text:
Page:SahaElectrodynamics.djvu/5

 We assume that this definition of synchronism is possible without involving any inconsistency, for any number of points, therefore the following relations hold :—

1. If the clock at B be synchronous with the clock at A, then the clock at A is synchronous with the clock at B.

2. If the clock at A as well as the clock at B are both synchronous with the clock at C, then the clocks at A and B are synchronous.

Thus with the help of certain physical experiences, we have established what we understand when we speak of clocks at rest at different stations, and synchronous with one another ; and thereby we have arrived at a definition of synchronism and time.

In accordance with experience we shall assume that the magnitude

We have defined time essentially with a clock at rest in a stationary system. On account of its adaptability to the stationary system, we call the time defined in this way as "time of the stationary system."

§ 2. On the Relativity of Length and Time.
The following reflections are based on the Principle of Relativity and on the Principle of Constancy of the velocity of light, both of which we define in the following way :—

1. The laws according to which the nature of physical systems alter are independent of the manner in which these changes are referred to two co-ordinate systems

title: Page:SahaElectrodynamics.djvu/5


text:
Page:SahaElectrodynamics.djvu/6

 which have a uniform translatory motion relative to each other.

2. Every ray of light moves in the "stationary co-ordinate system" with the same velocity c, the velocity being independent of the condition whether this ray of light is emitted by a body at rest or in motion. Therefore

$$\text{velocity} = \frac{\text{Path of Light}}{\text{Interval of time}},$$

where, by 'interval of time,' we mean time as defined in § 1.

Let us have a rigid rod at rest ; this has a length l, when measured by a measuring rod at rest ; we suppose that the axis of the rod is laid along the X-axis of the system at rest, and then a uniform velocity v, parallel to the axis of X, is imparted to it. Let us now enquire about the length of the moving rod ; this can be obtained by either of these operations.—

(a) The observer provided with the measuring rod moves along with the rod to be measured, and measures by direct superposition the length of the rod : — just as if the observer, the measuring rod, and the rod to be measured were at rest.

(b) The observer finds out, by means of clocks placed in a system at rest (the clocks being synchronous as defined in § 1), the points of this system where the ends of the rod to be measured occur at a particular time t. The distance between these two points, measured by the previously used measuring rod, this time it being at rest, is a length, which we may call the "length of the rod."

According to the Principle of Relativity, the length found out by the operation a), which we may call "the

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Page:SahaElectrodynamics.djvu/6


text:
Page:SahaElectrodynamics.djvu/7

 length of the rod in the moving system" is equal to the length l of the rod in the stationary system.

The length which is found out by the second method, may be called  'the length of the moving rod measured from the stationary system'. This length is to be estimated on the basis of our principle, and we shall find it to be different from l.

In the generally recognised kinematics, we silently assume that the lengths defined by these two operations are equal, or in other words, that at an epoch of time t, a moving rigid body is geometrically replaceable by the same body, which can replace it in the condition of rest.

Relativity of Time.
Let us suppose that the two clocks synchronous with the clocks in the system at rest are brought to the ends A, and B of a rod, i.e., the time of the clocks correspond to the time of the stationary system at the points where they happen to arrive ; these clocks are therefore synchronous in the stationary system.

We further imagine that there are two observers at the two watches, and moving with them, and that these observers apply the criterion for synchronism to the two clocks. At the time tA, a ray of light goes out from A, is reflected from B at the time tB, and arrives back at A at time t'A. Taking into consideration the principle of constancy of the velocity of light, we have

$$t_{B}-t_{A}=\frac{r_{AB}}{c-v}$$ ,

and

$$t'_{A}-t_{B}=\frac{r_{AB}}{c+v}$$ ,

title: Page:SahaElectrodynamics.djvu/7


text:
Page:SahaElectrodynamics.djvu/8

 where $$r_{AB}$$ is the length of the moving rod, measured in the stationary system. Therefore the observers stationed with the watches will not find the clocks synchronous, though the observer in the stationary system must declare the clocks to be synchronous. We therefore see that we can attach no absolute significance to the concept of synchronism ; but two events which are synchronous when viewed from one system, will not be synchronous when viewed from a system moving relatively to this system.

§ 3. Theory of Co-ordinate and Time-Transformation from a stationary system to a system which moves relatively to this with uniform velocity.
Let there be given, in the stationary system two co-ordinate systems, i.e., two series of three mutually perpendicular lines issuing from a point. Let the X-axes of each coincide with one another, and the Y and Z-axes be parallel. Let a rigid measuring rod, and a number of clocks be given to each of the systems, and let the rods and clocks in each be exactly alike each other.

Let the initial point of one of the systems (k) have a constant velocity in the direction of the X-axis of the other which is stationary system K, the motion being also communicated to the rods and clocks in the system (k). Any time t of the stationary system K corresponds to a definite position of the axes of the moving system, which are always parallel to the axes of the stationary system. By t, we always mean the time in the stationary system.

We suppose that the space is measured by the stationary measuring rod placed in the stationary system, as well as by the moving measuring rod placed in the moving

title: Page:SahaElectrodynamics.djvu/8


text:
Page:SahaElectrodynamics.djvu/9

 system, and we thus obtain the co-ordinates (x, y, z) for the stationary system, and (ξ, η, ζ) for the moving system. Let the time t be determined for each point of the stationary system (which are provided with clocks) by means of the clocks which are placed in the stationary system, with the help of light-signals as described in § 1. Let also the time τ of the moving system be determined for each point of the moving system (in which there are clocks which are at rest relative to the moving system), by means of the method of light signals between these points (in which there are clocks) in the manner described in § 1.

To every value of (x, y, z, t) which fully determines the position and time of an event in the stationary system, there correspond a system of values (ξ, η, ζ, τ) ; now the problem is to find out the system of equations connecting these magnitudes.

Primarily it is clear that on account of the property of homogeneity which we ascribe to time and space, the equations must be linear.

If we put x'=x-vt, then it is clear that at a point relatively at rest in the system K, we have a system of values (x' y z) which are independent of time. Now let us find out τ as a function of (x',y,z,t). For this purpose we have to express in equations the fact that τ is not other than the time given by the clocks which are at rest in the system k which must be made synchronous in the manner described in § 1.

Let a ray of light be sent at time $$\tau_{0}$$ from the origin of the system k along the X-axis towards x' and let it be reflected from that place at time $$\tau_{1}$$ towards the origin of moving co-ordinates and let it arrive there at time $$\tau_{2}$$ ; then we must have

$$\frac{1}{2}(\tau_{0}+\tau_{2})=\tau_{1}$$

title: Page:SahaElectrodynamics.djvu/9


text:
Author:Frances Jane Crosby

Works

 * "All the Way My Saviour Leads Me"
 * "Blessed Assurance, Jesus Is Mine"
 * "Close to Thee"
 * "Hear Us, O Saviour"
 * "I Am Thine, O Lord"
 * "Jesus Is Tenderly Calling"
 * "Jesus, Keep Me Near the Cross"
 * "Pass Me Not, O Gentle Saviour"
 * "Praise Him! Praise Him"
 * "Rescue the Perishing, Care For the Dying"
 * "Safe in the Arms of Jesus"
 * "Some Day the Silver Chord"
 * "Though Your Sins Be as Scarlet"

title: Author:Frances Jane Crosby


text:
Page:SahaElectrodynamics.djvu/10

 If we now introduce the condition that $$\tau$$ is a function of co-ordinates, and apply the principle of constancy of the velocity of light in the stationary system, we have

It is to be noticed that instead of the origin of coordinates, we could select some other point as the exit point for rays of light, and therefore the above equation holds for all values of (x', y, z, t).

A similar conception, being applied to the y- and z-axis gives us, when we take into consideration the fact that light when viewed from the stationary system, is always propagated along those axes with the velocity $$\sqrt{c^{2}-v^{2}}$$, we have the questions:

From these equations it follows that $$\tau$$ is a linear function of x'  and t. From equations (1) we obtain

where $$a$$ is an unknown function of v.

With the help of these results it is easy to obtain the magnitudes (ξ, η, ζ), if we express by means of equations the fact that light, when measured in the moving system is always propagated with the constant velocity c (as the principle of constancy of light velocity in conjunction with the principle of relativity requires). For a

title: Page:SahaElectrodynamics.djvu/10


text:
Page:SahaElectrodynamics.djvu/11

 time $$\tau = 0$$, if the ray is sent in the direction of increasing ξ, we have

Now the ray of light moves relative to the origin of k with a velocity c-v, measured in the stationary system ; therefore we have

Substituting these values of t in the equation for ξ, we obtain

In an analogous manner, we obtain by considering the ray of light which moves along the y-axis,

where $$\frac{y}{\sqrt{c^{2}-v^{2}}}=t,\ x'=0$$.

Therefore $$\eta=a\frac{c}{\sqrt{c^{2}-v^{2}}}y,\ \zeta=a\frac{c}{\sqrt{c^{2}-v^{2}}}z$$.

If for x', we substitute its value x—tv, we obtain

 * $$\tau=\phi\ (v)\cdot\beta\left(t-\frac{vx}{c^{2}}\right)$$,

 * $$\xi=\phi\ (v)\cdot\beta\left(x-vt\right)$$,

 * $$\eta=\phi\ (v)\ y$$,

 * $$\zeta=\phi\ (v)\ z$$,

where $$\beta=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$$, and $$\phi(v)=\frac{\alpha c}{\sqrt{c^{2}-v^{2}}}=\frac{\alpha}{\beta}$$ is a function of v.

title: Page:SahaElectrodynamics.djvu/11


text:
Page:SahaElectrodynamics.djvu/12

 If we make no assumption about the initial position of the moving system and about the null-point of t, then an additive constant is to be added to the right hand side.

We have now to show, that every ray of light moves in the moving system with a velocity c (when measured in the moving system), in case, as we have actually assumed, c is also the velocity in the stationary system ; for we have not as yet adduced any proof in support of the assumption that the principle of relativity is reconcilable with the principle of constant light-velocity.

At a time $$\tau = t = 0$$ let a spherical wave be sent out from the common origin of the two systems of co-ordinates, and let it spread with a velocity c in the system K. If (x, y, z), be a point reached by the wave, we have

with the aid of our transformation-equations, let us transform this equation, and we obtain by a simple calculation,

Therefore the wave is propagated in the moving system with the same velocity c, and as a spherical wave. Therefore we show that the two principles are mutually reconcilable.

In the transformations we have got an undetermined function $$\phi(v)$$, and we now proceed to find it out.

Let us introduce for this purpose a third co-ordinate system k', which is set in motion relative to the system k, the motion being parallel to the $$\xi$$-axis. Let the velocity of the origin be (—v). At the time $$t = 0$$, all the initial co-ordinate points coincide, and for $$t=x=y=z=0$$, the time t'  of the system k' = 0. We shall say that (x, y z t) are the co-ordinates measured in the system k', then by a

title: Page:SahaElectrodynamics.djvu/12


text:
Page:SahaElectrodynamics.djvu/13

 two-fold application of the transformation-equations, we obtain

$$t'=\phi(-v)\beta(-v)\left\{ \tau+\frac{v}{c^{2}}\xi\right\} =\phi(v)\phi(-v)t$$,

$$x' = \phi(v)\beta(v)(\xi + v \tau) = \phi(v)\phi(-v)x$$, etc.

Since the relations between (x, y, z, t), and (x, y, z, t) do not contain time explicitly, therefore K and k'  are relatively at rest.

