this case on the ratio of the month to the year. The periods in the actual case, however, must be corrected by terms depending on squares and higher powers of the sun's eccentricity and the two parameters defining the amplitudes.

In a paper in the Acta Mathematica,' vol. viii., Dr. Hill finds the period of a small oscillation of the first of the two kinds mentioned. His method involves the consideration of an infinite determinant. He states that there can hardly be a doubt that the determinant is convergent, but M. Poincaré has submitted the question to a rigid investigation. He concludes that an infinite determinant, when the constituents of the leading diagonal are all unity, converges absolutely and uniformly if the sum of all the other elements is finite. Any determinant can be reduced so that the elements of the leading diagonal are all unity, provided that the product of these elements is finite. Dr. Hill's determinant satisfies these conditions when the length of the month is sufficiently small. To complete the proof it is necessary to notice that M. Poincaré, in his Mécanique Céleste,' proves that the series defining Dr. Hill's variation curve converge for sufficiently small values of the length of the month.

At the conclusion of his paper, Dr. Hill solved his infinite determinantal equation, and obtained the principal part of the motion of the apse with great arithmetical accuracy. The value he obtains differs in the fourth significant figure from that calculated from Delaunay's series; it also agrees well with the observed value, thus verifying a prediction of Delaunay's, as far as the apse is concerned, that the remainder of his series would bring calculation into agreement with observation. Dr. Hill has lately calculated an algebraic value to eleven terms for the principal part of the motion of the perigee. He concludes by replacing the ratio of the month to the year by another parameter, empirically determined so as to increase the convergence of the last terms calculated. This last step, however, does not appear to be in any degree useful, as the convergency of the series near its tenth term throws no light on the convergency of the remainder.

The question of convergency of the series obtained in the lunar theory had hardly been investigated before Poincaré and Lindstedt. Formal solutions to the seventh order and arithmetical solutions have been obtained, but it cannot be assumed from the close agreement of the two that the coefficients can be represented by the algebraic series. Poincaré has shown, however, that in certain cases periodic solutions must exist, and as a special case the series for the coefficients of the variational terms must converge for sufficiently small values of the ratio of the month to the year. The motion of the node, so far as it depends on the ratio of the mean motions only, had been investigated by Adams before Dr. Hill's work on the perigee was published. Adams also obtained an infinite determinant. In the arithmetical work, however, he used a different value of the ratio of the mean motions to that used by Delaunay and by Dr. Hill. It is an illustration of the almost unnecessary accuracy of the numerical work that it should have been carried to fifteen decimal places, whereas the ratio of the mean motions, certainly by far the easiest quantity to determine by observation, can only be depended upon to seven places. I have recomputed the principal part of the motion of the node using Dr. Hill's numbers. It may be noticed that the arithmetical value in this case does not, as in the case of the perigee, justify Delaunay's prediction that the remainder of his series would account for the discrepancy between theory and

observation.

Dr. Hill's method of procedure is to use rectangular co-ordinates, the axes of reference rotating round the ecliptic with a velocity equal to the sun's mean motion. The calculation of the variation terms by this method is perhaps not so short as it would be by some other method-possibly the best way to obtain them would be by Delaunay's methods, the variables being reduced to two-but undoubtedly no theory is so simple for the calculation of the higher inequalities. For each new set of coefficients the problem can be quickly reduced to the solution of a system of linear simultaneous equations. The principal parts of the motions of the perigee and node are given by infinite determinants: the further approximations appear as

1 Bulletin de la Société Mathématique de France, xiv. pp. 77-90.

additional unknown quantities to be determined by the simultaneous linear equations. The solution has to proceed by continued approximation, and is exceedingly laborious. In an admirable paper in the current number of the American Journal, Prof. E. W. Brown has shown how the new part of the motion of the perigee and node can in all cases be evolved from the terms previously calculated. This consideration not only shortens very considerably the labour of the continued approximations, but it enables us to regard one of the simultaneous equations as an equation of verification. Professor Brown's paper-undoubtedly the most valuable of all the papers that are based upon Dr. Hill's researches-concludes with some extensions of Adams's theorems connecting the mean value of the parallax with the motions of the node and perigee. These extensions possess an analytical interest, but as applied to the development of a solution of the problem of three bodies in series, they only provide some equations of verification of a value far inferior to. those investigated in the earlier part of his paper.

The following advances have been made towards a complete development of the problem of three bodies. Dr. Hill calculated the variation terms; Professor Brown the terms depending on the ratio of the parallaxes, the terms depending on the first, and subsequently the second and third powers of the moon's eccentricity; also the terms depending on the first power of the sun's eccentricity, and also the product terms containing the first powers of both the eccentricities. These latter are the only product terms hitherto calculated by Dr. Hill's methods. The convergence of the series Delaunay obtains in his literal development is exceedingly slow, and the arithmetical values show a residue in some of Delaunay's series of over one second. I have calculated terms depending on the first three powers of the inclination. Besides this, Dr. Hill has obtained the principal part of the motion of the perigee, and Adams the principal part of the motion of the node. Professor Brown has calculated the correction to the motion of the perigee depending on the square of the eccentricity, and I have calculated the correction to the motion of the node depending on the square of the inclination.

