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On the Calculation of the Coefficient of Mutual Induction of a Circle and a Coaxial Helix, and of the Electromagnetic Force between a Helical Current and a Uniform Coaxial Circular Cylindrical Current Sheet." By Professor J. VIRIAMU JONES, F.R.S. Received November 12,-Read December 9, 1897.

§ 1. In measuring electrical resistance by the method of Lorenz we have to determine the coefficient of mutual induction of a helix of wire and the circumference of a rotating circular disc placed coaxially with it, the mean planes of the helix and the disc being coincident. In a paper presented to the Physical Society in November, 1888, I gave a method of calculating this coefficient; but subsequent consideration of the problem in connection with the Lorenz apparatus recently made for the McGill University, Montreal, has led me both to a simplification of the method previously described, and also to a more general solution.

§ 2. If M is the coefficient of mutual induction of we have

any

two curves

where the distance between two elements ds, ds' ; and

angle between the se elements.

€ = the

Let the equations to the circle and

and it may be readily seen by substituting for 2-pin the second and fourth integrals that

which is the coefficient of mutual induction of the circle, and a helix beginning in the plane of the circle of axial length, p.

VOL. LXIII.

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where x = p = the axial length of the helix, reckoned from the plane of the circle.

We may now proceed in two ways-either by expanding the logarithmic expression in powers of a/a, which leads to a series of limited application since it is convergent only so long as < A-a; or by integration by parts which leads to an expression applicable for all values of x.

§ 3. The first method I developed in the paper above mentioned. We have

The following properties of these elliptic integrals are perhaps worthy of notice :

(iv),

Pm

(2m+1) c'Qm+1 = 2m (1+c ́2) Qm-(2m-1)Qm-1

(2m+1) c'2Pm+1 = 2m (1+c'2) P (2m−3) (2m+1) P

c'2Qm = Qm−1+cQm

2m-1

where c'21 =

- c2, and the dotting of a function denotes differentia

tion with regard to c.

-1

It will be observed that Qo and Q_1 are respectively the complete elliptic integrals (F and E) of the first and second kinds with regard to modulus c.

Then we can find by a double application of (v) a relation between K+1, Km, and K-1, viz. :

This formula renders the calculation of successive terms of the

series sufficiently easy.

§ 5. Hence to find Me, given A, a, and x, we have to calculate the following quantities in order :

K2, K3, K., &c., by successive applications of (4),

(−1)Km,

and finally Me = (A+a) c2 (−1) Km.

§ 6. An example may be useful in showing the magnitudes of the various quantities concerned.

If the circle is in the mean plane of a helix of axial length 2x, the coefficient of mutual induction will be 2Me, or in case of the above dimensions,

M = 2Me = 18056 364 inches 45862 332 cm.

The value of Me given above was obtained in 1896 by