various ages, the corresponding line of Table I will give the information, while if the smoothed values are wanted, a similar line in the chart of isogens will give them after being smoothed, not in one dimension only but in two dimensions. Similarly, as regards the birthrates for a father of any specified age and for mothers of various ages, by following the vertical columns instead of the horizontal lines. In conclusion, I would remark that, though the method of isogens applied to Kőrösi's tables fully discusses the distribution of mean birth-rates, those tables do not enable us to determine the second postulate of paramount importance, namely, the degree of conformity of individual cases to the means of many cases. We know nothing thus far about the facility of error at the various positions in the chart, whether or no it conforms to the normal law of frequency; still less, what is its modulus, or whether the modulus is constant throughont the chart or varies in accordance with some definite law. The answer to these questions admits of being obtained by a moderate amount of work on the original observations, selecting at first a few squares for exploratory purposes, such as are (1) distributed evenly about the chart, and (2) contain each of them not less than some 300 observations, and (3) whose means accord with the smoothed isogens that pass over the squares, thereby affording satisfactory centres of reference. IV. “Appendix to a Communication entitled • The Mechanical Equivalent of Heat.'»* By E. H. GRIFFITHS, M.A. Communicated by R. T. GLAZEBROOK, F.R.S. Received December 7, 1893. Section 1. In a communication which I had the honour of making to the Royal Society in the spring of this year, the following statement occurs (p. 420):-“We are (with the help of Mr. Callendar) now entering on a careful direct comparison of thermometer Em with a new form of air thermometer, which, there is every reason to believe, will give very accurate results, but we are unable to assign any definite limit to the time that this investigation may take." A great number of comparisons have been made during the summer of this year by Mr. Callendar and myself between the mercury thermometer Em used by me for determining the temperature of the calorimeter, the Tonnelot thermometer, No. 11,048, described in the above paper (pp. 426—433), the platinum thermometer N, by which the mercury thermometer Em had been previously standardised, * 'Phil. Trans.,' vol. 184 (1893), A, pp. 361–504. and two air thermometers constructed by Hicks under Mr. Callendar's direction.* The indications of these air thermometers are independent of external pressure, and no difficulty was experienced in obtaining the temperature of the bulb to 1/1000° C. The results were, however, not entirely satisfactory. It was found impossible to maintain the exterior portion of the tank (where the comparisons had to be made) at a temperature constant to 1/1000° C, especially at higher temperatures. A reference to pp. 374—378 will show that the regulating apparatus was designed to maintain at a constant temperature the interior of the steel chamber there described, and this purpose it fulfilled admirably. Fluctuations, however, amounting to as much as 1/100° at 50° C. occurred in the surrounding water, and an element of uncertainty was thus introduced into our comparisons. I am now so modifying the apparatus as to eliminate these fluctuations in the exterior temperature, and thus render the tank more suitable for such observations. I believe Mr. Callendar proposes to make a communication to this Society in which the details of this comparison will be given, and he has, with this object in view, taken with him to Canada the records of our experiments. I therefore propose on this occasion to confine myself to results. I may, however, mention that extreme care was taken with the cleaning and drying of the air thermometers; observations were made with the thermometers filled with air, hydrogen, and nitrogen, and all the precautions observed which Mr. Callendar's considerable experience of air thermometers could suggest. The conclusions, as far as they affect my previous determinations of temperature, are that over the range through which the experiments were conducted (14° to 26° C.) the limit of error does not exceed 0:003° C. of the nitrogen thermometer. An error of such a magnitude would affect my final value of J by about 1 in 4000. Another possible cause of error, mentioned on p. 424, is the difference caused by unequal lag of the rising mercury thermometer at the beginning and end of the temperature range, and I have pointed out on p. 424 that a possible error of 0.008° C. might be due to this cause, I have recently performed the experiments by which I hoped to throw some light on this point-using as a thermometer a naked platinum wire immersed in pentane. The experiments are difficult to conduct, and I do not regard the results as entirely satisfactory. They agree with my former experiments in indicating that the lag is greater at the begiuning than at the end of the range by a quantity between 0:002° and 0·009° C. The mean result of my observations gives 0:004° C. * For a description of these air thermometers see · Roy. Soc. Proc.,' January, 1891. 1 as the difference, which would diminish our temperature range by about 1 in 3000, and would increase the value of J by a proportionate amount, although it would not affect the temperature coefficients of the capacity for heat of water or the specific heat of the calorimeter. I do not, however, feel that this point is sufficiently established to make it advisable to apply the correction to the previously published results. Section II. I regret to state that we have discovered a serious error in the arithmetic. · On pp. 407–410 is given an account of the comparison of the coils in the resistance-box with the B.A. standards, and Table XI gives in full the numbers actually obtained during the comparison. In order to simplify our work we constructed a table for our own use which gave the value of each coil in terms of a legal ohm, and afterwards transferred them (see p. 410) into true ohms. Unfortunately the 10-ohm coil in the bridge arm was entered in this private table as 10:0077, whereas it ought to have been 9.9977—an error of 1 in 1000, baving its origin in a mistake in addition. The experimental numbers actually given in Table XI will show that the ratio of the bridge arms was 9-9734/997-87, that is, 0.0099947, or, if