In this case the preceding is the most complete and only solution But if and itc., be such that the numerator and denominator of the preceding both vanish, the general solution is sqx = (is - 1) + (is - 2) n, + (n - 3) C, . . . . + 0, . . C„-, B.-1 + (I + Co + Co CI + • • • • + Co • • • co—f) For demonstration and extension, see the article cited in the Eneyo. Isletrop.; §i 192.193, and 235.
PhRIODS OF REVOLUTION. In the present article we simply describe the names, commencements, lengths, and uses of those periods which it is most requisite the reader should find distinctly explained in a work of reference, premising some explanation of the way in which comparison of different periods gives now periods.
By a period wo mean a definite portion of time, beginning from a given epoch, or are,' which, being repeated again and again, will serve to divide all time subsequent to the epoch (or precedent, if the repe tition, be also carried backwards from the epoch) into equal parts, for the purposes of common reckoning and historical chronology. A period is then a finite portion of time used for measurement, just as a foot or mile is used for measurement of length.
Periods may be divided into natural and artificial ; the former imme diately suggested by some recurrence of astronomical phenomena ; the latter arbitrarily chosen. Since, however, time cannot. be preserved and handed down as if it were material, it is natural that the know ledge of artificial periods should be preserved by representing the number of natural periods which they oontain ; nor is it to be supposed that artificial periods were ever invented in a perfectly arbitrary manner, or were indeed over anything more than convenient collections of natural periods.
When one period is contained an exact number of times in another, each recommencement of the larger ono is also a recommencement of the smaller one ; thus, the day being exactly twenty•four hours, if any one day begin at the beginning of an hour, all days will do the samo. But if the smaller period be not a measure of the larger, a longer period may be imagined, which in this article wo shall call a eye/e, consisting of the interval between the two nearest moments at which the smaller and larger periods begin together. Thus, a week of seven days and a month of thirty days give a cycle of seven months or thirty weeks, these two periods being equal. If, however, the two periods can be measured by a larger number of days, the cycle may be made smaller ; thus, a mouth of 30 days and a year of 365 daya, or a month of G times five days, and a year of 73 times fire days, would give a oycle of 6 x 73 times five days, that is of 6 years or 73 months.
When two natural periods are expressed by complicated fractions of days, the method explained in Fnacrioxs, CONTINUA), will serve to show nearly how many of one period make up an exact number of the other. Thus the tropical, or common year being 36524224 days, and the lunation being 29.53059 days, both approximate, it appears that 36,524,224 lunar months would be 2,953,059 years nearly. To reduce
this long cycle to others more convenient for use, and as accurate as the number of figures employed will permit, proceed as in the article cited with the fraction 2.953,059 86,524,224 The quotients obtained are 12, 2, 1, 2, 1, 1, 17, dm, at whieli we stop, because the appearance of so large a quotient. as 17, shelve that tho result of the preceding quotient,' is extremely near. The eucceseive approximations derived from the first nix quotients are 1 2 3 8 11 19 25 37 99 136 235Or 235 lunationa mako 19 years very nearly.
Tho period in which all others are expressed is the day, which is not, as many suppose, the simple time of revolution of the earth, but [DAs) the average time between noon and noon. To distinguish it front other days it is called the man solar day.
The year, or the time between two vernal equinoxes,is not a uniform period, nor does the average of one long period give precisely the same as another. [YEAR.] For chronological purposes, however, it is useless to take account of this variation, and 3652422114 days, the • These terms are used synonymously by most chronologers : bat some mean by are the whole of the time which is measured from the epoch. Tho ear of Cho reader wilt perhaps be familiar with both of the phrases, 5th century of the Christian )era," and " the 5th century after the Chrtstien era." year of astronomers in our day, may be considered as more than suffi ciently exact for any time. In fact the year is made to consist, in the long run, of 365'2425 days, and a cycle of 400 years is necessary to the complete explanation of this fraction. Supposing the years from A.D. 2001 to A.D.. 2400, both inclusive, each fourth year is leap-year, beginning with 2004, except only 2100, 2200, and 2300, which gives in the 400 years 365 days to each year, and 97 intercalated days; while adding 97 days to 400 years, adds on the average 97-400ths or of a day, to each year. As it is of considerable importAce distinctly to comprehend an intercalated cycle, that is one in which fractions are disregarded until they amount to a unit, when they are corrected, to use a common phrase, in the lump, we put down the effect of the correction which is made in the year 1S40, being leap-year. In 1836, immediately after the last intercalation was made, the sun was in the vernal equinox at about 39 minutes after 1 P.M. ou the 20th of March, and the equinoxes then took place as follows :— The intercalation of 1840 (but for which the sun would have come on the equinox at 41 minutes past noon on the twenty-first) has over done the correction, bringing the equinox nearer to noon than in 1836 by 58 minutes. Now this over-correction of nearly an hour in four years is set nearly right by leaving out the correction three times in 400 years ; a irovision the necessity of which may be imagined, though its exactness cannot be appreciated from the preceding rough calculation.