If C is a given curve in a Riemannian space Vn, a system of co-ordinates (z1, z2, …, zn) can be set up at each point P of C, thus generalising moving axes along a twisted curve. If in these co-ordinates a point Q is defined relative to the point P, then Q traces some curve as P moves along C; any curve C' can be defined in this way by setting up a (1, 1) correspondence between points of C and C′. We shall take the relative co-ordinates at P to be normal co-ordinates with origin at P, the parametric directions at P being the directions of an orthogonal ennuple defined at points of C. In general, this work is too heavy except for the consideration of points within a certain distance from the curve C, and we shall therefore consider only points the cube of whose distance from C may be neglected. This is sufficient for the application to such problems as the motion of a small rigid body in space-time.

Although the physical problem which originally led to the Mathieu equation

is now well over sixty years old, and although other problems studied in more recent years, such as the scattering of electric waves by an elliptic cylinder, depend essentially on the same equation, no thorough investigation of detail has hitherto been possible owing to the lack of tables of the functions defined by the equation. In fact, until 1924 all attempts to reduce the functions to a form suitable for calculation had failed. In that year a successful method of attack was discovered by the present writer, who then undertook the computation of the tables which are here published.

In any discussion of the zeros and turning points of the elliptic-cylinder function it is not necessary to define them more precisely than as functions of period π or 2π which satisfy the Mathieu equation

for the corresponding characteristic values of a. To simplify the following discussion, it is convenient merely to impose the condition that y(0)>0 if y(x) is even, and y′(0)>0 if y(x) is odd.

The injection of adrenaline hydrochloride into mature female mice is able to inhibit the occurrence of oestrus.

The injection of adrenaline hydrochloride into immature female mice prevents the onset of sexual maturity.

Adrenaline does not interfere with the action of α-hormone (œstrin) on the uterus and vagina; it has an inhibitory effect on the action of gonadotropic hormones (prepared from pregnancy urine) on the ovary.

I should like to thank Mr P. G. Marshall for the supply of gonadotropic preparation from pregnancy urine used in these experiments.