line of relative motion of $$S_1$$ and $$S_2$$, the units of measurement employed being those belonging to $$S_1$$. Likewise suppose that an experimenter C on $$S_2$$ constructs a ball $$B_2$$ of the same material, also of unit volume, and puts it in motion with unit velocity in a direction perpendicular to the line of relative motion of $$S_1$$ and $$S_2$$; we suppose that the measurements made by C are with units belonging to $$S_2$$. Assume that the experiment has been so planned that the balls will collide and rebound, over their original paths, the path of each ball being thought of as relative to the system to which it belongs.

Now the relation of the ball $$B_2$$ to the system $$S_1$$ is the same as that of the ball $$B_1$$ to the system $$S_2$$, on account of the perfect symmetry which exists between the two systems of reference in accordance with the results of the previous paper (already referred to). Therefore the change of velocity of $$B_2$$ relative to its starting point on $$S_2$$ as measured by A is equal to the change of velocity of $$B_1$$ relative to its starting point on $$S_1$$ as measured by C. Now velocity is equal to the ratio of distance to time: and in the direction perpendicular to the line of relative motion of the two systems the units of length are equal; but the units of time are unequal. Hence to either of the observers the change of velocity in the two balls, each with respect to its starting point on its own system, will appear to be unequal.

To A the time unit on $$S_2$$ appears to be longer than his own in the ratio $$\sqrt{1-\beta^{2}}:1$$ (see previous paper, theorem IV., p. 167). Hence to A it must appear that the change in velocity of $$B_2$$ relative to its starting point is smaller than that of $$B_1$$ relative to its starting point in the ratio $$\sqrt{1-\beta^{2}}:1$$. But the change in velocity of each ball multiplied by its mass gives its change in momentum. From postulate $$C_1$$ it follows

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 that these two changes of momentum are equal. Hence to A it appears th at the mass of the ball $$B_1$$ is smaller than that of the ball $$B_2$$ in the ratio $$\sqrt{1-\beta^{2}}:1$$.

Similarly, it may be shown that to C it appears that the mass of the ball $$B_2$$ is smaller than that of $$B_1$$ in the ratio $$\sqrt{1-\beta^{2}}:1$$.

From the general theorems concerning the measurement of length (in the theory of relativity) it follows that if the ball which has been constructed by A were transferred to $$C$$'s system it would be impossible for C to distinguish $$A$$'s ball from his own by any considerations of shape and size. Likewise, as A looks at them from his own system he is similarly unable to distinguish them. It is therefore natural to take the mass of $$C$$'s ball as that which $$A$$'s would have if it had the velocity v with respect to $$S_1$$ of the system $$S_2$$. Thus we obtain a relation existing between the mass of a body in motion and at rest.

Now, "mass" as we have measured it above is the transverse mass of our definition. From the argument just carried out we are forced to conclude that the transverse mass of a body in motion depends (in a certain definite way) on the velocity of that motion. The result may be formulated as follows:

I. ''Let $$m_0$$ denote the mass of a body when at rest relative to a system of reference S. When it is moving with a velocity v relative to S denote by $$t\left(m_{v}\right)$$ its transverse mass, that is, its mass in a direction perpendicular to its line of motion. Then we have''

where $$\beta=v/c$$ and c is the velocity of light $$\scriptstyle(MVLRC_{1})$$.

In the statement of this theorem we have tacitly assumed that the mass of a body at rest relative to S, when measured by means of units belonging to S, is independent of the direction in which it is measured. If this assumption were not true we should have a means of detecting the motion of S, a conclusion which is in contradiction to postulate M.

In order to find the longitudinal mass of a moving body we first find the relation which exists between longitudinal mass and transverse mass. We employ for this purpose the elegant method of Bumstead.

Let us as usual consider two systems of reference $$S_1$$ and $$S_2$$ moving with a relative velocity v, observers A and B being stationed on $$S_1$$ and $$S_2$$ respectively. Suppose that B performs the following experiment:

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 He takes a rod of two units length, whose mass is so small as to be negligible, and attaches to its ends two balls of equal mass. Then he suspends this rod by a wire so as to form a torsion pendulum. We assume that the line of relative motion of the two systems is perpendicular to the line of this wire.

Let us consider the period of this torsion pendulum in the two cases when the rod is clamped to the wire so as to be in equilibrium in each of the following two positions: (1) With its length perpendicular to the line of relative motion of $$S_1$$ and $$S_2$$; (2) with its length parallel to this line of motion.

As B observes it the period must be the same in the two cases; for, otherwise, he would have a means of detecting his motion by observations made on his system alone, contrary to postulate M. Then from the relation of time units on $$S_1$$ and $$S_2$$ it follows that the two periods will also appear the same to A. As observed by B the apparent mass of the balls is the same in both cases. We inquire as to how they appear to A. Let $$m_1$$ and $$m_2$$ be the apparent masses, as observed by A, in the first and second cases respectively. It is obvious that $$m_1$$ is the longitudinal mass and $$m_2$$ the transverse mass of the balls in question.

When the pendulum is in motion it appears to B that each ball traces a circular arc. From the relations between units of length in the two systems it follows that to A it appears that the balls trace arcs of an ellipse whose semi-axes are 1 and $$\sqrt{1-\beta^{2}}$$ and lie perpendicular and parallel, respectively, to the line of relative motion of the two systems.

Let us now determine the period of each of these two pendulums as they are observed by A. By equating the expressions for these periods we shall find the relation which exists between $$m_1$$ and $$m_2$$.

Let x and y be the cartesian coordinates of a point as determined by A, the axes of reference being the major and minor axes of the ellipse in which the balls move. Let $$x'$$ and $$y'$$ be the coordinates of the same point as determined by B. Then the circular path of motion, as determined by B, has the equations

$$x'=\cos\theta,\ y'=\sin\theta$$

the angle θ being measured from the major axis of the ellipse. The equations of the ellipse, as determined by A, are

$$x=\cos\theta,\ y=\sqrt{1-\beta^{2}}\sin\theta$$

In the first case — when the rod is perpendicular to the line of relative motion of $$S_1$$ and $$S_2$$ — the amount of twisting in the wire when the ball is in a given position is the absolute value of the corresponding angle θ;

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 and therefore the potential energy is proportional to θ², say that it is $$\tfrac{1}{2}k\theta^{2}$$. Now from the values of y and x above we have

For small oscillations we have x = 1 and tan θ = θ; and therefore

Hence the potential energy is

and the equation of motion of the particle becomes

Hence the period $$T_1$$ of oscillation is

In the second case — when the rod is parallel to the line of relative motion of S_1 and $$S_2$$ — the amount of twisting in the wire for a given position of the balls is the absolute value of $$(\pi/2)-\theta$$. The potential energy is $$\tfrac{1}{2}k\left[(\pi/2)-\theta\right]^{2}$$. We have

For small oscillations we have

Hence the potential energy is $$\tfrac{1}{2}kx^{2}$$ the period $$T_2$$ of oscillation is therefore

Equating the two periods of oscillation found above we have

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 Remembering that $$m_1$$ and $$m_2$$ are the longitudinal mass and transverse mass, respectively, and making use of theorem I., we have the following result:

II. ''Let $$m_0$$ denote the mass of a body when at rest relative to a system of reference S. When it is moving with a velocity v relative to S denote by $$l\left(m_{v}\right)$$ its longitudinal mass, that is, its mass in a direction parallel to its line of motion. Then we have''

$$l\left(m_{v}\right)=\frac{m_{0}}{\left(1-\beta^{2}\right)^{\frac{3}{2}}}$$

where $$\beta=v/c$$ and c is the velocity of light $$\scriptstyle\left(MVLRC_{1}C_{2}\right)$$.

