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 . At the request and for the satisfaction of three Persons of Quality, Aug. 8, 1671,' London, 1687, 4to. An earlier edition was published, sine loco [1684], 4to. His conference with Stillingfleet gave rise to the publication of several controversial pamphlets, and 'The Summ of a Conference on Feb. 21, 1686, between Dr. Clagett and Father Gooden, about the point of Transubstantiation,' was published in 1689-90 by William Wake, D.D., afterwards archbishop of Canterbury. It is reprinted in 'Seventeen Sermons,' &c. by William Clagett, D.D., 3rd edit., London, 1699, 8vo, vol. i.

 GOODENOUGH, EDMUND (1786–1845), dean of Wells, youngest son of [q. v.], bishop of Carlisle, by his wife, Elizabeth, eldest daughter of Dr. James Ford, physician extraordinary to Queen Charlotte, was born at Ealing, Middlesex, on 3 April 1785. At an early age he was sent to Westminster School, where in 1797, when only twelve years old, he was elected into college. In 1801 he obtained his election to Christ Church, Oxford, where he took honours in Easter term 1804, and graduated B.A. 1805, M.A. 1807, B.D. 1819, and D.D. 1820. Having taken orders, Goodenough became tutor and censor of Christ Church, and in 1810 was appointed curate of Cowley, Oxford. In 1811 he was chosen by the university as one of the mathematical examiners, and in 1816 filled the office of proctor. In Michaelmas term 1817 he was appointed select preacher to the university, and in the following year was instituted vicar of Warkworth, Northumberland. In 1819 Goodenough was appointed head-master of Westminster School and subalmoner to the king, in succession to Dr. Page. On 23 June 1824 he was made a prebendary of York, on 22 April 1826 a prebendary of Carlisle, and on 1 June 1827 a prebendary of Westminster. In 1828 he retired from the head-mastership, and was succeeded by Dr. Williamson. Towards the end of Goodenough's rule the numbers of the school steadily declined. On 6 Sept. 1831 he was nominated dean of Wells, in the place of the Hon. Henry Ryder, bishop of Lichfield, who succeeded to Goodenough's stall at Westminster. Goodenough was prolocutor of the lower house of convocation for a short time. He died suddenly at Wells, while walking in the fields near his house, on 2 May 1845, aged 59, and was buried in the Lady Chapel of Wells Cathedral, where there is a brass to his memory. He married, on 31 May 1821, Frances, daughter of Samuel Pepys Cockerell of Westbourne House, Paddington, by whom he had [q. v.] and many other children. His widow, dying of cholera at Malaga on 5 Aug. 1855, was buried there. A portrait of Goodenough hangs in the dining-room of the head-master of Westminster School. Goodenough was an excellent scholar, and a man of much general culture. He was elected on the council of the Royal Society in 1828. He published the three following sermons:
 * 1) 'A Sermon [on 1 Cor. xiv. 33] preached at … Lambeth [12 Nov. 1820], at the Consecration of … W. Carey, … Bishop of Exeter,' London, 1821, 4to.
 * 2) 'A Sermon [on Deut. xxxiii. 9] preached … [13 May 1830] at the Festival of the Sons of the Clergy,' &c., London, 1830, 4to.
 * 3) 'A Sermon [on Luke xii. 47 and part of 48] preached in the Abbey Church, Bath [24 Jan. 1832] at the Anniversary Meeting of the Bath Diocesan Association of the Society for the Propagation of the Gospel,' &c., London, 1832, 8vo.

 GOODENOUGH, JAMES GRAHAM (1830–1875), commodore, son of [q. v.], dean of Wells, and grandson of [q. v.], bishop of Carlisle, was born on 3 Dec. 1830, at Stoke Hill, near Guildford, Surrey. The close connection of his godfather, Sir James Graham, with the admiralty had fixed his profession from the beginning, and after three years at school at Westminster, he entered the navy in May 1844 on board the Collingwood, commanded by Captain Robert Smart, and carrying the flag of Rear-admiral Sir [q. v.] as commander-in-chief in the Pacific. On the Collingwood's paying off, in the summer of 1848, Goodenough was appointed to the Cyclops on the coast of Africa, from which, towards the end of 1849, he was permitted to return home in order to pass his examination and compete for the lieutenant's commission in a special course at the college at Portsmouth. This commission he obtained in July 1861, and in September was appointed to the Centaur, carrying Rear-admiral Henderson's flag


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Rh and interesting changes, and produced many unusual types of local communities. More than that, there are nascent neighborhoods and neighborhoods in process of dissolution. Consider, for example, Fifth Avenue, New York, which probably never had an improvement association, and compare with it 135th Street in the Bronx, (where the negro population is probably more concentrated than in any other single spot in the world) which is rapidly becoming a very intimate and highly organized community. It is important to know what are the forces which tend to break up the tensions, interests, and sentiments which give neighborhoods their individual character. In general these may be said to be anything and everything that tends to render the population unstable, to divide and concentrate attentions upon widely separated objects of interest.

 * What part of the population is floating?

 * Of what elements, i.e. races, classes, etc., is this population composed?

 * How many people live in hotels, apartments, and tenements?

 * How many people own their own homes?

 * What proportion of the population consists of nomads, hobos, gypsies?

On the other hand, certain urban neighborhoods suffer from isolation. Efforts have been made at different times to reconstruct and quicken the life of city neighborhoods and to bring it in touch with the larger interests of the community. Such is in part the purpose of the social settlements. These organizations and others which are attempting to reconstruct city life have developed certain methods and a technique for stimulating and controlling local communities. We should study, in connection with the investigation of these agencies, these methods and this technique, since it is just the method by which objects are practically controlled that reveals their essential nature, that is to say, their predictable character (Gesetztmässigkeit).

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Rh characteristic is that they are composed of persons of the same race, or of persons of different races but of the same social class, is East London, with a population of 2,000,000 laborers.

"The people of the original East London have now overflowed and crossed the Lea, and spread themselves over the marshes and meadows beyond. This population has created new towns which were formerly rural villages, West Ham, with a population of nearly 300,000; East Ham, with 90,000; Stratford, with its "daughters," 150,000; and other "hamlets" similarly overgrown. Including these new populations we have an aggregate of nearly two millions of people. The population is greater than that of Berlin or Vienna, or St. Petersburg, or Philadelphia.

"It is a city full of churches and places of worship, yet there are no cathedrals, either Anglican or Roman; it has a sufficient supply of elementary schools, but it has no public or high school, and it has no colleges for the higher education and no university; the people all read newspapers, yet there is no East London paper except of the smaller and local kind.... In the streets there are never seen any private carriages; there is no fashionable quarter .... one meets no ladies in the principal thoroughfares. People, shops, houses, conveyances—all together are stamped with the unmistakable seal of the working class.

"Perhaps the strangest thing of all is this: in a city of two millions of people there are no hotels! That means, of course, that there are no visitors."

In the older cities of Europe, where the processes of segregation have gone farther, neighborhood distinctions are likely to be more marked than they are in America. East London is a city of a single class, but within the limits of that city the population is segregated again and again by racial and vocational interests. Neighborhood sentiment, deeply rooted in local tradition and in local custom, exercises a decisive selective influence upon city population and shows itself ultimately in a marked way in the characteristics of the inhabitants.

What we want to know of these neighborhoods, racial communities, and segregated city areas, existing within or on the outer edge of great cities, is what we want to know of all other social groups.

 * What are the elements of which they are composed?

 * To what extent are they the product of a selective process?

 * How do people get in and out of the group thus formed?

 * What are the relative permanence and stability of their populations?

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Rh holds so far as the individual man finds in the chances, the diversity of interests and tasks, and in the vast unconscious co-operation of city life, the opportunity to choose his own vocation and develop his peculiar individual talents. The city offers a market for the special talents of individual men. Personal competition tends to select for each special task the individual who is best suited to perform it.

"The difference of natural talents in different men is, in reality, much less than we are aware of; and the very different genius which appears to distinguish men of different professions, when grown up to maturity, is not upon many occasions so much the cause, as the effect of the division of labour. The difference between the most dissimilar characters, between a philosopher and a common street porter, for example, seems to arise not so much from nature, as from habit, custom, and education. When they came into the world, and for the first six or eight years of their existence, they were perhaps very much alike, and neither their parents nor playfellows could perceive any remarkable difference. About that age, or soon after, they come to be employed in different occupations. The difference of talents comes then to be taken notice of, and widens by degrees, till at last the vanity of the philosopher is willing to acknowledge scarce any resemblance. But without the disposition to truck, barter, and exchange, every man must have procured to himself every necessary and conveniency of life which he wanted. All must have had the same duties to perform, and the same work to do, and there could have been no such difference of employment as could alone give occasion to any great difference of talent.....

"As it is the power of exchanging that gives occasion to the division of labour, so the extent of this division must always be limited by the extent of that power, or, in other words, by the extent of the market There are some sorts of industry, even of the lowest kind, which can be carried on nowhere but in a great town."

Success, under conditions of personal competition, depends upon concentration upon some single task, and this concentration stimulates the demand for rational methods, technical devices, and exceptional skill. Exceptional skill, while based on natural talent, requires special preparation, and it has called into existence the trade and professional schools, and finally bureaus for vocational guidance. All of these, either directly or indirectly, serve at once to select and emphasize individual differences.

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70 THE CONFORMAL TRANSFORMATIONS OF A SPACE OF FOUR DIMENSIONS AND THEIR APPLICATIONS TO GEOMETRICAL OPTICS

By.

[Received October 9th, 1908. — Read November 12th, 1908.]

1. The method of inversion which was first applied to problems in electrostatics by Lord Kelvin, and which forms the basis of his theory of electric images, has also been applied with success in other branches of mathematical physics, as, for instance, in hydrodynamics. In geometrical optics, however, the method has been seldom used, probably because the necessary developments are not to be found in books on geometrical optics. The object of this paper is to show that the method can be of real value in both geometrical and physical optics. It is found that the transformation which is really needed is an inversion in a space of four dimensions, the transition to three-dimensional space being made by replacing the fourth coordinate by ict, where t is the time and c the velocity of light.

The first part of the paper is devoted to the general conformal transformation of a space of four dimensions. Shortly after Lord Kelvin's discovery of the method of transforming electrostatical problems by means of inversion, Liouville obtained the most general transformation that can be used for three-dimensional problems in this way.

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1908.] The group of transformations of this kind is known as the group of conformal transformations of space, it preserves the angles between two surfaces and changes a sphere into either a sphere or a plane.

The property, however, upon which the applications to electrostatical problems depends is that the transformations enable us to pass from one solution of the equation

to another.

Now the group of conformal transformations in a space of four dimensions possesses the analogous property in connection with the two differential equations

which are fundamental in the wave theory of light.

This has been known for some time, but the analysis given in § 2 will be useful in indicating the procedure to be adopted to obtain the relation connecting the two solutions for any transformation of the group.

In § 3 a particular solution of the first of the above equations is

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72 expressed in terms of Riemann's general hypergeometric function, and new light is thrown upon the theory of the transformations of the hypergeometric equation into itself.

In § 4 the applications to geometrical optics are considered. When applied to a symmetrical optical instrument, the transformation reduces to a homographic transformation of the points on the axis.

2. The Conformal Transformations of a Space of Four Dimensions.

The study of the conformal transformations of a space of four dimensions is simplified by the introduction of the six homogeneous coordinates :

{{MathForm2|(1)|$$\left.\begin{array}{lll} l=x-iy, & m=z+iw, & n=x^{2}+y^{2}+z^{2}+w^{2}\\ \\\lambda=x+iy, & \mu=z-iw, & \nu=-1\end{array}\right\}$$}}

connected by the identical relation

A function F(x, y, z, w) can be expressed by means of them as a homogeneous function of arbitrary degree. For instance, we may write

In the first representation V is a homogeneous function of degree zero, and in the second U is a homogeneous function of degree -1. The coordinate n may be introduced into the representations by means of the identical relation (2), the homogeneity of the expression being thereby unaltered.