At the beginning of his last paper, referred to above, Professor Brown has collected the bibliography of the subject.

10. The Relation between the Morphological Symmetry and the Optical Symmetry of Crystals. By WILLIAM BARLOW.

Starting from the well-known facts of the influence of the presence of molecular matter generally on the velocity of light, and of the directional optical properties of crystals, the author reaches the conclusion that ether-movements: which take place in the same crystal in different directions experience different degrees of resistance and retardation, so that a state of things prevails roughly comparable to what would happen if a space occupied by a crowd of people were studded with posts arranged on parallel lines and evenly distributed; the movements of the crowd as it surged to and fro would be less impeded in some directions than in others, especially if the posts were not round, but of similar section sameways orientated. In the case of both the ether and the crowd what are compared are the collective resistances in each direction, differences in the retardation experienced by different particles or persons moving side by side in the same direction not being discriminated.

Even if the crystal employed belongs to the cubic system, and is therefore isotropic, the ether-movements must, as in the case of less symmetrical crystals, experience different retardation in different directions; and the necessary deduction from this is that if the influence of a homogeneous molecular structure on light depends on the arrangement of the molecular matter, it is an average effect, the velocity of a ray in any given direction depending, not merely on the resistanceto ether-movement experienced in some single direction definitely related to the direction of polarisation of the ray, but on that experienced in a number of different directions inclined to one another. The writer cites in support of this conclusion the fact that in crystals belonging to the less symmetrical crystal systems, in

which a change of velocity accompanies any continuous change of direction, this change of velocity is always a very smooth one, and not abrupt.

After remarking that if the velocity of a ray in any given direction were dependent equally on the resistances offered to ether-movement in every direction, this velocity would in all cases be entirely independent of any particular direction or directions in the structure, which would in all cases be isotropic, he says that the experimental facts show that the truth lies between the two extremes indicated; that the velocity of a ray depends neither on all the resistances to ethermovement experienced in all directions taken equally, nor on the resistance experienced in a single solitary direction, but depends equally, or almost equally, on the resistances afforded in all the directions included within some wide limits of angular inclination, this being the only kind of relation which would be in harmony with the great smoothness of the change of velocity presented when a continuous change of direction is made.

He then suggests that the simplest sort of relation which the velocity can be conceived to bear to the resistances offered by the structure to ether-movement is for the resistance whose direction is that of the polarisation of the ray-i.e., the direction in which the algebraically deduced wave-vibration takes place to exert a maximum influence, and the effect of the resistances in directions inclined to this to diminish as the inclination increases, the decrement of influence for directions near the direction which furnishes the maximum effect being, however, very small indeed.

He points out that if this simple kind of relation obtains, the velocity figure— i.e., the figure whose radii express the different velocities proper to different directions of polarisation for rays traversing a crystal-must exhibit a smoothed curvature derived indeed, but having a very different aspect, from that of the corrugated surface whose radii would express the relative facility of ethermovement taking place in different directions in the same crystal; and that the simplest conceivable result of such a smoothing or averaging will be for the velocity-figure to approximate as closely as we please to the result obtained by treating the velocity appropriate to any direction of polarisation whatever as the resultant of three components acting in some particular three widely separated directions, each component, in harmony with the averaging referred to above, being greater or less as the direction of the resultant which is being resolved lies nearer to or further from its direction, and being zero when the resultant lies in the plane of the remaining two components. The relative lie of the three directions will, of course, depend on the nature of the crystal structure. The reason for taking three directions is that this is the least number which can be employed consistently with generality.

He proceeds to show that the simplest figure thus obtainable is an ellipsoid, of which the three axes are conjugate diameters, and calls attention to the fact that the number of the sets of three axes which will fulfil the requisite conditions in any given case is unlimited.

From the fact that the velocity-figure is in all crystals found to be an ellipsoid (specialised, indeed, in some of the crystal systems), he finally argues that the velocity of a ray is an average effect of the different resistances to ether-movement offered in different directions of the nature above explained; and that the combination or averaging by which so simple a figure as the ellipsoid is reached must not only extend over a wide range of resistances for each velocity, but also that it must be so nearly uniform in its application throughout some considerable portion of this range as to preclude entirely all merely local effects of the structural features of the crystal on the contour of the velocity-figure.