It appears that the systems K and k' are identical.

$$\therefore\phi(v)\ \phi(-v)=1$$,

Let us now turn our attention to the part of the y-axis between ($$\xi = 0, \eta = 0, \zeta = 0$$), and ($$\xi = 0, \eta = 1, \zeta = 0$$). Let this piece of the y-axis be covered with a rod moving with the velocity v relative to the system K and perpendicular to its axis ;—the ends of the rod having therefore the co-ordinates

$$\left. \begin{array}{lll} x_{1}=vt, & y=\frac{l}{\phi(v)},    & z_{1}=0\\ x_{2}=vt, & y_{2}=\frac{l}{\phi(v)}, & z_{2}=0 \end{array} \right\}$$

Therefore the length of the rod measured in the system K is $$\frac{l}{\phi(v)}$$. For the system moving with velocity (-v), we have on grounds of symmetry,

$$\frac{l}{\phi(v)}=\frac{l}{\phi(-v)}$$

$$\therefore\phi(v)=\phi(-v),\ \therefore\phi(v)=1$$

title: Page:SahaElectrodynamics.djvu/13


text:
Page:American Journal of Sociology Volume 1.djvu/233

 Rh {| style="width:80%; height:300px" border="1" ! align="center"| !! align="center"| !! align="center"| !! align="center"|
 * + COMPARATIVE TABLE OF CONTENTS
 * align="center"| I.
 * Introduction.
 * Earlier Public Policy as to Combinations.
 * Introduction.
 * align="center"| II.
 * The Structure of Industry before Machinery.
 * The Condition of Affairs before the Interstate Commerce Law and the Anti-Trust Legislation.
 * Conditions of Development.
 * align="center"| III.
 * The Order of Developopment of Machine Industry.
 * The Forms of organization.
 * Early Corporate and State Regulation of Industry.
 * align="center"| IV.
 * The Structure of Modern Industry.
 * The Objects of Organization.
 * Individual Industry.
 * align="center"| V.
 * The Formation of Monopolies in Capital.
 * Nature and Effect of Trusts.
 * Trade Unions.
 * align="center"| VI.
 * Economic Powers of the Trust.
 * The Latest phase of corporation Law and its effect on the form and Nature of Combinations.
 * Position of Women.
 * align="center"| VII.
 * Machinery and Industrial Depression.
 * Public Opinion and the Combinations.
 * Coöperation.
 * align="center"| VIII.
 * Machinery and Demand for Labor.
 * Conclusions.
 * Municipal Control.
 * align="center"| IX.
 * Machinery and the Quality of Labor.
 * Modern State Control.
 * align="center"| X.
 * The Economy of High Wages.
 * Industrial Training.
 * align="center"| XI.
 * Some Effects of Modern Industry upon the Workers as Consumers.
 * Modern Industrial Guilds.
 * align="center"| XII.
 * Women in Modern Industry.
 * Industrial Integration.
 * align="center"| XIII.
 * Machinery and the Modern Town.
 * align="center"| XIV.
 * Civilization and Industrial Development.
 * }
 * Industrial Training.
 * align="center"| XI.
 * Some Effects of Modern Industry upon the Workers as Consumers.
 * Modern Industrial Guilds.
 * align="center"| XII.
 * Women in Modern Industry.
 * Industrial Integration.
 * align="center"| XIII.
 * Machinery and the Modern Town.
 * align="center"| XIV.
 * Civilization and Industrial Development.
 * }
 * align="center"| XIII.
 * Machinery and the Modern Town.
 * align="center"| XIV.
 * Civilization and Industrial Development.
 * }
 * align="center"| XIV.
 * Civilization and Industrial Development.
 * }
 * }
 * }
 * }


text:
Page:SahaElectrodynamics.djvu/14

§ 4. The physical significance of the equations obtained concerning moving rigid bodies and moving clocks.
Let us consider a rigid sphere (i.e., one having a spherical figure when tested in the stationary system) of radius R which is at rest relative to the system (K), and whose centre coincides with the origin of K then the equation of the surface of this sphere, which is moving with a velocity v relative to K, is

At time t = 0 the equation is expressed by means of (x, y, z, t,) as

$$\frac{x^{2}}{\left(\sqrt{1-\frac{v^{2}}{c^{2}}}\right)^{2}}+y^{2}+z^{2}=R^{2}$$.

A rigid body which has the figure of a sphere when measured in the moving system, has therefore in the moving condition — when considered from the stationary system, the figure of a rotational ellipsoid with semi-axes

Therefore the y and z dimensions of the sphere (therefore of any figure also) do not appear to be modified by the motion, but the x dimension is shortened in the ratio $$1:\sqrt{1-\frac{v^{2}}{c^{2}}}$$ ; the shortening is the larger, the larger is v. For v = c, all moving bodies, when considered from a stationary system shrink into planes. For a velocity larger than the velocity of light, our propositions become

title: Page:SahaElectrodynamics.djvu/14


text:
Page:SahaElectrodynamics.djvu/15

 meaningless ; in our theory c plays the part of infinite velocity.

It is clear that similar results hold about stationary bodies in a stationary system when considered from a uniformly moving system.

Let us now consider that a clock which is lying at rest in the stationary system gives the time t, and lying at rest relative to the moving system is capable of giving the time τ ; suppose it to be placed at the origin of the moving system k, and to be so arranged that it gives the time τ. How much does the clock gain, when viewed from the stationary system K? We have,

$$\tau=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\left(t-\frac{v}{c^{2}}x\right)$$, and $$x=vt$$,

$$\therefore\tau-t=\left[1-\sqrt{1-\frac{v^{2}}{c^{2}}}\right]t$$.

Therefore the clock loses by an amount $$\frac{1}{2}\frac{v^{2}}{c^{2}}$$ per second of motion, to the second order of approximation.

From this, the following peculiar consequence follows. Suppose at two points A and B of the stationary system two clocks are given which are synchronous in the sense explained in § 3 when viewed from the stationary system. Suppose the clock at A to be set in motion in the line joining it with B, then after the arrival of the clock at B, they will no longer be found synchronous, but the clock which was set in motion from A will lag behind the clock which had been all along at B by an amount $$\frac{1}{2}t\frac{v^{2}}{c^{2}}$$, where t is the time required for the journey.

title: Page:SahaElectrodynamics.djvu/15


text:
Page:SahaElectrodynamics.djvu/16

 We see forthwith that the result holds also when the clock moves from A to B by a polygonal line, and also when A and B coincide.

If we assume that the result obtained for a polygonal line holds also for a curved line, we obtain the following law. If at A, there be two synchronous clocks, and if we set in motion one of them with a constant velocity along a closed curve till it comes back to A, the journey being completed in t-seconds, then after arrival, the last mentioned clock will be behind the stationary one by $$\frac{1}{2}t\frac{v^{2}}{c^{2}}$$ seconds. From this, we conclude that a clock placed at the equator must be slower by a very small amount than a similarly constructed clock which is placed at the pole, all other conditions being identical.

§ 5. Addition-Theorem of Velocities.
Let a point move in the system k (which moves with velocity v along the x-axis of the system K) according to the equation

$$\xi=w_{\xi}\tau,\ \eta=w_{\eta}\tau,\ \zeta=0$$,

where $$w_{\xi}$$ and $$w_{\eta}$$ are constants.

It is required to find out the motion of the point relative to the system K. If we now introduce the system of equations in § 3 in the equation of motion of the point, we obtain

$$x=\frac{w_{\xi}+v}{1+\frac{vw_{\xi}}{c^{2}}},\ y=\frac{\left(1-\frac{v^{2}}{c^{2}}\right)^{\frac{1}{2}}w_{\eta}t}{1+\frac{vw_{\xi}}{c^{2}}},\ z=0$$.

title: Page:SahaElectrodynamics.djvu/16


text:
Page:SahaElectrodynamics.djvu/17

 The law of parallelogram of velocities hold up to the first order of approximation. We can put

$$U^{2}=\left(\frac{\partial x}{\partial t}\right)^{2}+\left(\frac{\partial y}{\partial t}\right)^{2},\ w^{2}=w_{\xi}^{2}+w_{\eta}^{2}$$,

and

$$\alpha=\tan^{-1}\frac{w}{w_{\xi}}$$

i.e., $$\alpha$$ is put equal to the angle between the velocities v, and w. Then we have—

$$U=\frac{\left[(v^{2}+w^{2}+2vw\ \cos\ \alpha)-\left(\frac{vw\ \sin\ \alpha}{c}\right)^{2}\right]^{\frac{1}{2}}}{1+\frac{vw\ \cos\ \alpha}{c^{2}}}$$

It should be noticed that v and w enter into the expression for velocity symmetrically. If w has the direction of the ξ-axis of the moving system,

$$U=\frac{v+w}{1+\frac{vw}{c^{2}}}$$

From this equation, we see that by combining two velocities, each of which is smaller than c, we obtain a velocity which is always smaller than c. If we put $$v = c - \chi$$, and $$w = c - \lambda$$ where $$\chi$$ and $$\lambda$$ are each smaller than c,

$$U=c\frac{2c-\chi-\lambda}{2c-\chi-\lambda+\frac{\chi\lambda}{c^{2}}}<c$$.

It is also clear that the velocity of light c cannot be altered by adding to it a velocity smaller than c. For this case,

$$U=\frac{c+v}{1+\frac{cv}{c^{2}}}=c$$

title: Page:SahaElectrodynamics.djvu/17


text: Author talk:Frances Jane Crosby

title: Author talk:Frances Jane Crosby


text:
Page:SahaElectrodynamics.djvu/18

 We have obtained the formula for U for the case when v and w have the same direction ; it can also be obtained by combining two transformations according to section § 3. If in addition to the systems K, and k, we introduce the system k', of which the initial point moves parallel to the ξ-axis with velocity w, then between the magnitudes, x, y, z, t and the corresponding magnitudes of k', we obtain a system of equations, which differ from the equations in §3, only in the respect that in place of v, we shall have to write,

$$(v+w)/\left(1+\frac{vw}{c^{2}}\right)$$

We see that such a parallel transformation forms a group.

We have deduced the kinematics corresponding to our two fundamental principles for the laws necessary for us, and we shall now pass over to their application in electrodynamics.

On the nature of the Electromotive Force caused by motion in a magnetic field.
The Maxwell-Hertz equations for pure vacuum may hold for the stationary system K, so that

$$\frac{1}{c}\frac{\partial}{\partial t}[X,\ Y,\ Z]=\left|\begin{array}{ccc} \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\ \\L & M & N\end{array}\right|$$,

title: Page:SahaElectrodynamics.djvu/18


text:
Page:SahaElectrodynamics.djvu/19

 and

where [X, Y, Z] are the components of the electric force, L, M, N are the components of the magnetic force.

If we apply the transformations in § 3 to these equations, and if we refer the electromagnetic processes to the co-ordinate system moving with velocity v, we obtain,

$$\left|\begin{array}{ccc} \frac{\partial}{\partial\xi} & \frac{\partial}{\partial\eta} & \frac{\partial}{\partial\zeta}\\ \\L & \beta\left(M+\frac{v}{c}Z\right) & \beta\left(N-\frac{v}{c}Y\right)\end{array}\right| ,$$

and

$$\frac{1}{c}\frac{\partial}{\partial\tau}\left[L,\beta\left(M+\frac{v}{c}Z\right),\ \beta\left(N-\frac{v}{c}Y\right)\right]$$

The principle of Relativity requires that the Maxwell-Hertzian equations for pure vacuum shall hold also for the system k, if they hold for the system K, i.e., for the vectors of the electric and magnetic forces acting upon electric and magnetic masses in the moving system k,

title: Page:SahaElectrodynamics.djvu/19


text:
Page:SahaElectrodynamics.djvu/20

 which are defined by their pondermotive reaction, the same equations hold, ... i.e. ...