Conversely, any homogeneous function of the six variables (l, m, n, λ, μ, ν) can be expressed as a function of (x, y, z, w). We shall now consider under what circumstances such a function can satisfy one of the differential equations—

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1908.] First, let V be a homogeneous function of degree zero. We evidently have

But, since V is a homogeneous function of degree zero,

and ν = -1, therefore

If, instead of (l, m, n, λ, μ, ν), we use the usual hexaspherical coordinates defined by the relations

the equation takes the more symmetrical form

This relation shows that a homogeneous function of degree zero, which is a solution of

i.e., of

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74 is a solution of

when expressed in terms of x, y, z, w.

Next, let U be a homogeneous function of degree -1 in (l, m, n, λ, μ, ν), then we can show in a similar way that

Hence, if U is a solution of

i.e., of

it is a solution of

when expressed in terms of (x, y, z, w).

When ($$a_{1},\ a_{2},\ \dots,\ a_{6}$$) are interpreted as the coordinates of a point in a space of six dimensions, the expressions

remain unaltered in form after any change of rectangular axes in which the origin remains the same. Any change of this kind corresponds to a transformation in the (x, y, z, w) space, enabling us to pass from one solution of the equation

to another, and a similar remark applies to the equation

To illustrate the method of formation of the transformation, we may consider the effect of simply interchanging n and -ν. The functions V and U of formulæ (8) and (4) then transform into

and

respectively, r² being written in place of x² + y² + z² + w².

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1908.] Accordingly, from a solution V = F(x, y, z, w) of the equation

we may derive a second solution

and from the solution U = f(x, y, z, w) of

we may derive another solution

Putting w = ict, where c is the velocity of light and t is the time, the equations take the well known form It may be mentioned here that Lorentz's fundamental equations of the electron theory, viz.,

may be reduced to a symmetrical form by writing s = ict and putting

The four mutually orthogonal vectors (A, B, C, D) whose components are respectively

(0, r, -q, -p), (-r, 0, p, -q), (q, -p, 0, -r), (p, q, r, 0)

satisfy the equations

where

$div\ M\equiv\frac{\partial M_{1}}{\partial x}+\frac{\partial M_{2}}{\partial y}+\frac{\partial M_{3}}{\partial z}+\frac{\partial M_{4}}{\partial s}$ if $M\equiv\left(M_{1},\ M_{2},\ M_{3},\ M_{4}\right)$|undefined

Again, if we put

and introduce four new vectors $$A_{1},\ B_{1},\ C_{1},\ D_{1}$$ whose components are respectively

(S, -Z, Y, -X) (Z, S, -X, -Y) (-Y, X, S, -Z) (X, Y, Z, S),

we find

where

Finally, if X, Y, Z, S can be derived from a potential function n so that

we can form four mutually orthogonal vectors θ, Φ, ψ, χ whose components are respectively

(n, r, -q, -p). (-r, n, p, -q), (q, -p, n, -r), (p, q, r, n),

and the equations then take the simple form

and the transformation may be written

where now

The study of this transformation will be taken up later.

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76 A second transformation of some interest is obtained by interchanging m with n and μ with ν. This changes

into

that is

f(x, y, z, w)

into

Putting w = ict and changing the sign of z, the formulæ for the transformation are

where, now,

If V = F(x, y, z, t) is a solution of

the function F(X, Y, Z, T) is also a solution, and, if U = f(x, y, z, t) is

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1908.] a solution of

the function

is also a solution.

In following up the connection between different solutions, it is convenient to use polar coordinates. Putting

we obtain the relations

There is a similar transformation for Laplace's equation.

If

a solution f(x, y, z) corresponds to a second solution

Putting

the formulæ of transformation become

The transition from one solution of Laplace's equation to another is now easily effected.

The effects of combining the different transformations belonging to a group of conformal transformations is most easily studied by interpreting

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78 the transformation as a change of axes in a space in which the coordinates are the spherical coordinates $$a_r$$. It is important to notice that the angle between two manifolds in this space is equal to the angle between the corresponding manifolds in the space to which the conformal transformations are applied. In the case of a space of four dimensions, we have, in fact,

from which the result easily follows.

A change in the sign of $$a_6$$ corresponds to an inversion, a change in the sign of $$a_6$$ coupled with a change in the sign of $$a_4$$ corresponds to the other transformation we have mentioned. It is evident that each of these transformations is of period 2. In general, a reflexion in a linear manifold in the a space corresponds to an inversion with regard to the corresponding circle, sphere, or hypersphere, in the space of four dimensions. A displacement of period n in the a space may be obtained by taking successive reflexions in two plane five-folds which cut at an angle π/n. It evidently corresponds to a periodic conformal transformation made up of inversions with regard to two hyperspheres which cut at an angle π/n.

3. The Relation between Riemann's General Hypergeometric Function and the Group of Conformal Transformations of a Space of Four Dimensions.

We shall now endeavour to satisfy the differential equation

by means of a function of the form

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1908.] where P is a function of the ratio of any two of the quantities lλ, mμ, nν. The quantity U will then be a solution of the equation

if the relation

is satisfied, for it will then be a homogeneous function of degree -1 in (l, m, n, λ, μ, ν).

Let us put

where a, b and c are arbitrary constants. The relation

is then satisfied, and P becomes a function of z alone. We may thus write

the particular functional form in terms of ξ, η, ζ being chosen to facilitate the calculations. H is clearly a homogeneous function of degree zero in ξ, η, ζ and therefore in l, m, n, λ, μ, ν.

On differentiating equation (1), we obtain

The differential equation

will thus be satisfied, if

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80 Now

Hence

Also, since

we have

this relation being obtained in the same way as the one above.

The last expression may be written

i.e.,

or

Now

and

consequently,

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1908.] The differential equation thus reduces to

This is Papperitz's form of the differential equation satisfied by Riemann's general hypergeometric function

hence we have the result that

is a homogeneous function of (l, m, n, λ, μ, ν) of degree -1, satisfying the equation

When expressed in terms of x, y, z and w, it will thus be a solution of the equation

The various transformations of the general hypergeometric function are easily obtained from this result. If we write U in the form

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82 where

we see that

is a multiple of

Again, if we write

we have

so that

This shows that P is the same function of the quantities a', b', c', z' as it is of a, b, c, z; that is

Hence the general hypergeometric function is unaltered if the quantities a, b, c, z are replaced by quantities a', b', c', z' which are derived from them by the same homographic substitution.

4. Applications to Geometrical Optics.

Let us consider a series of waves of light traversing a homogeneous or heterogeneous medium, and let

be the reduced path from a standard orthotomic surface or wave front to the point (x, y, z). Let us suppose, moreover, that V is expressed only in

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1908.] terms of the coordinates (x, y, z), and the constants of the standard wave front. Then V satisfies the differential equation

and is, in fact, the characteristic function introduced by Hamilton. If it is expressed as a function of x, y, z, and the coordinates of the initial point $$x_{0}y_{0}z_{0}$$, it is the Eikonal according to the nomenclature of Bruns.

Since V is proportional to the time this differential equation may be replaced by

where C is the velocity of radiation at the point (x, y, z).

Now suppose that the surfaces t = const, are obtained by solving an equation

for t; then, since

the function F must satisfy the differential equation

Confining ourselves to the case in which C is constant, we may use the results of § 2 to obtain new solutions of this differential equation.

Let

be the formulæ giving a transformation which enables us to pass from one solution of the above equation to another; then

when expressed in terms of x, y, z, t, is a second solution of the equation, and if the equation

be solved for t, the surfaces t = const, will form a system of parallel wave

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84 surfaces and t considered as a function of (x, y, z) will be the characteristic function for them.

The transformation

is of special importance because it makes the standard wave surface t = 0 in the original system correspond to a standard wave surface t = 0 in the new system; also, since the equations of the surfaces are

and

respectively, it is clear that one is the inverse of the other with regard to a unit sphere whose centre is at the origin. Our theorem tells us that if the surfaces parallel to the first are given by

the surfaces parallel to the second are given by

Applying this result to the family of right circular cones parallel to a given one, we may obtain the family of Dupin's cyclides parallel to a given one.

To obtain a geometrical interpretation of the transformation we describe a sphere of radius ct round the point (x, y, z) as centre. The inverse sphere is then of radius cT and its centre is at the point (XYZ).

A more general result is that the sphere

corresponds in the transformation to the sphere

the centres of the two spheres being corresponding points at the times $$t_{0},\ T_{0}$$ respectively.

To show that the laws of reflection and refraction remain unchanged in the transformation, we take the surface at which the light is incident as the standard one from which the time is measured. Let $$\left(x_{0},\ y_{0},\ z_{0}\right)$$ be a point on this surface; then we must associate with this point the time $$t_{0}=0$$.

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1908.] Consider a ray of light travelling from $$\left(x_{0},\ y_{0},\ z_{0}\right)$$ in a direction (l, m, n) with velocity c. At time t the wave has reached a point (x, y, z) on the ray, where

The corresponding point (X, Y, Z) derived from this by the transformation also travels along a straight line, for

where

The corresponding ray thus passes through the inverse point $$\left(X_{0}\ Y_{0}\ Z_{0}\right)$$ on the inverse surface at which it may be supposed to be incident. Its direction cosines (L, M, N) are connected with those of the former ray by means of the equations

These relations establish a correspondence between the sheafs of rays through the points $$\left(x_{0},\ y_{0},\ z_{0}\right)$$, $$\left(X_{0},\ Y_{0},\ Z_{0}\right)$$ respectively. This correspondence is such that the angle between two rays (l, m, n), (l', m', n') is equal to the angle between the two corresponding rays (L, M, N), (L', M', N'), for we have identically

Since the transformation enables us to derive the surfaces which are parallel to one surface from the surfaces which are parallel to the inverse surface, it is natural to expect that the above relation between the direction cosines will make the normals to the two surfaces correspond.

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86 Now

hence, if $$\left(l_{0},\ m_{0},\ n_{0}\right)$$ are the direction cosines of a tangent to the first surface, the direction cosines of the tangent for the corresponding displacement on the inverse surface are given by

The tangents to the two surfaces can thus be paired with one another in the correspondence.

Now, if (l, m, n) are the direction cosines of the normal to one surface, $$\left(l_{0},\ m_{0},\ n_{0}\right)$$ those of a tangent,

It follows, then, that for the corresponding ray (L, M, N)

This ray, being perpendicular to all the tangents to the inverse surface, is the normal at $$\left(X_{0},\ Y_{0},\ Z_{0}\right)$$ to this surface.

Since the angles between lines remain unaltered by the transformation, it follows that if (l, m, n), (l', m', n'), (p, q, r) are the direction cosines of an incident ray, the refracted ray and the normal at a point $$\left(x_{0},\ y_{0},\ z_{0}\right)$$ on a surface f, the corresponding quantities (L, M, N), (L', M', N'), (P, Q, R) are the direction cosines of an incident ray, refracted ray and the normal at the point $$\left(X_{0},\ Y_{0},\ Z_{0}\right)$$ of the inverse surface F. This is also easy to verify from the analytical formulæ.

The method of inversion can thus be applied to problems in geometrical optics. A medium of refractive index μ inverts into a medium of refractive index μ, a ray through the origin corresponds to a ray through the origin, and a ray through a fixed point (x, y, z) not on the surface of separation of the two media corresponds to a ray intersecting the line joining (x, y, z) to the origin.

A spherical shell of radiation emitted by an electron whose velocity was suddenly changed at time -t, corresponds to a shell of radiation emitted by an electron whose velocity was suddenly changed at time -T.

This method of inversion promises to be of great importance in the theory of radiation. It will be noticed that the spheres of radii ct whose centres are at the points

all touch one another at the point $$\left(x_{0},\ y_{0},\ z_{0}\right)$$ which may be regarded as a node of the radiation.

