It is well known that Euclidean metric geometry is a special case of projective geometry, inasmuch as it refers to transformations which leave the line at infinity and the circular points invariant. Every projective theorem will thus have one or more Euclidean counterparts according to which line and which points on it are taken to be the line at infinity and the circular points. It is, on the other hand, also possible to translate any theorem into projective language by removing every trace of metrical properties. A theorem of projective geometry is thus obtained of which the original theorem is a special case. This principle will be illustrated by applying it to the following elementary

There are many methods of solving algebraic equations whose coefficients are real and have given numerical values. In practice it very often happens that the coefficients are not given numerically, but are expressed in terms of certain parameters, and that what is wanted is not the roots of the equation for special values of the parameters, but an indication of the way in which the roots behave as these parameters vary between certain limits. For example, if the equation is the period equation of a mechanical or electrical system, we may wish to know how its roots vary as certain components of the system are changed, not merely the values of the roots for definite values of the components. The method sketched below has been found very useful for studying cubics from this point of view and applies quite well to quartics. It also provides interesting elementary examples in the theory of equations.

Suppose that a particle of mass m is attached to a fixed point O, by means of a light inextensible string of length a, and when hanging at rest is suddenly-given a velocity u0 at right angles to the string. The earlier phases of the motion are dealt with in standard textbooks, but I do not recall having seen a complete analysis of the later stages in the case when the particle leaves its circular path. It is well known that if u02 ≤2ag, the particle executes oscillations like a simple pendulum (this will be referred to later as the case of simple oscillations), while if u02≥5ag, it makes complete revolutions in a circle about O. If 2ag<02< 5ag, the particle leaves its circular path and then travels in a parabolic path until the string becomes taut again. The textbooks do not follow the motion beyond this point, perhaps because it is realised that no real string fulfils the condition of being inextensible. If, however, we assume that the ideal string exists, the resulting motion has some rather surprising features. Owing to the obvious loss of energy, the particle when it first passes through the lowest point again will have a smaller velocity that it had initially, and so will either execute simple oscillations or will leave its circular path at a lower point than previously and lose further energy when the string tightens a second time. It would at first appear that ultimately the energy must decrease to such a value that the particle will finally be moving with simple oscillations. This is not always true, as the following analysis will show.

1. This perennial question has come up, once again, in a Note (No. 1805, February 1945) by the President for the year.

The object of this contribution is to try to state something like a full case (the writer seems to have been working at this for most of his academic life) for the traditional view of the subject, for which the President contends : while at the same time indicating the complementary importance of the other view. (As in all such cases, strong difference of view by competent authorities implies important truth on both sides (which it is necessary to synthesise; and this is, in fact, a fundamental mathematical case of the kind).)

The idea of the “reversed effective force” or “centrifugal force” has been widely used to facilitate the solution of certain types of dynamical problems concerning the motion of particles. But the method can also be applied, as will be shown below, to investigate the constrained motion of a rigid body.

The locus of the centres of spherical curvature of a singly infinite family of geodesics which pass through a regular point O on a surface S, one in each direction in the tangent plane there, is, in general, a twisted curve. It will be proved that the only real surfaces, at all points of which (excluding umbilics) this locus is a plane curve, are quadrics of revolution and general cylinders.