Clearly both the systems of equations (2) and (3) developed for the system k shall express the same things, for both of these systems are equivalent to the Maxwell-Hertzian equations for the system K. Since both the systems of equations (2) and (3) agree up to the symbols representing the vectors, it follows that the functions occurring at corresponding places will agree up to a certain factor $$\psi(v)$$, which depends only on v, and is independent of $$(\xi, \eta, \zeta, \tau)$$. Hence the relations,

Then by reasoning similar to that followed in §(3), it can be shown that $$\psi(v)=1$$.

title: Page:SahaElectrodynamics.djvu/20


text:
Page:SahaElectrodynamics.djvu/21

 For the interpretation of these equations, we make the following remarks. Let us have a point-mass of electricity which is of magnitude unity in the stationary system K, i.e., it exerts a unit force upon a similar quantity placed at a distance of 1 cm. If this quantity of electricity be at rest in the stationary system, then the force acting upon it is equivalent to the vector $$(X, Y, Z)$$ of electric force. But if the quantity of electricity be at rest relative to the moving system (at least for the moment considered), then the force acting upon it, and measured in the moving system is equivalent to the vector $$(X', Y', Z')$$. The first three of equations (1), (2), (3), can be expressed in the following way :—

1. If a point-mass of electric unit pole moves in an electro-magnetic field, then besides the electric force, an electromotive force acts upon it, which, neglecting the numbers involving the second and higher powers of v/c, is equivalent to the vector-product of the velocity vector, and the magnetic force divided by the velocity of light (Old mode of expression).

2. If a point-mass of electric unit pole moves in an electro-magnetic field, then the force acting upon it is equivalent to the electric force existing at the position of the unit pole, which we obtain by the transformation of the field to a co-ordinate system which is at rest relative to the electric unit pole [New mode of expression].

Similar theorems hold with reference to the magnetic force. We see that in the theory developed the electromagnetic force plays the part of an auxiliary concept, which owes its introduction in theory to the circumstance that the electric and magnetic forces possess no existence independent of the nature of motion of the co-ordinate system.

title: Page:SahaElectrodynamics.djvu/21


text:
Page:SahaElectrodynamics.djvu/22

 It is further clear that the assymetry mentioned in the introduction which occurs when we treat of the current excited by the relative motion of a magnet and a conductor disappears. Also the question about the seat of electromagnetic energy is seen to be without any meaning.

§ 7. Theory of Döppler's Principle and Aberration.
In the system K, at a great distance from the origin of co-ordinates, let there be a source of electrodynamic waves, which is represented with sufficient approximation in a part of space not containing the origin, by the equations :—

$$\left. \begin{array}{c} X=X_{0}\sin\Phi\\ Y=Y_{0}\sin\Phi\\ Z=Z_{0}\sin\Phi\end{array} \right\} \left. \begin{array}{c} L=L_{0}\sin\Phi\\ M=M_{0}\sin\Phi\\ N=N_{0}\sin\Phi\end{array} \right\} \phi=\omega\left(t-\frac{lx+my+nz}{c}\right)$$

Here $$(X_0, Y_0, Z_0)$$ and $$(L_0, M_0, N_0)$$ are the vectors which determine the amplitudes of the train of waves, $$(l, m, n)$$ are the direction-cosines of the wave-normal.

Let us now ask ourselves about the composition of these waves, when they are investigated by an observer at rest in a moving medium k :— By applying the equations of transformation obtained in §6 for the electric and magnetic forces, and the equations of transformation obtained in § 3 for the co-ordinates, and time, we obtain immediately :—

title: Page:SahaElectrodynamics.djvu/22


text:
Page:SahaElectrodynamics.djvu/23

 where

$$\omega'=\omega\beta\left(1-\frac{lv}{c}\right),\ l'=\frac{l-\frac{v}{c}}{1-\frac{lv}{c}},\ m'=\frac{m}{\beta\left(1-\frac{lv}{c}\right)},\ n'=\frac{n}{\beta\left(1-\frac{lv}{c}\right)}$$.

From the equation for $$\omega'$$ it follows :— If an observer moves with the velocity v relative to an infinitely distant source of light emitting waves of frequency $$\nu$$, in such a manner that the line joining the source of light and the observer makes an angle of $$\Phi$$ with the velocity of the observer referred to a system of co-ordinates which is stationary with regard to the source, then the frequency $$\nu'$$ which is perceived by the observer is represented by the formula

$$\nu'=\nu\frac{1-\cos\Phi\frac{v}{c}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$$.

This is Doppler's principle for any velocity. If $$\Phi=0$$ then the equation takes the simple form

$$\nu'=\nu\left(\frac{1-\frac{v}{c}}{1+\frac{v}{c}}\right)^{\frac{1}{2}}$$.

We see that —contrary to the usual conception— $$\nu=\infty$$, for $$v = -c$$.

If $$\Phi'$$=angle between the wave-normal (direction of the ray) in the moving system, and the line of motion of the observer, the equation for l' takes the form

$$\cos\Phi'=\frac{\cos\Phi-\frac{v}{c}}{1-\frac{v}{c}\cos\Phi}$$.

title: Page:SahaElectrodynamics.djvu/23


text:
Page:SahaElectrodynamics.djvu/24

 This equation expresses the law of observation in its most general form. If $$\Phi=\frac{\pi}{2}$$, the equation takes the simple form

We have still to investigate the amplitude of the waves, which occur in these equations. If A and A' be the amplitudes in the stationary and the moving systems (either electrical or magnetic), we have

If $$\Phi=0$$, this reduces to the simple form

From these equations, it appears that for an observer, which moves with the velocity c towards the source of light, the source should appear infinitely intense.

§ 8. Transformation of the Energy of the Rays of Light. Theory of the Radiation-pressure on a perfect mirror.
Since $$\frac{A^{2}}{8\pi}$$ is equal to the energy of light per unit volume, we have to regard $$\frac{A}{8\pi}$$ as the energy of light in

title: Page:SahaElectrodynamics.djvu/24


text:
Page:SahaElectrodynamics.djvu/25

 the moving system. $$\frac{A'^{2}}{A^{2}}$$ would therefore denote the ratio between the energies of a definite light-complex "measured when moving" and "measured when stationary," the volumes of the light-complex measured in K and k being equal. Yet this is not the case. If $$l, m, n$$ are the direction-cosines of the wave-normal of light in the stationary system, then no energy passes through the surface elements of the spherical surface

$$(x-clt)^{2}+(y-cmt)^{2}+(z-cnt)^{2}=R^{2}$$,

which expands with the velocity of light. We can therefore say, that this surface always encloses the same light-complex. Let us now consider the quantity of energy, which this surface encloses, when regarded from the system k, i.e., the energy of the light-complex relative to the system k.

Regarded from the moving system, the spherical surface becomes an ellipsoidal surface, having, at the time $$tau=0$$, the equation :—

$$\left(\beta\xi-l\beta\frac{v}{c}\xi\right)^{2}+\left(\eta-m\beta\frac{v}{c}\xi\right)^{2}+\left(\zeta-n\beta\frac{v}{c}\xi\right)^{2}=R^{2}$$

If $$S=\text{volume of the sphere}$$, $$S'= \text{volume of this ellipsoid}$$, then a simple calculation shows that:

$$\frac{S'}{S}=\frac{\beta}{\sqrt{1-\frac{v}{c}\cos\Phi}}$$

If E denotes the quantity of light energy measured in the stationary system, E' the quantity measured in the

title: Page:SahaElectrodynamics.djvu/25


text:
Page:SahaElectrodynamics.djvu/26

 moving system, which are enclosed by the surfaces mentioned above, then

If $$\Phi=0$$, we have the simple formula :—

It is to be noticed that the energy and the frequency of a light-complex vary according to the same law with the state of motion of the observer.

Let there be a perfectly reflecting mirror at the co-ordinate-plane $$\xi=0$$, from which the plane-wave considered in the last paragraph is reflected. Let us now ask ourselves about the light-pressure exerted on the reflecting surface and the direction, frequency, intensity of the light after reflexion.

Let the incident light be defined by the magnitudes $$A cos \Phi$$, v (referred to the system K). Regarded from k, we have the corresponding magnitudes :

title: Page:SahaElectrodynamics.djvu/26


text:
Page:SahaElectrodynamics.djvu/27

 For the reflected light we obtain, when the process is referred to the system k :—

By means of a back-transformation to the stationary system, we obtain K, for the reflected light :—

$$\left. \begin{array}{c} A'=A\frac{1+\frac{v}{c}\cos\Phi''}{\sqrt{1-\frac{v^{2}}{c^{2}}}}=A\frac{1-2\frac{v}{c}\cos\Phi+\frac{v^{2}}{c^{2}}}{1-\frac{v^{2}}{c^{2}}},\\ \\\cos\Phi'=\frac{\cos\Phi+\frac{v}{c}}{1+\frac{v}{c}\cos\Phi''}=-\frac{\left(1+\frac{v^{2}}{c^{2}}\right)\cos\Phi-2\frac{v}{c}}{1-2\frac{v}{c}\cos\Phi+\frac{v^{2}}{c^{2}}},\\ \nu'=\nu\frac{1+\frac{v}{c}\cos\Phi''}{\sqrt{1-\frac{v^{2}}{c^{2}}}}=\nu\frac{1-2\frac{v}{c}\cos\Phi+\frac{v^{2}}{c^{2}}}{1-\frac{v^{2}}{c^{2}}},\end{array} \right\}$$

The amount or energy falling upon the unit surface of the mirror per unit of time (measured in the stationary system) is $$\frac{A^{2}}{8\pi(c\ \cos\Phi-v)}$$. The amount of energy going away from unit surface of the mirror per unit of time is $$A'^{2}/8\pi(-c\cos\Phi+v)$$. The difference of these two expressions is, according to the Energy principle, the amount of work exerted, by the pressure of light per unit of time. If we put this equal to $$P \cdot v$$, where $$P=\text{pressure of light}$$, we have

title: Page:SahaElectrodynamics.djvu/27


text:
Page:SahaElectrodynamics.djvu/28

 As a first approximation, we obtain

which is in accordance with facts, and with other theories.

All problems of optics of moving bodies can be solved after the method used here. The essential point is, that the electric and magnetic forces of light, which are influenced by a moving body, should be transformed to a system of co-ordinates which is stationary relative to the body. In this way, every problem of the optics of moving bodies would be reduced to a series of problems of the optics of stationary bodies.

§ 9. Transformation of the Maxwell-Hertz Equations.
Let us start from the equations :—

$$\left. \begin{array}{c} \frac{1}{c}\left(\rho u_{x}+\frac{\partial X}{\partial t}\right)=\frac{\partial N}{\partial y}-\frac{\partial M}{\partial z}\\ \\\frac{1}{c}\left(\rho u_{y}+\frac{\partial Y}{\partial t}\right)=\frac{\partial L}{\partial z}-\frac{\partial N}{\partial x}\\ \\\frac{1}{c}\left(\rho u_{z}+\frac{\partial Z}{\partial t}\right)=\frac{\partial M}{\partial x}-\frac{\partial L}{\partial y} \end{array} \right\} \left. \begin{array}{c} \frac{1}{c}\frac{\partial L}{\partial t}=\frac{\partial Y}{\partial z}-\frac{\partial Z}{\partial y}\\ \\\frac{1}{c}\frac{\partial M}{\partial t}=\frac{\partial Z}{\partial x}-\frac{\partial X}{\partial z}\\ \\\frac{1}{c}\frac{\partial N}{\partial t}=\frac{\partial X}{\partial y}-\frac{\partial Y}{\partial x} \end{array} \right\}$$

where $$\rho=\frac{\partial X}{\partial x}+\frac{\partial Y}{\partial y}+\frac{\partial Z}{\partial z}$$, denotes $$4 \pi$$ times the density of electricity, and $$(u_x, u_y, u_z)$$ are the velocity-components of electricity. If we now suppose that the electrical masses are bound unchangeably to small, rigid bodies

title: Page:SahaElectrodynamics.djvu/28


text:
Page:SahaElectrodynamics.djvu/29

 (Ions, electrons), then these equations form the electromagnetic basis of Lorentz's electrodynamics and optics for moving bodies.

