experiment, I was unable to effect any material change in my final result, I am compelled to see, in the divergent results of MM. Wilh. Weber, F. Kohlrausch, and L. Lorenz (who moreover conducted the investigation each according to one method only), values affected with errors of observation.

I. Determination of the Absolute Value of the Siemens Resistance-Unit on the basis of the Laws of Magneto-Induction.

As my first method of experiment for the determination of the value of the unit in question I chose a procedure which had already been employed by Wilhelm Weber on the introduction of absolute measurements of resistance *; and I managed it so that it could be carried out under two different conditions.

Two exactly equal, extremely regularly wound cylindrical spirals were connected with a multiplier so that their axes fell into one and the same horizontal straight line, which was perpendicular to the magnetic meridian. The inner radius of the spirals was 144-43 millims.; the outer radius amounted to 184.46; the depth of the space occupied by the turns amounted consequently to 40-03 millims. ; its breadth was 53.64 millims. ; and each spiral numbered 691 turns. A most powerful parallelepipedal magnet (its length, breadth, and height were 80, 20.1, and 21.1 millims. respectively) was placed with its centre exactly in the axis of the two spirals, and as nearly as possible in the middle between the central planes of the latter; it was supported by a thin brass wire of about 3 metres length. The stated dimensions of the multiplier and the magnet are of such a magnitude that, in the calculation of the mutual action between multiplier and magnet, in the place of the latter a system of two magnetic poles of equal magnetic moment could be put.

If a magnet within a multiplier be rotated a small angle from its position of equilibrium and left to the forces acting upon it, it will describe isochronous oscillations, the amplitudes of which diminish in a geometrical progression. multiplier "open"

the oscillation-period

With the

According to the law of magneto-induction, with the mul

tiplier "closed"

the oscillation-period

In these equations, K denotes the moment of inertia, and M the magnetic moment, of the magnet; H the horizontal component of the earth's magnetic force; B the torsion-moment of the suspension-wire; A the rotation-moment with which the wire and the surrounding medium act upon the magnet moved with the angular velocity 1; G the electromagnetic force with which the multiplier, when the current 1 flows through it, acts on the magnetic unit of mass concentrated in one polar point; and w the absolute value of the resistance of the multiplier (in electromagnetic measure).

From equations (1) and (2) result the further equations

and from these we get, for the absolute resistance w the expression

which, according to equation (1), can be replaced by

multiplier has been found equal to n Siemens mercury units,

Phil. Mag. S. 5. Vol. 5. No. 28. Jan. 1878.

D

G=

then the absolute value of one Siemens mercury unit (1 S. M. U)
in electromagnetic measure is :-

Strictly, in the development of the absolute value for w account would also have to be taken of the fact that the current induced primarily by the movement of the magnet is variable with the time, and in consequence of this acts inducingly on its own path. The carrying-out of the calculation shows that the influence of this induction of the induced current is so small in comparison with the other conditioning moments, that the expression above given for w is, in consequence thereof, only increased by (in round numbers) 20000. Since, in the measurements cited below, none of the quantities to be determined could be measured with such accuracy that an additional 20000 of its value could have been safely estimated, the influence of the induction on the part of the primarily induced variable current might be completely ignored.

For the determination of the absolute value of the S. M. U. by means of this procedure there are consequently seven different quantities to be measured.

The five quantities M, No2, T1, (1+0), and (M)

A2,

were determined by the method introduced by Gauss. The value of G was calculated, by means of the fundamental law of electromagnetic action at a distance, from the dimensions and form of the multiplier:

Here n denotes the number of the turns of the multiplier, R the mean radius of the turns, 2D the distance between the central planes of the two spirals, 2h the height, and 26 the breadth, of the cross section of the space occupied by the turns, p the quantity R2+D2, and 27 the distance between the poles of the oscillating magnet.

In deriving this expression it was presupposed that in place of the spiral turns circular turns might be put continuously filling the space occupied by the multiplier; further, the angle u of the deflection of the magnet was taken as so small that one might put cos u=1 and 5 sin2 u vanishingly small in comparison with 1. In the observations carried out u never exceeded the value 2°. The cylindrical spirals were so constructed and set up that the lengths R, D, h, and b could be accurately measured to within 0.1 millim. directly with the cathetometer.

The number n of the Siemens units which represented the resistance of the multiplier at the time of each observation was determined by aid of a bridge arrangement, which most carefully excluded all errors that might happen from extra currents, variations of temperature, dissimilar positions of the measuring-wire, the presence of transitory resistances, &c.

Eighteen series of experiments were carried out, according to this process, on 18 different days. The following was always the order of the operations:-determination of the and ; then determination of

number n; ascertaining of ( (H)

the values T1, A1, A2 from twelve successive series of observations with the multiplier alternately "open" and "closed;"

and, lastly, repetition of the measurement of (H), 1, and n.

The temperature of the observation-room never varied during any one series of experiments more than 0°-6 at the most, and was of course closely followed.

In order to get some light upon the trustworthiness of the results obtained by this method, two groups of experiments were instituted. In the first group the two spirals were pushed as near together as the suspension-wire of the magnet permitted (to the distance D=39-2 millims.); with this the difference - proved to be, on the average, 0·0296. At the same time the term

in the above-given general expression for G had here a value (about 2 per cent.) which together with the initial term 1 added considerably to its importance.

The mean value of these six series is

1 S. M. U. =0.95535 x 1010.

In the second group of trials the spirals were pushed so far asunder that the distance between their central planes was as closely as possible equal to the mean radius of their windings. For this position of the spirals (2D=1644 millims. nearly) the difference of the logarithmic decrements amounted to only about 0-0172; at the same time the expression of G was approximately independent of the pole-distance of the magnet: for the case that D=

G=

R

2'

16π.n
1 h2 312 36 62 31 h2
+
55.RL 15 R2 4
p2

1

and the value of the last member within the square brackets amounts to only -0.00028.

The results found with this arrangement of the experiments

were:

April 12, 1876, 1 S. M. U. =0·9531 × 1010

(millim.).

=0·9528 × 1010

The mean value amounts to

1 S. M. U. =0.95388

1010 (millim.).

During the summer of 1876 the multiplier was taken to pieces; in the autumn I once more subjected all the dimensions of both spirals to a cathetometric examination, and again

put

the spirals together so as to form a multiplier of the sort last described. The moment of the magnet had, in consequence of continual use at different times, become so considerably diminished that the difference of the logarithmic decrements λ2-λ now amounted to only about 0·0161.