expressed in legal ohms, 9.9977/1000-30 instead of 10·00771000:30, as given on line 13, P. 410. The mistake is obvious to anyone who compares the numbers given in Table XI with the conclusion drawn on p. 410. The whole of the arithmetic was carefully revised by both Mr. Clark and myself, but an error of this kind in simple addition is precisely the one to escape observation. The annoying circumstance is that a similar mistake in any of the other coils would have had no appreciable effect on our conclusions, but occurring as it did in the ratio of the bridge arms it affects all succeeding tables. I would venture to add that this incident shows how advisable it is, in work of this kind, to give in full all the experimental numbers on which the conclusions are based. In consequence of this discovery I have carefully again revised nearly the whole of the calculations, but I am glad to say that with the exception of two or three obvious misprints I am unable to detect any further arithmetical mistakes. The effect of the correction thus rendered necessary is to decrease the value of R, in all the tables where the reduction to true ohms is given, by 1 in 1000; hence the value of T in Tables XXXVII, XL, XLI must be increased in the same proportion. The resulting correction is a simple one, for, as the value of J varies directly as T, it has only to be increased by 1 in 1000. Fortunately the temperatures as obtained by the platinum thermometers are independent of the ratio of the bridge arms, and are, therefore, unaffected. The values of the temperature coefficients, of the capacity for heat of water and of the specific heat of the calorimeter, remain practically unaltered, as the correction only affects the sixth significant figure. The corrected value of J in terms of a thermal unit at 15° C. is thus (4:1940+0.0042) x 10' = 4.1982 x 10', and I estimate the limit of error due to the causes mentioned in Section I of this Appendix as +0.0020. Hence (assuming that g= 981.17), J = 427.88 kilogramme-metres in latitude of Greenwich. F V. “On the Reflection and Refraction of Light.” By G. A. Schott, B.A. (Cantab.), B.Sc. (Lond.), formerly Scholar of (Abstract.) supposing the transition between different refractive media to be effected continuously through a thin variable layer, to deduce expressions for the amplitudes and changes of phase of the reflected and refracted light, and to compare them with the results of experiments hitherto made on that subject. The theories examined are the elastic solid theories, both those assuming large velocities for the pressural wave, including Green's, Voigt's, and K. Pearson's theories, and also Lord Kelvin's contractile ether theory, and then the electromagnetic theory, in the form given by Hertz, which, it may be remarked, leads to the same equations as the contractile ether theory. The medium being continuously variable, the displacements and stresses, or the electric and magnetic force components, are everywhere continuous. The method thus avoids all hypotheses as to boundary conditions at surfaces of discontinuity. For convenience, the first constant portion of the medium, from which the incidental light comes, is called the first medium, the second constant part, into which the light is in part refracted, is called the second medium, the thin variable part is called the variable layer, and the arbitrarily chosen planes, which include the whole of the variable layer, are called the boundaries of the layer. Since at those planes the medium is continuous, the displacements and stresses have the same values on both sides of each plane. In the first medium are assumed an incident, a reflected, and a pressural wave, and in the second a refracted and a pressural wave, the pressural waves, of course, disappearing on the contractile ether and electromagnetic theories. Taking account both of vibrations in and perpendicular to the plane of incidence, there are 6 amplitudes and 6 retardations of phase, in. all, 12 constants, to be determined. From the continuity of the motion at the two boundary planes of the variable layer are obtained 12 pairs of equations, of which 6 pairs determine the motion inside the variable layer, and the remaining ones the 6 pairs of constants. In the actual work, imaginaries are used in the usual way, reducing the equations by one half, the requisite number of equations being obtained at the end by changing the sign of the imaginary. From the equations of vibration in the variable layer and one half of the bonndary conditions, solutions are obtained in ascending powers of the thickness of the layer, which, on substitution in the remaining boundary conditions, give sufficient eqnations to determine the amplitudes and phases by simple though long algebraic analysis. The solutions are found to be convergent, provided 20/1 is less than the reciprocal of the greatest value of the refractive index occurring in the variable layer, d being its thickness, and a the wavelength in vacuo of the light employed. This condition gives values for d which are less than the thickness of a soap film producing a red of the first order. Thus the films for which the theory holds at all cannot possibly produce colours of thin plates, which has been stated as a possible objection by Lord Rayleigh. The solutions are taken as far as is necessary to give the values of the amplitudes and phases correct to squares of d. The elastic solid theories lead to Green's formulæ, but the contractile ether and electromagnetic theories give modifications of Fresnel's formulæ, somewhat of Cauchy's type. The corrections to the amplitudes are of order d’, whilst the phases are of order d. If Mo, Mi, je are the refractive indices of the media and the layer, then these terms involve certain constants which are functions of Mos My and of the mean valaes of u?, 1/4* and certain combinations of u?, 1/4?. The effect is considered of supposing the velocities of the pressural waves to be large, but finite and different, the refractive index for the pressural wave from the first to the second medium having any value. The resulting formulæ deviate from Green's chiefly in that д° — 0° Hi-? m,? where HP + we? Mi? +H? 2(° + Mo?) sin i, sin i mo, m, are the large ratios of the pressural to the light velocity in the two media ; and this value of M holds for values of io, ¿y so large that sini, > 1/m, and sin i; > 1/m. The effect is always to increase M, and thus Haughton's proposal to substitute for Green's M a becomes |