§ 3. .
Denote the fundamental measurable physical entities mass, length and time by M, L and T respectively. Then the definition of derived entities gives rise to the so-called dimensional equations. Thus if B denote velocity, then from the definition of velocity we have the dimensional equation

$$V=\frac{L}{T}$$.

That such equations must be useful in obtaining the relations of a derived unit in two systems of reference is obvious. Thus from the above dimensional equation for V we may at once derive the fundamental result (see previous paper, theorem VI., p. 170) concerning the relation of units of length in the line of relative motion of two systems not at rest relatively to each other. For this purpose it is sufficient to employ postulate V. and theorem IV. of the previous paper. The reader can easily supply the argument. Or, conversely, if one knows the relations which exist between units of length and units of time in two systems one concludes readily to the truth of postulate V.

Likewise, from the dimensional equation

acceleration = $$\frac{L}{T^{2}}$$

one may readily determine the relations which exist between units of acceleration on two systems, it being assumed that the relations of time units and length units are known. Making this assumption, then, the two dimensional equations above give us the following theorem:

III. ''Let two systems $$S_1$$ and $$S_2$$ move with a relative velocity v in the direction of a line l, and let $$\beta=v/c$$ where c is the velocity of light. Then to an observer on $$S_1$$ it appears that the unit of velocity [acceleration] on''

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 $$S_1$$ bears to the unit of velocity [acceleration] on $$S_2$$ the ratio $$1:1\left[1:\sqrt{1-\beta^{2}}\right]$$ or $$1:\sqrt{1-\beta^{2}}\left[1:1-\beta^{2}\right]$$ according as the motion is parallel to I or perpendicular to I (MVLR).

Let us use F to denote force. Then from the dimensional equation

$$F=\frac{ML}{T^{2}}$$,

we shall be able to draw an interesting conclusion concerning the measurement of force.

Suppose that an observer B on a system $$S_2$$ carries out some observations concerning a certain rectilinear motion, measuring the quantities M', L', T', so that he has the equation

$$F'=\frac{M'L'}{T'^{2}}$$.

Another observer A on a system $$S_1$$ (having with respect to $$S_2$$ the velocity v in the line l) measures the same force, calling it F. Required the value of F in terms of $$F'$$, when the motion is parallel to l and when it is perpendicular to l, the estimate being made by A.

When the motion is perpendicular to l — that is, when the force acts in a line perpendicular to l — we have

$$F_{1}=\frac{ML}{T^{2}}=\frac{M'\sqrt{1-\beta^{2}}\cdot L'}{T'^{2}\left(1-\beta^{2}\right)}=\frac{F'}{\sqrt{1-\beta^{2}}}$$

When the motion is parallel to l we have

$$ F_{2} =\frac{ML}{T^{2}} =\frac{M'\left(1-\beta^{2}\right)^{\frac{3}{2}}\cdot L\sqrt{1-\beta^{2}}}{T'^{2}\left(1-\beta^{2}\right)}=\left(1-\beta^{2}\right)F'.$$

These results may be stated in the following theorem:

IV. ''In the same systems of reference as in theorem III., let an observer on $$S_2$$ measure a given force F' in a direction perpendicular to l and in a direction parallel to l, and let $$F_1$$ and $$F_2$$ be the values of this force as measured in the first and second cases respectively by an observer on $$S_1$$. Then we have''

It is obvious that a similar use may be made of the dimensional equation of any derived unit in determining the relation which exists between this unit in two relatively moving systems of reference.

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§ 4. .
The problem of determining sets of postulates logically equivalent to those which have been used as a basis of the theory of relativity is obviously important. So far as a justification of the theory is concerned it is, however, unnecessary to have complete logical equivalents; all that is essential is to find a set of postulates, experimentally demonstrable, by means of which it is possible to demonstrate the characteristic conclusions of relativity concerning the relations of units of time and units of length in two systems of reference. Such a set of postulates we shall call essential equivalents of the postulates of relativity. The object of this section is to determine essential equivalents of postulate R, that is, such postulates as may be taken in connection with postulates M, V, L, so that the new set shall be essentially equivalent to M, V, L, R.

For this purpose let us first consider the relation between the transverse mass of a moving body and its mass at rest as given in theorem I. Let us suppose that this theorem is true (whether proved experimentally or otherwise); and let us seek its consequences. Suppose that the experiment by means of which we proved theorem I. is now repeated. If we again assume the law of conservation of momentum and equate the two observed changes in momenta, it is clear that we shall have a relation between measurements of time as carried out in the two systems of reference, and that this relation will be precisely the same as in the usual theory of relativity. Having this relation concerning time units we can then proceed as in the first paragraph in § 3 to derive the usual relations between units of length. Hence we have the following result:

V. If $$m_0$$ and $$t\left(m_{v}\right)$$ have the same meaning as in theorem I. and if for any particular kind of matter whatever we have the relation

$$t\left(m_{v}\right)=\frac{m_{0}}{\sqrt{1-\beta^{2}}}$$

then this fact and postulates $$\scriptstyle\left(MVLC_{1}\right)$$ form an essential equivalent of postulates $$\scriptstyle\left(MVLRC_{1}\right)$$.

