8. On Mersenne's Numbers.

By Lt. Col. ALLAN CUNNINGHAM, R.E., Hon. Fell. King's Coll. Lond.

A Mersenne's number is one of form N = (21-1), where q is a prime. Divisors of these are difficult to discover. Their prime divisor (p), when N is composite, must be of form p = 2eq + 1, and also of one of forms p=8i± 1, and 2 must be a residue of order e.

Simple rules (due to Legendre, Gauss,' and Jacobi,) are given for finding directly divisors (p)-when such exist--for the cases of 2c=2, 6, 8, 16, 24. An indirect method (due to Mr. C. E. Bickmore) is also given for the case when p = 2ee'. q + 1, where 2e has any of the above values, and e' = an odd number >3.

A table of divisors (p)—the greater part of which is believed to be originalis given this is believed to be complete for all primes p<15,000 for the simple cases of 2c=2, 6, 8, 16, 24. The following thirteen new instances were discovered by the indirect method quoted, and are the outcome of much labour.

One of these, viz. (21971), is among those stated by Mersenne in 1,664, but without proof, to be composite; proof of this is now supplied in the discovery of a divisor (p=7,487).

Nineteen of the Mersenne's numbers stated by Mersenne to be composite (viz., when q=71, 89, 101, 103, 107, 109, 137, 139, 149, 157, 163, 167, 173, 181, 193, 199, 227, 229, 241), and three of those stated by him to be prime (viz., when q = 67, 127, 257), remain still unverified. The author has tried all prime numbers <50,000 without finding a divisor for any of them.

There are only ten prime Mersenne's numbers known, viz., when q= 1, 2, 3, 5, 7, 13, 17, 19, 31, 61; the establishment of any more is very difficult. It is worth noting that these ten values of q, as also three more (g = 67, 127, 257) conjectured by Mersenne to yield prime values of N, all fall under one of the four forms q = 2* ± 1, or 2+3; but it is not true that such values of q necessarily make N prime.

9. Recent Developments of the Lunar Theory. By P. H. COWELL, M.A.

Kepler discovered that the motion of planets about the sun and the moon about the earth took place approximately in ellipses, and Newton showed that motion in an ellipse according to Kepler's well-known laws was the consequence of the law of gravitation.

For nearly two centuries after Newton, the lunar theorists based their investigations of the moon's motion upon Newton's discovery. Their reason for doing this was that the elliptic inequality is by far the largest of all the lunar inequalities. The other inequalities were then calculated as disturbances due to the sun. One modification had, however, to be made. In order to agree with observations, displacements increasing with the time had to be assigned to the apse and node. The orbit thus modified no longer satisfied the undisturbed equations of motion, and the velocities of the node and apse, as well as the remaining inequalities, had to be found in most theories-by continued approximation.

In performing the algebraical calculations, however, it appeared that the terms constituting the inequality known as the variation had first to be calculated, and

rules.

The author is indebted to Mr. C. E. Bickmore for the communication of these

that subsequently the terms containing powers and products of the eccentricities, inclination, and ratio of the parallaxes could be calculated in turn, the lower orders being taken first. This point must have presented itself to Laplace and Pontécoulant in their respective theories; but it is brought out more clearly in those treatises where the object is not to obtain a complete analytical development, but to exhibit the method of procedure. In Delaunay's theory this point does not present itself, but by modifying Delaunay's theory so as to reduce the number of variables from six to two the variational terms might be calculated independently, whereas by no process could the other terms be calculated before the variational terms.

These considerations point to the curve indicated by the variational terms being treated as the intermediary orbit in preference to the ellipse; but it was not till the year 1877 that this idea was developed. Dr. G. W. Hill then published three papers in the first volume of the American Journal,' papers which Poincaré describes as containing the germ of the greater part of the progress that astronomy has since made.

Relatively to the sun's mean place the variation terms define a closed curve which the moon under suitable initial conditions could describe, if the sun's parallax and eccentricity were zero. The curve is symmetrical in all four quadrants, and remains symmetrical about the line of syzygies, when the sun's parallax is taken into account. Dr. Hill has drawn the variation curve for different ratios of the month to the year. The curve for small values of this ratio does not differ much from a circle, but is slightly elongated in quadrature. This elongation increases with the length of the month, and when there are only 1·78265 months to the year the curve has cusps on the line of quadratures. Such an orbit must certainly be unstable, and by a discussion of Jacobi's equation of relative energy Dr. Hill makes it appear probable that instability sets in for a much smaller value of the ratio. No numerical limit has, however, been obtained for stability. M. Poincaré has succeeded in obtaining the general shape of these curves when continued beyond the cusped curve. The orbit first crosses the line of quadratures obliquely, and then recrosses at a greater distance at right angles, then returns to the first intersection, thus forming a loop, and ultimately forms a closed curve with two loops, six intersections with the line of quadratures and two with that of syzygies. Dr. Hill has also calculated algebraically and numerically and with extreme accuracy the coefficients of the different periodic terms that define the variation curve.

In the older lunar theories the physical meaning of the quantity that in undisturbed motion denoted the eccentricity disappeared upon the introduction of the disturbance with the ellipse upou which it depended for its definition. At the conclusion of the theories it was defined analytically by equating a given function of it to the coefficient of one of the periodic terms. Such a definition is merely analytical, and has no physical interpretation. The quantity, in fact, is a mere constant of integration. It is not even correct to say that it reduces to the eccentricity when the sun's mean motion is put zero. The eccentricity of the ellipse obtained by putting the sun's mean motion zero is a function of the constants of integration and of the position of the sun's apse, and of the sun's parallax. By Dr. Hill's investigations, however, a physical meaning is restored. It is a parameter defining the amplitude of the oscillation that takes place about that state of steady motion relatively to the sun that Dr. Hill has shown can take place, provided only the sun's eccentricity be negligible. With the notion of eccentricity, the notion of the apse of the older theories becomes indistinct. The so-called apse is certainly not a point where the moon is moving at right angles to its radius vector. According to Dr. Hill's theory, the period of revolution with respect to the apse now becomes the period of the oscillation about steady motion. In like manner the inclination of the orbit may be considered as another oscillation about steady motion, the inclination constant of integration defines its amplitude, and the period of revolution with respect to the node, which is not accurately the point of intersection with the ecliptic, is now the period of this second oscillation. In accordance with the general theory of small oscillations, the two modes of oscillation can co-exist in complete independence so long as the squares of the amplitudes are negligible. The two periods in such a case depend only on the circumstances of steady motion, in

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 resistance to 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— 2.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 the directions of the principal axes of the velocity-ellipsoid will not