-+- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Table S15: The Polynomial Coefficients for DT -720.0 to 2016.0 v. 2018 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Row Years Polynomial Coefficients i K_i K_{i+1} a_0 a_1 a_2 a_3 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1 -720.0 400.0 20483.158 -20530.285 10289.578 -3726.121 2 400.0 1000.0 6516.330 -5962.228 -255.072 1244.067 3 1000.0 1300.0 1543.097 -1370.086 869.282 -378.827 4 1300.0 1500.0 663.466 -512.002 -118.755 169.232 5 1500.0 1600.0 201.941 -120.908 97.235 -65.127 6 1600.0 1650.0 113.140 -60.910 -24.537 17.734 7 1650.0 1720.0 45.427 -79.494 56.185 -9.757 8 1720.0 1800.0 12.360 4.118 35.151 -33.363 9 1800.0 1810.0 18.267 -3.209 -1.015 1.752 10 1810.0 1820.0 15.795 0.017 4.240 -3.582 11 1820.0 1830.0 16.471 -2.247 -6.504 3.097 12 1830.0 1840.0 10.816 -5.965 2.787 -0.014 13 1840.0 1850.0 7.625 -0.432 2.745 -0.594 14 1850.0 1855.0 9.343 1.637 0.241 -0.870 15 1855.0 1860.0 10.352 -0.490 -2.368 1.547 16 1860.0 1865.0 9.040 -0.586 2.272 -2.472 17 1865.0 1870.0 8.254 -3.457 -5.143 2.717 18 1870.0 1875.0 2.371 -5.593 3.008 -0.913 19 1875.0 1880.0 -1.126 -2.315 0.270 -0.039 20 1880.0 1885.0 -3.210 -1.893 0.152 0.563 21 1885.0 1890.0 -4.388 0.101 1.842 -1.439 22 1890.0 1895.0 -3.884 -0.531 -2.474 1.871 23 1895.0 1900.0 -5.017 0.134 3.138 -0.232 24 1900.0 1905.0 -1.977 5.715 2.443 -1.257 25 1905.0 1910.0 4.923 6.828 -1.329 0.720 26 1910.0 1915.0 11.142 6.330 0.831 -0.825 27 1915.0 1920.0 17.479 5.518 -1.643 0.262 28 1920.0 1925.0 21.617 3.020 -0.856 0.008 29 1925.0 1930.0 23.789 1.333 -0.831 0.127 30 1930.0 1935.0 24.418 0.052 -0.449 0.142 31 1935.0 1940.0 24.164 -0.419 -0.022 0.702 32 1940.0 1945.0 24.426 1.645 2.086 -1.106 33 1945.0 1950.0 27.050 2.499 -1.232 0.614 34 1950.0 1953.0 28.932 1.127 0.220 -0.277 35 1953.0 1956.0 30.002 0.737 -0.610 0.631 36 1956.0 1959.0 30.760 1.409 1.282 -0.799 37 1959.0 1962.0 32.652 1.577 -1.115 0.507 38 1962.0 1965.0 33.621 0.868 0.406 0.199 39 1965.0 1968.0 35.093 2.275 1.002 -0.414 40 1968.0 1971.0 37.956 3.035 -0.242 0.202 41 1971.0 1974.0 40.951 3.157 0.364 -0.229 42 1974.0 1977.0 44.244 3.198 -0.323 0.172 43 1977.0 1980.0 47.291 3.069 0.193 -0.192 44 1980.0 1983.0 50.361 2.878 -0.384 0.081 45 1983.0 1986.0 52.936 2.354 -0.140 -0.166 46 1986.0 1989.0 54.984 1.577 -0.637 0.448 47 1989.0 1992.0 56.373 1.649 0.709 -0.277 48 1992.0 1995.0 58.453 2.235 -0.122 0.111 49 1995.0 1998.0 60.677 2.324 0.212 -0.315 50 1998.0 2001.0 62.899 1.804 -0.732 0.112 51 2001.0 2004.0 64.082 0.675 -0.396 0.193 52 2004.0 2007.0 64.555 0.463 0.184 -0.008 53 2007.0 2010.0 65.194 0.809 0.161 -0.101 54 2010.0 2013.0 66.063 0.828 -0.142 0.168 55 2013.0 2016.0 66.917 1.046 0.360 -0.282 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - The above table of polynomial coefficients enables evaluation of DT in seconds (s) and its derivative (the length of day lod) in milliseconds (ms) for any epoch between $-720$ and 2016. It is not valid outside the specified range of years. For the year and fraction Y, extract the coefficients a_0, a_1, a_2, a_3 from row i, where K_i <= Y <= K_{i+1} and form t = (Y - K_i)/(K_{i+1} - K_i), where 0 <= t < 1, and thus calculate DT = a_0 + a_1 t + a_2 t^2 + a_3 t^3 seconds lod = (a_1 + 2 a_2 t + 3 a_3 t^2) / (K_{i+1}-K_i) / 0.36525 ms - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - These coefficients reproduce the spline approximation discussed by L.V. Morrison, F.R. Stephenson and C.Y. Hohenkerk, in ``Historical Changes in UT/lod from Eclipses", Proceedings of the Journees des Systemes de Reference et de la Rotation Terrestre, 2017, https://web.ua.es/journees2017/ (in preparation). This is an update of our Royal Society paper (2016)(see below) for the inclusion of three hitherto excluded Chinese eclipses in 198 BC, AD 306 and AD 616, and two critical Spanish observations in AD 1239 and strengthened the case for the Chinese observation AD 1361. The details of the observations and their reduction are available in our Royal Society paper and in the supplementary material: ``Measurement of the Earth's Rotation: 720 BC to AD 2015'', Stephenson, F.R., Morrison, L.V., and Hohenkerk, C.Y., published by Royal Society Proceedings A, and available from their website at http://rspa.royalsocietypublishing.org/lookup/doi/10.1098/rspa.2016.0404 -+- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -