THE EARTH.

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And it has been found that the results of the most carefultheoretical investigations show, that only about one-half of theobserved effect is actually due to a real, but very minute,quickening-, or acceleration of the Moon ’s movement, caused bya very slow change in the form of its orbit around the Sun.It has therefore been asked:—Can any cause be suggestedwhich, by affecting the length of the Earth ’s day, wouldaccount for the other half of the Moon ’s apparent accelera-tion as the result of the way in which we measure our time ?Is there, for instance, any friction acting upon the Earth ’ssurface which would very slightly check its rotation ?

It has been replied that the Tidal Wave, as it sweepsround the Earth twice in every twenty-four hours, must by itsfriction produce some such result. The effect day by day, it istrue, would be excessively minute, and very careful theoreti-cal calculations indicate that it does not exceed jrmrvmvof a second in a day. Nevertheless, this would be enoughto cause a difference of twelve seconds, in The course of asingle century, in the time at which the Moon would occupyany given position. It is shown in our foot-note that thisresult may be deduced by the summation of an arithmeticalprogression,* each term of which exceeds the precedingby the above-mentioned minute fraction of a second ; the

* An arithmetical progression is one in which each term exceeds, orfalls short of, the preceding by the same quantity. The sum of such aprogression is found by multiplying the sum of its first and last terms bya number equal to one-half of the whole number of terms in the series—e.g., if we take the arithmetical progression 1, 3, 5, 7, 9, 11, in which thereare 6 terms, then, in order to get their sum, we must multiply the sumof 1 and 11 by one-half of 6, the result being 3 times 12, or 36. Ifthe first term, as in the case referred to in the text, be the fractionTrimnmr °f a second of time, it is so small that we may neglect it informing the sum which is to be multiplied by one-half of the number ofterms in the series. If, also, each term exceed the preceding by thesame small fraction of a second, as we have explained that it will in theabove case, we shall have for the sum of 36,525 terms (which is thenumber of days in a century) very nearly one-half of 36,525 times thelast term, which last term will be yVin/jmrF of a second. The result(in seconds) is therefore 36525 multiplied by itself and divided by twice57,000,000; or 1,334,075,625 divided by 114,000,000; the quotient ofwhich evidently amounts to rather more than 12 seconds.

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