be difficult. Harmonic analysis of the curve resulted in the following 7-term approximation: Further analysis by Blagg resulted in the following simplified formula which yields less accurate results (provided in the paper in a non-normalized form, but given here in normalized form, i.e. with values from 0 to 1): where Neither of these formulas for function f are used in the calculations below. These calculations are based on a graph of function f which was drawn based on observed data. Her paper appeared in the Monthly Notices of the Royal Astronomical Society for 1913, and was forgotten until 1953, when A. E. Roy at Glasgow University Observatory came across it while researching another problem. He noted that Blagg herself had suggested that her formula could give approximate mean distances of other bodies still undiscovered in 1913. Since then, six bodies in three systems examined by Blagg had been discovered: Pluto, Jupiter IX Sinope, X Lysithea, XI Carme, XII Ananke, and Uranus V Miranda. Roy found that all six fitted very closely. This might have been an exaggeration: out of these six bodies, four were sharing positions with objects that were already known in 1913; concerning the two others, there was a ~6% overestimate for Pluto; and later, a 6% underestimate for Miranda became apparent. Another of Blagg's predictions was confirmed: that some bodies were clustered at particular distances. Her formula also predicted that if a transplutonian planet existed, it would be at ~68 AU from the Sun. Comparison of the Blagg formulation with observation Bodies in parentheses were not known in 1913, when Blagg wrote her paper. Some of the calculated distances in the Saturn and Uranus systems are not very accurate. This is because the low values of constant B in the table above make them very sensitive to the exact form of function f. Richardson formulation In 1945 D. E. Richardson independently arrived at the same conclusion as Blagg, that the progression ratio was not 2, but 1.728: where is an oscillatory function of , represented by distances from an off-centered origin to angularly varying points on a "distribution ellipse". Historical inertia Nieto, who conducted the first modern comprehensive review of the Titius–Bode Law, noted that "The psychological hold of the Law on astronomy has been such that people have always tended to regard its original form as the one on which to base theories." He was emphatic