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1908.] From a study of the nodes in the ether produced by a vibrating atom or molecule, it appears that various systems of nodes may be obtained from one another by inversions which correspond to the same frequencies of vibration. The investigation will be reserved, however, for another paper.

5. Application of the Preceding Results to a Symmetrical Optical Instrument.

Let QR (Fig.) be the incident ray, Q'R the refracted ray, OR the normal to the spherical interface, and let C be the centre of inversion.

We shall suppose that the incident ray makes a small angle with the axis.

Let

and let the velocity of light for the first medium be represented by 1/μ, the times corresponding to the points Q, Q', O, A respectively may then be taken to be -μx, -μx', -μa, 0, respectively, and the corresponding quantities ct are simply the reduced distances (-x, -x', -a, 0).

Let $$Q_{1},\ Q'_{1},\ O_{1},\ A_{1}$$ be the points corresponding to Q, Q', O, A in the transformation, $$x_{1},\ x'_{1},\ z_{1}-a_{1},\ z_{1}$$ their distances from C.

Now the sphere centre Q and radius QA inverts into the sphere centre $$Q_1$$ and radius $$QA_1$$; hence, if the radius of inversion be equal to k³, we have for the points in which these spheres meet the axis

We also have the relations

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88 The linear magnification in the new system is

Let c be the distance of the second interface from the first; then we associate the length -c with the second interface B, and the formulæ of the transformation become

The last of which may be written

i.e.,

where z and x' are measured, as before, from the first surface. This is exactly the former relation between two corresponding points; consequently the whole course of a ray in one instrument corresponds in the transformation to that of the corresponding ray in the corresponding instrument.

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1908.] The linear magnification is given by

hence

which is of the same form as before.

If M and $$M_1$$ be the total linear magnifications for the two instruments, we have

The relation between two corresponding points is evidently a homographic one; hence we have the following theorem:—

''If the points on the axis of a symmetrical optical instrument be transformed by means of a homographic transformation, any pair of conjugate points for the instrument are transformed into a pair of points which are conjugate with regard to a second instrument. The centres of curvature of the interfaces and the points in which the interfaces meet the axis correspond in the two instruments, and the refractive indices of corresponding media are the same.''

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text:
Executive Order 10438

By virtue of the authority vested in me by the Constitution and laws of the United States, and as President of the United States and Commander in Chief of the armed forces of the United States, it is hereby ordered as follows:

Dwight D. Eisenhower The White House, March 13, 1953. [FR Doc. 53–????? Filed 3–??–53; 8:45 am] Billing Code ????–??–?

title: Executive Order 10438


text: Author:Edmund Goodenough

title: Author:Edmund Goodenough


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Page:Dictionary of National Biography volume 24.djvu/426

 obtained about 1480 the estates of Ellam and Maystreet in Kent. Thomas's father, William Harman, added to these estates the manor of Mayton or Maxton in the same county. As his father's heir, Thomas inherited all this property, and lived at Cray ford, Kent, continuously from 1547. He writes that he was 'a poore gentleman,' detained in the country by ill-health. He found some recreation in questioning the vagrants who begged at his door as to their modes of life, and paid frequent visits to London with the object of corroborating his information. He thus acquired a unique knowledge of the habits of thieves and beggars. Occasionally his indignation was so roused by the deception practised by those whom he interrogated at his own door that he took their licenses from them and confiscated their money, distributing it among the honest poor of his neighbourhood. Before 1566 Harman had composed an elaborate treatise on vagrants, and came to London to superintend its publication. He lodged at 'the Whitefriars within the Cloister,' and continued his investigation even while his book was passing through the press. Of the first edition, issued in 1566 or very early in 1567, no copy is known. Its popularity was at once so great that Henry Bynneman and Gerrard Dewes were both fined by the Stationers' Company in 1567 for attempting to circulate pirated copies. Of the second edition two copies, differing in many particulars, are extant. One is in the Bodleian Library (dated 8 Jan. 1567-8), and the other belongs to Mr. A. H. Huth (dated 'Anno Domini 1567'). The former is doubtless the earlier of the two, neither of which seems to have been published till early in 1568. Both were issued by William Griffith. The title ran in the later copy, 'A Caueat or Warening for commen cvrsetors Yvlgarely called Vagabones.' A dedication by Harman to his neighbour, Elizabeth, countess of Shrewsbury, and 'the epistle to the reader' is followed by exhaustive little essays on each class of the thieves' and tramps' fraternity to the number of twenty-four, and by a list of names of the chief professors of the art 'lyuinge nowe at this present.' A vocabulary of 'their pelting speche' or cant terms concludes the volume, which is embellished by a few woodcuts, including one of 'an upright man, Nicolas Blunt,' and another of 'a counterfeit cranke, Nicolas Genynges.' Harman borrowed something from 'The Fraternitye of Vacabondes,' by John Awdelay [q. v.], which was probably first issued in 1561, although the earliest edition now known is dated 1575; but Harman's information is far fuller and fresher than Awdelay's, and was very impudently plagiarised by later writers. 'The Groundworke of conny-catching' (1592), very doubtfully assigned to Robert Greene, reprints the greater part of Harman's book. Thomas Dekker, in his 'Belman of London' (1608), made free use of it, and Samuel Rowlands exposed Dekker's theft in his 'Martin Mark-all, Beadle of Bridewell' (Lond. 1610). Dekker, in the second part of his 'Belman,' called 'Lanthorne and Candlelight' (1609), conveyed to his pages Harman's vocabulary of thieves' words, which Richard Head incorporated in his 'English Rogue' (1671-80). Harman's vocabulary is the basis of the later slang dictionaries (cf. among others, that forming the appendix to 'Memoires of John Hall' (d. 1707) [q. v.], 1708). Another edition of Harman's 'Caueat' appeared in 1573, and this was reprinted by Machell Stace in 1814. A carefully collated edition of the second edition was edited by Dr. Furnivall and Mr. Edward Viles for the Early English Text Society in 1869, and re-issued by the New Shakspere Society in 1880.

 HARMAR or HARMER, JOHN (1555?–1613), professor of Greek at Oxford, was born, probably of humble parentage, at Newbury in Berkshire about 1555. Through the influence of the Earl of Leicester, he was elected to St. Mary's College, Winchester, in 1569, at the age of fourteen; in 1572 he obtained a scholarship at New College, Oxford, where he matriculated on 10 Jan. 1575, being described as 'plebei filius' (Oxf. Univ. Reg., Oxf. Hist. Soc., n. ii. 60), and was admitted perpetual fellow. He graduated B. A. on 21 Jan. 1577 (ib. iii. 64), and M. A. 18 Jan. 1582. He was reckoned a 'subtle Aristotelian,' was well read in patristic and scholastic theology, and was a 'most noted Latinist and Grecian '. About this period he appears to have gone abroad, being assisted by the Earl of Leicester, and to have held disputations at Paris with the 'great doctors of the Romish party' (ib.) In 1585 the earl obtained his appointment as regius professor of Greek at Oxford, and on 26 April 1587 he was elected one of the proctors. From 1588 to 1595 he was head-master of Winchester, and in 1596 became warden of St. Mary's College, and held that office until his death. He was also rector of Droxford in Hampshire, and a prebendary of Winchester. In 1604 he was appointed one of the translators of the New Testament, and had a 'prime hand' in that work. On 16 May 1605 he was admitted B.D. He died 11 Oct. 1613, and was buried in the chapel of New


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1. Introduction.
The numerous experimental and theoretical investigations that have followed Bradley's discovery of aberration have led finally to the enunciation of a principle of relativity. This principle which is being widely used as a working hypothesis is gradually attaining a position analogous to that held by the second law of thermodynamics, inasmuch as it gives a satisfactory account of the experimental results and is not in direct contradiction with any known facts.

The mathematical analysis on which the principle is based was introduced by Lorentz in 1892. It consists of a certain space time transformation of the coordinates in which the fundamental equations of the electrodynamical field are unaltered in form.

The principal features of the transformation are the introduction of a local time, and the hypothesis that a body moving through the aether with a constant velocity v suffers a contraction $$\sqrt{1-v^{2}/c^{2}}:1$$ in the direction of motion: c being the velocity of light. With the help of this hypothesis

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224 a satisfactory explanation of the results of the Michelson-Morley experiment can be given on the supposition of a stationary aether. The results of the experiment seem at first sight to indicate that the aether is carried along with the earth, but on such a supposition a satisfactory explanation of aberration would still be wanting.

The theory of Lorentz's transformation has been developed very considerably by Einstein, Planck, and Minkowski. The transformation has become a powerful instrument of research, inasmuch as it provides a means of transition from the mathematical expressions of physical quantities connected with a system at rest to the corresponding quantities for a similar system in uniform motion. The transformation of Lorentz only enables us to pass from a system at rest to one in uniform motion, but it has been postulated that there is a more general transformation which can be applied to systems moving in a more general manner. The case of a system moving with a constant acceleration has, in fact, been discussed by Einstein.

The object of the present paper is to find all the transformations for which the electrodynamical equations are invariant. In the case of the simpler equations of the theory of electrons, it is proved that the transformations belong to a certain group which is isomorphic with the group of conformal transformations of a space of four dimensions. It is assumed, however, that the transformation is such that the total charge on a system of particles is unaltered.

I have great pleasure in thanking Mr. E. Cunningham for the stimulus which he gave to this research by the discovery of the formula of transformation in the case of an inversion in the four-dimensional space. These formulæ suggested the more general formulæ in terms of Jacobians, and the transition to the integral forms led at once to the present analysis. The two integral forms of the second order had been introduced previously by Mr. Hargreaves in another connection; they enable us to give a very concise expression of the electrodynamical equations which promises to be of considerable importance in future developments. The integrals that occur in the equations differ from the usual surface and volume integrals by the fact that the quantities in the integrand are calculated at different points of space at different times, these times being specified by an arbitrary known law.

The invariance of the two integral forms of the second order leads at once to the formulae of transformation of the electric and magnetic force,

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1909.] and to a number of identical relations which are shown to imply that the differential equation

is invariant. This is the differential equation of the characteristics of the electromagnetic equations and gives the form of an elementary wave front. Since this is a sphere, we have called the group of transformation the group of spherical wave transformations. The differential equation also expresses the condition that two neighbouring particles should be in a position to act on one another. The transformations which leave the electrodynamical equations for ponderable bodies unaltered in form must depend upon the form of the constitutive relations connecting the magnetic induction and electric force with the magnetic force and electric displacement when the bodies are in motion. Minkowski, Einstein, and Laub have proceeded in the opposite way and constructed a set of constitutive relations for bodies in uniform motion, by assuming that the transformations are Lorentzian transformations and transforming the known constitutive relations for bodies at rest.

This method, however, does not give the constitutive relations for the case of a dielectric whose motion is not uniform. The general spherical wave transformation can be applied to obtain a certain type of accelerated motion, but the dimensions and shapes of bodies are continually altering in the transformed system.

An attempt has been made to discover whether there are any types of constitutive relations which are invariant for the general space time transformation. It is shown in § 7 that it is possible to construct such relations on the assumption that a certain quadratic form is invariant for the transformation. These relations may be supposed to correspond to a special type of configuration and state of motion which preserves its character after any space time transformation, which satisfies certain limitations. The general theory of space time transformations is discussed with the help of two theorems on the transformation of integral forms. It is shown that when a transformation of variables is performed, and the forms equivalent to two given integral forms are known, then the integral forms may be multiplied together by the rules of Grassmann's calculus of extension, and the resulting integral form is equal to the one obtained by multiplying the equivalent integral forms in the same way. Secondly, it is shown that when a quadratic form and its equivalent are known, a pair of equivalent integral forms may be obtained from a given pair by a process of analysis analogous to reciprocation, the new integral forms being of order n — m, where n denotes the number of variables and m the order of the original integral forms. These theorems are very

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226 useful for finding invariants. Various invariants are calculated with a view of obtaining the general form of the equations of motion. Abraham's expression of the reaction of radiation on a moving electron is shown to be an invariant for the group of spherical wave transformations, provided it is multiplied by a certain factor.

