The present paper is concerned with a simple type of dynamical system, a system having only two freedoms and of “separable” type. The essential simplification introduced by this restriction is that the motions in the two coordinates can be discussed to some extent independently of each other. In Part I a method of classification of the possible orbits is explained. Four constants are needed to define the motion completely, but the general nature of the orbit depends on two constants suitably chosen, and the classification is effected by reference to a plane in which the two constants are taken as Cartesian coordinates. It is shewn that the plane of reference is divided into regions by critical curves, and that the orbits represented by points in the same region (each point representing a set of orbits, not a single orbit) are of the same general type. The various possible types of orbits, and their stability, in a special sense, are discussed. In Part II the theory is applied by way of illustration to a somewhat trivial example, the orbits for a particle under a central attraction proportional to the inverse (n + 1)th power of the distance. The most striking thing here is that the possible types of orbits are essentially the same for all values of n greater than 2. In Part III the theory is applied to the classical problem of a particle subject to an attraction of Newtonian type to two fixed centres, a problem of some interest in relation to the general problem of three bodies, and to some current questions in atomic dynamics. The problem is of course not a new one, but it seems that a satisfactory classification of all the possible orbits has not previously been given. It must be confessed that the complete classification for this problem is very laborious.

Let σs(n) denote the sum of the sth powers of the divisors of n,

and let , where ζ(s) is the Riemann ζfunction. Ramanujan, in his paper “On certain arithmetical functions*”, proves that, if

and

then , whenever r and s are odd positive integers. He conjectures that the error term on the right of (1·1) is of the form

for every ε > 0, and that it is not of the form . He further conjectures that

for all positive values of r and s; and this conjecture has recently been proved to be correct.

This problem was originally attacked by Stokes, and the formula obtained by Stokes for the case of steady motion (given at the end of § 1 below) is now generally used in experimental work. However, this formula throws no light on the way in which the velocity of the sphere approaches its steady value: and the only known discussion of this problem was given by Boggio (his results will be found in § 2 below). Unfortunately, when the specific gravity of the sphere exceeds ⅝ (or the constant σ, defined later, is less than 4) the solution so obtained cannot be converted into numerical results, because the constants α, β become complex numbers; and even when α, β are real, the formula is not very convenient for large values of t (as the velocity approaches the steady value).

The function τ (n), defined by the equation

is considered by Ramanujan* in his memoir “On certain arithmetical functions”

The associated Diriċhlet series

converges when σ = > σ0, for a sufficiently large positive σ0.

In this short note my chief object is to prove the following theorem:

If

(1) f(z) is a regular function of z(= ρeiψ) in the angle |ψ| ≤ α, where ;

(2) f(z)= 0(eαρ), where K is a positive constant, throughout this angle;

(3) f(z) is not identically zero;

then

(4)

is a continuous function of ψ for |ψ| < α.

It is well established by observation that in the ordinary flow of a stream there is a slow transverse flow outwards from the middle near the bottom, and inwards near the top; and such a circulation seems to be required to explain the fact that the filament of maximum onward velocity is depressed some distance below the centre of the free surface. Qualitative explanations of the phenomenon have been offered, depending on the extra friction near the sides, but it seems that the true explanation must depend on something more fundamental. For any explanation to be satisfactory it must show uniform linear flow to be impossible; and actually it can at least be rendered highly probable that such flow can exist over a very wide range of conditions.

The object of this paper is to give proofs of formulae, previously obtained by Schubert (for lines) and Palatini (for planes) by degeneration methods, by a direct application of the coincidence formulae of Pieri and Severi which are referred to below. In the case of planes the essential part of the proof is the deduction of the relation (10) below, from which the required formulae follow directly. The formulae in question have as their object the expression of a product of two incidence conditions as the sum of a number of simple ones.

The object of this note is to prove the following results, all of which hold when |a| < 1.

(2) If r is any positive integer other than zero, the

The formulae of this paper are chiefly concerned with multiple secants and tangents to surfaces in spaces of dimension four to seven; they are obtained by the functional method of Cayley and Severi, which was first applied to scrolls by C. G. F. James. The majority of the results have already been given for scrolls in James's third paper; but whereas the functional equations in this case involve only two variables, those of the present paper contain four, and the work is considerably more complicated. It is also to be noted that, though the formulae for scrolls might perhaps have been expected to follow as particular examples from the corresponding formulae for surfaces, this is not always the case. The reason for the discrepancy is not always clear, although in some instances theoretical reasons for disagreement can be assigned.

1. Young's criterion for the convergence of a Fourier series may be stated in the following form:

The integral

tends to zero as ω → ∞ whenever

(A) ø(t) is absolutely integrable in (0, a),

(B) ø(t) → 0 in mean as t → 0,

(C) tø(t) is of bounded variation to the immediate right of t = 0 and, as t → 0,

The surfaces considered in this note are Jacobians of four quadric forms (“quadrics”) in [4], where one or more of the quadrics are space-pairs. It is believed that, with one exception, they have not been noticed previously; and so little is known of non-rational surfaces in higher space that it seems worth while putting them on record.

The investigations of von Karman dealing with the unsymmetrical double row of vortices in an infinite sea of liquid are well known. He found that the unsymmetrical double row is stable when, and only when, cosh2πa/b = 2, where 2a is the distance between the two rows and 2b is the distance between consecutive vortices on the same row. A detailed account of the stability of the Karman street and of the symmetrical double row has been given by Lamb, and it has been shown that the symmetrical double row is unstable for all values of the ratio a/b. The object of this paper is to investigate the stability of a double row of vortices of arbitrary stagger. We define a double row of stagger 2l to be the system formed by positive vortices at the points (2nb + l, a) and negative vortices at (2mb − l, − a), where m and n assume all integral values from − ∞ to + ∞. The vortices are thus neither exactly “in step” nor exactly “out of step.” When l = 0 the system reduces to the symmetrical double row and when the system is the unsymmetrical double row.

1. The following investigations were undertaken in connection with Cayley's problem of seven lines on a quartic surface.

Any one who has dealt with the convergence of multiple series soon becomes aware that simple and interesting examples are very scarce. It may therefore be worth while placing on record the following result which is not altogether trivial.

1. The power possessed by water and air in motion of lifting and carrying solids considerably denser than themselves is familiar to all; but it does not seem to have been pointed out that classical hydrodynamics provides a simple explanation. If a solid rests on the bottom of a stream, the points of contact are points of zero velocity; and the velocity just above the solid, by the equation of continuity, must be greater than the general velocity. Hence the velocities produce high pressures under the solid and low ones above it, and the difference tends to lift the solid up. If the resulting thrust exceeds the weight of the solid in the liquid, the solid will be raised, and will be unable to rest in equilibrium on the floor of the stream.

In a paper* of some time back I gave the following theorem:

If f(z) is a regular function of z for z inside and on a circle Τ, and C is any convex curve inside Τ, and λ>0, then

1. The principal object of this note is to give a new derivation of the coincidence formula in the general correspondence of points in a rational n-space. It is supposed that the correspondence is algebraical and that there are ∞n pairs of corresponding points P, P′ (in the geometrical sense of ∞n). Denote by the symbol (i) (n − i)′ the number of pairs P, P′ such that P lies in a general i-space and at the same time P′ in a general (n − i)-space. When (P, P′) tends to a coincidence of the correspondence, the join PP′ tends to some limiting position, which may depend on the way in which (P, P′) tends to the coincidence. In general the limiting lines will lie on a conical figure—a star—with the double point as vertex. In the simplest case of an isolated double point, the star fills the whole n-space. Denote by the symbol ε(i, n) (n − i) the number of times it happens that a double point lies on a general (n − i)-space while the limiting line PP′ meets a general i-space(and lies in the n-space).

If we have one birational representation of a threefold locus V on an ordinary space S3, and if we can project V birationally on to another space S3′, then clearly there will be a Cremona correspondence between the two spaces S3 and S3′. This paper deals, in the first place, with threefold loci in higher space which can be represented birationally on ordinary space by means of cubic surfaces; and Cremona transformations for ordinary space are then obtained, as indicated above, by projecting such loci. In particular it will be shown that most of the more important cubic transformations and their reverses can be obtained very simply by this method.

The theorem given by Professor Mordell in the preceding paper is capable of a slight generalisation. I first prove that the series

where α, β, γ, … are positive, converges only when κ > Σ (a′/α), the summation referring to those exponents a′ of a, b, c, … for which a ≥ 0; or, if there are no such exponents, the series converges only when κ > a/α, k > b/β, …

The stability and the stream lines of the symmetrical and unsymmetrical double rows in a channel have already been discussed by the author. It is the object of this paper to investigate the possibility of the existence of a double row whose axis is parallel to, and not necessarily coincident with, the axis of the channel. It will be seen that, if the system of vortices is to move forward as a whole, then the only possible types of double row are those that have been discussed previously, that is, the symmetrical and unsymmetrical systems, and in both of these cases the centre lines of the row and channel are coincident. It was shown previously that the symmetrical row is always unstable and that the unsymmetrical system is stable within a particular domain of stability.