173. The successive periods of Recurring Decimals, taken separately, form geometrical progressions, having, respec .231231 &c. = 3 3 + + &c. in infinitum. 231 231 231 + + + &c. in infinitum. 103 106 109 Their sums may therefore be expressed by vulgar fractions, by means of the general formula of Art. 170; but in practice the most convenient methods are those explained in the Arithmetic, putting S for the value of the required fraction. The generality of these methods may be shewn as follows. (1) To transform .PPP... in infinitum, where P contains p digits, to its equivalent vulgar fraction. (2) To transform .PQQQ... in infinitum, where P and Q contain p and q digits respectively, into its equivalent vulgar fraction. We shall here introduce the proofs of the rules for the Multiplication and Division of Decimals. 174. To prove the rule for the Multiplication of Decimals. Let .M be the multiplier and .N the multiplicand, and let them contain m and n decimal places respectively; then from which we deduce the general conclusion, that the multiplication of decimals is performed as in whole numbers; and that the product contains as many decimal places as the multiplier and multiplicand together. 175. To prove the rule for the Division of Decimals. Let .M be the dividend and .N the divisor, and let them contain m and n decimal places respectively; then = = .M M N M 10n M 1 .N 10m 10n 10m N N 10m-n N according as m is greater or less than n. M = X or × 10n-m, First, let m be greater than n; then it is evident, that after the division is effected as in integers, the quotient must contain m n decimal places. Next, let m be less than n, then it appears that we must affix to the quotient n and that the result will be an integer. m ciphers, HARMONICAL PROGRESSION. 176. An Harmonical Progression is a series of quantities of which, if any three consecutive terms be taken, the first is to the third as the difference between the first and second is to the difference between the second and third. Thus, if a, a, a, a, &c. be the consecutive terms of an harmonical progression, 177. The reciprocals of quantities in harmonical progres sion are in arithmetical progression. For, let a, a, a, a,, &c. be the quantities; then α 178. Hence, if a be the nth term of an harmonical 1 progression, will be the nth term of the corresponding an arithmetical progression: and any number of harmonical means between a, and a, will be the reciprocals of the same number of arithmetical means between and 1 An VARIATIONS, PERMUTATIONS, AND COMBINATIONS. 179. The different orders in which any given number of quantities can be arranged, when only part of them are taken together, are called their Variations. Thus, the variations of a, b and c, taken two and two together, are ab, ac, ba, bc, ca and cb. 180. The different orders in which any number of quantities can be arranged, when the whole are taken together, are called their Permutations. Thus the permutations of a and b are ab and ba; of a, b and c are abc, acb, bac, bca, cab and cba. Without attending to the above distinction, the terms Variations, Permutations, Changes, &c. are frequently used promiscuously. 181. The different collections that can be made of any given number of quantities, without regard to the order in which they are placed, are called their Combinations. Thus, ab, ac and be are the combinations of the three letters a, b, c, taken two and two; ab and ba, though different variations, forming the same combination. Q 182. The number of variations of n things, taken r and r together, is denoted thus, "V,; the number of permutations of n things thus, "P; and the number of combinations, when they are taken r and r together, thus, "C,. 183. To find the number of variations of n different things, taken r and r together. Let a, a, a, ...a, represent the n things whose variations are required. Then the number of variations, when the quantities are taken separately, is clearly n. Also, the number, when they are taken two and two together, is n (n - 1); for a, may be placed before each of the rest, and thus form n-1 variations, two and two, in which a, stands first: and the same may be done with а2, а4 when there will be n-1 variations in which each stands first; and therefore on the whole there will be n (n - 1) such variations; or "V2: = n (n − 1). '3' ... ... ་ Again, the number of variations, when they are taken three and three, is n (n - 1) (n-2); for, since it appears that there are n(n−1) variations of n things taken two and two, there are (n − 1) (n−2) variations of the (n-1) things taken two and two together; and by prefixing a, to each of these, there will be (n-1) (n-2) variations of n things, taken three and three, of which α stands first and since there are the same number of variations in which a, a, ... a, respectively occupy the first place, there are on the whole n (n−1) (n−2) variations of n things taken three and three; or "V3= n (n − 1) (n − 2). By a similar process of reasoning, we should obtain for the number of variations of n things taken four and four together, n(n-1) (n-2) (n-3); where there are four |