2. The Integral Equations of the Theory of Electrons.
Let the electromagnetic units be those adopted by Lorentz, and let the units of time and length be so chosen that the velocity of radiation is unity.

Let curl H denote the vector whose components are

when the axes of x, y, z are right-handed and the vector whose components are

when the axes are left-handed. Then the fundamental equations of the theory of electrons may be written

{{MathForm2|(I)|$$\left.\begin{array}{l} \mathsf{curl}\ H=\frac{\partial E}{\partial t}+\rho w\\ \\\mathsf{curl}\ E=-\frac{\partial H}{\partial t}\\ \\\mathsf{div}\ E=\rho\\ \\\mathsf{div}\ H=0\end{array}\right\} ,$$}}

where E is the electric force, H the magnetic force, ρ the volume density of electricity, and ρw the convection current.

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1909.] These equations may be replaced by the two integral equations

provided the integrals receive suitable interpretations. The interpretation that first suggests itself is obtained by regarding (x, y, z, t) as the coordinates of a point in a space of four dimensions. Let any closed two-dimensional manifold $$S_2$$ in this space be assigned by equating x, y, z, t to one-valued differentiate functions of two parameters $$\alpha, \beta$$, and let $$S_2$$ be the boundary of a three-dimensional manifold $$S_3$$ in which the coordinates are like functions of three parameters $$\alpha, \beta, \gamma$$, of which $$\gamma=0$$ on $$S_2$$, and $$\gamma<0$$ on $$S_3$$. Then any term such as $$\iint H_{x}dy\ dz$$ may be interpreted to mean $$\iint H_{x}\frac{\partial(y,z)}{\partial(\alpha,\beta)}d\alpha\ d\beta$$ taken over $$S_2$$, and any term such as $$\iiint\rho\ dx\ dy\ dz$$ may be interpreted to mean $$\iiint\rho\frac{\partial(x,y,z)}{\partial(\alpha,\beta,\gamma)}d\alpha\ d\beta\ d\gamma$$ taken over $$S_3$$.

The relations (II) and (III) may now be obtained with the aid of (I) by applying the generalized Green-Stokes theorem as given by Baker, Poincaré, and others.

In order that equations (II) and (III) may be equivalent to (I), the axes must form a right-handed system. If we wish to use left-handed axes we must change the sign of H in (II) and (III).

We shall now endeavour to give a simpler interpretation to the integrals occurring in equations (II) and (III).

Let S be an arbitrary closed surface in the (x, y, z) space, and let t be expressed in terms of x, y, z by an arbitrary law $$t=t(x, y, z)$$, which must be chosen, however, in such a way that t is a single-valued function which is finite together with its derivatives with regard to x, y, z at all points within S and on S itself. Let the coordinates of points on S be expressed in terms of two parameters $$\alpha, \beta$$.

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228 A term such as $$\int\int E_{x}dx\ dt$$ may now be understood to mean

and a term such as $$\int\int\int\rho w_{x}dy\ dz\ dt$$ to mean

The equation (II) may now be written

Transforming the left-hand side by means of Green's theorem, we get

$$\begin{array}{ll} \int\int\int\left[\frac{d}{dx}\left(E_{x}-H_{y}\frac{\partial t}{\partial z}+H_{z}\frac{\partial t}{\partial y}\right)\right. & +\frac{d}{dy}\left(E_{y}-H_{z}\frac{\partial t}{\partial x}+H_{x}\frac{\partial t}{\partial z}\right)\\ \\ & \left.+\frac{d}{dz}\left(E_{z}-H_{x}\frac{\partial t}{\partial y}+H_{y}\frac{\partial t}{\partial x}\right)\right]dx\ dy\ dz.\end{array}$$

Now

hence the above integral may be written

$$\begin{array}{l} \int\int\int\left[\left(\frac{\partial E_{x}}{\partial t}-\frac{\partial H_{z}}{\partial y}+\frac{\partial H_{y}}{\partial z}\right)\frac{\partial t}{\partial x}+\left(\frac{\partial E_{y}}{\partial t}-\frac{\partial H_{x}}{\partial z}+\frac{\partial H_{z}}{\partial x}\right)\frac{\partial t}{\partial y}\right.\\ \qquad\left.+\left(\frac{\partial E_{z}}{\partial t}-\frac{\partial H_{y}}{\partial x}+\frac{\partial H_{x}}{\partial y}\right)\frac{\partial t}{\partial z}+\left(\frac{\partial E_{x}}{\partial x}+\frac{\partial E_{y}}{\partial y}+\frac{\partial E_{z}}{\partial z}\right)dx\ dy\ dz,\right]\end{array}$$

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1909. and this is equivalent to

$-\iiint\left(\rho w_{x}\frac{\partial t}{\partial x}+\rho w_{y}\frac{\partial t}{\partial y}+\rho w_{z}\frac{\partial t}{\partial z}-\rho\right)dx\ dy\ dz,$

if equations (I) are satisfied and the axes form a right-handed system. In the same way, it can be shown that the integral on the left-hand side of (II) is zero, if equations (I) are satisfied.

Conversely, if equations (II) and (III) are satisfied for every closed surface S and for every law $$t = t (x, y, z)$$, such that $$\partial t/\partial x,\ \partial t/\partial y,\ \partial t/\partial z$$ are finite and continuous within S and on its surface, then equations (I) are also satisfied, provided, of course, that the quantities $$E_{x},\dots,H_{x,}\dots$$ possess derivatives which behave in such a way that an application of Green's theorem is permissible.

It seems natural to regard equations (II) and (III) as the fundamental equations of the theory of electrons, because they do not require the assumptions that the medium is continuous and that vectors E, H can be associated with every point in space; all that is required is that the quantities $$E_{x},\dots,\rho w_{x,}\dots$$ shall be integrable.

It is clear that in the general case the quantities occurring in these equations are evaluated at different points of space at different times. The integrals are thus more general than the usual surface and volume integrals, and seem to be better adapted for purposes of measurement, the difficulty of measuring quantities at different points of space at the same time being avoided.

Equation (III) may be regarded as the definition of the electric charge associated with a system of particles. The triple integral represents the total charge on the particles. This may be proved by showing that the triple integral remains invariant during the motion of a system of electrons.

Let an electron which was at the point (x, y, z) at time t be in a new position (x', y', z') at time t', where

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

and t is a small quantity which is a function of x, y, z, t. The integral form is transformed into

$\rho'w'_{x}dy'dz'dt'+\rho'w'_{y}dz'dx'dt'+\rho'w'_{z}dx'dy'dt'-\rho'dx'dy'dz',$

where

$\rho w_{x}=\rho'w'_{x}\frac{\partial(y',z',t')}{\partial(y,z,t)}+\rho'w'_{y}\frac{\partial(z',x',t')}{\partial(y,z,t)}+\rho'w'_{z}\frac{\partial(x',y',t')}{\partial(y,z,t)}-\rho'\frac{\partial(x',y',z')}{\partial(y,z,t)},$ $\dots\qquad\dots\qquad\dots\qquad\dots\qquad\dots\qquad\dots\qquad\dots$

or

$\rho'w'_{x}\frac{\partial(x',y',z',t')}{\partial(x,y,z,t)}=\rho w_{x}\frac{dx'}{dx}+\rho w_{y}\frac{dx'}{dy}+\rho w_{z}\frac{dx'}{dz}+\rho\frac{dx'}{dt}.$ $\dots\qquad\dots\qquad\dots\qquad\dots\qquad\dots\qquad\dots\qquad\dots$

This gives

$\begin{array}{ll} \rho'w'_{x}\frac{\partial(x',y',z',t')}{\partial(x,y,z,t)}= & \rho w_{x}\left[1+\epsilon\frac{\partial w_{x}}{\partial x}+w_{x}\frac{d\epsilon}{\partial x}\right]+\rho w_{y}\left[\epsilon\frac{\partial w_{x}}{\partial y}+w_{x}\frac{d\epsilon}{\partial y}\right]\\ \\ & +\rho w_{z}\left[\epsilon\frac{\partial w_{x}}{\partial z}+w_{x}\frac{d\epsilon}{\partial z}\right]+\rho\left[\epsilon\frac{\partial w_{x}}{\partial t}+w_{x}\frac{d\epsilon}{\partial t}\right]\end{array}$|undefined $\dots\qquad\dots\qquad\dots\qquad\dots\qquad\dots\qquad\dots\qquad\dots$

$\rho'\frac{\partial(x',y',z',t')}{\partial(x,y,z,t)}=\rho w_{x}\frac{d\epsilon}{\partial x}+\rho w_{y}\frac{d\epsilon}{\partial y}+\rho w_{z}\frac{d\epsilon}{\partial z}+\rho\left[1+\frac{\partial\epsilon}{\partial t}\right].$

Multiplying the last equation by $$w_x$$, and subtracting from the first, we get

$\rho'\frac{\partial(x',y',z',t')}{\partial(x,y,z,t)}\left(w'_{x}-w_{x}\right)=\rho\epsilon\left[\frac{\partial w_{x}}{\partial t}+w_{x}\frac{\partial w_{x}}{\partial x}+w_{y}\frac{\partial w_{x}}{\partial y}+w_{z}\frac{\partial w_{x}}{\partial z}\right].$|undefined

Putting

$\frac{d}{dt}=\frac{\partial}{\partial t}+w_{x}\frac{\partial}{\partial x}+w_{y}\frac{\partial}{\partial y}+w_{z}\frac{\partial}{\partial z},$

we see that the equation is satisfied if

$\left(1+\frac{d\epsilon}{dt}\right)\left(w'_{x}-w_{x}\right)=\epsilon\frac{dw_{x}}{dt},$|undefined

and if $$\frac{d\epsilon}{dt}$$ is a small quantity of the first order

$w'_{x}-w_{x}=\epsilon\frac{dw_{x}}{dt},$|undefined

in other words $$\left(w'_{x},w'_{y},w'_{z}\right)$$ are the component velocities at time t + ε.

Since the integral form is an invariant, we may calculate its value by considering the swarm of electrons at a given time t. The temporal terms then disappear, and the integral reduces simply to

$-\rho\ dx\ dy\ dz$

i.e., the total negative charge on the system of electrons. Each particle is supposed to be within

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280 the closed surface S at the moment when its charge is evaluated, but since the charges on the particles are evaluated at different times, the particles need not all be within the closed surface at a given time. This explains why the total charge is not

The double integral expresses the total charge as a surface integral, but here again the particles over which the integration extends are not all on the surface at the same time, but at different times.

In the same way equation (II) may be regarded as a general way of expressing the fact that there is no free magnetism.

If Φ is the scalar electromagnetic potential and $$\left(A_{x},A_{y},A_{z}\right)$$ the components of the electromagnetic vector potential, we have

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1909.] These equations are equivalent to the single integral equation

provided a suitable interpretation is given to the integrals.