If these equations which hold in the system K, are transformed to the system k with the aid of the transformation-equations given in § 3 and § 6, then we obtain the equations :—

where

Since the vector $$(u_{\xi}, u_{\eta}, u_{\zeta})$$ is nothing but the velocity of the electrical mass measured in the system k, as can be easily seen from the addition-theorem of velocities in § 4—so it is hereby shown, that by taking

title: Page:SahaElectrodynamics.djvu/29


text:
Page:SahaElectrodynamics.djvu/30

 our kinematical principle as the basis, the electromagnetic basis of Lorentz's theory of electrodynamics of moving bodies correspond to the relativity-postulate. It can be briefly remarked here that the following important law follows easily from the equations developed in the present section :— if an electrically charged body moves in any manner in space, and if its charge does not change thereby, when regarded from a system moving along with it, then the charge remains constant even when it is regarded from the stationary system K.

§ 10. Dynamics of the Electron (slowly accelerated).
Let us suppose that a point-shaped particle, having the electrical charge e (to be called henceforth the electron) moves in the electromagnetic field ; we assume the following about its law of motion.

If the electron be at rest at any definite epoch, then in the next "particle of time," the motion takes place according to the equations

$$m\frac{d^{2}x}{dt^{2}}=eX,\ m\frac{d^{2}y}{dt^{2}}=eY,\ m\frac{d^{2}z}{dt^{2}}=eZ$$

Where $$(x, y, z)$$ are the co-ordinates of the electron, and m is its mass.

Let the electron possess the velocity v at a certain epoch of time. Let us now investigate the laws according to which the electron will move in the 'particle of time' immediately following this epoch.

Without influencing the generality of treatment, we can and we will assume that, at the moment we are considering,

title: Page:SahaElectrodynamics.djvu/30


text:
Page:SahaElectrodynamics.djvu/31

 the electron is at the origin of co-ordinates, and moves with the velocity v along the X-axis of the system. It is clear that at this moment ($$t=0$$) the electron is at rest relative to the system k, which moves parallel to the X-axis with the constant velocity v.

From the suppositions made above, in combination with the principle of relativity, it is clear that regarded from the system k, the electron moves according to the equations

$$m\frac{d^{2}\xi}{dt^{2}}=eX',\ m\frac{d^{2}\eta}{dt^{2}}=eY',\ m\frac{d^{2}\zeta}{dt^{2}}=eZ'$$,

in the time immediately following the moment, where the symbols ($$\xi, \eta, \zeta, \tau, X', Y', Z'$$) refer to the system k. If we now fix, that for $$t=v=y=z=0$$, $$\tau = \xi = \eta = \zeta=0$$, then the equations of transformation given in § 3 (and 6) hold, and we have :

$$\left. \begin{array}{c}\tau=\beta\left(t-\frac{v}{c^{2}}x\right),\ \xi=\beta(x-vt),\ \eta=y,\ \zeta=z,\\ \\X'=X,\ Y'=\beta\left(Y-\frac{v}{c}N\right),\ Z'=\beta\left(Z+\frac{v}{c}M\right) \end{array} \right\}$$

With the aid of these equations, we can transform the above equations of motion from the system k to the system K, and obtain :—

{{MathForm1|(A)|$$\left. \begin{array}{c} \frac{d^{2}x}{dt^{2}}=\frac{e}{m}\frac{1}{\beta^{3}}X,\ \frac{d^{2}y}{dt^{2}}=\frac{e}{m}\frac{1}{\beta}\left(Y-\frac{v}{c}N\right),\ \\ \\\frac{d^{2}z}{dt^{2}}=\frac{e}{m}\frac{1}{\beta}\left(Z+\frac{v}{c}M\right)\end{array} \right\}$$}}

title: Page:SahaElectrodynamics.djvu/31


text:
Page:SahaElectrodynamics.djvu/32

 Let us now consider, following the usual method of treatment, the longitudinal and transversal mass of a moving electron. We write the equations (A) in the form

$$\left. \begin{array}{c} m\beta^{3}\frac{d^{2}x}{dt^{2}}=eX=eX'\\ \\m\beta^{2}\frac{d^{2}y}{dt^{2}}=e\beta\left[Y-\frac{v}{c}N\right]=eY'\\ \\m\beta^{2}\frac{d^{2}z}{dt^{2}}=e\beta\left[Z+\frac{v}{c}M\right]=eZ'\end{array} \right\}$$

and let us first remark, that $$eX', eY', eZ'$$ are the components of the ponderomotive force acting upon the electron, and are considered in a moving system which, at this moment, moves with a velocity which is equal to that of the electron. This force can, for example, be measured by means of a spring-balance which is at rest in this last system. If we briefly call this force as "the force acting upon the electron," and maintain the equation :—

$$\text{Mass-number} \times \text{acceleration-number} = \text{force-number}$$, and if we further fix that the accelerations are measured in the stationary system K, then from the above equations, we obtain :—

$$\text{Longitudinal mass} = \frac{m}{\left(\sqrt{1-\frac{v^{2}}{c^{2}}}\right)^{\frac{3}{2}}}$$

$$\text{Transversal mass} = \frac{m}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$$

Naturally, when other definitions are given of the force and the acceleration, other numbers are obtained for the

title: Page:SahaElectrodynamics.djvu/32


text:
Page:SahaElectrodynamics.djvu/33

 mass ; hence we see that we must proceed very carefully in comparing the different theories of the motion of the electron.

We remark that this result about the mass hold also for ponderable material mass ; for in our sense, a ponderable material point may be made into an electron by the addition of an electrical charge which may be as small as possible.

Let us now determine the kinetic energy of the electron. If the electron moves from the origin of co-ordinates of the system K with the initial velocity 0 steadily along the X-axis under the action of an electromotive force X, then it is clear that the energy drawn from the electrostatic field has the value $$\int eXdx$$. Since the electron is only slowly accelerated, and in consequence, no energy is given out in the form of radiation, therefore the energy drawn from the electro-static field may be put equal to the energy W of motion. Considering the whole process of motion in questions, the first of equations A) holds, we obtain :—

$$W=\int\ eXdx=\overset{v}{\underset{0}{\int}\ }m\beta^{3}vdv=mc^{2}\left[\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}-1\right]$$.

For v=c, W is infinitely great. As our former result shows, velocities exceeding that of light can have no possibility of existence.

In consequence of the arguments mentioned above, this expression for kinetic energy must also hold for ponderable masses.

We can now enumerate the characteristics of the motion of the electrons available for experimental verification, which follow from equations A).

title: Page:SahaElectrodynamics.djvu/33


text:
Page:SahaElectrodynamics.djvu/34

 1. From the second of equations A); it follows that an electrical force Y, and a magnetic force N produce equal deflexions of an electron moving with the velocity v, when $$Y=\frac{Nv}{c}$$. Therefore we see that according to our theory, it is possible to obtain the velocity of an electron from the ratio of the magnetic deflexion Am, and the electric deflexion Ae, by applying the law :—

$$\frac{A_{m}}{A_{e}}=\frac{v}{c}$$.

This relation can be tested by means of experiments because the velocity of the electron can be directly measured by means of rapidly oscillating electric and magnetic fields.

2. From the value which is deduced for the kinetic energy of the electron, it follows that when the electron falls through a potential difference of P, the velocity v which is acquired is given by the following relation :—

$$P=\int\ Xdx=\frac{m}{e}c^{2}\left[\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}-1\right]$$.

3. We calculate the radius of curvature R of the path, where the only deflecting force is a magnetic force N acting perpendicular to the velocity of projection. From the second of equations A) we obtain :

$$-\frac{d^{2}y}{dt^{2}}=\frac{v^{2}}{R}=\frac{e}{m}\frac{v}{c}N\sqrt{1-\frac{v^{2}}{c^{2}}}$$,

or

$$R=\frac{mv\beta c}{eN}$$

These three relations are complete expressions for the law of motion of the electron according to the above theory.

title: Page:SahaElectrodynamics.djvu/34


text: File:SahaSpaceTime.djvu

title: File:SahaSpaceTime.djvu


text: Index:Metamorphoses (Ovid, 1567).djvu

title: Index:Metamorphoses (Ovid, 1567).djvu


text:
Page:More Tales from Tolstoi.djvu/6

or, “The Turks in Hungary.” By Maurus Jókai.

By. Illustrated by.

or, “The Scourge of God.” By. Translated by. With Preface by.

By.

Translated from the Russian by. With Biography brought up to date.

By.

By.

A Romance of Tennyson-Land. By.

By.

Translated from the Russian by. With Biography of.

Translated from the Russian of by.

By. Translated by.

By. Translated by.

title: Page:More Tales from Tolstoi.djvu/6


text: Page:More Tales from Tolstoi.djvu/1

title: Page:More Tales from Tolstoi.djvu/1


text:
Page:Folk-lore - A Quarterly Review. Volume 4, 1893.djvu/358

 350 I may as well take this opportunity to record the following:

In Innisbofin, co. Galway, the people have a very firm belief in fairies. Mr. Allies, who resides there, informed me that one old man told him that he saw a number of fairy girls, dressed in brown, around him one day when he (Mr. Allies) was shooting rabbits. Mr. Allies offered £50 if a fairy could be shown to him, and £100 if he took a photograph of one. Mr. Allies has not yet paid away any money. Mr. Allies and his brother were quarrying a rock by the side of the harbour, and at last the men refused to work at it any longer, as it was so full of the "good people" as to be hot. This was two or three years ago. Mr. Lane gives an amusing instance of the solicitude of the old women for Mrs. Allies' baby (see p. 358).

My first batch of folk-notes are those contributed by Miss Emily Fitzgerald, with the aid, for Donegal, of Miss Sinclair:—

Valencia.—In illness the " old people" say any improvement taking place on Friday or Sunday is unlucky. Not likely to last.

Cure for Erysipelas (Kerry, Valencia).—To arrest erysipelas, the name of the patient must be written round the part affected in the blood of a black cat, a cat that has not a single white hair.

"Febrifuge" (Valencia).—The first egg laid by a little black hen, eaten the very first thing in the morning, will keep you from fever for the year.

Cure for Erysipelas (Donegal).—Rub the part affected with butter made from the milk of the cows belonging to a married couple, who both had the same name before their marriage. —, Bonny Glen, Donegal.

Cure for Erysipelas (Donegal, Arranmore).—Send the son or daughter of a couple who each had the same name before their marriage to the bog for bog-water, and bathe the part affected with it.—.

Apply the blood of anyone of the name of McCaul to the affected part.—.


text:
Page:Folk-lore - A Quarterly Review. Volume 4, 1893.djvu/359

 Rh For Ulcerated Sore-throat (Donegal).—Take the patient by the two ears and "shake the devil out of him or her".—.

(This Miss S. knows to be a fact, for it was done to one of their labourer's sons.)

Dried fox's tongue has many virtues; e.g., it will draw thorns however deep.—.

Cure for the Evil.—A robin's breast rubbed on the place.—Idem.

Cure for a Sick Cow (Donegal).—Cut off the piece of turf on which the cow first treads when getting up, and hang it on the wall, and the cow will recover.—.

Cure for a Sore Mouth (Donegal).—A posthumous child will cure a sore mouth.—.

Cure for Whooping-Cough (Kerry).—Some milk to be poured into a saucer, a ferret to drink some of it, and the rest to be given to the patient.—, Waterville.

St. John's Eve Fires (Kerry).—Fires were (and are still in a less degree) lighted all over the country on St. John's Eve, especially little fires across the road; if you drove through them it brought you luck for the year. Cattle were also driven through the fires.

When anyone is lying dead in a room the walls must be hung with sheets, and the door left open (because the spirit hovers in the room after it has left the body, and must have free egress), five candles must be round the coffin, one of which is not to be lighted. As the coffin is being taken out of the door the sheets are to be taken down.—Mrs. , Darrynane.

