still a residue of gas. This gas is partially absorbed by bromine, not entirely. The residue appears to be chloride of methyle. The liquids condensed both on the surface of the water and in the U-tube were obviously for the most part olefines. They began to boil below zero (Cent.), and probably boil at all temperatures up to about 60°, when the distilling vessel was found to be dry. They appeared to be a mixture of various olefines. This reaction apparently resembles closely that with the oxalate of amyle.

London Institution.

IV. On the Explanation of Stewart and Tait's Experiments on the Heating of a Disk rotating in a Vacuum. By OSCAR EMIL MEYER*.

IN

N a previous paper† I have already briefly discussed the experiments on the heating of a disk in an exhausted space which Messrs. Stewart and Tait laid, in June 1865, before the Royal Society of London‡; and I then expressed the opinion that the agitations which are communicated to the rotating disk by the wheelwork are the chief cause of the heating. I revert to the subject once more, because the gentlemen in question have published another paper on the subject§, in which it is proved that the cause of the heating is to be sought neither in terrestrial magnetism, nor in conduction, nor radiation of heat, nor in the surrounding air. I think, as I have already said, that the cause lies in the agitations caused by the wheelwork.

I should not consider it worth while to elucidate this subject to the readers of the Annalen, if, with the aid of the explanation in question, the experiments did not enable us to calculate the coefficient of thermal radiation of the disk in absolute measure. We find in this manner a number which agrees most completely with a formula which Professor Neumann, of Königsberg, has most kindly communicated to me. This formula is based on the observations of Dulong and Petit on the law of cooling ||, and an observation which he himself has made. The agreement between the results obtained in these different ways affords the conviction not only that the explanation of Stewart and Tait's observation is correct, but also that the value in absolute measure obtained for the thermal radiation is undoubtedly accurate.

* Translated from a separate impression, communicated by the Author, of a paper published in Poggendorff's Annalen, vol. cxxxv. p. 285. † Pogg. Ann. vol. cxxvii. p. 380.

Proc. Roy. Soc. vol. xiv. p. 339. Phil. Mag. S. 4. vol. xxx. p. 314. § Proc. Roy. Soc. vol. xv. p. 290. Phil. Mag. S. 4. vol. xxxiii. p. 224. Ann. de Chim. et de Phys. vol. vii. (1817).

The agitations to which I ascribe the heating are communicated to the disk by the wheelwork; they are due to slight irregularities in the working of the axes and wheels, and are snch that the rotating disk and its axis, within the play left to it, is continually moved backwards and forwards.

Such an oscillation cannot escape observation; for the radius of the disk amounts to 6 inches = 165 millims. If, therefore, the axis (which is certainly far shorter) moves only the hundredth of a millimetre in its bearings, there must be a shaking of the edge, and in rapid rotation an apparent increase in the thickness, of the disk.

Stewart and Tait have, it is true, noticed this phenomenon ; they observed a rising and sinking of the aluminium disk used (which was inch thick) of 0.015 inch, or 0.38 millim., on both sides of the edge*. They explain this, it appears, on the assumption that disk and axis were not fastened to each other exactly at right angles. I consider it not less probable that an oscillation of the axis was the cause.

If, however, this assumption is correct, it is a necessary consequence that the oscillation must be the stronger the lighter the disk. This, in fact, was noticed by Stewart and Tait; for they found that while the disk of inch thickness deviated by 0·015 inch, that which was half as thick moved up and down as much as 0.02 inch.

It follows, moreover, from this assumption that the vis viva which was communicated to the disk by the wheelwork must have been the same in both cases. The quantities of heat resulting from these equal vires viva must have been equal; that is, the one of half the thickness must have been twice as hot as the one which was double. This, however, is exactly what has been observed by Stewart and Taitt.

After this confirmation of the hypothesis, it seemed worth while to calculate the magnitude of the vis viva which is changed into heat by agitations and impulses.

In this calculation we are concerned both with the number of the impulses and with their strength.

Since the wheelwork runs with constant velocity, the impulses occur regularly. The axis of the disk rolls, therefore, with regularity within the space which its ends have on their bearings. The axis describes a kind of conical surface. After each revolution it comes into the same position, or, at all events, into almost the same position; after each half revolution, into the opposite one. During each revolution, therefore, it is thrown once forward and once backward; or during each turn it experiences two impulses which change its position and direction.