Take any closed curve C and a surface $$\Sigma$$ bounded by this curve. Let t be expressed in terms of (x, y, z) by an arbitrary known law; then the line integral may be understood to mean

and can be transformed into the surface integral by means of Stokes's theorem. Since

$$\begin{array}{l} \frac{d}{dy}\left(A_{z}-\Phi\frac{\partial t}{\partial z}\right)-\frac{d}{dz}\left(A_{y}-\Phi\frac{\partial t}{\partial y}\right)\\ \\\qquad=\frac{\partial A_{z}}{\partial y}-\frac{\partial A_{y}}{\partial z}+\left(\frac{\partial A_{z}}{\partial t}+\frac{\partial\Phi}{\partial z}\right)\frac{\partial t}{\partial y}-\left(\frac{\partial A_{y}}{\partial t}+\frac{\partial\Phi}{\partial y}\right)\frac{\partial t}{\partial z}\\ \\\qquad=H_{x}-E_{z}\frac{\partial t}{\partial y}+E_{y}\frac{\partial t}{\partial z},\end{array}$$

the surface integral which is obtained is

$$\begin{array}{ll} \iint\left[\left(H_{x}-E_{z}\frac{\partial t}{\partial y}+E_{y}\frac{\partial t}{\partial z}\right)dy\ dz\right. & +\left(H_{y}-E_{z}\frac{\partial t}{\partial z}+E_{z}\frac{\partial t}{\partial x}\right)dz\ dx\\ \\ & \left.+\left(H_{z}-E_{y}\frac{\partial t}{\partial x}+E_{x}\frac{\partial t}{\partial y}\right)dx\ dy\right],\end{array}$$

or

Hence equation (IV) is established.

3. The Group of Point Transformations for which the Integral Equations of the Theory of Electrons are Invariant.
Let us consider a transformation of coordinates from (x, y, z, t) to (x', y', z', t') which is biuniform within a certain domain of values of (x, y, z, t). We shall suppose that the choice of a transformation is

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282 limited by the condition that the total charge on a system of particles is invariant. This is expressed analytically by the equation

provided the axes form a right-handed system in each case.

If the transformation is such that the integral equations of the theory of electrons are invariant, we must have

where $$\theta$$ is a constant.

These relations give two sets of equations connecting the quantities $$E_{x},\dots,H_{x},\dots$$ with $$E'_{x},\dots,H'_{x},\dots$$ viz.,

In order that these equations may be equivalent to one another we must

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1909.] have $$\theta^{2}=1$$ and eighteen relations of the types

There are clearly nine relations of the first type and nine relations of the second type. We shall now show that when $$\theta^{2}=1$$ these relations imply that there is a relation of the form

for this purpose we shall require the following lemma.

. — Let the sixteen quantities

be connected by the eighteen relations of type

which imply that conjugate minors of the determinant $$\left[\alpha_{1}\beta_{2}\gamma_{3}\delta_{4}\right]$$ are equal. The identity

then gives

or

Introducing the notation

we may obtain in the above way the equations

$$\begin{array}{ccc} \delta_{1}(\alpha\gamma)=\gamma_{1}(\alpha\delta), & & \delta_{3}(\alpha\gamma)=\gamma_{3}(\alpha\delta),\\ \\\delta_{2}(\alpha\gamma)=\gamma_{2}(\alpha\delta), & & \delta_{4}(\alpha\gamma)=\gamma_{4}(\alpha\delta),\\ \\\delta_{1}(\beta\gamma)=\gamma_{1}(\beta\delta), & & \delta_{3}(\beta\gamma)=\gamma_{3}(\beta\delta),\\ \\\delta_{2}(\beta\gamma)=\gamma_{2}(\beta\delta), & & \delta_{4}(\beta\gamma)=\gamma_{4}(\beta\delta).\end{array}$$

Hence, either

or all the quantities of type $$\gamma_{1}\delta_{2}-\gamma_{2}\delta_{1}$$ are zero. In the same way we can prove that either

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284 or all the quantities of type $$\beta_{1}\delta_{2}-\beta_{2}\delta_{1}$$ are zero, and that either

$(\beta\alpha)=(\beta\delta)=(\gamma\alpha)=(\gamma\delta)=0$

or all the quantities of type $$\alpha_{1}\delta_{2}-\alpha_{2}\delta_{1}$$ are zero. It follows from this that either (1) the six quantities (αβ) are zero or (2) that the thirty-six quantities $$\left(\alpha_{1}\delta_{2}-\alpha_{2}\delta_{1}\right),\ \left(\beta_{1}\delta_{2}-\beta_{2}\delta_{1}\right),\ \left(\gamma_{1}\delta_{2}-\gamma_{2}\delta_{1}\right),\ \dots$$ are all zero or (3) that there is a set of relations

It is easy to see, however, that in the latter case we also have

$(\gamma\delta)=(\alpha\beta)=0.$

Hence, in all cases,

$(\alpha\gamma)=(\alpha\beta)=(\alpha\delta)=(\beta\gamma)=(\beta\delta)=(\gamma\delta)=0.$

Again, we have

$\left

hence, since the last term is zero,

$\left

This gives

$\begin{array}{l} \left(\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}+\alpha_{4}^{2}\right)\left(\delta_{1}^{2}+\delta_{2}^{2}+\delta_{3}^{2}+\delta_{4}^{2}\right)=\end{array}\left

say, and there are similar equations in $$\alpha,\beta,\alpha\gamma,\beta\gamma,\beta\delta,\gamma\delta.$$. It follows that

$\begin{array}{cl} \alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}+\alpha_{4}^{2} & =\beta_{1}^{2}+\beta_{2}^{2}+\beta_{3}^{2}+\beta_{4}^{2}\\ & =\gamma_{1}^{2}+\gamma_{2}^{2}+\gamma_{3}^{2}+\gamma_{4}^{2}=\delta_{1}^{2}+\delta_{2}^{2}+\delta_{3}^{2}+\delta_{4}^{2}=\pm\lambda^{2}\end{array}$

These conditions, combined with the previous set, imply that the sixteen quantities $$\alpha_{1},\beta_{2},\dots$$ are the elements of an orthogonal matrix.

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1909.] If the signs of the α's are all changed, the relations take the form

$\beta_{2}\gamma_{3}-\beta_{3}\gamma_{2}=-\left(\alpha_{1}\delta_{4}-\alpha_{4}\delta_{1}\right),$

but the final result is unaltered; hence a set of relations of this type also imply that the quantities α are the elements of an orthogonal matrix.

Taking the quantities $$\alpha,\beta,\dots$$ to be the elements of the matrix

$\left

we have the result that, if eighteen relations of the type

$\frac{\partial(y',z')}{\partial(x,y)}=\frac{\partial(x',t')}{\partial(z,t)},\ \frac{\partial(y',z')}{\partial(x,t)}=-\frac{\partial(x',t')}{\partial(y,z)}$

or of the type

$\frac{\partial(y',z')}{\partial(x,y)}=-\frac{\partial(x',t')}{\partial(z,t)},\ \frac{\partial(y',z')}{\partial(x,t)}=\frac{\partial(x',t')}{\partial(y,z)}$

are satisfied, then the matrix is an orthogonal one, and this implies that

$dx'^{2}+dy'^{2}+dz'^{2}-dt'^{2}=\lambda^{2}\left[dx^{2}+dy^{2}+dz^{2}-dt^{2}\right];$

in other words, that the differential equation

$(dx)^{2}+(dy)^{2}+(dz)^{2}-(dt)^{2}=0$

is an invariant. The converse is also true, and may be easily verified.

This differential equation expresses the condition that two neighbouring particles should be in a position to act on one another; it also indicates that an elementary wave starting from a given point will have the form of a sphere. For this reason we shall call a transformation which leaves the differential equation invariant, a spherical wave transformation. Our analysis shows that a transformation which leaves the integral equations of the theory of electrons invariant is necessarily a spherical wave transformation. We must next inquire whether all spherical wave transformations are relevant for our purpose. If we call t' the local time, we must exclude transformations which make the local time run backwards as t increases. Hence, in order that a transformation may be relevant, the condition

$\frac{\partial t'}{\partial t}>0$

must be satisfied.

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Rh CIVILISING THE BLACKS. this column we propose to narrate a few incidents illustrating the relations between whites and blacks in Queensland. We shall take every possible care that nothing appears here but what is true—the plain unexaggerated statement of something that has actually occurred. But it is not always easy to get a true description of these incidents. It is seldom, for instance, that any white spectator is allowed to witness the manner in which the native police transact the business we pay them to do, and they are strictly forbidden by their official superiors either to report their proceedings or to describe in print the simplest occurrence with which they have been officially connected. Moreover, men are naturally reluctant to confess in public deeds which, though sanctioned by our Government and Parliament, and by the public sentiment of a section of our colonists, are yet of a nature which would according to the common law, bring them to the gallows. We do not, therefore, propose to give names of people or localities. It would be unfair to single out individual perpetrators of crime which we, as a community, tolerate and even encourage. Our readers must trust to the care which we may fairly claim to have always exercised in investigating the facts we have laid before them, and believe our assurance that we shall be particularly careful not to add unnecessarily to the disgraceful tale we have to tell. And we regret to add that for the reason we have given there is good cause to believe that the real truth far exceeds anything they will read here. All that these columns will contain may be regarded as mere echoes of the shameful tragedy that has been played since this colony was first founded, and which is continuing actively at the present day, even at the very moment we write.

With these preliminary remarks we shall proceed to tell the story of Toby. This was the name of a black lad, one of a tribe that had its headquarters on a well-known Western station. In that particular tract of country rational humanity had quieted the blacks. The tribe in question, finding how useless it was to contend with the white man's weapons, and not being unnecessarily molested, soon "came in," and their main camp was at the head station. The superintendent, a sensible man, had made an arrangement with one of the blacks, a sort of head man in the tribe. If one of them was guilty of any depredation he warned the head man that it had occurred and demanded the surrender of the offender. He was always produced, even if the tribe had some difficulty in hunting him up, for all knew the value to them of keeping on good terms with the whites. When caught he was duly whipped in presence of the whole assemblage of blacks and whites, the head man of the former agreeing as a sort of assessor with the superintendent as to the amount of punishment. The system answered admirably, and although there were sometimes hundreds of blacks on the run they neither disturbed the cattle camps nor touched the shepherds' huts. But the station passed into other hands, and a new superintendent, who had none of these "d———d Exeter Hall notions," took charge. It happened that Kanakas were introduced on the station about the time, and the blacks at first hardly knew what to make of them, but afterwards became quite friendly with them. One evening Toby, who was a bright young lad, and very useful in working the cattle, but who also possessed a full share of the mischievousness common to growing lads, black and white, took into his head to go with two other black lads to the hut of a Kanaka shepherd, about a mile from the head station. What happened there is not very clear. The Kanaka came into the head station declaring that they asked him for rations, and were going to kill him; but as he arrived with his boots on, and not at all blown, it is not likely that three active young blacks would have let him escape so easily if they had really meant mischief. It is probable that he was timid, and the boys had asked him for sugar, and that he became frightened, and they thought it a great bit of fun. On examination, it appeared that his ration of sugar was gone—or at least so much of it as a Kanaka would be likely to have left near the end of the week. Undoubtedly Toby had offended, and the former superintendent would have given him a crack with his whip when next he saw him. Toby evidently expected something of the kind, and, as was his custom when he had been in mischief, absented himself for a week or two. On his return the blacks told Mr. ——— that he was in the camp, probably expecting that he would go down and give him his fair deserts. The superintendent, however, telling the storekeeper to accompany him, went to the camp, and, seeing Toby there, spoke to him mildly, half jokingly, and warned him not to do such a thing again. Then, as if he had dismissed the subject, he told him to get the horses, for he would be wanted to go down to the cattle run. Toby cheerfully obeyed, and the three started in the direction indicated, poor Toby, in his joy

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286 Secondly, we must exclude transformations in which the axes are changed from a right-handed system to a left-handed system, for then equation (1) would imply a change from positive electricity to negative electricity. Such a transformation, however, becomes relevant when equation (1) is replaced by

equation (2) by

and equation (3) by

These relations also imply that the transformation is a spherical wave transformation, and so a transformation which changes a right-handed system of axes into a left-handed system is relevant, but the formulae of transformation of the components of the electric and magnetic force are not the same as before. The sign of θ will be determined later by the condition that a right-handed set of axes is transformed into a lefthanded set. If

corresponding sets of axes are both right-handed or both left-handed. If, on the other hand,

the axes are right-handed in one system and left-handed in the other.

