pulsions, which act inversely as the square of the distance, is named, irrespective of the sign, the reciprocal potential of the system of points. As it is advisable to have a convenient name* for the case in which the attractions and repulsions are governed by any law whatever, or, more generally still, for every case in which the work accomplished in an infinitely small motion of the system may be represented by the differential of any magnitude dependent only on the space-coordinates of the points, I propose to name the magnitude whose differential represents the negative value of the work, from the Greek word epyov (work), the ergal of the system. The theorem of the equivalence of vis viva and work can then be expressed very simply; and in order to exhibit distinctly the analogy between this theorem and that respecting the virial, I will place the two in juxtaposition:

(1) The sum of the vis viva and the ergal is constant.
(2) The mean vis viva is equal to the virial.

In order to apply our theorem to heat, let us consider a body as a system of material points in motion. With respect to the forces which act upon these points we have a distinction to make: in the first place, the elements of the body exert upon one another attractive or repulsive forces; and, secondly, forces may act upon the body from without. Accordingly we can divide the virial into two parts, which refer respectively to the internal and the external forces, and which we will call the internal and the external virial.

Provided that the whole of the internal forces can be reduced to central forces, the internal virial is represented by the formula above given for a system of points acting by way of attraction or repulsion upon one another. It is further to be remarked that, with a body in which innumerable atoms move irregularly but in essentially like circumstances, so that all possible phases of motion occur simultaneously, it is not necessary to take the mean value of ro(r) for each pair of atoms, but the values of ro(r) may be taken for the precise position of the atoms at a certain moment, as the sum formed therefrom does not importantly differ from their total value throughout the course of the individual motions. Consequently we have for the internal virial the expression

Σεφ (r).

As to the external forces, the case most frequently to be considered is where the body is acted upon by a uniform pressure normal to the surface. The virial relative to this can be expressed

*The term force-function, besides some inconvenience, has the disadvantage of having been already used for another magnitude, which stands to the one in question in a relation similar to that in which the potentialfunction stands to the potential.

very simply; for, p signifying the pressure, and v the volume of the body, it is represented by

pv.

Denoting, further, by h the vis viva of the internal motions (which we call heat), we can form the following equation:h=1Σr$(r)+3pv.

We have still to adduce the proof of our theorem of the relation between the vis viva and the virial, which can be done very easily.

The equations of the motion of a material point are :—

The terms of this equation may now be integrated for the time from 0 to t, and the integral divided by t; we thereby obtain

occurring in the above equation, represent, if the duration of time t

is properly chosen, the mean values of

dx12

(de) and Xx, which were

dt

For a periodic motion the

duration of a period may be taken as the time t; but for irregular motions (and, if we please, also for periodic ones) we have only to consider that the time t, in proportion to the times during which the point moves in the same direction in respect of any one of the directions of coordinates is very great, so that in the course of the time t many changes of motion have taken place, and the above expressions of the mean values have become sufficiently constant.

The last term of the equation, which has its factor included in the square brackets, becomes, when the motion is periodic,

d(x2)

=0 at the end of each period, as at the end of the period dt

resumes the initial value

d(x2)
dt

When the motion is not periodic, but irregularly varying, the factor in brackets does not so regularly become =0; yet its value cannot continually increase with the time, but can only fluctuate within certain limits and the divisor t, by which the term is affected, must accordingly cause the term to become vanishingly small with very great values of t. Hence, omitting it, we may write

As the same equation is valid also for the remaining coordinates, we have

m

2

[(金)+(金)+(金)]
(dz ) ] = − ( X x + Y y + Zz),

or, more briefly,

m

2

v2 = − 4 (Xx+Yy+Z≈),

and for a system of any number of points we have the perfectly corresponding one

Hence our theorem is demonstrated; and at the same time it is evident that it is not merely valid for the whole system of material points, and for the three directions of coordinates together, but also for each material point and for each direction separately.

XVII. Proceedings of Learned Societies.

ROYAL SOCIETY.

[Continued from p. 68.]

April 28.-Dr. William Allen Miller, Treasurer and Vice-
President, in the Chair.

THE following communication was read :—

TH

"On a Cause of Error in Electroscopic Experiments." By Sir Charles Wheatstone, F.R.S.

To arrive at accurate conclusions from the indications of an electroscope or electrometer, it is necessary to be aware of all the sources of error which may occasion these indications to be misinterpreted.

