The three partial fractions will be obtained by dividing these values of P, Q, and R, by 1+ax, 1+bx, and 1+cx, respectively. The two preceding examples, as well as some of the exercises that follow, will serve to illustrate the nature of an important branch of analysis, the decomposition of rational fractions; it is impossible to enter further on the subject here. 2. Convert -X- -x2 into a series. Ans. 1+3x+4x2+7x3+11x1+18x5+29x6 &c. (1-x) into a series. 20 x2 3.2.3 3.5x4 3.5.7.x5 Ans. 1 into a series. a' + bx + c'x 244 CHAP. XI.-PROPORTION AND PROGRESSION. ART. 153. DEF. I. Ratio is the relation which two quantities of the same kind have to one another, with respect to the number of times the one is contained in the other. Ratio is expressed by two dots, or by the sign of division thus, the ratio of a to b is expressed by a:b, : 154. Cor. A ratio is not altered by multiplying or dividing its terms by the same quantity (Art. 67). 155. II. The terms of a ratio are called the antecedent and consequent. 156. III. If the second of four quantities is contained in the first as often as the fourth in the third, the first is Isaid to have to the second the same ratio as the third to the fourth. 157. IV. When the second of four quantities is contained in the first oftener than the fourth in the third, the first is said to have to the second a greater ratio than the third to the fourth. Thus, 4:2 is a greater ratio than 4 : 3, and 8: 5 a less ratio than 7:4; because 2>, and 33<35. 158. V. The ratio of one quantity to another is said to be compounded of two or more ratios, when the number which is the measure (Art. 46) of the ratio in question is equal to the product of the measures of the other ratios. Thus, the ratio of 3 to 108 is compounded of 6:3, 8:4, 15:5, 12: 4, or of any others whose measures are factors of 36. Also, the ratio ace : bdf is compounded of a: b, c:d, e:f.* 159. VI. A compound ratio whose measure is the product of two equal factors, that is, the second power, of the measure of another ratio, is called the duplicate of that ratio. In like manner, the ratio of two quantities is said to be the triplicate ratio of two other quantities, when its measure is the third power of the measure of the ratio of two other quantities. Thus, 3:75 is duplicate of 4: 20, and 3: 192 is triplicate of 4:16. 160. VII. Proportion, or analogy, is the equality of ratios, and quantities having the same ratio are called proportionals. Thus, if a: b be equal to c: d, these quantities constitue a proportion, which is written ab::c:d, * For if b be contained x times in a, and d be contained y times in c, and ƒ be contained z times in e, bdf will be contained xyz times in ace. Hence we might define compound ratio thus: In any number of ratios, if the product of the antecedents be taken, as also that of the consequents, the products are said to have to one another a ratio compounded of the other ratios. † When there are four quantities, such that the difference of the first and second is the same as that of the third and fourth, the quantities are called equidifferent. The term arithmetical proportion is sometimes applied to such quantities; and for the sake of distinction, proportion as defined above, is called geometrical proportion. It follows, from the definition of equidifferent quantities, that the sum of the means is equal to that of the extremes; for if a, b, c, d, be four such quantities, then a-b-c-d; therefore, a+d=b+c. Also, if a, b, C, be three equidifferent quantities, then b-a-c-b; a+c whence, b = that is, the middle term is half the sum of the extremes. This is generally called an arithmetical mean. 161. VIII. A continual proportion, sometimes called a geometrical progression, is a series of terms such that each term bas the same constant ratio to the one next following. Thus, a, ar, ar2, ar3, art, ar5,.... ar-1, is a series of continual proportionals, in which r is the common ratio, and arn-1 the last term.* 162. Cor. If three quantities be continual proportionals, the ratio of the first to the third is duplicate of the ratio of the first to the second; and if there be four quantities, the ratio of the first to the fourth is triplicate of the ratio of the first to the second. For, let there be the two series, ar2, ar, a; ar3, ar2, ar, a; then the measure of ar2: a is r2, and of ar2: ar is r; also, the measure of ar3: a is r3, and of ar3 : ar2 is r. Hence, the ratio of the squares is duplicate, and of the cubes triplicate, of the ratio of the first powers.† 163. IX. Three or four quantities are said to be in harmonical proportion, when the first has the same ratio to the last that the difference of the first and second has to the difference of the last and last but one Thus, a, b, c, are in harmonical proportion, when a:c::a-b : b—c ; and a, b, c, d, are in harmonical proportion, when a:d:, a-b: c-d. * The terms arithmetical and geometrical, commonly applied to proportions and progressions, are improper, as they lead to a misconception of the nature and connexion of these series. The terms equidifferences, and continual proportionals, are much more expressive of their nature. † The ratio of the square roots is called sub-duplicate, and of the cube roots sub-triplicate of the ratio of the first powers. |