Table 9 shows the fitted value for each entry of table 1. As expected, the fit is very good. The sum of squares of the residuals for 100 values is 0.0109. This gives a root mean square deviation per value of 0.0104. 6. Interpolation We mentioned earlier that the final fit should be adequate not only in reproducing the values of the original table, but also as an interpolation formula. In table 10, a comparison is made between values of F as given by the Biometrika Tables, and those given by our empirical fit, for combination of P, v1, and v1⁄2 not included in table 1 (the basis for our formula). Note, in particular, the values for P = 2.5 percent, a level that was totally absent from table 1. The sum of squares of residuals for these 80 values is 0.01812. Thus, the root mean square deviation per value is 0.048. Interpolation would of course be better if a larger table of F values had been used for the fitting process. The total number of parameters is the sum of 10 (one for each ✪ or T), and 4 × 17 (four for each of the three eigenvectors v, and four for each of the 14 vectors occuring in eqs (5), (6), and (7)); i.e., 78. As mentioned in Part II, this number can be somewhat reduced through algebraic manipulation, but this is unnecessary for a fitting process carried out on a programmable calculator or on a computer. We finally repeat our previous assertion (see also Part II) that these 78 parameters fit not only a table of 100 observations (Table 1), (which would be a waste of time) but actually any F value, for P between 1 and 25 percent, and for v1 and v1⁄2 between 4 and ∞. 7. Conclusion for Part III Through repeated application of the procedure given in Part II, it is possible to fit functions of more than two arguments, provided the data appear as a complete factorial. This is accomplished by first combining all combinations of two or more factors into one factor until a two-way table is obtained. The parameter vectors of the SVD of this table are then expressed as two-way tables themselves and further SVD's are carried out. The procedure is simple in principle but can become quite cumbersome in practice. It is not recommended for functions of more than three arguments, unless no other appropriate fitting procedure is available. 8. References [1] Pearson, E. S., and H. O. Hartley, Editors, Biometrika Tables for Statisticians, (Cambridge University Press, London, 1970). JOURNAL OF RESEARCH of the National Bureau of Standards Vol. 86, No. 1, January-February 1981 The Refractivity of Air Frank E. Jones* National Bureau of Standards, Washington, DC 20234 July 23, 1980 The air density equation of Jones, Edlén's dispersion formula for standard air, and Edlén's empiricallyderived expressions for the effects of Co2 abundance and water vapor partial pressure on refractivity have been combined into a simplified equation for the refractivity of air, and estimates have been made of uncertainties in calculated refractivity. Under ambient conditions typical of metrology laboratories, the agreement between the simplified equation and Edlén's formulation is well within the uncertainty in each. The simplified equation is valid in the visible region. Key words: Air density; index of refraction of air; refractivity of air; wavelength of light in air. 1. Introduction 101325 Pa and a CO2 abundance of 0.0003 by volume. Edlén [1] expressed the refractivity, (n-1),, of dry air at temperature t (in °C) and pressure p (in torr) as (n−1) = K1 Dip (2) where K, [3] is a dispersion factor which is independent of t and p, and the density factor, Dp, is D1 = p (1 + €, p)/ {(1 + œt)[1 − _(n−1)1p_]}, (3) In metrological applications of wavelengths of light in air, it is necessary to calculate the wavelength at ambient conditions of temperature (T), pressure (P), effective water vapor partial pressure (e), and CO2 abundance (xco2), using the refractive index of air under these conditions. The relation between Avac, the vacuum wavelength, Xair, the wavelength in air, and n, the refractive index of air, is Avac = n λair. Edlén [1]1 has derived a dispersion formula for standard air (T = 288.15K, P = 101325 Pa, e' = 0, xco2 = 0.0003 by volume) and a formulation for the refractivity of ambient air, (n-1)pf. Edlén's formulation is in general use in metrology. Jones [2] has recently published a reformulation of the equation for the density of air and applied it to the transfer of the mass unit. It is the purpose of the present D1 =p [1+p (0.817-0.0133 t) × 10-6]/(1+0.0036610 t). (4) paper to combine the air density equation, Edlén's disper 6 where a = 1/273.15 and e, is a factor which multiplies p in an expression for the nonideality of the gas. By substituting suitable values, (3) becomes sion formula for standard air, and Edlén's empirically. For air with a CO2 abundance of x by volume, Edlén derived derived expressions for the effects of CO, abundance and water vapor partial pressure on refractivity, and in so doing to develop a simpler formulation and to estimate uncertainties in the calculated refractivity. (n 1)x = [1 + 0.540 (x and, for the difference in refractive index of moist air holding h |