burner used in this experiment gives an extremely constant quality of light *.

In comparison with the other curves shown in fig. 2, the writer's visibility curve for the thirteen subjects is slightly

FIG. 3

more contracted. The maximum visibility occurs at wavelength 553 in agreement with Ives's curve, as against 555 for Nutting's curve.

* Standardized burners may be obtained from the Research Laboratory, Eastman Kodak Company, Rochester, N.Y.

Fig. 3 shows the mean of the writer's results and Nutting's results on the five subjects who served as observers in both experiments. The average maximum visibility found by the writer is 555, and by Nutting 554.

This work was carried out at the suggestion of Dr. P. G. Nutting, and the writer wishes to thank him and the other members of the laboratory for their assistance.

Author's Note.-This paper had been completed before a paper by Coblentz and Emerson* appeared, so that their latest data have not been included.

XX. On the Hodographic Treatment and the Energetics of undisturbed Planetary Motion. By Professor ANDREW GRAY, F.R.S.†

1.

THE

HE following bit of Newtonian dynamics was suggested by some passages in a recent discussion in the Philosophical Magazine. It sets forth a mode of dealing with the elementary theory of planetary motion which may have some novelty, and states certain results, which I think are interesting, with regard to the energetics of such motion. Into problems of relativity I do not at present enter.

As a matter of scientific history there would seem to be no doubt that the first idea of the hodograph is due to August Ferdinand Möbius, the inventor of the barycentric calculus. In § 22 of his book, Die Elemente der Mechanik des Himmels, which was published in 1843, Möbius specifies a point H which moves so that its distance OH from a fixed point is continually equal in length to, and in the same direction as, the line which represents the velocity of a particle moving in a given path. He derives the result that the acceleration of the particle in the path is represented in magnitude and direction by the velocity of the point H. This of course is the whole idea of the hodograph very clearly stated. So far as I have been able to see, Möbius does not make any particular use of the idea; and he does not seem to be aware that the curve described by the point H is, for a planet moving in a conic section, a circle, with O as an eccentric point within it.

* Coblentz, W. W., Emerson, W. B. "Relative Sensibility of the Average Eye to Light of Different Colors and some Practical Applications to Radiation Problems." Scientific Paper 303, Bureau of Standards, issued Sept. 12, 1917.

+ Communicated by the Author.

2. Apparently Hamilton was ignorant of the work of Möbius when he published his paper on "The Law of the Circular Hodograph" in the Proceedings of the Royal Irish Academy in 1847. More than ten years after, in a letter of date March 3, 1858, to De Morgan, he says:"Do you know much about Möbius and his works? He wrote to me a couple of years ago . . . that he had been lecturing on my theory of the circular hodograph, to which he might, very plausibly, have put in a sort of claim, or at least a claim to the general conception." [Graves, ‘Life of Hamilton,' vol. iii. p. 543.]

That the hodograph for the motion of an undisturbed planet is a circle, was of course the great discovery, and without doubt this discovery was Hamilton's, and Hamilton's alone. That a name was given to the curve may have had something to do with the progress of discovery. When the curve had thus been endowed with a kind of personality, such questions naturally presented themselves as: What is the nature of the hodograph of a planet? What is the hodograph of an unresisted projectile? and so on. names given to electric and magnetic quantities have certainly helped to distinguish sharply between one idea and another closely related to it, and led to the evaluation of the individualities which the names emphasized.

The

3. I was surprised to see Professor Eddington's statement in the Philosophical Magazine for October that the theorem of the resolution of the orbital velocity of a planet into two components of constant amount, one of them also in a constant direction, seemed to be overlooked in dynamical textbooks. It was given in Frost's Newton' in 1854, and is stated on p. 147 of the fourth edition. It is to be found in Routh's Dynamics of a Particle,' § 397, and in a book on Dynamics by Dr. J. G. Gray and myself, § 134. In the last mentioned work there is a rather full treatment of the subject referred to recently in the Phil. Mag.—the motion of bodies of varying mass.

The existence of these two components of velocity, one of constant amount v1, say, and constant direction, the other of constant amount 2, directed always forward at right angles to the radius vector, affords the most elegant mode of passing from the circular hodograph to the orbit. I shall show first that the hodograph is a circle, and then pass from the hodograph to the orbit.

Let the planet, denoted by P and supposed to be of unit mass, be acted on by a force directed towards a fixed point S, and varying inversely as the square of the distance SP. If e

be the angular speed with which SP is turning about S, and r=SP, we have r2=h, where h is a constant. Thus =h/r2. From a centre C let a circle be described with some radius 2a in the plane of the path or in a parallel plane. Let a radius CH of the circle be always parallel to SP. The velocity of H is 2a0, and is therefore proportional to 1/2. If the force per unit mass on the planet at distance r be μ/r2, the acceleration is μė/h, and so

Thus, to the factor 2ah/u, the velocity of H represents the magnitude of the acceleration of the planet. Its direction is. at right angles to SP, and represents the direction that PS would have if SP were turned 90° about P in the opposite direction to that in which SP turns, as P moves in the orbit.

Drawing the chord H.H, which on the scale adopted represents the total change of velocity, between a previous point Ho, corresponding to a radius vector SP, and H, we see that, if we choose the proper point O in the plane of the circle, OH, and OH will represent the velocities at the beginning and end of the time t. It is clear that O must lie within the circle, for the vector OH, which represents the velocity, must turn through an angle 2 while the planet traverses the orbit once, which would not be the case if O were outside the circle.

The position of O must be independent of the value of t, otherwise the speed for the radius vector SP would depend on the choice of the initial radius vector SP。. Thus O is a definite point. This result, taken along with the representation of the acceleration by the motion of a point in the circle, shows that the hodograph of the planet is a circle. The velocities are represented in magnitude and direction by the radii vectores drawn to the circle from the eccentric point O.

OH, is perpendicular to the direction of motion at P。, and OH to the direction of motion at P. The members of the family of lines drawn from 0 to the sequence of points H are perpendiculars to the corresponding tangents to the path.

4. It is obvious that OH may be resolved into the two components OC, CH, that is into two components of fixed amounts, v1, v2, at right angles respectively to the line OC

and to the radius vector SP. We can now instantaneously find the orbit.

Let be the angle which SP makes with the direction OC. The angular momentum of the planet about S is r( v2+ v1 cos 0), which has a constant value h. Thus if we write e for v1/v2, we get

that is the path is a conic section of which S is a focus. The length of the major axis is the sum of the lengths r1, 2 of r obtained by putting cos 0=1, cos 0=-1, respectively. If it is denoted by 2a we have

Thus the perihelion and aphelion distances are a (1—e) and a(1+e).

5. It is convenient to make the radius of the hodograph equal to the 2a just found. The length of the line OH, representing the velocity, is then equal to twice the length of the perpendicular let fall from O on a line which is parallel to the tangent to the orbit at the corresponding position of the planet. In fact a circle of radius 2a described from S as centre serves very conveniently as hodograph. The hodographic origin O is then coincident with the empty focus, as will easily be seen from the fact that the perpendicular from the empty focus to any tangent of the ellipse at a point P intersects the radius vector SP on the circle.

6. As has been seen in (2) the two components v1, v2 of velocity give an instantaneous integration of the differential equations of the orbit. The radial differential equation

shows this more clearly, and incidentally gives the value of h in terms of v. Since the component e, is at right angles to the major axis, and is directed towards the side of that axis to which the planet passes at perihelion, we have

where is the angle traversed by the radius vector from perihelion to the position considered. Hence