12. Involve a+b-c to the square. Ans. a2 +2 ab—2 ac+b2 2 be+c2. SIR ISAAC NEWTON'S RULE FOR INVOLUTION. Whereby any power of a given compound quantity may be obtained by an easy and expeditious mental operation. 54. For Binomials. To find the terms and indices. RULE I. Write down the leading quantity successively, as many times as there are units in the index of the required power. II. Over the first of these place the index of the power; over the second, the index decreased by 1; over the third, the index decreased by 2; and so on, making the index of each term always 1 less than that of the preceding term. III. Subjoin the following quantity to the second, and every succeeding term, of the above, and carry it one place beyond. IV. Make 1 (understood) the index of the first of these; 2 the index of the second; 3 of the third, and so on, constantly increasing by 1 to the last; the index of which will be that of the required power. Thus, if it be required to involve a-z to the fifth power, the quantities and indices will stand as follows. Both quantities connected... a3, a1z, a2z3, a2z3, az1, z3. To find the coefficients of the terms. RULE. I. The coefficient of the first term is always 1, (understood,) and that of the second term is always the index of the required power. d This rule is a branch of the celebrated Binomial Theorem, discovered by Sir Isaac Newton in 1669. The author first discovered it by induction, namely, by observing the law which the signs, coefficients, and indices invariably follow, in a Binomial actually involved to several different powers. It will be a profitable employ for the learner to make the same induction: let him compare the 8th example with Newton's rule, and he will see that the coefficients, indices, signs, &c. of the terms, in every one of the powers, observe invariably the law on which the rule is founded. II. Multiply the coefficient and index of the leading quantity in the second term together, divide the product by 2, and the quotient will be the coefficient of the third term. III. And in general, if the coefficient and index of the leading quantity in any term be multiplied together, and the product divided by the number denoting the place of that term, the quotient will be the coefficient of the next succeeding term. Thus, in the above example. The coefficient of the first term will be 1, understood. Thus, the terms connected with their coefficients in the above example, will be x, 5 x1z, 10x3z2, 10x1z3, 5 xz1, z3. To find the signs. When the signs of both terms of the Binomial are +, all the signs of the power will be +; but when the second term is —, all the odd terms (the 1st, 3rd, 5th, &c.) will be +, and the even terms (the 2nd, 4th, 6th, &c.) —. Thus in the example above proposed, where it is required to involve a—z to the fifth power; the second term being —, the 1st, 3rd, and 5th terms of the power will be +, and the 2nd, 4th, and 6th terms —. Wherefore -5 xaz+ 10 x3z2 — 10 x 2 z3 + 5 xz*—z3 will be the fifth power (signs, coefficients, and indices complete) of the given quantity x—z. b3, bs. 13. Involve a+b to the sixth power, by Newton's rule. The leading quantity a... a, a3, a*, at, a3, a2, a. The following quantity b..... b, b2, b3, Both connected. .... u, a3b, aab2, a3b3, a2b‡, ab3, bo. 6 x 5 15x4 20 x 3 15x2 6x1 The coefficients 1, 6, 2 3 4 5 Or, 1, 6, 15, 20, 15, 6, 1. 6 The signs of the terms are all +: Wherefore by connecting the signs, coefficients, and terms, a+b}3 = a + 6 a3b+ 15 a2b2 +20 a3b3 + 15 a2 b1 +6 ab3 + bˆ : required. 14. Involve a-z to the 9th power. Terms. =the power ao, asz, a7z2, az3, azt, atz3, a3z6, a2z7, az, z9. Or, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1. The signs of the odd terms are +, and the even terms —. Wherefore a—29=a9—9 a3z+36 a7z2 —84 aoz3 + 126 a3z1 — 126aa z3+84 a3z6—36 a2z7 +9 az8—z9=the power required. 