expression, with the sign of one of its terms changed; and repeat the operation in the same way, as long as there are surds, when the last result will be rational. 2. When the terms of the binomial surd are odd roots, the rule becomes more complicated; but for the sum or difference of two cube roots, which is one of the most useful cases, the multiplier will be a trinomial surd, consisting of the squares of the two given terms and their product, with its sign changed. sign of the binomial, and the lower with the lower; and the series continued to n terms. This multiplier may be derived from observing the quotient which arises from the actual division of the numerator by the denominator of the following fractions: thus, x n−1 +xn−2y+xn-3y2+, &c., ...........+yn-1 to n terms, x + y whether n be even or odd. II. xn — yn x + y Xn-1 terms, where n is an even number. III. x + y terms, when n is an odd number. — yn-1 — xn—2y+xn-3y2-, &c., ·xn−2y + xn−3y2—, &c., And, since x Y3 α b b; then x = "/a, y="√b, and these fractions b ab and n/ b n a + n b a + b wa+n/b N -1 N--2 X &c., also y2= b2, ท n./ an-1 — njan-2b+ n¡ an−3b2 —, &C., ton terms;. where the terms and an odd number; and the sign N-2 n/a nj b = n / an-1+njan-2b+njan-3b2+, &c. . . . + nj zn~ ton terms; where n may be any whole number whatever. And, a+b Wa+b +yn-1 to n ± Won → bn-1 have the sign +, when ʼn is when n is an even number. n b. a ra Now, since the divisor multiplied by the quotient gives the dividend, it appears from the foregoing operations, that, if a binomial surd of the form njan nhbe multiplied by nan-ib nan-2b+, &c.,...+n/bn-1, (n being any whole number whatever), the product will be a — tional quantity; and if a binomial surd of the form a+b be multiplied by nan—1 n/an-2b+ n an—2b2 —, &c., +"/bn-1, the product will be a+b, or a -b; according as the index n is an odd or an even number. See my Elementary Treatise on Algebra, Theoretical and Practical.-ED. N : n EXAMPLES. 1. To find a multiplier that shall render 5 + 3 rational. Given surd 5+ √3 Multiplier 5√3 Product 25 - 3 2. To find a multiplier that shall make √ 5 + √ 3 rational 5+ √3 Product 32, as required. 4 3. To find multipliers that shall make 5+ 4/3 rational Given surd 4/5+ 4√3 1st multiplier V5-V3 5. 2d product 4. To find a multiplier that shall make Given surd 3/7 + 3/3 3 Multiplier 72 — 3 (7 X 3) + V 3a 532, as required. 7+ 3 rational. 17 +3✓ (3 × 72) E-Comm − 3 (3 × 72) — 3/ (7 × 3a) Product 7 + 3 7 +3=10, as was required. 5. To find a multiplier that shall make √5 — √ ≈ rational. Ans. 5+ √∞. 6. To find a multiplier that shall make a + √b rational. Ans. a -√b. 7. To find a multiplier that shall make a + √b rational. Ans. a ✔b. 8. It is required to find a multiplier that shall make 1-3/2a rational. Ans. 1+2a+34a2. 3 9. It is required to find a multiplier that shall make 3/3-1/2 rational. Ans. 9+ 6 + 14. 10. It is required to find a multiplier that shall make ^√ (a) + √(b3), or a 4+brational. 3. 3 9 Ansa (a°b3) + 4/ (a3bo). - ^/ bo. 4. 1✅ \/ CASE XIV. To reduce a fraction, whose denominator is either a simple or a compound surd, to another that shall have a rational denominator. Or, when it is of the form Է 3/ 2 a RULE.-1.—When any simple fraction is of the form √ a' multiply each of its terms by va, and the resulting fraction b√ a will be a the result will be And for the general form bn✓ an-1 1. Reduce the fractions have rational denominators. 2 Here √3 2. Reduce shall be rational. 