8. Discussion of Methods. The condition which is theoretically necessary in a determination of the radiation constant is that both the source and the receiver should be full radiators or "black bodies." In the foregoing experiments this condition has been realized as far as is possible in experimental work of this description, by having constant temperature enclosures with good radiating walls and apertures small in comparison with the dimensions of the enclosure. Keene has been the only other observer using a similar method to determine the radiation constant who has endeavoured to obtain "black body" conditions in radiator and receiver. In his paper he states that owing to the watercooled diaphragm on the furnace being in the reverse direction to that used above, his mean value of 5.89 x 10-5 c.g.s. units may be as much as 2 per cent. high, through heat reflected from the bevel re-entering the receiver. A range of only 23° C. (1097°-1120°) was used, and the distance between the source and receiver was only varied by 0-023 cm. The receiver was an aniline Thermoscope of rather high thermal capacity which was calibrated by an electrical method. Coblentz has made a determination of this constant using a modified Angström Pyrheliometer. His value 5.722 × 10-5 c.g.s. units, is the mean of a large number of results using different receivers, but all of similar construction. The variation in the values of σ is about 4 per cent., after a correction for reflexion, separately determined, has been applied to the observed values. Other experimenters using different methods, such as the ratio of emissivities, have obtained a value for σ of the order of 5.7 x 10-5 c.g.s. units. These results have not been discussed in detail here as the principles of the methods used differ from those of the methods already mentioned. Lewis and Adams have used the theory of ultimate rational units in order to calculate the value of σ. Using data based upon the electronic charge e, the gas constant R and the Faraday equivalent F, they obtain the value 5.7 x 10-5 c.g.s. units for the radiation constant. More recently Millikan § using Planck's equation for the distribution of energy in the spectrum of a "black body," *Proc. Roy. Soc. lxxxviii. p. 49 (1913). + Bull. Bur. Stds. xii. p. 503 (1916). has calculated the value of σ. The value he obtains is 5·72034 × 10-5 c.g.s. units. σ From a critical examination of the values of a obtained by a variety of methods, it now seems well established that the value of this constant lies in the range 5'70 to 5.75 x 10-5 c.g.s. units. 9. Conclusion. The concordance of the results obtained with the Radio Balance under widely differing conditions shows that the method used for evaluating the absolute measure of radiation is extremely satisfactory, and that the instrument is well suited for this purpose. It has the further advantages of being easy to manipulate, quickness in working, and accuracy in the necessary measurements. I wish to express my gratitude to Professor Callendar for the continual advice he has given me, and the kindly interest he has shown throughout the course of the experiments. LXXXIII. Note on a Type of Determinantal Equation. By R. C. J. HOWLAND, M.A., M.Sc., University College, London *. occur frequently in vibration and other problems, and in many such problems not more than two, the highest, values of λ are of interest. When equation (1) is expanded, the coefficient of (−λ)n is unity; that of (-)-1 is the sum of the diagonal elements; while that of (-)-2 is the sum of n(n-1)/2 determinants of the second order, and is readily calculated. The remaining coefficients, however, appear as the sums of determinants of the third or higher orders. Experience shows that the calculation of such determinants, using a *Communicated by the Author. calculating machine, is not only laborious, but is of a type in which it is easy to fall into error. It is the object of the present note to show that the equation may also be solved by an application of the rootsquaring method. The calculations involved are about as extensive as those needed for the direct solution, but they are of a kind better adapted to the calculating machine, and for this reason the method may prove to have some advantage in practice. in order to form an equation whose roots are the squares of those of (1), we first form a new equation from (1) by changing the sign of λ and then multiply the two determinants in matrix fashion, rows by columns. This gives the new equation in the form Since A,, consists of a sum of simple products, its calculation is of a type to which a calculating machine is well adapted. The process is now repeated with equation (2), and continued until the roots are so far separated that the first two can be estimated from three terms only of the equation; these, as we have seen, are readily calculated. Consider, for example, the equation which occurred in a problem of the whirling of a shaft. Whittaker & Robinson, The Calculus of Observations,' pp. 106 et seq. (London, 1924). The first two roots of this equation were found to be 239.5 and 14.55. The first three terms of the expanded equation are If the highest root is estimated from the coefficient of 1 only, we obtain λ=259. If the second root is estimated from the ratio of the coefficients of X3 and X1, the result is 18.2. A first application of the root-squaring process to (3) gives 4840-X2, 8336, 9558, 8224, 14285, 8336, 9558, 8224, 4727 14398-x2, 16560, 14285, 8224 16560, 19125-X2, 16560, 9558 16560, 14398-X2, 8336 4727, 8224, 9558, 8336, 4840-λ2 The first three terms of this equation are X10-57601x8+1.350 × 107x6....=0. . . (5) From the coefficient of λ we now estimate the first root as while, the ratio of the coefficients of X and X8 leads to λε = √1350/5.76=15·3. The approximation is already quite good. In repeating the process, it is necessary to work to seven or eight figures if the coefficients are to be accurate to three or four. The resulting determinant is therefore rather long and will not be written down. It leads to the equation λ20-3.291 x 109161569 x 101412-...0.. From this we have λ1 = (3.291 × 10o)=239.5, λι (6) The error in A is about 1.6 per cent,, and, since the frequency depends on A, the error in this will be less than 1 per cent. Thus the first two roots are obtained with sufficient accuracy for most practical purposes. In practice it is necessary to judge the degree of approximation that has been reached by watching the Phil. Mag. S. 7. Vol. 6. No. 38. Suppl. Nov. 1928. 3 I convergence of the sequence of estimates given by the successive equations such as (4), (5), and (6). The three successive estimates of X, namely 18.2, 153, 14-78, indicate a rapid convergence to a value not very different from the last of them. Such an indication can usually be accepted with confidence, since it is generally known that the roots of the initial equation are real and well separated, conditions which are known to be sufficient to make the process a rapidly convergent one. LXXXIV. Heaviside's Formula for Alternating Currents in Cylindrical Wires. By T. J. I'A. BROMWICH *. IT T was certainly due to Heaviside † that two correlated problems of this type were fully solved in terms of Bessel-functions; but of late years text-books have usually given the much less convenient solutions, obtained later by Lord Kelvin in terms of the ber an bei functions. Lord Kelvin himself had certain numerical results tabulated from his formulæ, being (apparently) unaware that equivalent values had been previously tabulated by Heaviside in other respects Lord Kelvin was fully aware of the importance of Heaviside's work. It is, in fact, due to Heaviside's work that we obtain the (now commonplace) view that the seat of the electromagnetic energy is the surrounding medium: that this energy gradually soaks into the conducting wire, and is there used up in heating the wire. Hitherto (so far as I know) no proofs have been given of Heaviside's formulæ given in § 2 below; thus, although the results are some 40 years old, it may be of service to provide proofs, and at the same time to direct attention to the advantages obtained by using these formula. § 1. Preliminary Formula. In Heaviside's discussion of an alternating current flowing along a cylindrical wire, the fundamental equations are (writing p for dsdt) : ән 4π 1 d(rH2) *Communicated by the Author. , + Papers, vol. i. p. 362; and vol. ii. p. 97. Lord Kelvin himself refers repeatedly to the work done by Heaviside in this connexion. |