Laplaces improvements in theory were substantial, but they still left prediction in an approximate state. This position changed in the 1860s when the local circumstances of tidal phenomena were more fully brought into account by William Thomsons application of Fourier analysis to the tidal motions as harmonic analysis. Thomsons work in this field was further developed and extended by George Darwin, applying the lunar theory current in his time. Darwins symbols for the tidal harmonic constituents are still used.
Darwins harmonic developments of the tide-generating forces were later improved when A.T. Doodson, applying the lunar theory of E.W. Brown, developed the tide-generating potential (TGP) in harmonic form, distinguishing 388 tidal frequencies.  Doodsons work was carried out and published in 1921. Doodson devised a practical system for specifying the different harmonic components of the tide-generating potential, the Doodson numbers, a system still in use.
Since the mid-twentieth century further analysis has generated many more terms than Doodson's 388.  About 62 constituents are of sufficient size to be considered for possible use in marine tide prediction, but sometimes many fewer can predict tides to useful accuracy.  The calculations of tide predictions using the harmonic constituents are laborious, and from the 1870s to about the 1960s they were carried out using a mechanical tide-predicting machine, a special-purpose form of analog computer. More recently digital computers, using the method of matrix inversion, are used to determine the tidal harmonic constituents directly from tide gauge records.