It is easy to establish a relation between the two quantities

Since

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 we have

$\begin{array}{ll} \frac{\partial(x',y',z')}{\partial(x,y,z)}= & \theta\left[\frac{\partial t'}{\partial t}\left\{ \left(\frac{\partial x'}{\partial x}\right)^{2}+\left(\frac{\partial y'}{\partial x}\right)^{2}+\left(\frac{\partial z'}{\partial x}\right)^{2}-\left(\frac{\partial t'}{\partial x}\right)^{2}\right\} \right.\\ \\ & \qquad\left.-\frac{\partial x'}{\partial x}\left\{ \frac{\partial x'}{\partial x}\frac{\partial x'}{\partial t}+\frac{\partial y'}{\partial x}\frac{\partial y'}{\partial t}+\frac{\partial z'}{\partial x}\frac{\partial z'}{\partial t}-\frac{\partial t'}{\partial x}\frac{\partial t'}{\partial t}\right\} \right]\end{array}$

or

We also have

$\begin{array}{ll} \frac{\partial(x',y',z',t')}{\partial(x,y,z,t)} & =\frac{\partial(y',z')}{\partial(y,z)}\frac{\partial(x',t')}{\partial(x,t)}+\frac{\partial(z',x')}{\partial(y,z)}\frac{\partial(y',t')}{\partial(x,t)}+\dots\\ \\ & =\theta\left[\left\{ \frac{\partial(x',t')}{\partial(x,t)}\right\} ^{2}+\left\{ \frac{\partial(y',t')}{\partial(x,t)}\right\} ^{2}+\dots-\left\{ \frac{\partial(y',z')}{\partial(x,t)}\right\} ^{2}-\dots\right]\\ \\ & =-\theta\left[\left(\frac{\partial x'}{\partial x}\right)^{2}+\left(\frac{\partial y'}{\partial x}\right)^{2}+\left(\frac{\partial z'}{\partial x}\right)^{2}-\left(\frac{\partial t'}{\partial x}\right)^{2}\right]\\ \\ & \qquad\times\left\{ \left(\frac{\partial x'}{\partial t}\right)^{2}+\left(\frac{\partial y'}{\partial t}\right)^{2}+\left(\frac{\partial z'}{\partial t}\right)^{2}-\left(\frac{\partial t'}{\partial t}\right)^{2}\right\} \\ \\ & \qquad-\left\{ \frac{\partial x'}{\partial x}\frac{\partial x'}{\partial t}+\frac{\partial y'}{\partial x}\frac{\partial y'}{\partial t}+\frac{\partial z'}{\partial x}\frac{\partial z'}{\partial t}-\frac{\partial t'}{\partial x}\frac{\partial t'}{\partial t}\right\} \end{array}$

Therefore

$\begin{array}{ll} \frac{\partial(x',y',z',t')}{\partial(x,y,z,t)}= & -\theta\left[\left(\frac{\partial x'}{\partial x}\right)^{2}+\left(\frac{\partial y'}{\partial x}\right)^{2}+\left(\frac{\partial z'}{\partial x}\right)^{2}-\left(\frac{\partial t'}{\partial x}\right)^{2}\right]\\ \\ & \qquad\times\left[\left(\frac{\partial x'}{\partial t}\right)^{2}+\left(\frac{\partial y'}{\partial t}\right)^{2}+\left(\frac{\partial z'}{\partial t}\right)^{2}-\left(\frac{\partial t'}{\partial t}\right)^{2}\right]\end{array}$

This gives us the relation

which holds for any spherical wave transformation.

We shall now introduce the further restriction that the inequality

$(dx')^{2}+(dy')^{2}+(dz')^{2}<(dt')^{2}$

is a consequence of

$(dx)^{2}+(dy)^{2}+(dz)^{2}<(dt)^{2}$

This means that, if a particle is moving with a velocity less than that of light in one system of coordinates, it is also moving with a velocity less than that of light in the transformed system.

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 Putting

we see that λ² is positive, and therefore

is negative. This shows that $$\frac{\partial(x',y',z')}{\partial(x,y,z)}$$ and $$\frac{\partial t'}{\partial t}\frac{\partial(x',y',z',t')}{\partial(x,y,z,t)}$$ must have the same sign. Hence, if $$\frac{\partial t'}{\partial t}$$ is positive, $$\frac{\partial(x',y',z')}{\partial(x,y,z)}$$ must have the same sign as the Jacobian. Accordingly, a transformation which changes a right-handed system of axes into a right-handed system must have a positive Jacobian; a transformation which changes a right-handed system of axes into a left-handed system must have a negative Jacobian.

The sign of θ may now be determined from equation (A). Since

it is necessarily positive. Consequently θ must have the same sign as $$\frac{\partial(x',y',z')}{\partial(x,y,z)},$$ and therefore the same sign as the Jacobian.

We can now obtain the formulae of transformation in the two possible cases.

(i) When the Jacobian is positive,$$\theta=+1$$, and the formulae of transformation are

$$\begin{array}{rl} E_{x}= & E'_{x}\frac{\partial(y',z')}{\partial(y,z)}+E'_{y}\frac{\partial(z',x')}{\partial(y,z)}+E'_{z}\frac{\partial(x',y')}{\partial(y,z)}\\ \\ & \qquad-H'_{x}\frac{\partial(x',t')}{\partial(y,z)}-H'_{y}\frac{\partial(y',t')}{\partial(y,z)}-H'_{z}\frac{\partial(z',t')}{\partial(y,z)}\\ \\-H_{x}= & E'_{x}\frac{\partial(y',z')}{\partial(x,t)}+E'_{y}\frac{\partial(z',x')}{\partial(x,t)}+E'_{z}\frac{\partial(x',y')}{\partial(x,t)}\\ \\ & \qquad-H'_{x}\frac{\partial(x',t')}{\partial(x,t)}-H'_{y}\frac{\partial(y',t')}{\partial(x,t)}-H'_{z}\frac{\partial(z',t')}{\partial(x,t)}\\ \\\rho w_{x}= & \rho'w'_{x}\frac{\partial(y',z',t')}{\partial(y,z,t)}+\rho'w'_{y}\frac{\partial(z',x',t')}{\partial(y,z,t)}+\rho'w'_{z}\frac{\partial(x',y',t')}{\partial(y,z,t)}-\rho'\frac{\partial(x',y',z')}{\partial(y,z,t)},\\ \\-\rho= & \rho'w'_{x}\frac{\partial(y',z',t')}{\partial(x,y,z)}+\rho'w'_{y}\frac{\partial(z',x',t')}{\partial(x,y,z)}+\rho'w'_{z}\frac{\partial(x',y',t')}{\partial(x,y,z)}-\rho'\frac{\partial(x',y',z')}{\partial(x,y,z)}.\end{array}$$

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 These equations may be simplified by using the relations of type

$\frac{\partial(y',z',t')}{\partial(y,z,t)}=\theta\frac{\partial x'}{\partial t}\lambda^{2},$

which are proved in the same way as (A), the quantity λ² being defined by equation (C).

The new equations are

and these imply that

$\rho w_{x}dx+\rho w_{y}dy+\rho w_{z}dz-\rho dt=\lambda^{2}\left[\rho'w'_{x}dx'+\rho'w'_{y}dy'+\rho'w'_{z}dz'-\rho'dt'\right]$

The formulae connecting the electromagnetic potentials are obtained by putting

$A_{x}dx+A_{y}dy+A_{z}dz-\Phi dt=\theta\left[A'_{x}dx'+A'_{y}dy'+A'_{z}dz'-\Phi'dt'\right].$

Since $$\theta=+1$$, they are

$\begin{array}{rl} A_{x}= & A'_{x}\frac{\partial x'}{\partial x}+A'_{y}\frac{\partial y'}{\partial x}+A'_{z}\frac{\partial z'}{\partial x}-\Phi'\frac{\partial t'}{\partial x},\\ \\-\Phi= & A'_{x}\frac{\partial x'}{\partial t}+A'_{y}\frac{\partial y'}{\partial t}+A'_{z}\frac{\partial z'}{\partial t}-\Phi'\frac{\partial t'}{\partial t}.\end{array}$

(ii) When the Jacobian is negative and $$\partial t'/\partial t>0$$, the axes are righthanded in the original system and left-handed in the system specified by the dashed letters. To obtain the correct formulae of transformation we must change the sign of H' in equations (2) and (8) and put $$\theta=-1$$. There is also a doubt about the sign in the equation

$\begin{array}{l} \rho w_{x}dy\ dz\ dt+\rho w_{y}dz\ dx\ dt+\rho w_{z}dx\ dy\ dt-\rho dx\ dy\ dz\\ \qquad=\pm\left(\rho'w'_{x}dy'\ dz'\ dt'-\rho'w'_{y}dz'\ dx'\ dt'-\rho'w'_{z}dx'\ dy'\ dt'+\rho'dx'\ dy'\ dz'\right),\end{array}$

the sign depending upon whether positive electricity is transformed into negative electricity or positive electricity. If the negative sign be taken there must be a corresponding alteration in equation (3).

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4. Spherical Wave Transformations and the Group of Conformal Transformations of a Space of Four Dimensions.
The group of spherical wave transformations may be reduced to a known group by putting t = is. The quadratic form which remains invariant is then of the type

$\lambda^{2}\left(dx^{2}+dy^{2}+dz^{2}+ds^{2}\right),$

and so the transformation is a conformal one.

The group of conformal transformations in a space of four dimensions has been studied by Sophus Lie, who has shown that it is composed of reflexions, translations, rotations, magnifications, and inversions. The transformations which are of importance in the present case are the imaginary ones, and it should be noticed that by a combination of two imaginary inversions we can obtain a transformation of the type

which is quite different from an inversion or simple displacement. This corresponds to the real spherical wave transformation

An imaginary rotation in the four-dimensional space may be specified, in a particular case, by the equations

$\begin{array}{lrcc} x'= & x\ \cos iw+s\ \sin iw, & & y'=y,\\ \\s'= & -x\ \sin iw+s\ \cos iw, & & z'=z.\end{array}$

Putting

$\tanh w=v,$

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 we obtain Lorentz's transformation

The formulae of transformation of the electric and magnetic vectors are obtained at once from the general formulae; they are

These agree with the formulae of Einstein, Lorentz's formulae for the convection currents being slightly different.

In the case of an inversion with regard to a hypersphere whose centre is at the origin, and whose radius is a real quantity k, the formulas of transformation are

Putting $$s=it,\ x^{2}+y^{2}+z^{2}=r^{2}$$, we get the real spherical wave transformation

This transformation has a negative Jacobian; the formulae for transforming the components of the electric and magnetic force have been obtained by E. Cunningham.

It should be noticed that

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 and is positive; hence t' increases as t increases, if (x, y, z) are kept constant.

The transformation which corresponds to a reflexion in the four-dimensional space is also of considerable interest. In the particular case when the reflecting space passes through the plane x = 0, s = 0, the reflexion may be replaced by a rotation round the plane x = 0, s = 0, and a reflexion in the space x = 0. The corresponding transformation is thus made up out of a transformation of Lorentz

$x'=\frac{x-vt}{\sqrt{1-v^{2}}},\ y'=y,\ z'=z,\ t'=\frac{t-vx}{\sqrt{1-v^{2}}}.$|undefined

and a change in the sign of x'. Putting

$\frac{v}{\sqrt{1-v^{2}}}=\frac{2u}{1-u^{2}},\ \frac{1}{\sqrt{1-v^{2}}}=\frac{1+u^{2}}{1-u^{2}},$|undefined

and changing the sign of x', we get

$\begin{array}{l} x'=-\frac{1+u^{2}}{1-u^{2}}x+\frac{2u}{1-u^{2}}t,\\ \\y'=y,\\ \\z'=z\\ \\t'=\frac{1+u^{2}}{1-u^{2}}t-\frac{2u}{1-u^{2}}x.\end{array}$|undefined

The quantity u is introduced because the angle of rotation in the four-dimensional space is twice the angle between the reflecting space and the space x = 0. Its geometrical meaning in the case of the spherical wave transformation is indicated by the equation

$x'-ut'=-(x-ut)$

which implies that a plane moving with the constant velocity u is transformed into itself. Further, when x = ut, we have

$x'=x,\ y'=y,\ z'=z,\ t'=t;$

hence every point of the plane is transformed into itself.

