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THIS Essay aims at a complete calculation of the effects produced by the action of a planet on the motion of the moon under the following limitations and conditions: (1) The problem of the motion of the moon under the action of the sun (supposed to move round the centre of mass of the earth and moon in a fixed elliptic orbit) and the earth, is considered to have been completely solved.(2) All the bodies are supposed to attract in the same manner as particles of masses equal to their actual masses and situated at the centres of mass.(3) All the planets are supposed to move in fixed elliptic orbits, i.e., the effect of the action of a planet transmitted either through the earth or through another planet is neglected.(4) Perturbations of the first order with respect to the ratio of the mass of a planet to that of the sun are alone calculated.(5) The exception to the above limitations occurs in the periods of revolution of the apse and node of the moon's orbit. These periods are not exactly those arising from (1) but they are the observed periods or, what amounts to the same thing owing to the close agreement between the observed and calculated periods, the periods after all known causes have been included. The point is only of importance in terms of very long period.(6) All coefficients greater than 0 01 in longitude, latitude and parallax have been obtained. Many are also given which are less than 0 01 whenever they have been accurately calculated. There are, in addition, classes of terms of short period which run in series and which in the aggregate will add up at certain times to much more than 0 01: these have been found to be 0 002.(7) The maximum period considered is 3500 years, but as the sieve in Section iv. retained a few terms of longer period, these were also included in the general scheme

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