THE MOON’S MOTIONS.

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to perigeal regression, must tend to increase theeffect here considered, even though it might exerciseno direct influence on the motion of the perigee. Nowwe have seen that when the moon’s orbit is so placedthat the radial force causes the perigee either toadvance or recede, the tangential force causes theperigee to move in the same way, thus reinforcingthe effects due to the radial disturbing action.These effects due to the tangential action do indeedcounterpoise each other, so far as they are directlyconcerned; in other words, their direct effect is nil inthe long run. But insomuch as they reinforce thoseeffects which (as we have seen) cause the lunarorbit to linger in positions favourable to perigealadvance, and to shift quickly from positions favour-able to perigeal regression, they indirectly reinforcethe perigeal advance. Surprising though it mayseem, these indirect results of the tangential action,—these perturbations of perturbations as they may becalled,—actually exert so important an influence as todouble the mean rate of perigeal advance. Newtoneither overlooked this indirect action, or rather fellinto the mistake of supposing it might safely beneglected. Accordingly the only striking feature ofthe lunar perturbations which he was unable to explainin full, was the advance of the perigee. He couldaccount but for about one-half of the advance.Clairaut , who first applied analytical investigations(as distinguished from Newton’s geometrical method)to this question, deduced a result agreeing very