In the course of some experiments on electrical conduction and induction which I have recently resumed, I was frequently delayed by what at first appeared to be very puzzling results. Occasionally I found that I could not discharge the electrometer with my finger, or only to a certain degree, and that it was necessary, before commencing another experiment, to place myself in communication with a gas-pipe which entered the room. How I became charged I could not at that time explain; the following chain of observations and experiments, however, soon led me to the true solution.

I was sitting at a table not far from the fireplace, with the electrometer (one of Peltier's construction) before me, and was engaged in experimenting with disks of various substances. To ensure that the

one I had in hand, which was of tortoiseshell, should be perfectly dry, I rose and held it for a minute before the fire; returning and placing it on the plate of the electrometer, I was surprised to find that it had apparently acquired a strong charge, deflecting the index of the electrometer beyond 90°. I found that the same thing took place with every disk I thus presented to the fire, whether of metal or any other substance. My first impression was that the disk had been rendered electrical by heat, though it would have been extraordinary that, if so, such a result had not been observed before; but on placing it in contact with a vessel of boiling water, or heating it by a gas-lamp, no such effect was produced. I next conjectured that the phenomenon might arise from a difference in the electrical state of the air in the room and that at the top of the chimney; and to put this to the proof, I adjourned to the adjacent room, where there was no fire, and bringing my disk to the fireplace I obtained precisely the same result. That this conjecture, however, was not tenable was soon evident, because I was able to produce the same deviation of the needle of the electrometer by bringing my disk near any part of the wall of the room. This seemed to indicate that different parts of the room were in different electrical states; but this again was disproved by finding that when the position of the electrometer and the place where the disk was supposed to be charged were interchanged, the charge of the electrometer was still always negative. The last resource was to assume that my body had become charged by walking across the carpeted room, though the effect was produced even by the most

careful treading. This ultimately proved to be the case; for resuming my seat at the table and scraping my foot on the rug, I was able at will to move the index to its greatest extent.

Before I proceed further I may state that a gold-leaf electrometer shows the phenomena as readily.

When I first observed these effects the weather was frosty; but they present themselves, as I have subsequently found, almost equally well in all states of the weather, provided the room be perfectly dry.

I will now proceed to state the conditions which are necessary for the complete success of the experiments, and the absence of which has prevented them from being hitherto observed in the striking manner in which they have appeared to me.

The most essential condition appears to be that the boot or shoe of the experimenter must have a thin sole and be perfectly dry; a surface polished by wear seems to augment the effect. By rubbing the sole of the boot against the carpet or rug, the electricities are separated, the carpet assumes the positive state and the sole the negative state; the former being a tolerable insulator prevents the positive electricity from running away to the earth, while the sole of the foot, being a much better conductor, readily allows the charge of negative electricity to pass into the body.

So effective is the excitation, that if three persons hold each other by the hands, and the first rubs the carpet with his foot while the third touches the plate of the electrometer with his finger, a strong charge is communicated to the instrument.

Even approaching the electrometer by the hand or body, it becomes charged by induction at some distance.

A stronger effect is produced on the index of the instrument if, after rubbing the foot against the carpet, it be immediately raised from it. When the two are in contact, the electricities are in some degree coerced or dissimulated; but when they are separated, the whole of the negative electricity becomes free and expands itself in the body. A single stamp on the carpet followed by an immediate removal of the foot causes the index of the electrometer to advance several degrees; and by a reiteration of such stamps the index advances 30° or 40°.

The opposite electrical states of the carpet and the sole of the boot were thus shown: after rubbing, I removed the boot from the carpet, and placed on the latter a proof-plate (i. e. a small disk of metal with an insulating handle), and then transferred it to the plate of the electrometer; strong positive electricity was manifested. Performing the same operation with the sole of the boot, a very small charge was carried, by reason of its ready escape into the body.

The negative charge assumed by sole-leather when rubbed with animal hair was thus rendered evident. I placed on the plate of the electrometer a disk of sole-leather and brushed it lightly with a thick camel's-hair pencil; a negative charge was communicated to the electrometer, which charge was principally one of conduction, on account of the very imperfect insulating-power of the leather.

Various materials, as India-rubber, gutta percha, &c., were subPhil. Mag. S. 4. Vol. 40. No. 265. Aug. 1870.

Κ