The first child that dies in a family must be buried in the children's burial-ground (there are numbers of them about the country for unbaptised children), otherwise two others will follow if the first is buried in the churchyard.

Water that has been used to bathe the feet must be put outside the door at night for fear of fairies.

A gentleman I know at Listowel remembers, about eight years ago, being very much astonished when a cloud of dust was being blown along a road, seeing an old woman rush to the side and drag handfuls of grass out of the fence, which she threw in great haste into the cloud of dust. He inquired, and learned that this was


text:
Page:More Tales from Tolstoi.djvu/326

LIST OF FICTION. The Gray House of the Quarries. (Second Edition.) By Mary H. Norris. With etched Frontispiece by Edmund H. Garrett. . . . . ^ (J/- " Susanna is a splendid study. No person who takes up the book can resist its fascination." — Westminster Review, D i staff. (Suond Edition. ) By Marya Rodziewicz, Author of " Anima Vilis," etc. Translated from the Polish by Count S. C. de Soissons. With a finely engraved Portrait of the Author 6/- " A striking novel." — Spectator, The Captive of Pekin. (Fourth Edition.) A Realistic Story of Chinese Life and Manners. By Charles Hannan. With twenty-three graphic Illustrations from life, depicting the Chinese torture fiends, by A, J. B. Salmon 5/- " Told with great Tividness, a thrilling story dramatically told. The reader's interest does not flag from beginning to end." — The Times. " A powerfully written and absorbing story." — Morning Post. A Daughter of Mystery, {second Edition.) By R. Norman Silver. . . . 6/- plots make the brain reel. The book should be read, and will repay the most exacting lovers of the exciting." — Daily News. Wayfarers All. {Second Edition.) By Leslie Keith, Author of " Xisbeth " " My Bonnie Lady." (J/- "An extremely entertaining and sympathetic romance. The Misses Green are masterly characterisations, and so are Ruth's fascinating children." — Daily Telegraph. The Inn by the Shore. (Fifteenth Thousand.) By Florence Warden, Author of " The House on the Marsh,", etc. ..... 3/6 "A rattling story, told in a lively way, incident following on incident in rapid succession." — Daily Chronicle. Judy a Jilt. (TUrd Edition.) By Mrs. Conney, Author of "A Lady House Breaker," " Gold for Dross," etc. . . . 3/6 "Written in Mrs. Conney's happiest manner * Judy a Jilt' is a telling story throughout," — Daily Telegraph.
 * ' A pleasant story, full of aWlity." — Pall Mall Gazette.
 * ' It cannot comfortably be laid down, until it is finished. The plots and counter-

title: Page:More Tales from Tolstoi.djvu/326


text:
Page:More Tales from Tolstoi.djvu/325

 LIST OF nCTION. In Tight Places. (Third Eduum.) By Major Arthur Griffiths, Author of " For- bidcfen by Law," etc. 6'- " A liv«hf sod vaHcd Mrics of oofmopoliua Cfimc, with plenty of mixed advestore •ad MatAtioa. Sucfa ftoriei Aiways fuciiuUc, vA Mmjor Arthur Gnditltt iuwws well bow to taU i)aa:'—Pall MaUGmMtllt. St* Peter's Umbrella. [Third Eduum.) By KAlmAn MikszAth, Author of "The Good People of Palvez." Translated from the original Hungarian by W. B. Worswick. With Introduction by R. Nisbet Bain. A charming Photogravure Portrait of the Author and three illustrations. 6 '- "Tht frnbnvM, bigh tptriu, and batnour of Mikuub null* bim a ixi--^ a- * eompiuiioo. Hu p«a«aiit», pria«t«, aod gcfttlcfoUu arc uiuuiogf jr boBUui. :•!.... /^;;i U a Wn itory'ialiar." — Tfu SftctaUr. The Adventures of Cyrano de Bergerac. Captain Satan. {Founh Edu,oH,) From the French of Louis Gallet With specially engraved portrait of Cyrano de Ikrgcrac. . 6- " A dali|blful book. Sa viviiily daliaaMad arc tb< drMmatit f*rt»mm, to biia««- In^ an'l antbratUng are tb« m<;tU«niji in ih* tlair«i.>|jm«it of tiia uJa, ih;ii ii ta itii|. <»«ililr t.i >li|i ona l>-■ t tili^fafX. A Woman's Burden. [Thud tMu>n.) By Fergus Hume, Author of " The Mystery of a ll.insdm Cab," " The Lone Inn," etc., . Gl- "N'l-rv ^'.wA tr.wlltV); " —. I /^ -..-rNwl, "Siiuiily lull i'l diiiM* from oo*«r tocover."— /'•»A/ii4rfr," CircuUr, Vivian of Virginia. {SH*mi a.a/.<.h.) Ik'ing the Memoirs of Our First Rebellion, by John Vivi;m, of Middle Pl.intati^Mi, V'irt^ini.i. By Hulbert Fuller, Author of" God's Rebel." With ten charming Ilhistr.itiDiis by Frank T. Merrill. . . . O • "Thara ii not a tlall momant in iba muim:) -writteo aiory, advantnia fall-^^v nj advaninra, balding iba raadar in thrall ; whi>i ih. lov* loiaraat b (itUy MUbiu^. — G*HlUw*msM, Anima Vilis. [s*toNd EMum.) A t.ile of the Great Siberian Steppe. By Marya RODZIEWICZ. Translated from the Polish by Count S. C. de Soissons. With a fine Phot(^ravure Portrait of the Author. . . ... 6/- ' A •triking aaK^."—TAt rnmu. " Ha* both poarar and charm.'— /Lilma^itrt

title: Page:More Tales from Tolstoi.djvu/325


text:
Page:More Tales from Tolstoi.djvu/324

 FROM JARROLD &■ SONS' LIST ., The Lion of danina. Mfth Edition. " It is a fascinating story . . . It is a brilliant and lurid series of pictures drawn by a great master's hand. Maurus J6kai . . . paints with a big brush, with a masculinity, an audacity of touch that rivals Rubens. He has imagination, he has vision, he has great style. To say more of him were here superfluous! to say less were stupid," — Daily Chronicle.

"A striking specimen of Oriental fiction. It is a stern, terrible drama that is unfolded leaf by leaf. The scenes are full of life and movement. The scene in which Ali is described as making his last stand is very fine indeed."—The World.

Pretty Michal. Fourth Edition. " It is part of the high art of the story that the author has kept the whole atmosphere free from the complexity, the subtle motives of a later and more sophisticated age. Maurus Jdkai's novel is a book to be read." — Saturday Review. "The most vivid and enthralling of J6kai's later novels."— Daily News. "Amidst all the palpitating excitement of the numerous and thrilling adventures that so plentifully adorn this fascinating novel, the reader is conscious of the immense intellectual strength by which the book is dominated, and his admiration is stirred by the force and fertility of imagination, the grasp of character and the fund of special knowledge displayed. It is not only talent, but a touch of actual genius, that animates the work of Maurus Jokai." — The Speaker. 'Midst tiie Wild Carpathians. Fourth Edition. " Will enthral all English lovers of romance, and gratify the natural curiosity now general concerning the work of the Hungarian novelist." — Saturday Review. " Much and well as J6kai has written, his pen has done nothing more vivid than some of these descriptions, and nothing more lurid than this grim story of intrigue, battle, and treachery " — The Academy. "It has power to engage and to thrill. The descriptive pieces are very fine. It is powerful, it is vigorous, and what is more than all, it is fresh." — The ^un. The White Kose. This beautiful and picturesque tale of Oriental life reads like a chapter out of the " Arabian Nights." The heroine of the present story is a beautiful young Greek girl, who escapes the gilded dis- honour of the harem by feigning death and enduring torments, while the hero, an honest Mussulman Turk of the lower orders, loves the damsel for her own sake, and unexpectedly attains world-wide fame in his efforts to shelter and defend her. The scene of the story is Stambul, in the eighteenth century, and every phase of life in the great metropolis of the Ottoman Empire is described with singular fidelity. The chapter on the Tulip Gardens of the Grand Seignier and their fatal influence on character, is, moreover, an exceedingly fine piece of grim humour in the master's best style. London: Jarrold and Sons, loand ii, Warwick Lane, E>C.

title: Page:More Tales from Tolstoi.djvu/324


text:
Page:More Tales from Tolstoi.djvu/323

 Selections from ^ ^ Jarrold & Sons^ # # list of iflctloit # ^ Maurus Jokai's Famous Novels. AulHoriitd Edifittii. Crcvm iv, Art Lintn, 0/» uuk. Black Diamonds. (FiphE^.tun) By Maurus J6kai, Author of "The Green Book," " Poor Plutocrats," etc. Translated by I- ranees Gerard. With Special Preface by the Author " Foil of vigour . . . hb toiicb« of bumuur arvucriiani.''— JV«rmi'iV Aw/. " Ail IfilaraitlDg ilory." Tima. The Qreen Book. (Freh'om Under the Snow.) (Sixth EHUimt.) Hy Maurus J6kai. Translated by Mrs. U'au^'h With a finely engraved Portrait of Dr. J6kai. " BiUllully dxiwu. . . • book to hm ntA.'—DmUy CknnuU. " Tliur.>iiitbly ulculaiiil to ctMrm (ho noixl-r«uliii( pobUc by iu cxMlMk uai< intoi . . . from flm (o lui ih* IntwM nmr ngi. A mrk of lb< muM Mdilng IntanMi uxl luiwrb dvcrlpUoiu."— yOiUikmai. Pretty Michal. (Fo^nk sdinom.) By Maurus J6kai. Translated by R. Nisbet Haiu. With I specially engraved Photogravure Portrait of Dr. JdkaL " A bKinkttfiii Bowl.'*— T^f SptmMtr, " Hia woriun&ub^ b adnunblc, and b« powM * d«gtM of fympKtbatic Lniftginatioo aoi inqiiutad by any livtnc nowlut- Tb« actioa of ha itoriat u U£a- ltk«, autd fult of raov«intnl and loiarwL — IIVj/nt««/#r G^»4tt4. A Hungarian Nabob. (/i/"> a<av<««.) By Maurus J6kai. Translated by R. Nisbet Bain. With a fine Photogravure Portrait of Dr. J6kai. "Tk wotk of * guiiB.'— /°<» M*U (MtlU.
 * FhU of udiing indiltiiii u.J mMHrlr tindia ofchancur."— CmmCvraiW'.

title: Page:More Tales from Tolstoi.djvu/323


text:
Page:More Tales from Tolstoi.djvu/322

More Tales from Tolstoi truth when he said that next time he would make his head ache. Why, horny hands are nothing to this; work of this sort would pretty soon give one lumps on the head." And Ivan would have gone to see if much work had been done, when suddenly the ground opened and the old Devil plunged through the earth—only a hole remained.

Ivan scratched his head. "Ugh! the filthy creature! So it is he again, eh! Like children, like father—I'm well rid of him!" And Ivan is living to this day, and all the people crowded into his kingdom, and his brothers came to him, and he fed them. Any who like may come and say: "Feed us!" "All right!" says Ivan, "live with us; we have plenty of everything." There is only one settled custom in his kingdom: those who have horny hands sit at table, and those who have not must eat the scraps.

title: Page:More Tales from Tolstoi.djvu/322


text:
In re Commitment of Edward S.

Angela V. Passalacqua, Assistant Deputy Public Advocate, argued the cause for appellant (Alfred A. Slocum, Public Advocate, attorney).

James P. Lynch, Assistant Prosecutor, argued the cause for respondent (Samuel Asbell, Camden County Prosecutor, attorney).

