THE MOON’S MOTIONS.

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to which the magnitude of the perturbing forces areaffected. We have hitherto supposed the earth at hermean distance from the sun, or about 91,500,000miles from him. Let us now take the case when sheis in perihelion, or about 90,000,000 miles from him.The moon’s distance, 238,800, is contained a smallernumber of times in the smaller distance, in the pro-portion of 900 to 915 ; in other words, the perturbingforce represented by Mi A is a larger aliquot part ofthe sun’s direct influence, in the proportion of 915 to900. But the sun’s direct influence is itself increasedby the approach of the earth and her satellite, in theproportion of the squares of these numbers; or as(915) 2 to (900) 2 . Hence the actual amount of theperturbing force is increased in the proportion ofthe cubes of these numbers, or as (915) 3 to (900) 3 .Similarly when the earth is in aphelion, or 93,000,000miles from the sun, the sun’s perturbing influence isless than when the earth is at her mean distance, inthe proportion of (900) 3 to (930) 8 .

There is, however, a simpler method, sufficientlyaccurate for our purposes, of indicating these rela-tions. When we cube two numbers which are nearlyequal, we triple the proportional difference (approxi-mately). Thus if we cube 100 and 101 (whose dif-ference is l-100th of the former) we obtain thenumbers 1,000,000 and 1,030,301, which are to eachother very nearly as 100 to 103; so that their dif-ference is about 3-100ths of the former. Now theearth’s greatest, mean, and least distances from the