9. If the geometric mean between x and y : the harmonic mean:: m : n, then x: y:: m + √ (m2 — n3) : m · √ (m2 — n2). 10. If the geometric mean between a and b be to the harmonic as 4: 3; then a:b:: 4+ √T: 4 — √7. CHAPTER IX. BINOMIAL SURDS. Propositions. XXV. 1. The square root of a quantity cannot be partly rational, and partly a quadratic surd. = If possible, let √x = a + √y, then xa2+2a √y + y, rational quantity = an irrational quantity, which is impossible. 2. If a + √∞ = by be an equation between rational quantities and quadratic surds, then a b, and a + √y, and if b a be not = For √x = b 0, √x would be partly rational and partly a quadratic surd, which is impossible, by Proposition 1, .. b — a = 0, or b = a, .. √x = √y. 3. The product of two dissimilar surds is irrational. If possible, let ✓ x × √ y = ax, then xy=ax”, y = a2 x, :. √ y = a √x; that is, y and x may have the same surd factor, which is contrary to the supposition. 4. To extract the square root of a binomial consisting of a rational quantity and a quadratic surd. a · = a + √ (a2 - b) y = a— √ (a2 — b) ··√(a + √b) = √x + √y = 2 1 Extract the square root of 7 ± 4 √3. 2. Extract the fourth root of 49 +20 √6. Assume (49 + 20 √6) = √x + √y, then 49 +20 √6 = x + y + 2 √ xy. ..x + y = 49, 2 √xy = 20 √6, x2+2xy + y = 2401, y2 4xy = 2400, .. x2-2xy + y2 = 1, .. x - y = 11 x+y=49 =49} (49 +206) = √ x + √ y = Again, assume √(5 ± 2 √6) = √x ± then 5 ± 2 √6 = x + y ± 2 √xy, (49+20√6)=√(5±2√6)=√x±√y=√3±√2 x = = 3, 5) y = 2, Assume that √ √ y = √(0 + √ −1). 17 8 Extract the fourth root of 14+ 8 √3; and of 4 ✓2 3 ... ... XXVI. Let A + B + C x2 + = a + bx + c x2 + be an equation which holds true for any value whatever of x; then the coefficients of like powers of a shall be equal to each other; that is, A = a, B = b, c = c, &c. Now, if A ~a + (B a be not equal to 0, let it be equal to some fixed quantity M, then (B~ b) x + (C~ c) x2 + ... = M ; and A and a are invariable quantities, .. A a or M is invariable. But M may have various values dependent upon the variations of x, .. M is variable; that is, M is both variable and invariable, which is impossible; •. A ~ a = 0, or A = a. Again, Bb + (C ~ c) x + ... = 0, b, and C = c, &c. + A'y + B'xy + +a'y + b'xy + a + bx + c x2 + and if some fixed value may be shown, as above, that ... ... be given to a while + A" y2+ +a" y2+ ... 2 y is variable, it A = a, Bb, C= c, A' a', B' = b', C' = c', &c. Examples 2x 1. Resolve (222 + 1) (22 + 3) into its partial fraction-s |