For affirmance and remandment — Chief Justice WILENTZ and Justices CLIFFORD, HANDLER, POLLOCK, O'HERN, GARIBALDI and STEIN. Opposed — None. The opinion of the Court was delivered by WILENTZ, C.J. HANDLER, J., concurring. HANDLER, J., concurring in result.

title: In re Commitment of Edward S.


text: Index:SahaSpaceTime.djvu

title: Index:SahaSpaceTime.djvu


text:
Page:Folk-lore - A Quarterly Review. Volume 4, 1893.djvu/360

 352 in order to give something to the fairies that were flying along in the dust.

People had assured him, and no laughing could get them out of the belief, that they had seen a field full of fairies—little people two or three feet high.—Mr. .

A headless coach—that is, without horses—was said to career about the neighbourhood of Listowel when any misfortune was about to take place. Mr. Creagh remembers, as a boy, servants assuring him that they had seen it.

There was a common belief, though it is not much heard of now, that priests could turn people into hares.

Country people in Kerry don't eat hares; the souls of their grandmothers are supposed to have entered into them. (February 1891.)

The following notes were contributed in June, 1892, by Mr. G. C. Campbell, as nearly as possible in the actual words of his various informants: —

The Origin of the Fairies.—The. fairies are fallen angels. The time when Lucifer was head-angel, he was cast out of heaven. Pride put Lucifer down. There was wans o' the angels took part wi' the Almighty, and there was wans took part wi' Lucifer. The wans that sided wi' the Almighty, they stayed in heaven, an' the wans that sided wi' Lucifer they went straight to hell. But there was a third party, wans that kep' silent, an' the Almighty sent them out o' heaven into the rocks, an' sea, an' bushes, an' land; an' they are the gentry, the wee-folk. They say if there 's wan drap o' blood in them at the Judgment Day they'll be pardoned, but I don't believe they have wan drap o' blood in them.—Informant,, Londonderry, beggar.

Added to this.—They say some are hanging by the heels in the elements yet.—, co. Derry, farmer's wife.

Fairy Story.—There was a young married lady, an' she was very rich, an' the fairies took her away the night her first baby was born; so they could not find her no road. They had a coachman, an' he was always listenin' at the door of the fairies. So on Hallowe'en night he went back to the door; with that they opened the door, an' got him listenin', and let him in. So when


text:
Page:SahaSpaceTime.djvu/1

70 SPACE AND TIME

A Lecture delivered before the Naturforscher Versammlung (Congress of Natural Philosophers) at Cologne — (21st September, 1908).

Gentlemen,

The conceptions about time and space, which I hope to develop before you to-day, has grown on experimental physical grounds. Herein lies its strength. The tendency is radical. Henceforth, the old conception of space for itself, and time for itself shall reduce to a mere shadow, and some sort of union of the two will be found consistent with facts.

I

Now I want to show you how we can arrive at the changed concepts about time and space from mechanics, as accepted now-a-days, from purely mathematical considerations. The equations of Newtonian mechanics show a twofold invariance, (i) their form remains unaltered when we subject the fundamental space-coordinate system to any possible change of position, (ii) when we change the system in its nature of motion, i. e., when we impress upon it any uniform motion of translation, the null-point of time plays no part. We are accustomed to look upon the axioms of geometry as settled once for all, while we seldom have the same amount of conviction regarding the axioms of mechanics, and therefore the two invariants are seldom mentioned in the same breath. Each one of these denotes a certain group of transformations for the differential equations of mechanics. We look upon the existence of the first group as a fundamental characteristics of space. We always prefer to leave off the second group to itself, and with a light heart conclude that we can never decide from physical considerations whether the space, which is supposed to be

title: Page:SahaSpaceTime.djvu/1


text:
Page:SahaSpaceTime.djvu/2

Rh at rest, may not finally be in uniform motion. So these two groups lead quite separate existences besides each other. Their totally heterogeneous character may scare us away from the attempt to compound them. Yet it is the whole compounded group which as a whole gives us occasion for thought.

We wish to picture to ourselves the whole relation graphically. Let (x, y, z) be the rectangular coordinates of space, and t denote the time. Subjects of our perception are always connected with place and time. No one has observed a place except at a particular time, or has observed a time except at a particular place. Yet I respect the dogma that time and space have independent existences. I will call a space-point plus a time-point, i.e., a system of values x, y, z, t, as a world-point. The manifoldness of all possible values of x, y, z, t, will be the world. I can draw four world-axes with the chalk. Now any axis drawn consists of quickly vibrating molecules, and besides, takes part in all the journeys of the earth; and therefore gives us occasion for reflection. The greater abstraction required for the four-axes does not cause the mathematician any trouble. In order not to allow any yawning gap to exist, we shall suppose that at every place and time, something perceptible exists. In order not to specify either matter or electricity, we shall simply style these as substances. We direct our attention to the world-point x, y, z, t, and suppose that we are in a position to recognise this substantial point at any subsequent time. Let dt be the time element corresponding to the changes of space coordinates of this point [dx, dy, dz]. Then we obtain (as a picture, so to speak, of the perennial life-career of the substantial point), — a curve in the world — the world-line, the points on which unambiguously correspond to the parameter t from +∞ to -∞. The whole world appears to be

title: Page:SahaSpaceTime.djvu/2


text:
Page:SahaSpaceTime.djvu/3

72 resolved in such world-lines, and I may just deviate from my point if I say that according to my opinion the physical laws would find their fullest expression as mutual relations among these lines.

By this conception of time and space, the (x, y, z) manifoldness t = 0 and its two sides t < 0 and t > 0 falls asunder. If for the sake of simplicity, we keep the null-point of time and space fixed, then the first named group of mechanics signifies that at t = 0 we can give the x, y, and z-axes any possible rotation about the null-point corresponding to the homogeneous linear transformation of the expression

$x^{2}+y^{2}+z^{2},$

The second group denotes that without changing the expression for the mechanical laws, we can substitute $$ x - \alpha t ,\, y - \beta t ,\, z - \gamma t, t$$ for (x, y, z) where (α, β, γ) are any constants. According to this we can give the time-axis any possible direction in the upper half of the world t > 0. Now what have the demands of orthogonality in space to do with this perfect freedom of the time-axis towards the upper half?

To establish this connection, let us take a positive parameter c, and let us consider the figure

$c^{2}t^{2}-x^{2}-y^{2}-z^{2}=1.$

According to the analogy of the hyperboloid of two sheets, this consists of two sheets separated by t = 0. Let us consider the sheet, in the region of t > 0, and let us now conceive the transformation of x, y, z, t in the new system of variables; (x', y', z', t') by means of which the form of the expression will remain unaltered. Clearly the rotation of space round the null-point belongs to this group of transformations. Now we can have a full idea of the transformations which we picture to ourselves from a particular

title: Page:SahaSpaceTime.djvu/3


text:
Page:SahaSpaceTime.djvu/4

 transformation in which (y, z) remain unaltered. Let us draw the cross section of the upper sheets with the plane of the x-and t-axes, i.e., the upper half of the hyperbola $$c^{2}t^{2}-x^{2}=1$$, with its asymptotes (vide fig. 1).

Then let us draw the radius rector OA', the tangent A' B' at A', and let us complete the parallelogram OA' B' C'; also produce B' C' to meet the x-axis at D'. Let us now take Ox', OA' as new axes with the unit measuring rods OC' = 1, OA'= $$\tfrac{1}{c}$$; then the hyperbola is again expressed in the form $$c^2t'^2-x'^2=1, t'> 0$$ and the transition from (x, y, z, t) to (x' y' z' t) is one of the transitions in question. Let us add to this characteristic transformation any possible displacement of the space and time null-points; then we get a group of transformation depending only on c, which we may denote by Gc.

Now let us increase c to infinity. Thus $$\frac{1}{c}$$ becomes zero and it appears from the figure that the hyperbola is gradually shrunk into the x-axis, the asymptotic angle becomes a straight one, and every special transformation in the limit changes in such a manner that the t-axis can have any possible direction upwards, and x' more and more approximates to x. Remembering this point it is clear that the full group belonging to Newtonian Mechanics is simply the group Gc, with the value of c = ∞. In this state of affairs, and since Gc is mathematically more intelligible than G∞, a mathematician may, by a free play of imagination, hit upon the thought that natural phenomena possess an invariance not only for the group G∞, but in fact also for a group Gc, where c is finite, but yet

title: Page:SahaSpaceTime.djvu/4


text:
Page:SahaSpaceTime.djvu/5

 exceedingly large compared to the usual measuring units. Such a preconception would be an extraordinary triumph for pure mathematics.

At the same time I shall remark for which value of c, this invariance can be conclusively held to be true. For c, we shall substitute the velocity of light c in free space. In order to avoid speaking either of space or of vacuum, we may take this quantity as the ratio between the electrostatic and electro-magnetic units of electricity.

We can form an idea of the invariant character of the expression for natural laws for the group-transformation Gc in the following manner.

Out of the totality of natural phenomena, we can, by successive higher approximations, deduce a coordinate system (x, y, z, t); by means of this coordinate system, we can represent the phenomena according to definite laws. This system of reference is by no means uniquely determined by the phenomena. ''We can change the system of reference in any possible manner corresponding to the abovementioned group transformation Gc, but the expressions for natural laws will not be changed thereby. ''

For example, corresponding to the above described figure, we can call t' the time, but then necessarily the space connected with it must be expressed by the manifoldness (x' y z). The physical laws are now expressed by means of x', y, z, t', — and the expressions are just the same as in the case of x, y, z, t. According to this, we shall have in the world, not one space, but many spaces, — quite analogous to the case that the three-dimensional space consists of an infinite number of planes. The three-dimensional geometry will be a chapter of four-dimensional physics. Now you perceive, why I said in the beginning

title: Page:SahaSpaceTime.djvu/5


text:
Page:SahaSpaceTime.djvu/6

 that time and space shall reduce to mere shadows and we shall have a world complete in itself.

Now the question may be asked, — what circumstances lead us to these changed views about time and space, are they not in contradiction with observed phenomena, do they finally guarantee us advantages for the description of natural phenomena?

Before we enter into the discussion, a very important point must be noticed. Suppose we have individualised time and space in any manner; then a world-line parallel to the t-axis will correspond to a stationary point; a world-line inclined to the t-axis will correspond to a point moving uniformly; and a world-curve will correspond to a point moving in any manner. Let us now picture to our mind the world-line passing through any world point x, y, z, t; now if we find the world-line parallel to the radius vector OA' of the hyperboloidal sheet, then we can introduce OA' as a new time-axis, and then according to the new conceptions of time and space the substance will appear to be at rest in the world point concerned. We shall now introduce this fundamental axiom: —

The substance existing at any world point can always be conceived to he at rest, if we establish our time and space suitably. The axiom denotes that in a world-point the expression

$c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2},$

shall always be positive or what is equivalent to the same thing, every velocity V should be smaller than c. c shall therefore be the upper limit for all substantial velocities and herein lies a deep significance for the

title: Page:SahaSpaceTime.djvu/6


text:
Page:SahaSpaceTime.djvu/7

 quantity c. At the first impression, the axiom seems to be rather unsatisfactory. It is to be remembered that only a modified mechanics will occur, in which the square root of this differential combination takes the place of time, so that cases in which the velocity is greater than c will play no part, something like imaginary coordinates in geometry.

The impulse and real cause of inducement for the assumption of the group-transformation Gc is the fact that the differential equation for the propagation of light in vacant space possesses the group-transformation Gc. On the other hand, the idea of rigid bodies has any sense only in a system mechanics with the group G∞. Now if we have an optics with Gc, and on the other hand if there are rigid bodies, it is easy to see that a t-direction can be defined by the two hyperboloidal shells common to the groups G∞, and Gc, which has got the further consequence, that by means of suitable rigid instruments in the laboratory, we can perceive a change in natural phenomena, in case of different orientations, with regard to the direction of progressive motion of the earth. But all efforts directed towards this object, and even the celebrated interference-experiment of Michelson have given negative results. In order to supply an explanation for this result, H. A. Lorentz formed a hypothesis which practically amounts to an invariance of optics for the group Gc. According to Lorentz every substance shall suffer a contraction $$1:\left(\sqrt{1-\frac{v^{2}}{c^{2}}}\right)$$ in length, in the direction of its motion $$\frac{l}{l'}=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\qquad l'=l\left(1-\frac{v^{2}}{c^{2}}\right).$$.

title: Page:SahaSpaceTime.djvu/7


text:
Page:SahaSpaceTime.djvu/8

 This hypothesis sounds rather phantastical. For the contraction is not to be thought of as a consequence of the resistance of ether, but purely as a gift from the skies, as a sort of condition always accompanying a state of motion.