Next, let us suppose that for some particular kind of matter we have the relation

$$t\left(m_{v}\right)=\left(1-\beta^{2}\right)l\left(m_{v}\right)$$

where $$t\left(m_{v}\right)$$ and $$l\left(m_{v}\right)$$ denote the transverse mass and the longitudinal mass, respectively, as above. Then repeat the experiments by means of

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 which we proved theorem II. As before the balls will appear to B to move on arcs of the circle

$$x'=\cos\theta,\ y'=\sin\theta.$$

Suppose that to A they appear to move along arcs of the ellipse

$$x=\cos\theta,\ y=\rho\sin\theta$$

where ρ is a constant to be determined. As before, without the use of postulate R, it may be shown that to A the periods will be the same in the two cases. Then determine the periods as in the preceding discussion. The expressions for the period will contain ρ; in fact on equating them we shall find

$$m_{2}=\rho^{2}m_{1}.$$

But $$m_{1}=l\left(m_{v}\right)$$ and $$m_{2}=t\left(m_{v}\right)$$ whence on account of the relations between $$l\left(m_{v}\right)$$ and $$t\left(m_{v}\right)$$, we have at once

$$\rho=\sqrt{1-\beta^{2}}$$

This, in connection with postulate L, leads readily to the usual relations concerning the units of length in two systems of reference. Having these relations of length units, the dimensional equation

$$V=\frac{L}{T}$$

taken in connection with postulate V leads at once to the usual relation of time units, provided we take the motion along the line of relative motion of the two systems. Hence we have the following theorem:

VI. ''If $$l\left(m_{v}\right)$$ and $$t\left(m_{v}\right)$$ have the same meaning as in theorems L and II. and if for any particular kind of matter whatever we have the relation''

$$t\left(m_{v}\right)=\left(1-\beta^{2}\right)\cdot l\left(m_{v}\right)$$

then this fact and postulates $$\scriptstyle\left(MVLC_{1}C_{2}\right)$$ are essential equivalents of postulates $$\scriptstyle\left(MVLRC_{1}C_{2}\right)$$.

§ 5. .
Our postulates V, L, $$C_1$$ have been universally accepted as part of the basis of the classical mechanics. Many persons have found no difficulty in accepting postulate M; certain it is at least that we have absolutely no evidence to contradict it. We have seen in theorem V, that these four postulates, taken in connection with the formula for transverse mass, form

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 an essential equivalent of $$\scriptstyle\left(MVLRC_{1}\right)$$; in other words, the experimental demonstration of the formula for transverse mass carries with it the experimental proof of the theory of relativity, provided that postulates $$\scriptstyle\left(MVLC_{1}\right)$$ are accepted as experimentally proved.

Bucherer has carried out some investigations which have been supposed to furnish this experimental verification for the formula of transverse mass, and hence for the whole theory of relativity. In order to draw this conclusion from Bucherer's direct results it is necessary to make use of a law which we have not yet employed, namely, the law of conservation of electricity which we have stated as postulate $$C_2$$. Since this law has customarily been accepted and has not yet led to contradictions it should certainly still be supposed to hold. Accepting it, then, we have in Bucherer's results a partial experimental confirmation of the theory of relativity, as we now show.

Bucherer's investigations have to do with the mass of a moving electron. There seems to be no means at hand for a direct measurement of this mass, and Bucherer resorted to the expedient of determining the ratio of charge to mass. Let us denote the charge by e, which we suppose to be constant, in accordance with postulate $$C_3$$. As before let $$m_0$$ and $$t\left(m_{v}\right)$$ denote the mass at rest and the mass when moving with velocity v, of the electron in consideration. Bucherer's experiments were carried out to determine the relation which exists between $$e/m_{0}$$ and $$e/t\left(m_{v}\right)$$. The measurements agreed in a remarkable way, not only as to general characteristics but also as to exact numerical results, with the formula

$$\frac{e}{t\left(m_{v}\right)}=\frac{e\sqrt{1-\beta^{2}}}{m_{0}}$$

Taking this formula as thus experimentally demonstrated we have at once our fundamental relation for transverse mass: $$\sqrt{1-\beta^{2}}t\left(m_{v}\right)=m_{0}$$.

From this it follows that the experimental demonstration of the theory of relativity is complete when we have proved M, V, L, $$C_1$$ and $$C_3$$, provided that one accepts Bucherer's proof of the above relation between $$e/m_{0}$$ and $$e/t\left(m_{v}\right)$$. That is, the essentials of the theory of relativity flow from

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 principles for each of which there is strong experimental confirmation. This important conclusion has often been pointed out.

To the present writer, however, it seems that one point especially should be subjected to further examination. Is it in fact true that the charge of a moving electron is independent of the velocity with which it moves? Let $$e_0$$ be the charge of the electron when at rest and denote by $$t\left(e_{v}\right)$$ its apparent charge when in motion with velocity v, the charge being measured by means of tests in which the line of action is perpendicular to the line of motion of the charge. In the above work we have assumed, in accordance with the usual practice, that $$e_{0}=t\left(e_{v}\right)$$. Suppose however that the true relation were different from this, that, in fact, we have

then Bucherer's experiment would lead to the conclusion that $$t\left(m_{v}\right)=m_{0}$$, and thus the whole theory of relativity would be overturned. Furthermore, if any relation other than $$e_{0}=t\left(e_{v}\right)$$ is the true one, some modification at least of the theory of relativity would have to be made or else one would have to give up postulate $$C_1$$ which asserts the law of conservation of momentum. This result brings to notice the great importance of the question of the constancy of electric charge on the electron. We shall treat this matter further in the next section.

§ 6. .
Just as theorem V. was used for the theoretical basis of Bucherer's (partial) experimental demonstration of the theory of relativity so theorem VI. may be employed as the theoretical basis of a new experimental investigation which has not yet been carried out, one which bears the same essential relation as that of Bucherer to the confirmation or disproof of the entire theory of relativity. The object of this section is to indicate the nature of this experiment.

Let $$e_0$$ denote the charge of an electron when at rest with respect to a given system of reference. When it is in motion with a velocity v let $$t\left(e_{v}\right)$$ and $$l\left(e_{v}\right)$$ be the apparent charge when measured by means of tests whose lines of action are perpendicular and parallel, respectively, to the line of motion of the electron.

If we employ, postulate $$C_3$$ we conclude that $$e_{0}=t\left(e_{v}\right)=l\left(e_{v}\right)$$. We shall first assume the truth of one of these relations, namely, $$t\left(e_{v}\right)=l\left(e_{v}\right)$$, and we shall denote the common value of these two quantities by e. Now let us suppose that some means are found for measuring both the

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 quantities $$e/t\left(m_{v}\right)$$ and $$e/l\left(m_{v}\right)$$, where $$t\left(m_{v}\right)$$ and $$l\left(m_{v}\right)$$ denote as usual the transverse mass and the longitudinal mass respectively of the moving electron, whose velocity is v. Bucherer's methods furnish the means of measuring the first of these ratios; it will be necessary to devise a way to determine the value of the second ratio.

Or, instead of finding a means of measuring the two quantities $$e/t\left(m_{v}\right)$$ and $$e/l\left(m_{v}\right)$$ it will be sufficient if one determines only their ratio, as will be obvious from the discussion following.

Suppose now that we find the relation predicted by the theory of relativity:

This equation leads to the relation $$t\left(m_{v}\right)=\left(1-\beta^{2}\right)\cdot l\left(m_{v}\right)$$. According to theorem VI. this would give a new experimental confirmation of the theory of relativity. The importance of such a result is apparent.