3 Here √ ɔ̃−√2 3. Reduce be rational. b 3⁄4 a' 3 √ 5 −√2 b na 2 √3 2√3 3 1 × 5 5 X √5+√2 1 √2 result will be a 2. If it be a compound surd, find sucn a multiplier, by the last rule, as will make the denominator rational; and multiply both the numerator and denominator by it, and the result will be the fraction required. EXAMPLES 2 3 and 6 multiply by "a"-1, and the n multiply them by 3/ a2, and 3 24/5 4/125 the answer required. and ; 3 X 3/5 to others that shall 53 to a fraction whose denominator 4/53 3√5+3√2 3√5+3√2 √5+ √2 the answer required. to a fraction, whose denominator shall 10 9. Reduce tional denominator. 3√2+2 92 to a fraction that shall have a ra√(42) — √(18) Ans. 4 4 4/4+/5 + √2 the answer required. Jay to a fraction that shall have a rational x √ x Ans. Ans. 2+3 √2 7 9 to a fraction, the denominator of which 2a v b b Ans. 7. Reduce to a fraction that shall have a ra 3/5 3/7 tional denominator. Ans. 5 × [3 (49) + 3⁄4√ (35) +3/✓ (25)] Յ 8. Reduce 3 3 tional denominator. to a fraction that shall have a ra33/9 — 33/ (10) +3⁄4/ (300) 19 to a fraction that shall have a ra 3x + b PROGRESSION. Ans. 4 { — √ 10 − 2 √⁄2 + (2 + √ 5) ×√ 5 }. OF ARITHMETICAL PROPORTION AND ARITHMETICAL PRORORTION, is the relation which two quantities of the same kind, have to two others, when the difference of the first pair is equal to that of the second. Hence, three quantities are said to be in arithmetical proportion, when the difference of the first and second is equal to the difference of the second and third. Thus, 2, 4, 6, and a, a + b, a + 26, are quantities in arithmetical proportion. And four quantities are said to be in arithmetical proportion, when the difference of the first and second is equal to the difference of the third and fourth. Thus, 3, 7, 12, 16, and a, a+ b, c, c +1, are quantities in arithmetical proportion. ARITHMETICAL PROGRESSION, is when a series of quantities increase or decrease by the same common difference. Thus, 1, 3, 5, 7, 9, &c., and a, a + d, a + 2d, a + 3d, &c., are increasing series in arithmetical progression, the common differences of which are 2 and d. And 15, 12, 9, 6, &c., and a, a d, a 2d, a 3d, &c., are decreasing series in arithmetical progression, the common differences of which are 3 and d. T D The most useful properties of arithmetical proportion and progression are contained in the following theorems : 1. If four quantities are in arithmetical proportion, the sum of the two extremes will be equal to the sum of the two means. Thus, if the proportionals be 2, 5, 7, 10, or a, b, c, d, then will 2 + 10 = 5 +7, and a + d b+c. 2. And if three quantities be in arithmetical proportion, the sum of the two extremes will be double the mean. Thus, if the proportionals be 3, 6, 9, or a, b, c, then will 3 +9 = 2 × 6 = 12, and a + c 26. 3. Hence an arithmetical mean between any two quantities is equal to half the sum of those quantities. Thus, an arithmetical mean between 2 and 4 is 3; and between 5 and 6 it is 5. a + b And an arithmetical mean between a and b is * 5+6 2 2+4 ૭ 2 4. In any continued arithmetical progression, the sum of the two extremes is equal to the sum of any two terms that are equally distant from them, or to double the middle term, when the number of terms is odd. two arithmetical means between a and b, * If two, or more arithmetical means between any two quantities be required, they may be expressed as below. 2a + b a +26 Thus, and 3 3 a being the less extreme and b the greater, na+b (n−1) a +2b (n-2) a +36 3b a+nb And &c. to } n + 1' n+1 n + 1 n1 ber (n) of arithmetical means between a and b; where common difference; which, being added to a, gives the first of these means; and then again to this last, gives the second; and so on. b a is the n+1 any num |