I shall show in our figure that Lorentz's hypothesis is fully equivalent to the new conceptions about time and space. Thereby it may appear more intelligible. Let us now, for the sake of simplicity, neglect (y, z) and fix our attention on a two dimensional world, in which let upright strips parallel to the t-axis represent a state of rest and another parallel strip inclined to the t-axis represent a state of uniform motion for a body, which has a constant spatial extension (see fig. 1). If OA' is parallel to the second strip, we can take t' as the t-axis and x' as the x-axis, then the second body will appear to be at rest, and the first body in uniform motion. We shall now assume that the first body supposed to be at rest, has the length l, i.e., the cross section PP of the first strip upon the x-axis = l · OC, where OC is the unit measuring rod upon the x-axis — and the second body also, when supposed to be at rest, has the same length l, this means that, the cross section Q'Q' of the second strip has a cross-section l·OC, when measured parallel to the x'-axis. In these two bodies, we have now images of two Lorentz-electrons, one of which is at rest and the other moves uniformly. Now if we stick to our original coordinates, then the extension of the second electron is given by the cross section QQ of the strip belonging to it measured parallel to the x-axis. Now it is clear since Q'Q' = l·OC', that QQ = l·OD'. If $$\frac{dx}{dt}=v$$, an easy calculation gives that

$OD'=OC\sqrt{1-\frac{v^{2}}{c^{2}}}$, therefore $\frac{PP}{QQ}=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$|undefined

title: Page:SahaSpaceTime.djvu/8


text:
Page:SahaSpaceTime.djvu/9

 This is the sense of Lorentz's hypothesis about the contraction of electrons in ease of motion. On the other hand, if we conceive the second electron to be at rest, and therefore adopt the system (x', t') then the cross-section P'P' of the strip of the electron parallel to OC' is to be regarded as its length and we shall find the first electron shortened with reference to the second in the same proportion, for it is,

$\frac{P'P'}{Q'Q'}=\frac{OD}{OC'}=\frac{OD'}{OC}=\frac{QQ}{PP}$

Lorentz called the combination t' of (t and x) as the local time (Ortszeit) of the uniformly moving electron, and used a physical construction of this idea for a better comprehension of the contraction-hypothesis. But to perceive clearly that the time of an electron is as good as the time of any other electron, i.e. t, i' are to be regarded as equivalent, has been the service of A. Einstein [Ann. d. Phys. 891, p. 1905, Jahrb. d. Radis... 4-4-1 1—1907] There the concept of time was shown to be completely and unambiguously established by natural phenomena. But the concept of space was not arrived at, either by Einstein or Lorentz, probably because in the case of the above-mentioned spatial transformations, where the (x', y') plane coincides with the x-t plane, the significance is possible that the x-axis of space some-how remains conserved in its position.

We can approach the idea of space in a corresponding manner, though some may regard the attempt as rather fantastical.

According to these ideas, the word "Relativity-Postulate" which has been coined for the demands of invariance in the group G, seems to be rather inexpressive for a true understanding of the group Gc, and for further progress.

title: Page:SahaSpaceTime.djvu/9


text:
Page:SahaSpaceTime.djvu/10

 Because the sense of the postulate is that the four-dimensional world is given in space and time by phenomena only, but the projection in time and space can be handled with a certain freedom, and therefore I would rather like to give to this assertion the name "The Postulate of the Absolute world" [World-Postulate].

III

By the world-postulate a similar treatment of the four determining quantities x, y, z, t, of a world-point is possible. Thereby the forms under which the physical laws come forth, gain in intelligibility, as I shall presently show. Above all, the idea of acceleration becomes much more striking and clear.

I shall again use the geometrical method of expression. Let us call any world-point O as a "Space-time-null-point." The cone

$c^{2}t^{2}-x^{2}-y^{2}-z^{2}=0$

consists of two parts with O as apex, one part having t < 0', the other having t > 0. The first, which we may call the fore-cone consists of all those points which send light towards O, the second, which we may call the aft-cone, consists of all those points which receive their light from O. The region bounded by the fore-cone may be called the fore-side of O, and the region bounded by the aft-cone may be called the aft-side of O. (Vide fig. 2).

On the aft-side of O we have the already considered hyperboloidal shell $$F=c^{2}t^{2}-x^{2}-y^{2}-z^{2}=l,\ t>0$$.

title: Page:SahaSpaceTime.djvu/10


text:
Page:Folk-lore - A Quarterly Review. Volume 4, 1893.djvu/361

 Rh he went in, he got his eye on this one (the lady), he got a hoult of her, an' took her out wi' him; he won her from them, an' took her home to her own house.

Says they, "Ye have her with ye now, but she'll not be much use to ye now, for she's both deaf an' dumb. . . ." That night twelvemonth the coachman went back to the door again, an' he heard them saying, "Well, this night twelve-month we lost a noble lady; she was not much service to them, though, for we left her both deaf an' dumb." "Well," says one fairy, speakin' out, "it wouldn't be hard for them to cure her, for if they would go to a spring-well where the water-grass grows, an' take some water-grass an' squeeze the juice out of it, an' put some of it in her ears, an' give her the rest to drink, it would cure her."

The coachman then went straight to a spring-well and got the water-grass, an' did just what they said, an' the lady got all right, an' was never bothered with the wee-folk again.—, Derry, pedlar.

Fairy Story.—I heard my mother tell of a young man, an' he lived up bye there. One Hallowe'en night he went out for a bit of a daunder; an' just as he was comin' off the lane into the road he saw a whole troop of fairies comin' along the road, an' what had they but a girl wi' them; an' he seen she wasn't one of the fairies, so he catched a hoult of her, an' at that they turned into everything—horses, and all that. But he wasn't feared o' any o' them, an' kep' a hoult of her until he got her right intil his mother's. An' the girl she could speak noan—for ye know the wee-folk puts a thmg in their mouth that they can't speak. The mother she came forrard an' shook hands wi' her, an' said she was right glad to see her, an' the girl she laughed, but said nothin'. She stayed wi' them, an' did all the work for them.

An' Hallowe'en night was a twelve-month. The young fellow he was goin' out, an' his mother she wasn't for him goin' out, but the girl she was glad like to see him goin', an' signed with her hand to him to go on. An' when he got forenenst the place he got the girl, he catched sight of the fairies again, an' he kep' back, an' he heard them talking, an' says one to the other: "This night


text:
Page:SahaSpaceTime.djvu/11

 The region inside the two cones will be occupied by the hyperboloid of one sheet

$-F=x^{2}+y^{2}+z^{2}-c^{2}t^{2}=k^2$,

where k² can have all possible positive values. The hyperbolas which lie upon this figure with O as centre, are important for us. For the sake of clearness the individual branches of this hyperbola will be called the "Interhyperbola with centre O." Such a hyperbolic branch, when thought of as a world-line, would represent a motion which for t = — ∞ and t = ∞ asymptotically approaches the velocity of light c.

If, by way of analogy to the idea of vectors in space, we call any directed length in the manifoldness x, y, z, t a vector, then we have to distinguish between a time-vector directed from O towards the sheet +F = 1, t > O and a space-vector directed from O towards the sheet -F = 1. The time-axis can be parallel to any vector of the first kind. Any world-point between the fore and aft cones of O, may by means of the system of reference be regarded either as synchronous with O, as well as later or earlier than O. Every world-point on the fore-side of O is necessarily always earlier, every point on the aft side of O, later than O. The limit c = ∞ corresponds to a complete folding up of the wedge-shaped cross-section between the fore and aft cones in the manifoldness t = 0. In the figure drawn, this cross-section has been intentionally drawn with a different breadth.

Let us decompose a vector drawn from O towards (x, y, z, t) into its components. If the directions of the two vectors are respectively the directions of the radius vector OR to one of the surfaces ±F = 1, and of a tangent RS

title: Page:SahaSpaceTime.djvu/11


text:
Page:SahaSpaceTime.djvu/12

 at the point R of the surface, then the vectors shall be called normal to each other. Accordingly

$c^{2}tt_{1}-xx_{1}-yy_{1}-zz_{1}=0,$

which is the condition that the vectors with the components (x, y, z, t) and $$\left(x_{1}\ y_{1}\ z_{1}\ t_{1}\right)$$ are normal to each other.

For the measurement of vectors in different directions, the unit measuring rod is to be fixed in the following manner; — a space-like vector from to -F = 1 is always to have the measure unity, and a time-like vector from O to +F = 1, t >0 is always to have the measure $$\tfrac{1}{c}$$.

Let us now fix our attention upon the world-line of a substantive point running through the world-point (x, y, z, t); then as we follow the progress of the line, the quantity

$d\tau=\frac{1}{c}\sqrt{c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}},$|undefined

corresponds to the time-like vector-element (dx, dy, dz, dt).

The integral $$\tau=\int d\tau$$, taken over the world-line from any fixed initial point P0 to any variable final point P, may be called the "Proper-time" of the substantial point at P0 upon the world-line. We may regard (x, y, z, t), i.e., the components of the vector OP, as functions of the "proper-time" τ; let $$\left(\dot{x},\ \dot{y},\ \dot{z},\ \dot{t}\right)$$ denote the first differential-quotients, and $$\left(\ddot{x},\ \ddot{y},\ \ddot{z},\ \ddot{t}\right)$$ the second differential quotients of (x, y, z, t) with regard to τ, then these may respectively

title: Page:SahaSpaceTime.djvu/12


text:
Page:SahaSpaceTime.djvu/13

 be called the Velocity-vector, and the Acceleration-vector of the substantial point at P. Now we have

{{Center|$$\left.\begin{array}{l} c^{2}\dot{t}^{2}-\dot{x}^{2}-\dot{y}^{2}-\dot{z}^{2}=c^{2}\\ c^{2}\dot{t}\ddot{t}-\dot{x}\ddot{x}-\dot{y}\ddot{y}-\dot{z}\ddot{z}=0\end{array}\right\} $$}}

i.e., the ’Velocity-vector’ is the time-like vector of unit measure in the direction of the world-line at P, the ’Acceleration-vector’ at P is normal to the velocity-vector at P, and is in any case, a space-like vector.

Now there is, as can be easily seen, a certain hyperbola, which has three infinitely contiguous points in common with the world-line at P, and of which the asymptotes are the generators of a 'fore-cone' and an 'aft-cone.' This hyperbola may be called the "hyperbola of curvature" at P (vide fig. 3). If M be the centre of this hyperbola, then we have to deal here with an 'Inter-hyperbola' with centre M. Let P = measure of the vector MP, then we easily perceive that the acceleration-vector at P is a vector of magnitude $$\frac{c^2}{\rho}$$ in the direction of MP.

If $$\ddot{x},\ \ddot{y},\ \ddot{z},\ \ddot{t}$$ are nil, then the hyperbola of curvature at P reduces to the straight line touching the world-line at P, and ρ = ∞.

IV

In order to demonstrate that the assumption of the group Gc for the physical laws does not possibly lead to any contradiction, it is unnecessary to undertake a revision of the whole of physics on the basis of the assumptions underlying this group. The revision has already been successfully made in the case of

title: Page:SahaSpaceTime.djvu/13


text:
Page:SahaSpaceTime.djvu/14

 "Thermodynamics and Radiation," for "Electromagnetic phenomena", and finally for "Mechanics with the maintenance of the idea of mass."

For this last mentioned province of physics, the question may be asked: if there is a force with the components X, Y, Z (in the direction of the space-axes) at a world-point (x, y, z, t), where the velocity-vector is $$\left(\dot{x},\ \dot{y},\ \dot{z},\ \dot{t}\right)$$, then how are we to regard this force when the system of reference is changed in any possible manner? Now it is known that there are certain well-tested theorems about the ponderomotive force in electromagnetic fields, where the group Gc is undoubtedly permissible. These theorems lead us to the following simple rule; if the system of reference be changed in any way, then the supposed force is to be put as a force in the new space-coordinates in such a manner, that the corresponding vector with the components

$\dot{t}X,\ \dot{t}Y,\ \dot{t}Z,\ \dot{t}T,$

where

$T=\frac{1}{c^{2}}\ \left(\frac{\dot{x}}{\dot{t}}X+\frac{\dot{y}}{\dot{t}}Y+\frac{\dot{z}}{\dot{t}}Z\right)=\frac{1}{c^{2}}$|undefined

(the rate of which work is done at the world-point), remains unaltered. This vector is always normal to the velocity-vector at P. Such a force-vector, representing a force at P, may be called a moving force-vector at P.

Now the world-line passing through P will be described by a substantial point with the constant mechanical mass m. Let us call m-times the velocity-vector at P as the

title: Page:SahaSpaceTime.djvu/14


text:
Page:SahaSpaceTime.djvu/15

 impulse-vector, and m-times the acceleration-vector at P as the force-vector of motion, at P. According to these definitions, the following law tells us how the motion of a point-mass takes place under any moving force-vector :

''The force-vector of motion is equal to the moving force-vector. ''

This enunciation comprises four equations for the components in the four directions, of which the fourth can be deduced from the first three, because both of the above-mentioned vectors are perpendicular to the velocity-vector. From the definition of T, we see that the fourth simply expresses the "Energy-law." Accordingly c²-times the component of the impulse-vector in the direction of the t-axis is to be defined as the kinetic-energy of the point-mass. The expression for this is

$mc^{2}\frac{dt}{d\tau}=mc^{2}/\sqrt{1-\frac{v^{2}}{c^{2}}}$|undefined

i.e., if we deduct from this the additive constant mc², we obtain the expression ½mv² of Newtonian-mechanics up to magnitudes of the order of $$\frac{1}{c^2}$$. Hence it appears that the energy depends upon the system of reference. But since the t-axis can be laid in the direction of any time-like axis, therefore the energy-law comprises, for any possible system of reference, the whole system of equations of motion. This fact retains its significance even in the limiting: ease c = ∞, for the axiomatic construction of Newtonian mechanics, as has already been pointed out by T. R. Schütz.

title: Page:SahaSpaceTime.djvu/15


text:
Page:SahaSpaceTime.djvu/16

 From the very beginning, we can establish the ratio between the units of time and space in such a manner, that the velocity of light becomes unity. If we now write $$\sqrt{-1}t=l$$, in the place of t, then the differential expression

$$d\tau^{2}=-\left(dx^{2}+dy^{2}+dz^{2}+dl^{2}\right),$$

becomes symmetrical in (x, y, z, l); this symmetry then enters into each law, which does not contradict the world-postulate. We can clothe the essential nature of this postulate in the mystical, but mathematically significant formula

3·105 km = $$\sqrt{-1}$$ Sec.

The advantages arising from the formulation of the world-potulate are illustrated by nothing so strikingly as by the expressions which tell us about the reactions exerted by a point-charge moving in any manner according to the Maxwell-Lorentz theory.

Let us conceive of the world-line of such an electron with the charge (e), and let us introduce upon it the "Proper-time" τ reckoned from any possible initial point. In order to obtain the field caused by the electron at any world-point P1 let us construct the fore-cone belonging to P1 (vide fig. 4). Clearly this cuts the unlimited world-line of the electron at a single point P, because these directions are all time-like vectors. At P, let us draw the tangent to the world-line, and let us draw from P1 the normal to this tangent. Let r be the measure of P1Q. According to the definition of a fore-cone, r/c is to be reckoned as the measure of PQ. Now at the world-point P1,

title: Page:SahaSpaceTime.djvu/16


text:
Page:SahaSpaceTime.djvu/17

 the vector-potential of the field excited by e is represented by the vector in direction PQ, having the magnitude $$\frac{e}{cr}$$, in its three space components along the x-, y-, z-axes; the scalar-potential is represented by the component along the t-axis. This is the elementary law found out by A. Lienard, and E. Wiechert.

If the field caused by the electron be described in the above-mentioned way, then it will appear that the division of the field into electric and magnetic forces is a relative one, and depends upon the time-axis assumed; the two forces considered together bears some analogy to the force-screw in mechanics; the analogy is, however, imperfect.

I shall now describe the ponderomotive force which is exerted by one moving electron upon another moving electron. Let us suppose that the world-line of a second point-electron passes through the world-point P1. Let us determine P, Q, r as before, construct the middle-point M of the hyperbola of curvature at P, and finally the normal MN upon a line through P which is parallel to QP1. With P as the initial point, we shall establish a system of reference in the following way: the t-axis will be laid along PQ, the x-axis in the direction of QP1. The y-axis in the direction of MN, then the z-axis is automatically determined, as it is normal to the x-, y-, z-axes. Let $$\ddot{x},\ \ddot{y},\ \ddot{z},\ \ddot{t}$$ be the acceleration-vector at $$P,\ \dot{x}_{1},\ \dot{y}_{1},\ \dot{z}_{1},\ \dot{t}_{1}$$ be the velocity-vector at P1. Then the force-vector exerted by the first electron e, (moving in any possible manner)

title: Page:SahaSpaceTime.djvu/17


text:
Page:SahaSpaceTime.djvu/18

 upon the second electron e, (likewise moving in any possible manner) at P1 is represented by

$-ee_{1}\left(\dot{t}_{1}-\frac{\dot{x}_{1}}{c}\right)\mathrm{F}$|undefined

For the components $$\mathrm{F}_{x},\ \mathrm{F}_{y},\ \mathrm{F}_{z},\ F_{t}$$ of the vector $$F$$ the following three relations hold: —

$c\mathrm{F}_{t}-\mathrm{F}_{x}=\frac{1}{r^{2}},\ \mathrm{F}_{y}=\frac{\ddot{y}}{c^{2}r},\ \mathrm{F}_{z}=0$|undefined

''and fourthly this vector F is normal to the velocity-vector P1, and through this circumstance alone, its dependence on this last velocity-vector arises. ''

I£ we compare with this expression the previous formulæ giving the elementary law about the ponderomotive action of moving electric charges upon each other, then we cannot but admit, that the relations which occur here reveal the inner essence of full simplicity first in four dimensions; but in three dimensions, they have very complicated projections.

In the mechanics reformed according to the world-postulate, the disharmonies which have disturbed the relations between Newtonian mechanics, and modern electrodynamics automatically disappear. I shall now consider the position of the Newtonian law of attraction to this postulate. I will assume that two point-masses m and m1 describe their world-lines; a moving force-vector is exercised by m upon m1, and the expression is just the same as in the case of the electron, only we have to write +mm1 instead of -ee1. We shall consider only the special case in which the acceleration-vector of m is always zero;

title: Page:SahaSpaceTime.djvu/18


text:
Page:SahaSpaceTime.djvu/19

 then t may be introduced in such a manner that m may be regarded as fixed, the motion of m is now subjected to the moving-force vector of m alone. If we now modify this given vector by writing $$\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$$ instead of $$\dot{t}$$ ($$\dot{t}=1$$ up to magnitudes of the order $$\tfrac{1}{c^2}$$), then it appears that Kepler's laws hold good for the position (x1, y1, z1), of m1 at any time, only in place of the time t1, we have to write the proper time τ1 of m1. On the basis of this simple remark, it can be seen that the proposed law of attraction in combination with new mechanics is not less suited for the explanation of astronomical phenomena than the Newtonian law of attraction in combination with Newtonian mechanics.

Also the fundamental equations for electro-magnetic processes in moving bodies are in accordance with the world-postulate. I shall also show on a later occasion that the deduction of these equations, as taught by Lorentz, are by no means to be given up.

The fact that the world-postulate holds without exception is, I believe, the true essence of an electromagnetic picture of the world; the idea first occurred to Lorentz, its essence was first picked out by Einstein, and is now gradually fully manifest. In course of time, the mathematical consequences will be gradually deduced, and enough suggestions will be forthcoming for the experimental verification of the postulate; in this way even those, who find it uncongenial, or even painful to give up the old, time-honoured concepts, will be reconciled to the new ideas of time and space, — in the prospect that they will lead to pre-established harmony between pure mathematics and physics.

title: Page:SahaSpaceTime.djvu/19


text: Page:South Carolina Declaration and Ordinance of Secession.djvu/4


text: Index:South Carolina Declaration and Ordinance of Secession.djvu


text:
Page:American Journal of Sociology Volume 1.djvu/240

 228 approaching large and satisfying improvements of social conditions. This is indicated in his closing paragraph. "It is my belief that the future belongs neither to the prophets of individualism, nor to the ideals of the social democrats. Its next phases belong to social reorganization. And the probability is that this will show a corporate character, and will be sustained and controlled by public supervision."

I have not given so much space to the expression of a favorable judgment upon these books because they are altogether free from inaccuracies as to fact, or because I indorse their conclusions. Taken together the three volumes set a high standard for investigators of social conditions. They exhibit a worthy conception of what is involved in qualification for judgment either about the quality of present industrial relations or about the directions in which or the means by which we should seek for change. The books may be used together as a very valuable concrete exhibit and application of the abstract principles of societary exposition which some of the most sagacious contemporary social philosophers have adopted as parts of their working hypothesis.

Industrial Evolution of the United States. By. 12mo., pp x. + 362. Flood & Vincent, the Chautauqua-Century Press, 1895.

often is a valuable popular book written as is this by one whose past work is the chief original authority for much of the contents. In preparing this volume, Mr. Wright gets much of his best material from his past reports as chief for many years of the Massachusetts and later of the United States departments of labor statistics and from his other special researches. This popular condensation of such investigations by the leading labor statistician of America, and probably of the world, will be of great value not only to the general public but to many specialists. The author describes the development in colonial and subsequent epochs, and the magnitude as revealed in our various census reports of some of our leading manufacturing industries.

Many interesting facts are marshaled to prove that wages were much higher in both money and purchasing power in 1890 than in 1860 or 1840, or any previous period, though it is conceded that amid such natural resources and inventions, labor should have gained still more. The common claim of the wage-worker that machinery displaces labor and increases the number of the involuntary idle, is met


text:
Space and Time (Saha)

Raum und Zeit (Minkowski)

title: Space and Time (Saha)


text:
Page:South Carolina Declaration and Ordinance of Secession.djvu/5

The People of the State of South Carolina, in Convention assembled, on the 26th day of April, A. D., 1852, declared that the frequent violations of the Constitution of the United States, by the Federal Government, and its encroachments upon the reserved rights of the States, fully justified this State in then withdrawing from the Federal Union; but in deference to the opinions and wishes of the other slaveholding States, she forbore at that time to exercise this right. Since that time, these encroachments have continued to increase, and further forbearance ceases to be a virtue.

And now the State of South Carolina having resumed her separate and equal place among nations, deems it due to herself, to the remaining United States of America, and to the nations of the world, that she should declare the immediate causes which have led to this act.

In the year 1765, that portion of the British Empire embracing Great Britain, undertook to make laws for the government of that portion composed of the thirteen American Colonies. A struggle for the right of self-government ensued, which resulted, on the 4th July, 1776, in a Declaration, by the Colonies, "that they are, and of right ought to be, FREE AND INDEPENDENT STATES;