The effect of the inertia of the Earth's core on the bodily tides and related phenomena is examined. Previous discussions, which have neglected this inertia, are found to be sufficiently accurate for the semidiurnal, fortnightly, and semiannual tides, but to need modification for those depending on terms of the forms zx, zy in the gravitational potential. A theory of these is developed for an Earth model of Wiechert type, with a fluid core, and applied to the free variation of latitude and the lunar nutation. It was known that with a rigid shell a fluid core would produce a shortening of the free period, in contradiction with observation; but it is fouhd that with a shell of reasonable rigidity (1.9 × 10 12 dynes/cm. 2 ) the motion induced in the core is much less than with a perfectly rigid shell, and that agreement between theory and observation can be obtained with a rigidity differing little from what was found satisfactory when the inertia of the core was neglected. Fluidity of the core has been seen to be capable of producing a reduction of the amplitude of the lunar nutation about three times as large as is needed to account for the present discrepancy between theory and observation. Allowance for elasticity of the shell is found to reduce this; allowance for the considerable obliquity of the ecliptic further reduces the correction for the nutation in latitude, but increases that for the nutation in longitude. Even for the nutation in latitude the correction remains too large, but possibly not too large for the excess to be due to inaccuracies of the Wiechert model, and there is a need for a theory that takes account of the fuller knowledge now available on the distribution of density and rigidity in the Earth.