But we should also have more than this. Having now concluded that the theory of relativity is confirmed and this result having been reached without the use of a relation between $$e_0$$ and $$t\left(e_{v}\right)$$ we may now use the experiment of Bucherer to draw further conclusions concerning electric charges in motion. In particular, it is obvious that we should have a proof of the fundamental relation

That is to say, having assumed that $$t\left(e_{v}\right)$$ and $$l\left(e_{v}\right)$$ are equal we conclude further on experimental evidence that each of these is equal to $$e_0$$. Now it is difficult to conceive how $$t\left(e_{v}\right)$$ and $$l\left(e_{v}\right)$$ could be different, for this would imply that the notion of electric charge is in need of essential modification. In fact, if the charged body is moving, the notion of charge would be indefinite in meaning until we had assigned the direction along which such charge is to be measured. Thus, if the experiment should turn out as surmised above, we should not only have the strongest sort of experimental confirmation of the theory of relativity but we should also have a valuable verification of the fact that an electric charge does not vary in amount with the velocity of the body which carries it.

Suppose, on the other hand, that we make no assumption concerning the relation of $$t\left(e_{v}\right)$$ and $$l\left(e_{v}\right)$$ or of $$t\left(m_{v}\right)$$ and $$l\left(m_{v}\right)$$. On carrying out the experiments a relation of the form

will be obtained where k is a constant or a variable depending on v. If

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 it is found that k is different from unity we shall be forced to the conclusion that either our conception of mass in the classical mechanics or our conception of charge in the classical electrical theory is in need of essential modification. Again, if k = I and if we assume, as is natural, that $$t\left(e_{v}\right)=l\left(e_{v}\right)$$ then we have an experimental disproof of the theory of relativity. In fact we have such a disproof unless $$k=1/\left(1-\beta^{2}\right)$$, provided of course that we assume $$t\left(e_{v}\right)=l\left(e_{v}\right)$$.

From these remarks it is obvious that, whatever may be the result of the experiments, they will certainly lead to important conclusions of a fundamental nature; that is, we have here a crucial experiment, one that cannot fail to lead somewhither. It is to be hoped that some laboratory worker will soon perform the requisite experiments; the writer, who is a mathematician, can only regret that he cannot conveniently carry out the work himself.

A variation of the experiment of Bucherer would seem to be sufficient for the purpose here. Bucherer's results were obtained by subjecting the moving electron to a magnetic field and also to an electric field each at right angles to the line of motion. A variation of the direction of these fields relative to the line of motion of the electron would probably afford a means of making the necessary measurements for the experimental proof of the relations requisite for use in the preceding discussion.

§ 7. .
If, as is frequently done, we employ for the definition of the kinetic energy E the relation (compare §1)

it is clear that for the mass m we should take the longitudinal mass $$l\left(m_{v}\right)$$. Then let $$m_0$$ denote the mass of the body at rest, $$E_0$$ its energy when at rest (that is, the energy due to its internal activity), and $$E_v$$ its energy when moving at the velocity v. Then clearly $$E=E_{v}-E_{0}$$, so that in view of theorem II. we have

whence, on integration, we have

Hence for the kinetic energy of a moving body we have

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 or, to a first approximation only,

$$E=\frac{1}{2}m_{0}v^{2}$$

Therefore the usual formula for kinetic energy in the classical mechanics is only a first approximation.

Since relation (1) is to be true for all velocities v it is obvious that we have

$$E_{v}=\frac{m_{0}c^{2}}{\sqrt{1-\beta^{2}}}+k,\ E_{0}=m_{0}c^{2}+k,$$

where k is a constant, that is, a quantity independent of v. From the first of these equations we conclude further that

$$E_{v}=c^{2}\cdot t\left(m_{v}\right)+k$$

so that the total energy of a body, decreased by the constant k, is directly proportional to its transverse mass. In case the body is at rest its mass in one direction is the same as in another; hence $$m_{0}=t\left(m_{0}\right)$$. Bearing this in mind we have the following theorem:

VII. ''Let $$m_0$$ be the mass of a body when at rest with respect to a given system of reference and let $$t\left(m_{v}\right)$$ denote its transverse mass when it is moving with the velocity v (the case v = 0 is not excluded). Then the total energy $$E_v$$ which it possesses is $$c^{2}t\left(m_{v}\right)+k$$, where k is a constant.''

The following relations are immediate consequences of equations written out above:

$$\frac{E_{v}-E_{0}}{t\left(m_{v}\right)-m_{0}}=c^{2},\ E_{v}-k=\frac{E_{0}-k}{\sqrt{1-\beta^{2}}}$$

Now, suppose that an experimenter contributes to a body which is at rest a known amount of energy and determines the velocity which this causes the body to acquire. If the two measurements are made with sufficient accuracy one will be able, by substituting the results in the first of the above equations, to determine in this way the velocity of light. Actually to carry out this remarkable method for measuring c would doubtless be very difficult; but the obvious great importance of the result is certainly such as to justify a careful consideration of the problem. If the value of c determined in this way should agree well with its value as otherwise found, this would give us an interesting confirmation of the theory of relativity.

Let us consider the mass of a rotating top, the mass being measured along the axis of rotation. According to our results this mass should be different from that of the same top when at rest, and the difference should bear a definite relation to the amount of energy which is involved

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 in the rotation. If the measurements here involved could be made with sufficient accuracy we would have another means, independent of light itself, for the measurement of the light-velocity c. Again, this experiment would afford us a measure of transverse mass and in that way could lead to a confirmation of the theory of relativity, provided that we assume c as known from independent measurements; and this confirmation, it is to be noticed, would be independent of electrical considerations.

It seems to be impossible to determine the constant k which enters into the above discussion. But in the absence of any evidence to the contrary it would appear natural tentatively to assume that k is zero. On the basis of this assumption we should have the following remarkable conclusions: The mass of a body at rest is simply the measure of its internal energy. The transverse mass of a body in motion is the measure of its internal energy and its kinetic energy taken together. Its longitudinal mass is its total energy multiplied by a simple factor. (The longitudinal mass, therefore, bears a simple ratio to the total energy.) One can hardly resist the conclusion that the transverse mass of a body depends entirely on its energy, and therefore that matter is merely one manifestation of energy.

§ 8. .
In the preceding section we have seen that in the theory of relativity the classical formula $$E=\tfrac{1}{2}mv^{2}$$ for the measure of kinetic energy is true only as a first approximation. This is due to the fact that mass is a variable quantity. But the conclusion does not appear to necessitate our surrender of the law of conservation of energy.

The same causes which lead to a modification in the formula for E will also require a corresponding modification in the value of the action A as defined in § I. The question arises as to whether the principle of least action is left intact. I cannot enter upon the investigation here; but the problem seems to me to be of importance, and consequently I am stating it in the hope that some one will be led to consider its solution.

Undoubtedly the principle of least action is one which should be given up only when there are strong reasons for it. It is a mathematical formulation of the law that nature accomplishes her ends with the least possible expenditure of labor, so to speak. Certainly this law is one which appeals to our minds with strong force. There is something about it which is aesthetically satisfying in a high degree. It seems to me, however, that a fresh study of it should be made in the light of the theory of relativity.