The formulae of transformation of the electromagnetic vectors are found from the first set of equations for a transformation with negative Jacobian.

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 The general infinitesimal spherical wave transformation is

where the coefficients are all constants, and ε is a quantity whose square may be neglected. Since there are fifteen arbitrary constants the group is a fifteen parameter group.

We have

If (x, y, z) is kept constant as t varies, the corresponding point (x', y', z') moves along a parabola, but in one type of transformation it moves along a straight line with constant acceleration. This is the case, for example, when

$\begin{array}{lll} x'=x+\epsilon\left(y^{2}+z^{2}-x^{2}-t^{2}\right), & & y'=y(1-2\epsilon x),\\ \\z'=z(1-2\epsilon x), & & t'=t(1-2\epsilon x),\end{array}$

for, since quantities of order ε² may be neglected, the first equation may be written

$x'=x+\epsilon\left(y^{2}+z^{2}-x^{2}-t^{2}\right)$

Hence, if (x, y, z) are kept constant, the point (x', y', z') moves with constant acceleration γ given by

$\gamma=-2\epsilon.$

Substituting for γ, we get

$\begin{array}{lll} x'=x-\frac{1}{2}\gamma\left(y^{2}+z^{2}-x^{2}-t^{2}\right), & & y'=y(1+\gamma x),\\ \\z'=z(1+\gamma x), & & t'=t(1+\gamma x).\end{array}$

The last equation agrees with the one obtained by Einstein.

In the case of the general infinitesimal transformation, the expression

$(x'-x)^{2}+(y'-y)^{2}+(z'-z)^{2}-(t'-t)^{2},$

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 is of order ε², but there is one type of transformation in which it is of order ε³ at least. The formulae of transformation are then

where

$l^{2}+m^{2}+n^{2}=1,$

and signify that (x', y', z'), (x, y, z) are successive positions of a particle which is moving with the velocity of light.

Real spherical wave transformations may be obtained geometrically in the following way. Let the space time point (x, y, z, t) be represented by a sphere of radius t having its centre at the point (x, y, z). Then, if we apply a real conformal transformation of space to these representative spheres, the new set of spheres may be taken as the representative spheres of a new set of space time points (x', y', z', t'). These are connected with the original set by a system of equations which define a spherical wave transformation. This theorem has already been established in the case of an inversion, and since any real conformal transformation of space can be built up from inversions, it follows that the general transformation obtained in the above way is a spherical wave transformation.

It should be noticed that the general spherical wave transformation cannot be obtained in this way, because t' = 0 always corresponds to t = 0. If, however, we combine these transformations with the real spherical wave transformation obtained by increasing or decreasing the radii of all the representative spheres by the same amount, it is possible to obtain any spherical wave transformation by a suitable combination. The proof of this will be left to the reader.

When we use the representative spheres the differential equation

$(dx)^{2}+(dy)^{2}+(dz)^{2}-(dt)^{2}=0$

admits of a very simple interpretation, as it implies that the two consecutive representative spheres specified by (x, y, z, t) and (x+dx, y+dy, z+dz, t+dt) touch one another internally.

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5. The Transformation of Integral Forms.
The general theory of the transformation of physical problems by means of a change of coordinates can be developed in a convenient manner by studying a number of integral forms, examining the way in which they are related to one another, and obtaining the formulae by means of which they can be transformed.

We shall commence by studying the simple case of a transformation in two variables from (x, y) to (x', y'). Suppose that

then

and it is easy to see that

$aB-Ab=(a'B'-A'b')\frac{\partial(x',y')}{\partial(x,y)}.$

This implies that

Now this relation can be obtained from the previous pair by the process of multiplication used in Grassmann's calculus of extension. In this calculus, the sign of a product depends on the order of the terms; thus

$dx\ dy=-dy\ dx$ and $dx\ dx=0$

This rule can also be applied to the case in which the quantities a and b are differential operators; thus, if

a relation which may be obtained directly by means of Green's theorem.

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 Next, suppose that the formula of transformation of a quadratic differential form is known, e.g.,

then taking two independent differentials dx, dy, δx, δy, and writing $$dx+\lambda\delta x,\ dy+\lambda\delta y,\dots$$ in place of dx, dy, we get the formula of transformation of the bilinear form

We may multiply this equation by itself, multiplying both sets of differentials according to Grassmann's rule. The resulting equation is

This gives

or

Multiplying (6) by (1) according to Grassmann's rule, we obtain

$$\begin{array}{l} \left[(Eb-Fa)\delta x-(Ga-Fb)\delta y\right]dx\ dy\\ \qquad=\left[(E'b'-F'a')\delta x'-(G'a'-F'b')\delta y'\right]dx'\ dy'.\end{array}$$

This gives the formula of transformation of a linear form

which may be called the reciprocal of the first.

Multiplying this equation by

respectively, we get

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 These formulae are well known in the differential geometry of a surface.

We shall now show that the above theory may be extended to transformations in any number of variables. In the first place we must show that the law of multiplication still holds. Let there be n variables $$x_{1},\dots,x_{n}$$ and suppose that

then by multiplication we may obtain an integral form of the second order

the multiplication being performed by Grassmann's rule. To verify this we have only to replace $$dx_{r}dx_{s}$$ by $$\frac{\partial\left(x_{r},x_{s}\right)}{\partial(\alpha,\beta)}d\alpha\ d\beta$$, and notice that the last equation may be written

$$\left|\begin{array}{ccc} \sum a_{r}\frac{\partial x_{r}}{\partial\alpha}, & & \sum b_{r}\frac{\partial x_{r}}{\partial\alpha}\\ \\\sum a_{r}\frac{\partial x_{r}}{\partial\beta}, & & \sum b\frac{\partial x_{r}}{\partial\beta}\end{array}\right|\begin{array}{c} d\alpha\ d\beta=\end{array}\left|\begin{array}{ccc} \sum a'_{r}\frac{\partial x'_{r}}{\partial\alpha}, & & \sum b'_{r}\frac{\partial x'_{r}}{\partial\alpha}\\ \\\sum a'_{r}\frac{\partial x'_{r}}{\partial\beta}, & & \sum b'_{r}\frac{\partial x'_{r}}{\partial\beta}\end{array}\right|\begin{array}{c} d\alpha\ d\beta.\end{array}$$

Similarly, if we take three integral forms,

and multiply them together by Grassmann's rule, we obtain the integral

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 form of the third order

$$\begin{array}{c} \sum\limits _{r,s,t}\end{array}\left|\begin{array}{ccc} a_{r} & a_{s} & a_{t}\\ b_{r} & b_{s} & b_{t}\\ c_{r} & c_{s} & c_{t}\end{array}\right|\begin{array}{c} dx_{r}dx_{s}dx_{t}=\sum\limits _{r,s,t}\end{array}\left|\begin{array}{c} \begin{array}{ccc} a'_{r} & a'_{s} & a'_{t}\\ b'_{r} & b'_{s} & b'_{t}\\ c'_{r} & c'_{s} & c'_{t}\end{array}\end{array}\right|\begin{array}{c} dx'_{r}dx'_{s}dx'_{t}.\end{array}$$

It will be seen also that this equation may he obtained by multiplying the integral form of the second order derived from (α) and (β), by (γ). The question then arises whether integral forms of any order may be multiplied together by Grassmann's rule. To show that this is the case we shall consider the two integral forms

of orders k and m respectively. Choosing m + k variables $$\alpha,\beta,\dots,$$, we have $$\frac{(m+k)!}{m!k!}$$ relations of the type

and the same number of relations of the type

These may be arranged in conjugate pairs in such a way that the whole set of variables $$\alpha,\beta,\gamma,\delta,\dots$$ occur in each pair. Multiplying conjugate pairs together, attributing proper signs to each product, and adding, we find from the properties of the minors of a determinant that the coefficient of a term such as $$A_{s,t,\dots}B_{p,q,r,\dots}$$, is zero, unless the quantities s,t,p,q,r,... are all different. When these quantities are different the coefficient is simply the determinant $$\frac{\partial\left(x_{s},x_{t},\dots,x_{p},x_{q},x_{r},\dots\right)}{\partial\left(\alpha,\beta,\gamma,\dots\right)}$$ with a proper sign. When multiplied by $$d\alpha\ d\beta\ d\gamma\dots$$ this yields $$dx_{s},dx_{t},\dots,dx_{p},dx_{q},dx_{r},\dots$$ and it is clear from this that the multiplication of the integral forms by Grassmann's rule is justified.

Taking the case of a transformation from four variables (x, y, z, t) to (x', y', z', t'). If

$$\begin{array}{l} H_{x}dy\ dz+H_{y}dz\ dx+H_{z}dx\ dy+E_{x}dx\ dt+E_{y}dy\ dt+E_{z}dz\ dt\\ \qquad=H'_{x}dy'dz'+H'_{y}dz'dx'+H'_{z}dx'dy'+E'_{x}dx'dt'+E'_{y}dy'dt'+E'_{z}dz'dt',\end{array}$$

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 and

$$\begin{array}{l} E_{x}dy\ dz+E_{y}dz\ dx+E_{z}dx\ dy-H_{x}dx\ dt-H_{y}dy\ dt-H_{z}dz\ dt\\ \qquad=E'_{x}dy'dz'+E'_{y}dz'dx'+E'_{z}dx'dy'-H'_{x}dx'dt'-H'_{y}dy'dt'-H'_{z}dz'dt',\end{array}$$

we obtain, on multiplication,

$$\begin{array}{l} \left(E_{x}^{2}+E_{y}^{2}+E_{z}^{2}-H_{x}^{2}-H_{y}^{2}-H_{z}^{2}\right)dx\ dy\ dz\ dt\\ \qquad=\left(E_{x}^{'2}+E_{y}^{'2}+E_{z}^{'2}-H_{x}^{'2}-H_{y}^{'2}-H_{z}^{'2}\right)dx'dy'dz'dt',\end{array}$$

while, if either form be multiplied by itself, we obtain

These equations indicate the invariance of the property that at a surface of discontinuity, or in a spherical wave, the electric force is equal in magnitude to the magnetic force and perpendicular to it.

A space time transformation from the variables (x, y, z, t) to (x', y, z, t') can be used to transform the whole motion in one dynamical system into a corresponding motion in another, as far as the kinematics is concerned, provided the velocities $$\left(w_{x},w_{y},w_{z}\right),\ \left(w'_{x},w'_{y},w'_{z}\right)$$ of corresponding points are such that the equations

are a consequence of the relations

This condition is satisfied, if

Multiplying these equations together by Grassmann's rule, we get

This shows that there is an integral invariant of the form

This fact has already been used in § 3 and will be required again in § 7.

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 We shall now obtain some further properties of a general space time transformation.

When an expression has the same form in the dashed letters as in the original ones it will be convenient to say that it is an invariant.

Let us suppose that the bilinear form

is an invariant. If we multiply it by itself four times according to Grassmann's rule, we obtain the invariant

$$\left|\begin{array}{cccc} A & H & G & U\\ H & B & F & V\\ G & F & C & W\\ U & V & W & D\end{array}\right|\begin{array}{c} dx\ dy\ dz\ dt\ \delta x\ \delta y\ \delta z\ \delta t.\end{array}$$

Putting $$dx=\delta x,\dots,$$ and denoting the determinant by Δ, we have the invariant

Now, let

be an integral invariant of the third order.

