Coil 163 turns; resistance (hot) 0·15 ohm.

Coil 274 turns; resistance (hot) 0.6 ohm.

6. An advantage of the alternate-current electromagnet is the greater constancy in the pull which it exerts over a given range. If supplied at constant voltage a fairly constant pull on the armature can be obtained over a considerable range. The reaction set up by the core tends to set up a back-electromotive force the limiting value of which is equal to the impressed voltage. If the resistance of the coil were actually zero and the magnetic leakage also negligible, this limiting value would be attained. Now suppose that when acting upon its armature at a distance the self-inductive reaction chokes the current down to a certain value, the magnetization of the core going round cycles between definite maxima. Suppose the core to be drawn in nearer to the pole, the reluctance of the magnetic circuit is lessened, and fewer ampere-turns will suffice to produce an equal magnetization; but because the reluctance is lessened the coefficient of selfinduction is correspondingly increased, and the current choked in corresponding proportion. These two effects counterbalance one another, the magnetization going round cycles of practically the same amplitude as before. Hence the pull will be practically unchanged, save in so far as any change in the magnetic leakage at the different positions of the armature comes in to affect the question by weakening the field between the pole-faces and the arinature. Were the pole-faces made relatively large so as to obviate the tendency

to leakage and ensure the field being nearly uniform in the gap-spaces, the pull should be also very nearly constant throughout a range equal to at least one third of the breadth of the pole-face.

7. [Added April 10th, 1894.]-It is useful also to know the ratio between the different voltages that are needed, for a given electromagnet, to produce an equal number of ampereturns with alternating and continuous electromotive forces.

Let Va stand for the alternating volts and V. for the continuous volts which must be applied to the terminals of the coil in order to produce equal virtual amperes. Then

Va √ R2+p2L2

V

=

Ꭱ

(6)

When, as is generally the case, R is small compared with pL, we may take as a sufficient approximation

The alternate voltage ratio is proportional to the frequency and to the time-constant of the electromagnet.-For this ratio we may, on certain assumptions, find an expression in terms of the dimensions of the core and coil. Confining ourselves to the case where the magnetic circuit is closed we may find the value of L as follows:

Let ly be the length (in centimetres) of the iron magnetic circuit, A, its cross section, v1 its volume, m1 its mass in grammes, its permeability, and S the number of windings in the coil. Now my 7.7901. Then L (in henries) will be L=4πA1μS2/10°1 = 4πv1μS2/10/12 = 4πm1 μS2/7.79 x 109,2 Assuming that B does not exceed 6000 lines per so centimetre, we may take for ordinary wrought-iror μ=2000, and inserting this we obtain

Further, let l2 be the mean length of one turn winding, A, the cross section of the copper ductivity (mhos per centimetre cube =10 volume of copper (=SA22), and m, th (=8.8v2). Then we have for the resista R=12S/A,k=122S22/v2k=8° Inserting these values for L and

μ.

It is, however, doubtful, since the magnetic circuit contains two joints, whether so high a value as 2000 can be expected in ordinary work for u. In that case the coefficient 1·41 will be lower. With the electromagnet described above, at a frequency n=93, and with the armature in contact, the coefficient was 0-6 instead of 1:41. This is tantamount to saying that the working permeability was only 850. As so arranged, L being 0044, the inductance pL was 25-26 ohms, and R was 0.15 ohm. The alternate voltage ratio was therefore about 170.

As the armature was removed from proximity to the poles the self-induction, and therefore the impedance, fell, making the voltage ratio for equal currents lower. When the armature was at 317 millim. the voltage ratio fell to 34, and at 9.52 millim. to 21.5.

It was stated above that in ordinary cases the ratio pL/R might be taken instead of the more complete expression of equation (6). If it is desired to obtain a nearer approximation to R2+ p2L2/R than is afforded by simply neglecting the R under the square-root sign, we may take the value Va PL R = +

V R 2pL'

which, by reference to the preceding numerical instances, shows that the correcting term is really negligible, being in the first case, where pL/R was 170, only 340, and in the last case, where pL/R was 21-5, only 4, less than of 1 per

cent.

8. [Added April 27th.]—Lastly, a very simple expression can be found for the number of windings, in terms of any desired mean value of the magnetic flux N in the iron. For if the iron of cross-section A is subjected to cycles of magnetization in which the mean value of the permeation is B, the mean flux is N=AB, and if the frequency be n periods per second, the self-induced electromotive force in W windings surrounding the iron will be equal to 2πnWN÷108 volts. If, then, the resistance is negligibly small, this may be equated to V, the mean volts of supply, whence we obtain at once

For example:-Suppose that in the above magnet it were desired to obtain a permeation of 4000 lines to the square centimetre, the total number of lines N would equal the area A (6.5 square centim.) multiplied by 4000, or N=26,000. Taking the volts as 50, and the periods per second as 93, we find by formula (9), W=329.

Phil. Mag. S. 5. Vol. 37. No. 229. June 1894.

2 R

IN

LVII. The Second Law of Thermodynamics.

By S. H. BURBURY, F.R.S.*

66

I Na statistical system of molecules let T be the mean kinetic energy, and v1...v, certain controllable" COordinates, according to the notation used by Messrs. Larmor and Bryan in their Report on Thermodynamics. Let Q be the energy that must be supplied, or spent in external work, when T becomes T+T, and v1 becomes v1+dv1, &c. The object of this paper is to find the most general condition aq that will make a complete differential of a function of T, v... v. It will be found to be the same condition which is necessary to make the motion stationary with T, v1 ... v,

constant.

T

1. Let the state and position of a molecule be defined by then generalized coordinates ... with corresponding velocities, and the r controllable coordinates v... v. It is assumed that the v's vary so slowly that their velocities may be neglected, but their components of momentum shall be V1... V. Let U be the potential, 7 the kinetic energy of a molecule, E-U+T.

2. Let us assume provisionally that if the molecule be started from any position with any velocities, v... v, being maintained constant, its motion will be periodic; and there

dV

fore on average of time =0 for each v.

dt

3. Generally, if only the conservative forces act, v1...tr will vary. We must then apply external constraining forces P1... P, to prevent v... v, from increasing.

By Lagrange's equations,

4. For any given constant values of v... v, we may have many different motions of the molecule, all of this character, but with different values of 7, the mean kinetic energy. So with the same we may have different values of v1... v. If we wish to make the system pass from an original motion, in

* Communicated by the Author.

which these quantities are 7, v... v,, to a varied motion, in which they are 7+dī, v1+dv1, &c., we have (1) to supply energy E, (2) to do work ΣPov against the constraining forces. The whole energy required is E+ΣPav, or

dt

5. Now let there be a great number, N, of such molecules, all with the same values of v1... v,, but different values of and E. According to the provisional assumption above made, the N molecules are describing periodic motions with different periods, and with an infinite variety of phases. Owing to dV this variety of phases =0 for each V, not only on average of the time, but on average at any instant of all the molecules describing a given orbit. And if the whole motion be stationary, this property may hold even though the motions of individual molecules be not strictly periodic. We may then replace the provisional assumption, that each molecule is in periodic motion, by the assumption that =0 for each v.

dV

dt

6. Let Nf(x1...xn)dxı...dxn, or, shortly, Nfdo, be the number of molecules whose coordinates at any instant lie between the limits

Then fdo is the chance that any given molecule at any instant shall belong to that class.

Similarly let f'(x1...ï‚)dx1... dan, or, shortly, ƒ'do', be the chance that for a given molecule at any instant the velocities shall lie between the limits

1 and 1+dx1,

¿2 and 2+dx2,

&c.

(B)

Whether or not ƒ' be a function of the coordinates x1...Xn, we must have

Sf'do'=1.

The chance that a given molecule shall at any instant belong to both class A and write it, F do do'.

class B is ff' do do', or, as we will