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§ 9. Speculative Considerations.
From some results in the preceding discussion it has appeared that the transverse mass of a body is merely a manifestation of its total energy, that it is in fact a measure of that energy. It is then natural to suppose, on the other hand, that anything which possesses energy has mass; and we thus conceive of mass and energy as coextensive.

Now a beam of light possesses energy; whence we conclude naturally that it also has mass. But we have seen that no "material body" can have a velocity as great as that of light. How are these two facts to be reconciled? If we define "matter" as that which possesses mass (and this is probably the best definition) we shall perhaps best be able to represent to ourselves the nature of matter if we think of it as a strain in the ether. Then the two facts which we have to reconcile would be entirely consistent if we suppose that the beam of light sets up a strain in the ether (whence its mass) but that this strain as a whole is not propagated with the velocity of light. In fact, if it moves at all it is probably with a velocity relatively much smaller than that of light.

Again, if mass is merely a manifestation of energy in the form of a strain in the ether it would follow that gravitation is simply an interaction among these several strains. A strain principally localized in one place would have lines of strain going out from it in all directions, and the action of these lines of strain upon one another would afford the effective means by which gravitation acts.

Whether these strains should be thought of as static in the ether or as due to the relative motion of the parts of the ether would probably be determined differently by different minds. If one inclines to the latter form of representation, the necessary movement might be conceived of as vortex whorls in the ether, setting up strains and lines of strain by aid of which we are able to interpret observed phenomena.
 * November, 1912.
 * November, 1912.

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By R. D..

I. Introduction.
who look on physics from the outside not infrequently have the feeling that it has forgotten some of its philosophical foundations. And among physicists themselves this condition of their science has not entirely escaped notice.

The physicist who, above all other men, has to deal with space and time has fallen into conventions concerning them of which he is often not aware. It may be true that these conventions are exactly the ones which he should make. It is certain, however, that they should be made only by one who is fully conscious of their nature as conventions and not as absolute realities beyond the power of the investigator to modify.

Likewise, a question arises as to what element of convention is involved in our usual conceptions of mass, energy, etc.; that the question is not easily answered will become apparent on reflection.

These and many other considerations suggest the desirability of a fresh analysis of the foundations of physical science. Now it is a ground of gratulation for all those interested in this matter that there has arisen within modern physics itself a new movement capable of contributing most effectively to the construction of a more satisfactory foundation for its superstructure of theory. I refer to the recent widespread interest in the principle of relativity and the rapid developments to which it has led.

It is at once admitted that the theory of relativity is not yet established on an experimental basis which is satisfactory to all persons; in fact, some of those who dispute its claim to acceptance are among the most

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 eminent men of science of the present time. On the other hand there is a large and growing body of workers who are rapidly pushing forward investigations the inspiration for which is afforded by the theory of relativity.

This state of affairs will probably give rise to a considerable controversial literature. If the outcome of this controversy is the acceptance in the main of the theory of relativity, then this theory will afford just the means needed to arouse in investigators in the field of physics a lively sense of the philosophical foundations of their science. If the conclusions of relativity are refuted this will probably be done by a careful study of the foundations of physical science and a penetrating analysis of the grounds of our confidence in the conclusions which it reaches. This of itself will be sufficient to correct the present tendency to forget the philosophical basis of the science.

It follows that in any event the theory of relativity is certain to force a fresh study of the foundations of physical theory. If it accomplishes no more than this it will have done well.

The object of this paper is to point out some of the more immediate and fundamental conclusions of a philosophical nature to which the theory of relativity gives rise, especially those related to the foundations of physics. An attempt is made to present the discussion in such form as to be intelligible to one who is unacquainted with the theory of relativity; this is done for the convenience of those who have not hitherto been interested in the matter. It will probably be found desirable, however, to have as much previous knowledge of the subject as may be obtained from Comstock's paper in Science, N. S., 31 (1910), 767-772.

II. The Fundamental Basis of Relativity.
The fundamental basis of the theory of relativity may be set forth in a few statements each of which is a generalization from experiment. In so far as these statements are generalizations they are of course unproved experimentally. But they are nevertheless supposed to be the natural teachings of experiment; and as such are to be accepted with a good degree of confidence unless it can be shown that they are in contradiction to other known experimental facts.

The first of these statements gives expression to a law which is suggested by the famous Michelson-Morley experiment, the object of which was to try out a certain theory concerning the ether by ascertaining

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 whether a result predicted by it would be found by actual test. For a long time it had been supposed that the known facts about light, electricity and magnetism required for their explanation the theory that the ether of space is stationary. Such a conclusion led to the belief that the velocity of the earth through the ether could be determined by optical experiments. Thus it was predicted that the time which would be required for a beam of light to pass a given distance and return would be different in the two cases when the path of light was parallel to the direction of the motion and when it was perpendicular to this direction. The object of the Michelson-Morley experiment, as we have said, was to put this prediction to a crucial test.

The experiment was a bold one, seeing that the velocities to be measured were so little different; and yet it was carried out in such a brilliant way as to permit no serious doubt of the accuracy of the results. The difference of velocity predicted by theory was found by experiment not to exist; there was not the slightest difference of time in the passage of light along two paths of equal length, one in a direction parallel to the earth's motion and the other in a direction perpendicular to it.

There are different points of view from which one may look at this experiment. In the theory of relativity it is taken in the light of an attempt to detect the earth's motion through space by means of the effect of this motion on terrestrial phenomena. So far as the experiment goes, it indicates that such motion cannot be detected in this way. Furthermore, no one has yet been able to devise an experiment by means of which the earth's motion through space can be detected by observations made on the earth alone. The question arises: Is it possible to have any such experiment at all? In the theory of relativity this question is answered in the negative. The Michelson-Morley experiment and other experiments are thus generalized into one of the fundamental laws or postulates of relativity, which may be stated as follows:

A. The uniform translatory motion of any system cannot be detected by an observer traveling with the system and making observations on it alone.

Another part of the fundamental basis of the theory of relativity is a principle which has long been familiar in the theory of light and has never been found in disagreement with experimental facts. It may be stated thus:

B. The velocity of light in free space is independent of the velocity of the source of light.

By means of laws A and B, taken in connection with certain principles universally accepted in the classical mechanics, it may be shown that

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 the velocity of light is independent of the direction of motion of the observer. Thus we have:

C.. The velocity of light in free space as measured by any observer is independent of the direction of motion of that observer.

We are thus led to inquire further as to whether the velocity of light in free space is independent of the absolute value of the velocity of the observer. It appears to be impossible to prove that it is so independent; and yet one is unable to conceive of any way in which it could be dependent on the absolute value without at the same time depending on the direction of the velocity also. But, accepting A and B, we prove C which asserts that such dependence does not exist. There seems, then, nothing left to do but to assume that dependence on absolute value does not exist; and this is what is done in the theory of relativity. In the absence of experimental information in contradiction to this assumption it is undoubtedly the natural one to make. Any other procedure would be out of harmony with the usual methods of science. This assumption, therefore, is one which we make as the most natural teaching of experimental facts; and as such we treat it as a "law of nature" so long as fresh experiment does not invalidate it. This law may be stated as follows:

D. The velocity of light in free space as measured by any observer is independent of the absolute value of the velocity of the observer.

