is

Hence, for the whole conductor, the magnetic force at P

For a finite portion AB of the conductor, the contribution to the magnetic force at P will be

In the case of a thin straight wire of attracting matter, if p be the linear density, the attraction on unit mass at P,

due to an element of the wire, will be

Gp.ds

along r, where G is the constant of gravitation. Hence the component perpendicular to the wire is Gp.ds. sin 0/2 and that parallel to it Gp.ds. cos /r2.

Comparing these expressions with that for the magnetic field due to a current element, it will be seen that a similar method can be employed for their summation. Thus for an infinite wire the resultant force perpendicular to it is 2Gp/a, For a wire AB of finite length, the component perpendicular to it is obviously

Gp

a2

Gp

·Pq= (sin 2-sin 1),

a

Returning now to the magnetic field of a circuit, the magnetic force at any point due to a rectilinear circuit can be calculated by the method given, whereas, with the usual circular coil, the calculation for points not on the axis involves the use of spherical harmonics or elliptic integrals. In the case of a rectangular circuit, the expression for the magnetic force at any point in its plane was, I believe, first derived by the late Prof. G. M. Minchin *, in a form which does not appear to be usually given.

Let ABCD (fig. 2) be the rectangular circuit and P the given point. Through P draw the straight lines LM, NR,

Ibid. Sept. 6, 1895.

parallel to the sides. Then denoting LA and AN by a1 and ag and the angles they subtend at P by 1 and 2, the contribution of these adjacent portions to the magnetic force at P will be

where p is the perpendicular from P on LN.
Hence for the whole circuit the magnetic force at P is

If the point is outside the circuit as at P', it will be seen that two of the perpendiculars, p, and p3, will be negative. This result was obtained by Prof. Minchin from the consideration of the potential due to the equivalent magnetic shell, a more fundamental but less simple method than that given here, as it involves the rather troublesome calculation of the solid angle and the derivation of the magnetic force from the potential. Prof. Minchin also gave*, though without proof, a corresponding expression for the magnetic force at any point in the plane of a triangular coil, viz.,

where A, B, and C are the angles of the triangle, and P1, P2, and P3 the perpendiculars from the given point on the lines joining the feet of the perpendiculars from the point on the

sides.

But it can be easily shown that a similar expression holds for the magnetic force at a point in the plane of any rectilinear circuit.

For let BAC (fig. 3) be two adjacent sides of the circuit, PL and PN the perpendiculars on these two sides from the given point P, and PR the perpendicular on LN, the line joining the feet of these perpendiculars. Then the magnetic force at P due to the adjacent portions LA, AN of the circuit is

But, since the points PLAN are concyclic, we have

=i(si

(sin PLN.cos PNL + sin PNL. cos PLN

sin PAN. cos PAL+ sin PAL. cos PAN

PR

PLN)

PR

PAN)

Hence, for the whole circuit, the magnetic force at P may be written

The expressions for the rectangle and triangle are thus but particular cases of this more general theorem.

Phil. Mag. S. 7. Vol. 6. No. 34. July 1928.

Р

XVI. A new Method of determining the Mobility of Ions or Electrons in Gases. By R. J. VAN DE GRAAFF, B.Sc., Queen's College, Oxford *.

1. N many of the experiments that have been made to determine the velocity of ions or electrons in the direction of an electric force, the velocity that is measured is in some cases an upper limit of the mean velocity and in others a lower limit. This may be due to the fact that the velocities of all the ions are not the same, and also to the fact that a group of ions tends to diffuse in all directions. In order to avoid these and certain other difficulties I have devised a method for measuring the mean velocity of a group of ions moving in a gas under a steady and uniform electric force.

The principle of the method is the same as that used for the determination of the velocity of light by Fizeau, who employed a rotating toothed wheel as a periodic shutter to break up a beam of light into sections which travel a known distance and then again encounter the periodic shutter, which, depending on the time of arrival of the sections of the beam, either stops them or allows them to pass on. Thus the transmitted light may be observed as a function of the speed of rotation of the disk, and from this, taking into account the geometry of the apparatus, the velocity of light may be easily computed.

2. The apparatus used for the determination of the velocity of ions works in an analogous way. An oscillating potential in combination with grids gives the shutter effect corresponding with that of the rotating toothed wheel of Fizeau. The arrangement is shown diagrammatically in fig. 1. The plate A is maintained by the battery M at such a potential that a small glow-discharge is formed at the needle-point P, thus supplying to plate B ions of the sign whose velocity is to be measured. These, for the sake of clearness in description, may be assumed to be positive ions. Plates B, C, and D are provided, as shown, with central grids. The connexions to the batteries M and N maintain a steady potential of a few volts, opposing the passage of positive ions from plate B to plate C, and likewise from plate D to plate E. The plate E is connected to an electrometer for the measurement of the ionic current received through the

*Communicated by Prof. J. S. Townsend, F.R.S.