violating cyclic order by the use of arrangements such as b-c, a - c, a―b, or a — c, b — a, b — c. It will always be found that the work is rendered shorter and easier by following cyclic order from the beginning, and adhering to it throughout the question. Find the value of EXAMPLES XIII. h. 1. 2. + с + (a - b) (a − c)' (b − c) (b − a) ' (c − a) (c — b) 4. + + 5. 6. 7. 8. y (z − x) (z − y) x + y (x − y) (x − z) ' (y − z) (y — x) ' (≈ — x) (≈ − y) (a − b) (a − c)' (b − c) (b − a)' (c − a) (c − b) (a - b) (ac)' (b − c) (b − a) ' (c − a) (c — b) (p − q) (p − r) ' (q − r) (q − p) ' (r− p) (r− q) (a2 — b2) (a2 — c2)' (b2 — c2) (b2 — a2) (c2 — a2) (c2 — b2) x + y + x+y (p − q) (p − r) ' (q − r) (q − p) g+r + r+p (x − y) (x − z) ' (y − z) (y — x) + x + y + p+q CHAPTER XIV. COMPLEX FRACTIONS. MIXED EXPRESSIONS. 152. We now propose to consider some miscellaneous questions involving fractions of a more complicated kind than those already discussed. In the previous chapter, the numerator and denominator have been regarded as integers; but cases frequently occur in which the numerator or denominator of a fraction is itself fractional. 153. A Complex Fraction is one that has a fraction in the numerator, or in the denominator, or in both. In the last of these types, the outside quantities, a and d, are sometimes referred to as the extremes, while the two middle quantities, b and c, are called the means. 154. By definition (Art. 148) 8101018 is the quotient resulting from the division of by; and this by Art. 143 is 155. From the preceding article we deduce an easy method of writing down the simplified form of a complex fraction. Multiply the extremes for a new numerator, and the means for a new denominator. 156. The student should especially notice the following cases, and should be able to write off the results readily. 157. We now proceed to show how complex fractions can be reduced by the rules already given. NOTE. To ensure accuracy and neatness, when the numerator and denominator are somewhat complicated, the beginner is advised to simplify each separately as in the above example. In the case of complex fractions like the following, called Continued Fractions, we begin from the lowest fraction, and simplify step by step. |