580 HEAT and, because the emission is supplied by conduction from within, , ,f dv ev -=( j- I .
drj
Hence
- ,
v v< The value of e for a blackened globe hung within a hollow, with blackened bounding surface, is about -nnj-jj according to Macfarlane s experiments (Table VII. below), and considerably less for surfaces with any degree of polish. A r e may therefore take ^Vjr as a maximum value for e. The values of k for copper and iron at ordinary temperatures are, in C. G. S., approximately 95 and 18. Hence, if r = 5 cms. (or the diameter of the bar 10 cms., which is more than it is likely to be in any laboratory experiments), we find and " for iron. -lliO Hence the error will be practically nothing if we take r = r. Thus, and if we suppose k to be independent of temperature, (1) becomes dv k dv2e ,_, dv which is Fourier s equation for the conduction of heat along a bar or the circumference of a thin ring. Its solution to express simple harmonic variations of temperature produced in an infinitely long bar by properly varied thermal appliances at one end is v = R,f-9*cos(nt-fx + c) ..... (4), n, R, e being arbitrary constants, the "speed," the semi-range, and the epoch for x = ; and /, cj constants given by the formulas For iron and copper the values of c are respectively 95 and -845. Hence, with the previously used values of k for these metals, and with 1/4000 for c, we find K = 1 1 for copper and K= 2 for iron ; and for either, A = l/1700r nearly enough. Suppose, for example, r = 2 cm., this makes A = l/3400; and suppose the period to be 32 m. (the greatest of those chosen by Angstrom), this makes n=2ir/(GO x 32), or roughly ?t = l/310 and h/n = I/l7. Now when h/n is small, we have approximately and therefore with the assumed numbers , /nf l /= V 2^ 1 -3 by which we see that the propagation of the variation of tempera ture is but little affected by the lateral surface emissivity. Little as this effect is, it is very perfectly eliminated by the relation /I/ = 2 7 - ........ (6), which we find from (5). It is convenient to remark that g is the rate of diminu tion of the Napierian logarithm of the range, and / the rate of retardation of the epoch (reckoned in radians) per centimetre of the bar. Were there no lateral emissivity these would be equal, and the diffusivity might be calcu lated from each separately. This was done by William Thomson in his analysis of the Edinburgh underground temperature observations. But in the propagation of periodic variation of temperature along a b,ar (as of electric potential along the conductor of a submarine cable) lateral emissivity (or imperfect insulation) augments the rate of diminution of the logarithm of the amplitude, and diminishes the rate of retardation of the phase, leaving the product of the two rates unaffected, and allowing the diffusivity to be calculated from it by (6). This was carried out for copper and iron by Angstrom in Sweden, and the results communicated to the Royal Swedish Academy in January 1861. German and English editions of his paper have been published in PoggendorfTs Annalen for 1863, and the Phil Mag. for 1863 (first half year). The details of the apparatus and of the actual experiments, in which Angstrom had the assistance of Thalen, are sufficiently described in this paper, 1 and in a subsequent paper (Pogg. Annalen for 1863, p. 428), to allow us to feel perfect confidence in the very approximate accuracy of the results. Hence we have included them in our Table. 79. The question, Does thermal conductivity vary with temperature] was experimentally investigated by Forbes about thirty years ago ; and in a first provisional statement of results communicated to the British Association at Belfast in 1852 it was stated that the thermal conductivity of iron is less at high temperatures than at low. Forbes s investigation was conducted by an elaborate method of experimenting, in which the static temperature of a long bar of metal is observed after the example of the earlier experiments of Despretz, with a most important additional experiment and measurement by means of which the static result is reduced to give conductivity in absolute measure, and not merely as in Despretz s experiments to give comparisons between the conductivities of different metals. In 1861 and 1865 Forbes published results cal culated from his experiments, including the first deter mination of thermal conductivity of a metal (iron) in absolute measure, and a confirmation of his old result that the conductivity of iron diminishes with rise of tem perature. Forles s bars have been inherited and further utilized, and bars of copper, lead, and other metals have been made and experimented upon according to the same method, by his successor in the university of Edinburgh, Professor Tait. The investigation was conducted partly with a view to test whether the electric conductivities and the thermal conductivities of different metals, more or less approximately pure, and of metallic alloys, are in the same order, and, further, if their thermal conductivities are approximately in the same proportion as their electric conductivities. The following results quoted from his paper on " Thermal and Electric Conductivity " (Transactions li.S.U., 1878) are valuable as an important instalment, but expressly only an instalment, towards the answering of this interesting question: " Taking the inferior copper ( Copper C ) as unit both for ther mal and for electric conductivity, we find the following table of conductivities at ordinary temperatures, with the rough results as to specific gravity and specific heat referred to in 15 above : Thermal. Electric. Copper, Crown ........................................... 1 41 1-7 2 J C .................................................. 1-00 1-000 Forbes s iron ............................................. 29 C 2(>4 Lead ......................................................... 0-12 149 German silver ..................... . ....................... 014 0-117 1 The first paper is marred unhappily by two or three algebraic and arithmetical errors. One algebraic error is very disturbing to a careful reader, and might even to a hasty judgment seem to throw doubt on the validity of the experimental use which is made of the formula;. There is, however, no real foundation for any such doubt. The fol lowing little correction suffices to put the matter right. For the general term as printed in Angstrom s paper read with the following values for g it g . : / / ^ H:! _?L V V R 2 ^ + 4K-i? + 2K i fff- / / ** V V K 2 T a 4K a f 2 2Ki instead of these formulae without the i, as Angstrom gives them. Here we see that flyj = ^ ; and it is the product gj/ { that Angstrom uses in his experimental application, not the separate values of either g i or $. Hence no error is introduced by his having overlooked that
r/i is not equal to g^ except for t = l.