The law which we have stated as A above is often referred to as the first postulate of relativity. B, C and D taken together constitute the second postulate of relativity. We have broken this postulate into parts in order to state clearly just how it depends on experiment. Combining the parts we may state its essential content as follows:

E. The velocity of light is independent of the relative velocity of the source of light and the observer.

The theory of relativity consists of those conclusions which can be derived by logical process (that is, mathematically) from A and E in conjunction with certain principles which are universally accepted in the classical mechanics — at least, this may be taken as a tentative definition of the theory of relativity. If one agrees that experiment has been properly generalized in A, B and D one has therefore the alternative of accepting the conclusions of relativity or of giving up almost the whole of the usual system of mechanics. That one should take the first horn of the dilemma hardly admits question.

III. Fundamental Conclusions of the Theory of Relativity.
For the sake of convenience in stating some of the fundamental conclusions of the theory of relativity let us suppose that we have two observers

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 A and B placed on platforms moving with respect to each other. Let us suppose that A is on a platform denoted by $$S_1$$ and that B is on a platform denoted by $$S_2$$. Suppose that to A the platform $$S_2$$ appears to move with the velocity v in the direction indicated by the arrow at $$S_2$$; then to B the platform $$S_1$$ will appear to move with velocity v in the direction indicated by the arrow at $$S_2$$. Suppose further that the units of length employed by A and B are such that they arrive at the same numerical results in measuring the length MN, MN being perpendicular to the line of relative motion of $$S_1$$ and $$S_2$$. The platform $$S_1$$ and the instruments employed by A for measuring time and length will be spoken of as the system of reference $$S_1$$. Similarly, we shall speak of the system of reference $$S_1$$. For convenience we shall use β to denote the ratio v/c, where c is the velocity of light.

The two systems of reference $$S_1$$ and $$S_1$$ being thus defined, the question arises as to how the units of length and of time of $$S_1$$ are related to the corresponding units of $$S_2$$. If we accept as true the laws stated in A and E of the preceding section — as, in fact, is done in the theory of relativity — we are led to some very remarkable conclusions. We state first the relation of the time units:

To an observer A on $$S_1$$i the time unit of $$S_1$$ appears to be in the ratio $$\sqrt{1-\beta^{2}}:1$$ to that of $$S_2$$, while to an observer B on $$S_2$$ the time unit of $$S_2$$ appears to be in the ratio $$\sqrt{1-\beta^{2}}:1$$ to that of $$S_1$$.

Thus if A compares his clocks with those of B it will appear to A that $$A$$'s clocks are running faster than $$B$$'s clocks; on the other hand, if B compares his clocks with those of A it will appear to B that $$B$$'s clocks are running faster than $$A$$'s clocks. Thus the two observers are in hopeless disagreement as to the measurement of time. An analysis of this divergence is made below in section V.

Another matter of fundamental importance in the measurement of time will be brought to attention by the question: When are two events which happen at different places to be considered simultaneous? The nature of the difficulty can be seen from the following result in the theory of relativity, an analysis of which is to be found below in section VI.:

If an observer A on $$S_1$$ places two clocks at a distance d apart in a line parallel to the line of relative motion of $$S_1$$ and $$S_2$$ (say at P and Q respectively) and adjusts them so that they appear to him to mark the same time: then to an observer B on $$S_2$$ the clock on $$S_1$$ which is forward in point of motion appears to be behind in point of time by the amount

$$\frac{v}{c^{2}}\frac{d}{\sqrt{1-\beta^{2}}}$$.

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 Let us consider the units of length in $$S_1$$ and $$S_2$$. If A and B make measurements of length in the direction MN their results are in perfect agreement in the theory of relativity as in the classical mechanics; but the state of matters is very different when measurement is made in a line parallel to the line of relative motion of $$S_1$$ and $$S_2$$, as one sees from the following theorem:

''Let l denote a line parallel to the line of relative motion of $$S_1$$ and $$S_2$$. Then to an ob server A on $$S_1$$ the unit of length of $$S_1$$ along l appears to be in the ratio $$\sqrt{1-\beta^{2}}:1$$ to that of $$S_2$$ while to an observer on $$S_2$$ the unit of length of $$S_2$$ along l appears to be in the ratio $$\sqrt{1-\beta^{2}}:1$$ to that of $$S_1$$.''

Thus it appears that when A and B are measuring length in a line parallel to their line of relative motion they are in hopeless disagreement. What this result signifies we shall attempt to explain in section IV.

From the results stated above it may be shown that:

The velocity of light is a maximum which the velocity of a material body may approach but can never reach.

If one brings into consideration the mass of a moving body another result, essentially equivalent to that just given, may also be obtained; this may be stated as follows (see further discussion in section VIII. below) :

No finite force is sufficient to give a material particle a velocity as great as that of light.

In classical mechanics it is customary to assume that the mass of a given body is constant and that it is independent of the direction in which the mass is measured. But if the principle of relativity is accepted it follows that neither of these conclusions is valid. It turns out that the mass of a body depends on its velocity and also on the direction, relative to the line of motion of the body, along which that mass is measured. For convenience in distinguishing these measurements of mass we shall speak of the "transverse mass" as that with which we must reckon when we consider motion in a line perpendicular to the line of relative motion of the systems $$S_1$$ and $$S_2$$; when the motion is parallel to this line we shall speak of the "longitudinal mass" of the body.

The general conclusions of the theory of relativity concerning mass may now be stated as follows:

''Let $$m_0$$ denote the mass of a body when at rest relative to a system of reference S. When it is moving with a velocity v relative to S, denote by $$t_v$$ its transverse mass, that is, its mass in a line perpendicular to its line of motion. Similarly, denote by $$l_v$$ its longitudinal mass. Then we have''

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IV. The Notion of Length.
In the preceding section we saw that two observers A and B on relatively moving systems of reference $$S_1$$ and $$S_2$$ respectively are in disagreement as to units of length along a line l parallel to their line of relative motion. This disagreement is of a very peculiar character. To A it appears that $$B$$'s units are longer than his own. On the other hand, it seems to B that his units are shorter than $$A$$'s. In the two cases the apparent ratio is the same; more precisely, the unit which appears to either observer to be the shorter seems to him to have the ratio $$\sqrt{1-\beta^{2}}:1$$ to that which appears to him to be the longer. Although they are thus in disagreement there is yet a certain symmetry in the way in which their opinions diverge.

Let us suppose that these two observers now undertake to bring themselves into a closer agreement in measurements of length along the line l. Suppose that B agrees arbitrarily to shorten his unit so that it will appear to A that the units of A and B are of the same length. Then, so far as A is concerned, all difficulty has disappeared. How is B affected by this change? We see that the difficulty which he experienced is not disposed of; on the other hand it is greater than before. Already, it seemed to him that his unit was shorter than $$A$$'s. Now, since he has shortened his unit, the divergence appears to him to be increased. Moreover, the symmetry which we found in the former case is now absent.

Furthermore, if any other changes in the units of A and B are made we shall always find difficulties as great as or greater than those which we encountered in the initial case. There is no other conclusion than this: We are face to face with an essential difficulty — one that is not to be removed by any mere artifice. What account of it shall we render to ourselves?

This much is already obvious: The length of an object is not an absolute something; it depends upon the measurer in an essential way.

We have just spoken of the length of an object, a material object. One can hardly refrain from raising the question of the abstract notion of length as apart from any material thing having length. But this problem has new difficulties of its own. All lengths of which I have experience are lengths of material objects or lengths between material

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 objects. Can one formulate in the mind the notion of length as apart from these material things which are measured; and, if so, how can such a notion of length be applied experimentally to the measurement of material things?

Questions of this nature we shall lay aside and shall return to the consideration of the length of material objects.

We have certain intuitive notions concerning the nature of matter which it is necessary for us to examine now. We have usually supposed that to revolve a steel bar, for instance, through an angle of ninety degrees has no effect upon its length. Let us suppose for the moment that this is not so; but that the bar is shorter when pointing in some directions than in others, so that its length is the product of two factors one of which is its length in a certain initial position and the other of which is a function of the direction in which the body points relative to that in the initial position. Suppose that at the same time all other objects experience precisely the same change for varying directions. It is obvious that in this case we should have no means of ascertaining this dependence of length upon the direction in which the body points.

To an observer placed in a situation like this it would be natural to assume that the length of the steel bar is the same in all directions. In other words, in arriving at his definition of length he would make certain conventions to suit his convenience.

Now suppose that the system of such an observer is set in motion with a uniform velocity v relative to the previous state of the system; and that at the same time all bodies on his system undergo simultaneously a continuous dilatation or contraction. This observer would have no means of ascertaining that fact; and accordingly he would suppose that his steel bar had the same length as before. In other words, he would unconsciously introduce a new convention concerning his measurement of length.

There is no a priori reason why our actual universe should not be such as the hypothetical one just described. To suppose it so unless our experience demands such a supposition would be unnatural; because it would introduce an unnecessary inconvenience. But suppose that in our growing knowledge of the universe there should come a time when we could more conveniently represent to ourselves the actual facts of our experience by supposing that all material things are subject to some such deformations as those which we have indicated above; there is certainly no a priori reason why we should not conclude that such is the essential nature of the structure of the universe.

Naturally we would not come to this conclusion without due consideration.

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 We would first enquire earnestly if there fa not some more convenient way by which we can reconcile all experimental facts; and only in the event of a failure to find such a way would we be willing to so profoundly modify our views of the material world.

Now, if we agree to suppose that our actual universe is subject to a certain (appropriately defined) deformation of the general type discussed above it would follow that observers A and B on the respective systems $$S_1$$ and $$S_1$$ would be in just such disagreement as to units of length as that which exists, according to the theory of relativity. Therefore, that which at the outset seemed to be of such essential difficulty is easily enough explained, if we are willing to modify so profoundly our conception of the nature of material bodies.

Whether in the present state of science experimental facts demand such a radical procedure is a question which will be answered differently by different minds. To one who accepts the postulates of relativity there is indeed no other recourse; one who refuses to accept them must find some other satisfactory way to account for experimental facts. The Lorentz theory of electrons gives striking evidence in favor of supposing that matter is subject to some such deformations as those mentioned above; and this evidence is the more important and interesting in that the deformations (as conceived in this theory) were assumed to exist simply in order to be able to account directly for experimental facts.

V. The Measurement of Time.
That two observers in relative motion are in hopeless disagreement as to the measurement of length in their line of relative motion is a conclusion which is probably (at first) sufficiently disconcerting to most of us; and some no doubt have the feeling that any explanation of it so far offered is at best artificial and does not reach the root of the matter. But it is an even greater shock to intuition to conclude, as we are forced to do according to the theory of relativity, that there is a like ineradicable disagreement in the measurement of time. A discussion similar to that in the preceding section brings out the fact that our observers A and B cannot possibly arrive at consistent means of measuring intervals of time. The treatment is so far similar to the preceding discussion for length that we need not repeat it; we shall content ourselves with a brief discussion of conclusions to be drawn from the matter.

Why is this inability of A and B to agree in measuring time received

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 in our minds with such a distinct feeling of surprise and shock? It is doubtless because we have such a lively sense of the passage of time. It seems to be a thing which we know directly, and the conclusion in question is contrary to our unsophisticated intuition concerning the nature of time.

But what is it that we know directly? We have an immediate perception of what it is for two conscious phenomena to coexist in our mind; and consequently we perceive immediately the simultaneity of events in our minds. Further, we have a perfectly clear sense of the order of succession of events in our own consciousness. Is that not all that we know directly?

The difficulties which A and B experience in correlating their measurements of time grow out of two things, of neither of which we have direct perception.

In the first place there are two consciousnesses involved; and what reason have we to suppose that succession of events is the same for these two? This question we shall not treat, assuming that the principal matter can be put into such impersonal form as to obviate this difficulty altogether. (As a matter of fact, so far as anything characteristic of the theory of relativity is concerned this can be done.)

The other difficulty has to do with the measurement of time as opposed to the mere psychological experience of its passage. In this matter we are absolutely without any direct intuition to guide us. We have no immediate sense of the equality of two intervals of time. Therefore, whatever definition we employ for such equality will necessarily have in it an important element of convention. To keep this well in mind will facilitate our discussion.

Our problem is this: How shall we assign a numerical measure of length to a given time-interval; say to an interval in which a given physical phenomenon takes place? We shall arrive at the answer by asking another question: Why should we seek to measure time-intervals at all, seeing that we have no immediate consciousness of the equality of such intervals? There can be only one answer: we seek to measure time as a matter of convenience to us in representing to ourselves our experiences and the phenomena of which we are witnesses. In such a way we can render to ourselves a better account of the world in which we live and of our relation to it.

Now, since our only reason for attempting to measure time is in a matter of convenience, the way in which we measure it will be determined by the dictates of that convenience. The system of time measurement which we shall adopt is just that system by means of which the laws of nature may be stated in the simplest form for our comprehension.

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 Let us return to the case of the two observers A and B of section III. Suppose that each of them has chosen a system of measuring time that suits his convenience in the interpretation of the laws of nature on his system. There is no a priori reason why the two observers should measure time-intervals in the same way. In fact, since there is an arbitrary element in the case of each method of measurement and since the two systems are in a state of relative motion, it is not at all unnatural that the units of A and B should differ.