In closing, the writer remarks that the directions which give maximum or minimum velocity-ie, those of the principal axes of the ellipsoid—will not necessarily be directions of maximum or minimum facility of ether-movement, the indents and protuberances of the corrugated figure whose radii express the relative facility of ether-movement in different directions not being traceable as such on the velocity-figure.

Also that tue directions of the principal axes of the velocity-ellipsoid will not

be ascertainable from the morphological constants unless the degrees of resistance presented in different directions are known; except, however, the cases of the more symmetrical systems in which the positions of these axes are fixed by symmetrical considerations.

A similar observation applies to the absorption-figure for monochromatic light, which is also an ellipsoid.

The fact that the elasticity-figure of crystals is a surface of a higher order than an ellipsoid is due to its being the outcome of a compounding and averaging whose scope is more limited and not so uniform as that above referred to.

11. On a Species of Tetrahedron the Volume of any member of which can be determined without employing the proof of the proposition that Tetrahedra on equal bases and having equal altitudes are equal, which depends on the Method of Limits. By M. J. M. HILL, M.A., D.Sc., F.R.S., Professor of Mathematics at University College, London. The object of this communication is to prove the existence of the species of the tetrahedron mentioned in the title.

Art. 1. A proof is first given of the known proposition, that if the edges B A, CA, DA of the tetrahedron ABCD be produced through A to E, F, G respectively, so that BA-AE, CA=AF, DA=AG, then the tetrahedra ABCD, A EFG are of equal volume.

Art. 2. From the above proposition it is deduced that if the edge DA of the tetrahedron ABCD be perpendicular to the plane A B C, and if D A be produced to E, so that DA=AE, then the tetrahedra ABCD, ABCE are of equal volume.

Art. 3. Now let ABCD be a tetrahedron, and let DH, CK be drawn equal and parallel to B A.

Join H A, AK, KH, H C.

Then if B H be perpendicular to the plane A CD, it follows, by applying Art. 2 twice over, that the tetrahedra ABCD, A DCH are of equal volume.

In like manner if DK be perpendicular to the plane AC H, it follows that the tetrahedra A DCH, AHCK are of equal volume.

A

K

B

H

Hence the tetrahedron ABCD is one third of the prism, having the same base and altitude. The two conditions

(1) That BH is perpendicular to the plane A C D, and (2) that DK is perpen

dicular to the plane A CH-result in the expression of the lengths of the six edges of the tetrahedron in terms of two positive quantities a, k as follows:—

Hence o< k < No3.

AC=a√9-3k2;
AD=BC=2a;

AB=BD=DC = a √l + k2.

The faces B D A, B D C are equal isosceles triangles.

The faces AC B, ACD are equal scalene triangles.

The planes BCA, BCD are perpendicular to each other, and so are the planes ADB, ADC.

The planes AC B, ACD are inclined at an angle of 60°.

The planes ABC, ABD are inclined at the acute angle whose cosine is 3-k, and so are the planes CDA, CD B.

The planes BDA, BDC are inclined at the angle whose cosine is (k2 − 1), which is obtuse if o<<1, but acute if 1<k</3.

The volume of the tetrahedron is a3k2√3-k2.

12. On Absolute and Relative Motion. By Prof. J. D. EVERETT, F.R.S.

Though there is no test by which we can distinguish between absolute rest and uniform velocity of translation, D'Alembert's principle furnishes a test by which deviation from such uniformity can be detected. Every deviation produces the same effects which would be produced by bodily forces opposite to the actual changes of velocity. The intensity of the apparent bodily force is equal in each case to the absolute acceleration.

What is called centrifugal force is an apparent bodily force directed outwards from the centre of curvature of the body's path, and having an intensity equal to the distance from this centre, multiplied by the square of the absolute angular velocity. Angular velocity, unlike velocity of translation, involves acceleration; and by comparing the accelerations of different points of a rigid body we can measure the absolute angular velocity of the body. The slope of a conical pendulum and the concavity of the surface of the liquid in a revolving vessel are phenomena which depend on absolute velocity of horizontal rotation; and another measure of horizontal angular velocity is furnished by differences of pressure at different points in a horizontal tube full of liquid.

13. On the Magnetic Field due to a Current in a Solenoid.
By W. H. EVERETT, B.A.

The case of a solenoid of circular section is the only one hitherto investigated, and this has been done by considerations derived from magnetic shells. In this paper the problem is approached by a more direct method, and general solutions are obtained in a form which can be readily worked out to numerical values. Special application is made to the case of a rectangular (or polygonal) solenoid, the component forces being expressed in finite terms. For a very long solenoid of any form of section the longitudinal force in either of the end sections is shown to be exactly the same at all points, and in any solenoid the longitudinal force is shown to be more uniform in the end sections than in the medial section. As a particular case the method gives the component forces due to a plane circuit at any point in its field; and a simple expression is found for the force, at any interior point, due to a circular current.