15. Involve +3 to the fourth power. Terms x, x3 × 3, x2 ×32, x x 33, 34. Coefficients 1, 4, 6, 4, 1. Signs all +. Wherefore_x+4x3 × 3+ 6 x2 × 32 + 4 x × 33 +3a, that is, x1+ 12x+54x+108x+81, is the power required. 16. Involve a+b to the square. Ans. a2+2ab+b2. 17. Find x+y3. Ans. x+3x2+3xy2+y3. 18. Find m―n}*. 19. Find a 25. 2 Ans. m1-4 m3n + 6 m2 n2 —4 mn3 +n1. 20. Find z-17. Ans. z7-726 +21 z5-35 z++35 z3-21 z2+ 72-1. 21. Required the nth power of a+x? Ans. a 1 n +na RULE. Let two of the terms of the given trinomial be consi • The ingenious Mr. Abraham Demoivre has given a method, whereby any power or root of a multinomial, consisting of any number of terms whatever, may be found, which may be seen in the Philosophical Transactions, No. 230, с с 2 dered as one factor, and the remaining term as the other, and proceed as before. 22. Involve a+b+c to the third power. 2 Let a be one factor, and b+c the other; then will a+b+c3 =a+3a2. b+c+зa. b + c2+b+3, which by expanding the powers of b+c becomes a3+3a2. b+c+3a, b2 + 2 bc + c2 + b2 + 3 b c +3 bc2+c3. This by multiplication becomes a3+3a2 b+3a2c+3 ab2 +6 abc +3 ac2+b+3b+c+3 be2+c, the power required. 23. Involve x+y-z to the fourth power. Let x be one factor, and y-z the other; then will x+y−2}+ = x2+4x3.y−z +6 x 2.y—z)2 + 4x.y—z}3+y—z\*, which expanded and multiplied, becomes x+4x3y—4x3z+6 x2y2—12x2yz+6x2 z2 +4 xy3 — 12 xy2z+12 xyz2—4 xz3 +y*—4 y3z + 6 y2 z2 — 4 yz3 + z, the fourth power required. 3 Or thus. Let x+y be one factor, and -z the other; then will x+y. =x+y) +—4.x+y3.z+6x+y) 2.z2 — 4.x+y.z3+z+: this expanded and multiplied, becomes x++4x3y+6 x2 y3 +4 xy3+y+—4x3z— 12x2yz-12xyz-4y3z+6x2z2 + 12 xyz +6y'z' —4 xx3-4yz3 +z+, the fourth power as before. EVOLUTION. 56. To extract the roots of simple quantities. RULE I. Place the fractional index denoting the required root, over the given quantity. II. Extract the root of the coefficient, and the result will be the coefficient of the root. III. Multiply the index of each letter by the index of the root, and place the results each over its proper letter for the literal part of the root, to which prefix the coefficient f. f The rule, as it respects the numeral coefficient, is sufficiently plain, being merely the evolution of the root of a number, which is explained in the Arithmetical part, (from Art. 268. to Art. 285.) With respect to the literal part, we may consider evolution and involution as parts of the same general rule, differing in one particular only, namely, the indices; those employed in invo Note. If the given quantity be affirmative, all its roots will be +; but if negative, its odd roots will be and its even ones impossible ". EXAMPLES. 1. Extract the square root of 9 aa. OPERATION. 9a3a2, the root. Explanation. Here is the index of the square root, and is placed over 9 a4, then the square root of 9 is 3 for the coefficient, and 4 × = 2 is the index; wherefore 3a is the root, the sign of which is + . 57. To extract the square root of compound quantities. RULE I. Range the quantities according to the dimensions of some letter concerned, as in division, Art. 50. II. Find the root of the left hand term, (Art. 56.) set it in lution being integers, and those in evolution fractions; hence the process in both cases is the same as far as it respects the letters, as may be seen by comparing this rule with the rule for involution. This is evident, because no quantity multiplied into itself can possibly produce a negative product; for + into + produces +, and into produces +; and therefore a negative power can have no even root, or in other words, the even roots of all negative quantities are impossible. с с 3 |