Multiplying by (1) and rejecting the factor $$\sqrt{\Delta}dx\ dy\ dz\ dt$$, we obtain an invariant

where

{{MathForm2|(5)|$$\left.\begin{array}{l} \sqrt{\Delta}v_{x}=Au_{x}+Hu_{y}+Gu_{z}+Uu_{t}\\ \sqrt{\Delta}v_{y}=Hu_{x}+Bu_{y}+Fu_{z}+Vu_{t}\\ \sqrt{\Delta}v_{z}=Gu_{x}+Fu_{y}+Cu_{z}+Wu_{t}\\ \sqrt{\Delta}v_{t}=Uu_{x}+Vu_{y}+Wu_{z}+Du_{t}\end{array}\right\} $$}}

Multiplying (3) and (4), we obtain the invariant

hence the quantity in the square brackets divided by Δ is an invariant

Conversely, if we are given an integral form of the first order (4), we may obtain a reciprocal integral form of the third order (8), by multiplying

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 (4) by (1) three times and rejecting the factor $$dx\ dy\ dz\ dt\ \sqrt{\Delta}$$. The relations expressing the quantities $$u_x$$ in terms of the quantities $$u_x$$ are exactly the same as those obtained by solving equations (5).

Next, suppose we are given an integral form of the second order

which is an invariant. Then multiplying it twice by (1), and rejecting the factor $$\sqrt{\Delta}dx\ dy\ dz\ dt$$, we obtain a reciprocal integral invariant

where

{{MathForm2|(9)|$$\left.\begin{array}{rc} \sqrt{\Delta}D_{x} & =\left(BC+F^{2}\right)E_{x}+(FG-CH)E_{y}+(HF-BG)E_{z}\\ \\ & +(HW-VG)B_{x}+(BW-VF)B_{y}+(FW-CV)B_{z}\\ & \dots\qquad\dots\qquad\dots\qquad\dots\qquad\dots\qquad\dots\qquad\dots\qquad\\ \\-\sqrt{\Delta}H_{x} & =\left(HW-VG\right)E_{x}+(GU-AW)E_{y}+(AV-HU)E_{z}\\ \\ & +(AD-U^{2})B_{x}+(HD-UV)B_{y}+(GD-UW)B_{z}\\ & \dots\qquad\dots\qquad\dots\qquad\dots\qquad\dots\qquad\dots\qquad\dots\qquad\end{array}\right\} $$}}

The relation between these two invariants is evidently a mutual one.

Multiplying them together, we obtain the integral invariant

and the absolute invariant

The theory of reciprocal invariants can evidently be extended to the case in which there are n variables, and a bilinear integral form is known to be invariant.

If we suppose that a biquadratic integral form

is invariant for a transformation, we may multiply it by itself and obtain the invariant

$\left(\kappa_{11}\kappa_{44}+\dots+2\kappa_{23}\kappa_{14}+\dots\right)dx\ dy\ dz\ dt\ \delta x\ \delta y\ \delta z\ \delta t.$

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 Putting $$dx=\delta x$$ and extracting the square root, we obtain an invariant

Now, let

$B_{x}dy\ dz+B_{y}dz\ dx+B_{z}dx\ dy+E_{x}dx\ dt+E_{y}dy\ dt+E_{z}dy\ dt,$

be an invariant of the second order. Multiplying it by (12) and rejecting a factor $$\sqrt{\Theta}dx\ dy\ dz\ dt$$, we obtain an invariant

$D_{x}\delta y\ \delta z+D_{y}\delta z\ \delta x+D_{z}\delta x\ \delta y-H_{x}\delta x\ \delta t-H_{y}\delta y\ \delta t-H_{z}\delta z\ \delta t,$

where

$\begin{array}{c} \sqrt{\Theta}D_{x}=\kappa_{11}E_{x}+\kappa_{12}E_{y}+\kappa_{13}E_{z}+\kappa_{14}B_{x}+\kappa_{15}B_{y}+\kappa_{16}B_{z},\\ \dots\qquad\dots\qquad\dots\qquad\dots\qquad\dots\qquad\dots\qquad\dots\qquad\\ -\sqrt{\Theta}H_{x}=\kappa_{41}E_{x}+\kappa_{42}E_{y}+\kappa_{43}E_{z}+\kappa_{44}B_{x}+\kappa_{45}B_{y}+\kappa_{46}B_{z}.\end{array}$

The relation between the two invariants will be a mutual one if the coefficients $$\kappa_{11},\ \dots$$. are the elements of an orthogonal matrix.

The Invariants of a Spherical Wave Transformation.
Starting from the fundamental invariants

we may obtain a number of others by the methods of multiplication and reciprocation. It will be sufficient to enumerate these if we mention the equations from which they are derived,

$\begin{array}{ccc} (1\ \mathsf{and}\ 6) & \frac{1}{\lambda^{2}}\left[\rho w_{x}dx+\rho w_{y}dy+\rho w_{z}dz-\rho dt\right], & (7)\\ \\(5\ \mathsf{and}\ 4) & \left(E_{x}^{2}+E_{y}^{2}+E_{z}^{2}-H_{x}^{2}-H_{y}^{2}-H_{z}^{2}\right)dx\ dy\ dz\ dt & (8)\\ \\(4\ \mathsf{and}\ 4) & \left(E_{x}H_{x}+E_{y}H_{y}+E_{z}H_{z}\right)dx\ dy\ dz\ dt & (9)\\ \\(3\ \mathsf{and}\ 6) & \rho\left[A_{x}w_{x}+A_{y}w_{y}+A_{z}w_{z}-\Phi\right]dx\ dy\ dz\ dt & (10)\\ \\(6\ \mathsf{and}\ 7) & \frac{\rho^{2}}{\lambda^{2}}\left(1-w^{2}\right)dx\ dy\ dz\ dt, & (11)\\ \\(5\ \mathsf{and}\ 7) & \frac{\rho}{\lambda^{2}}\left[\left(E_{x}-w_{z}H_{y}+w_{y}H_{z}\right)dy\ dz\ dt+\left(E_{y}-w_{x}H_{z}+w_{z}H_{x}\right)dz\ dx\ dt\right. & (12)\\ & \left.+\left(E_{z}-w_{y}H_{z}+w_{x}H_{y}\right)dx\ dy\ dt-\left(w_{x}E_{x}+w_{y}E_{y}+w_{z}E_{z}\right)dx\ dy\ dz\right],\\ \\(12) & \frac{\rho}{\lambda^{4}}\left[\left(E_{x}+w_{y}H_{z}-w_{z}H_{y}\right)\delta x+\left(E_{y}+w_{z}H_{x}-w_{x}H_{z}\right)\delta y\right. & (13)\\ & \left.+\left(E_{z}+w_{z}H_{y}-w_{y}H_{x}\right)\delta z-\left(w_{x}E_{x}+w_{y}E_{y}+w_{z}E_{z}\right)\delta t\right],\\ \\(13\ \mathsf{and}\ 2) & \rho\ dx\ dy\ dz\ dt\left[\left(E_{x}+w_{y}H_{z}-w_{z}H_{y}\right)\delta x+\left(E_{y}+w_{z}H_{x}-w_{x}H_{z}\right)\delta y\right. & (14)\\ & \left.+\left(E_{z}+w_{x}H_{y}-w_{y}H_{x}\right)\delta z-\left(w_{x}E_{x}+w_{y}E_{y}+w_{z}E_{z}\right)\delta t\right].\end{array}$|undefined

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 These invariants represent quantities which are of considerable importance in the theory of electrons. Another invariant which is of some importance is obtained in the following way.

Let

so that

$w_{1}=\frac{dx}{ds},\ w_{2}=\frac{dy}{ds},\ w_{3}=\frac{dz}{ds},\ w_{4}=\frac{dt}{ds},$

Then

Hence

$\begin{array}{l} \left(\frac{dw_{1}}{ds}\right)^{2}+\left(\frac{dw_{2}}{ds}\right)^{2}+\left(\frac{dw_{3}}{ds}\right)^{2}-\left(\frac{dw_{4}}{ds}\right)^{2}\\ \\\qquad=\frac{\dot{w}^{2}}{\left(1-w^{2}\right)^{2}}+\frac{(w\dot{w})^{2}}{\left(1-w^{2}\right)^{3}}+\frac{w^{2}(w\dot{w})^{2}}{\left(1-w^{2}\right)^{4}}-\frac{(w\dot{w})^{2}}{\left(1-w^{2}\right)^{4}}\\ \\\qquad=\frac{\dot{w}^{2}}{\left(1-w^{2}\right)^{2}}+\frac{(w\dot{w})^{2}}{\left(1-w^{2}\right)^{3}}\end{array}$|undefined

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 Thus

$\begin{array}{l} \frac{d^{2}w_{1}}{ds^{2}}-w_{1}\left\{ \left(\frac{dw_{1}}{ds}\right)^{2}+\left(\frac{dw_{2}}{ds}\right)^{2}+\left(\frac{dw_{3}}{ds}\right)^{2}-\left(\frac{dw_{4}}{ds}\right)^{2}\right\} \\ \\\qquad=\frac{\ddot{w}_{x}}{\left(1-w^{2}\right)^{\frac{1}{2}}}+\frac{3\dot{w}_{x}(w\dot{w})}{\left(1-w^{2}\right)^{\frac{1}{2}}}+\frac{w_{x}}{\left(1-w^{2}\right)^{\frac{1}{2}}}\left\{ (w\ddot{w})+\frac{3(w\dot{w})}{1-w^{2}}\right\} \end{array}$|undefined

Now Abraham has given a formula for the reaction of radiation upon a moving electron, the x component of the reaction being

$K_{x}=\frac{2e^{2}}{3\left(1-w^{2}\right)}\left[\ddot{w}_{x}+\dot{w}_{x}\frac{3(w\dot{w})}{1-w^{2}}+\frac{w_{x}}{1-w^{2}}\left\{ (w\ddot{w})+\frac{3(w\dot{w})^{2}}{1-w^{2}}\right\} \right];$|undefined

accordingly,

$\begin{array}{ll} K_{x}dt & =\frac{2e^{2}}{3c^{2}}\left[\frac{d^{2}w_{1}}{ds^{2}}-w_{1}\left\{ \left(\frac{dw_{1}}{ds}\right)^{2}+\left(\frac{dw_{2}}{ds}\right)^{2}+\left(\frac{dw_{3}}{ds}\right)^{2}-\left(\frac{dw_{4}}{ds}\right)^{2}\right\} \right]ds\\ \\ & =K_{1}ds.\end{array}$|undefined

To complete the symmetry of the result, we define quantities $$K_t$$ and $$K_4$$ by the equations

$\begin{array}{ll} K_{t}dt & =\frac{2e^{2}}{3c^{2}}\left[\frac{d^{2}w_{4}}{ds^{2}}-w_{4}\left\{ \left(\frac{dw_{1}}{ds}\right)^{2}+\left(\frac{dw_{2}}{ds}\right)^{2}+\left(\frac{dw_{3}}{ds}\right)^{2}-\left(\frac{dw_{4}}{ds}\right)^{2}\right\} \right]ds\\ \\ & =K_{4}ds.\end{array}$|undefined

we then have

Laue has shown recently that Abraham's formula may be derived by means of the principle of relativity. We shall complete this result by showing that

$\left(K_{x}\delta x+K_{y}\delta y+K_{z}\delta z-K\ \delta x\right)dt$

is an integral invariant for the whole group of spherical wave transformations. It will be sufficient to prove this for the case of the transformation

$x'=\frac{x}{r^{2}-t^{2}},\ y'=\frac{y}{r^{2}-t^{2}},\ z'=\frac{z}{r^{2}-t^{2}},\ t'=\frac{t}{r^{2}-t^{2}}.$|undefined

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 Now, if

Hence

Now

therefore

The Electrodynamical Equations for Ponderable Bodies.
We shall consider a system of electrodynamical equations for moving ponderable bodies of the type used by Minkowski, Einstein and Laub.

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