* Article 20 (2).

+ Experiments XIII. and XX. Article 18.

At every impulse upon the axis one part of the vis viva present is lost; for at each impulse the position of the axis of rotation is changed; hence, of the vis viva present, only that part remains which corresponds to a rotation about the new axis; all the rest of the vis viva is lost, as far as rotation is concerned, and is used in heating the disk.

From this we can easily calculate the loss of vis viva and the gain in heat occurring in each second. If we denote the angular velocity of the disk by, the vis viva of the particles at the distance from the axis is

At this distance, however, there is an infinitely narrow zone of the breadth dr, and the thickness of the disk 8, which contains the mass

2πr Adr,

if ▲ denotes the density of the disk of aluminium. This zone has therefore the vis viva

and the entire disk the integral of this expression,

If, now, owing to one of the impulses in question, the axis of the disk is deviated through the angle y, the residual vis viva thereby becomes

MR22 cos2,

and that which is lost for rotation and changed into heat is

MR242 sin2 p.

This loss of vis viva and gain in heat occurs twice during each

2

rotation—in the unit of time, if T denotes the time of one rotation of the disk. The heat produced, therefore, in the unit of time is equivalent to the vis viva,

In this expression R sin & has a simple meaning; for it is nothing more than the magnitude of the alternate rising and sinking of the edge of the disk R, the value of which is 0.015 inch or 0.38 millim. By introducing this value the loss of vis viva may also be written

This vis viva of motion changed into heat is first of all consumed in raising the temperature of the disk, and is then imparted to the surrounding medium by radiation. Since after some time both the velocity of the rotation and also the excess of the temperature of the disk over that of the surrounding medium became constant, the heat lost in a second by radiation must be equivalent to vis viva transformed into heat during the same time. The first may be calculated from Newton's law of cooling, which, owing to the small amount of the heating, may be unhesitatingly accepted. If the constant excess of the temperature of the disk amounts to t degrees, the quantity of heat radiated in a second from both surfaces of the disk is

2ThR2t,

if the constant h denotes the heat which is radiated by the unit of surface for an increase of 1 degree. I obtain the mechanical work equivalent to this heat by multiplying by Qg, where g is the accelerating force of gravity, and Q the height to which the unit of mass can be raised by the unit of heat. The equivalent in work of that heat is therefore

2πhRetQg.

The work thus produced corresponds to the vis viva consumed —that is,

The first idea suggested by an inspection of this formula is a circumstance which apparently disagrees with observation. For Stewart and Tait have observed that the heating of the disk is inversely proportional to its thickness. From the above equation we might be tempted to conclude that the heating t increases proportionally to the mass M, and therefore also to the thickness of the disk. We must, however, remember that the oscillation must be the greater the less the thickness of the disk*. The os

* It is true that the above numbers do not accurately confirm this; but they are only approximate measurements.

cillation k is therefore inversely proportional to the thickness, and it follows that the heating of the disk must also increase inversely as the thickness.

All the magnitudes occurring in the formula are known from Stewart and Tait's measurements, or may easily be calculated from them, including the constant h which defines the thermal radiation. No direct statements have been published; so much the more interesting, therefore, does it appear to deduce their value from the observations in question.

If we introduce into the above formula the numerical values

and for g and their well-known values; and if, finally, we assume for the heating the mean value

t=0°.84 F. =0°46 C.,

which holds for the disk coated by lampblack, we get the thermal radiation

h=0·0017.

This number contains no arbitrary unit of heat, but is connected solely with the so-called absolute units (that is, the millimetre and the second of time), as well as the density of water as unit of specific gravity. It stands as for a surface blackened 9 by lampblack in a rarefied space in which there is a tension of 0.3 inch or 7.6 millims. of mercury.

An idea is obtained of the meaning of the number thus found by considering that a blackened surface of 1 square metre, which has been heated 1 degree above the surrounding rarefied air, loses in a second a quantity of heat which would raise a kilogramme through 0.72 metre.

The value found for the thermal radiation h is in remarkable agreement with the result which, with the kind aid of Professor Neumann of Königsberg, I was able to deduce from the observations of Dulong and Petit. I take this opportunity of thanking him publicly.

Those philosophers have combined the results of their observations on the cooling of a heated body in a rarefied space, in the law that the quantity of heat emitted in the unit of time by the unit of surface is expressed by the formula

ma (a'-1)+npetb.