Now it is to be noticed that each of the observers A and B is in just the situation in which we find ourselves. We have chosen a method of measuring time which seems to us convenient. In so far as that method depends on convenience it is relative to us who are observers, and therefore it has in it something which is arbitrary. There is no doubt that it would be desirable for us to know what it is which is arbitrary, which is relative to us who observe; but it is equally obvious that it must be difficult for us to determine what this arbitrary element is.

The theory of relativity makes a contribution to the solution of this problem. We suppose that two observers on different systems find the laws of nature the same as we find them; or, more exactly, we suppose that they find certain specific laws the same as we find them. Then we inquire as to their agreement in measuring time and see that they differ in a certain definite way. This difference is due to things which are relative to the two observers; and thus we begin to get some insight into the ultimate basis of our own method of measurement.

The matter will become clearer if we speak of the simultaneity of events which happen at different places; and therefore we turn to a discussion of this topic.

VI. Simultaneity of Events Happening at Different Places.
What shall we mean by saying that two events which happen at different places are simultaneous?

First of all it should be noticed that we have no direct sense of what this statement should mean. I have a direct perception of the simultaneity of two events in my own consciousness. I consider them simultaneous because they are so interlocked that I cannot separate them without mutilating them. If two things happen which are far removed from each other I do not have a direct perception of both of them in such way that I perceive them as simultaneous. When should I consider such events to be simultaneous?

To answer this question we are forced to the same considerations as those which we met in the preceding section. There can be no absolute

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 criterion by which we shall be able to fix upon a definition as the only appropriate one. We must be guided by the demands of convenience, and by this alone.

In view of these considerations there is nothing unthinkable about the conclusion (in the theory of relativity) concerning simultaneity which we have given in section III. An observer A on one system of reference regulates clocks so that they appear to him to be simultaneous. It is apparent that to him the notion of simultaneity appears to be entirely independent of position in space. His clocks, even though they are separated by space, appear to him to be running together, that is, to be together in a sense which is entirely independent of all considerations of space.

But when B from another system of reference observes the clocks of $$A$$'s system they do not appear to him to be marking simultaneously the same hour; and their lack of agreement is proportional to their distance apart, the factor of proportionality being a function of the relative velocity of the two systems.

Thus instants of time at different places which appear to A to be simultaneous in a sense which is entirely independent of all considerations of space appear to B in a very different light; namely, as if they were different instants of time, one preceding the other by an amount directly proportional to the distance between the points in space at which events occur which mark these instants. Even the order of succession of events is in certain cases different for the two observers, as one can readily verify.

It thus appears that the notion of simultaneity is relative to the system on which it is determined. In other words, there is no such thing as the absolute simultaneity of events which happen at different places. The only meaning which simultaneity can have is that which is given to it by convention.

Remark. — The difficulties which we have investigated in this and the preceding section are not peculiar to the theory of relativity. The fact that the quantitative measurement of time is relative to the observer who measures it has been insisted upon by philosophers, notably by Poincaré in his Value of Science already referred to. The considerations on which these conclusions have been based have been largely of a speculative character.

The interest of the subject from the point of view of the theory of relativity is that here we have an experimental basis for the conclusions which were previously reached by speculative considerations. Starting out from certain laws which we have (tentatively) accepted as demonstrated by experiment we have by logical processes alone reached these

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 remarkable conclusions concerning the nature of time as conceived by us. Thus we have experimental demonstration of important results previously reached only by speculative considerations.

In the very nature of things speculation must often outrun experiment. The philosopher vaguely and boldly conceives a truth which it requires years of patient labor to establish on a firm and satisfactory foundation.

It is one of the glories of modern science that things hitherto of a speculative nature are being brought under the domain of experimental fact. In the process many speculations are overturned and thrown to the winds; but that which is really of value we may believe will always be safe and will some day find its justification in results achieved in the laboratory.

VII. Time as a Fourth Dimension.
I have no intention of asserting that time is a fourth dimension of space in the sense in which we ordinarily employ the word "dimension"; such a statement would have no meaning. I wish to point out rather that it is in some measure connected with space, and that in many formulae it must enter as it would if it were essentially and only a fourth dimension.

This will come out clearly if we consider the Einstein formulae of transformation from one system of reference to another. These have been worked out in detail in several different places, and the result will be assumed here. It may be stated as follows:

Let us consider three mutually perpendicular axes Ox, Oy, Oz of which Ox is in the line of relative motion of two systems of reference S and $$S'$$. Likewise let Ox', Oy', Oz' be three mutually perpendicular axes parallel respectively to Ox, Oy, Oz. Let the first be fixed to S and the second to $$S'$$. At time t = 0 let O and $$O'$$ coincide. Then if t, x, y, z; t', x', y', z' are the time and space coordinates on S and $$S'$$ respectively, we have

$$\begin{array}{l} t'=\frac{1}{\sqrt{1-\beta^{2}}}\left(t-\frac{v}{c^{2}}x\right),\\ x'=\frac{1}{\sqrt{1-\beta^{2}}}\left(x-vt\right),\\ y'=y,\\ z'=z,\end{array}$$

where c is the velocity of light, v the relative velocity of S and $$S'$$, and $$\beta=v/c$$.

In these formulae the time variable t enters in a way precisely analogous to that in which the space variables x, y, z enter.

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 Suppose now that the law of some phenomenon as observed on $$S'$$ is given by the equation

and we desire to know the expression of this law on S. We substitute for x', y', z', t', their values in terms of x, y, z, t given above; and thus obtain an equation stating the law in question.

From these considerations it appears that in many of our problems, namely, in those which have to do at once with two or more systems of reference, the time and space variables taken together play the rôle of four variables each having to do with one dimension of a four-dimensional continuum.

This conclusion alone raises philosophical questions of profound importance concerning the nature of space and time; but into these we cannot enter here.

VIII. A Maximum Velocity for Material Bodies.
There are several ways by which it may be shown that a material body cannot have a velocity as great as that of light. One of the simplest is that which comes from a consideration of mass. Let us consider the equation

where $$m_0$$ is the mass of a body at rest relative to a given system of reference S, $$l_v$$ is the longitudinal mass of the body moving with a velocity v with respect to S. If we consider larger and larger values of the velocity v we see that $$l_v$$ increases and becomes infinite as v approaches c. This is equivalent to saying that the longitudinal mass of any material body becomes infinite as the velocity of that body approaches c. Therefore it would require an infinite force to give to a material body the velocity c; that is, c is a maximum velocity which the velocity of a material body may approach but can never reach.

We may obtain the same result in a different way, and thus arrive at a deeper understanding of the matter. From the first formula of transformation given in the preceding section we have

Now suppose that v/c > 1 and that t and x are real. Then $$t'$$ is imaginary. Hence, if two systems have a relative velocity greater than that of light, the time measurement in one of them is expressed as an imaginary

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