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where h is the radiation-constant and 0 is the temperature at the surface of the body.
The remainder, which is to include both true conductive and also convective loss, will be proportional to the temperature-gradient in the medium close to the surface of the body. It may on this assumption be represented by
is the temperature-gradient in the medium at
the bounding surface of the body, and c is a positive constant which will be referred to as the convection-conductivity.
The expression inside the brackets is the quantity called the "emissivity" in the usual treatment of the problem. On the above assumption, far from being a constant, it is seen to be a thing whose value will vary with every modification of the experiment by which it is sought to be determined.
To fix ideas, take the case of a cylindrical rod or wire of radius "a" heated uniformly at all points and maintained at constant temperature by mechanism the nature of which is of no consequence. Let it be surrounded by a coaxal cylindrical sheath of radius R maintained at a constant temperature, which will be taken as the zero of temperature, the intervening space being filled with conducting or pseudo-conducting material. The differential equation to be satisfied by the temperature in the medium is
d20 1 de
since with the above conditions everything is symmetrical with regard to, and uniform parallel to, the axis of the cylinder; and the solution of this which satisfies the boundary conditions is
log R-log r 0 = a log R-log a
This, then, is presumably an approximation to a theoretic formula for the case of experiments like those of Messrs. Ayrton and Kilgour referred to before; and the accompanying Table I. and curves (fig. 1) (pp. 272 & 273) show how closely, by a suitable choice of constants, it can represent the experimental values. The constants I determined by the method of least squares; this I considered it advisable to do, because the experimental values are not sufficiently precise to enable one to draw a smooth curve through them with anything like certainty. In particular I may mention that their own empirical formulæ fail to fit in with the experimental values yielded by the wire of radius 0037; and these values are similarly shunned by the formula which I give.
The first term in each of my formulæ should represent the radiation-constant. It must be noted, however, that for such very thin wires radiation forms only a small portion of the whole emission; and therefore, without having results of greater accuracy from which to deduce the formulæ, it is not. possible to assign the value of this term with definiteness. All the values for it are, however, higher than those obtained by observers who have experimented on the emission from surfaces in vacuo, as the following data show :—
Bottomley: Sooted globe in Sprengel | 000125 for excess of 83°.6.
Bottomley Silvered and highly po-0000354
As regards the other constant "b" (=·4343 c), we have
b='0696 × 10-3 c='16 x 10-3,
TABLE I.-Thermal Emissivities of Wires at different Temperatures obtained experimentally by Messrs. Ayrton and Kilgour, compared with calculated Values.
Thermal Emissivities of Wires at Different Temperatures.
Ordinates represent Emissivities.
Messrs. Ayrton and Kilgour's Experimental Values indicated thus:
Points on their Empirical Curves indicated thus:
Curves represented by formulæ given in this paper indicated thus:
Péclet's results were derived from experiments on five cylinders: only two of them are available for my purpose, since the others were either of different material or different surface. They were short cylinders of unpolished brass, terminated by hemispherical ends, allowance for which was made from the experiments on spheres. Péclet, therefore, considered his results to apply to infinitely long cylinders. Their radii were 5.1 and 6.9 cm. respectively, and measurements were made on them in a cylindrical enclosure 40 cm. radius and 100 cm. long. His results are :
In large cylinders like these radiation plays a more conspicuous part, and the value of the first term in my formula which represents it is in very good accordance with values. obtained by direct methods. The value of "b" is, however, very different from that obtained from Ayrton and Kilgour's experiments; and the conclusion is forced upon one that although to consider "b" a constant may serve as long as the enclosure remains the same, it will by no means suffice to take it the saine constant under such widely different conditions as obtain in these two sets of experiments. The character of the convective flow is evidently totally different in the two cases.
The same conclusion is arrived at by an examination of results obtained from experiments on spheres. The formula for a sphere, found by means of the same assumption as before, is
Péclet experimented on three spheres in the same enclosure; and assuming the enclosure to behave like a spherical one of 45 centim. radius, the formula becomes
The experimental and calculated values are here given: