ters taken singly, or two and two, three and three, &c. An important application of these principles will be seen in the next Section. Letters. Singly. 2 and 2.3 and 3.4 and 4.5 and 5.6 and 6.7 and 7.8 and 8.9 and 9. 10 and 10. SECTION XVI. INVOLUTION OF BINOMIALS. (262.) We have shown, in Art. 142, how to obtain any power of a binomial by actual multiplication. We now propose to develop a theorem by which this labor may be greatly abridged. Taking the binomial a+b, its successive powers found by actual multiplication are as follows: (a+b)1a +b, (a+b)'=a'+2ab+b2, (a+b)'=a'+3a2b+3ab' +b3 (a+b)'=a*+4a3b+6a2b2 +4ab3 +-b', (a+b)'=a*+5a'b+10a3b2+10a*b*+5ab* +b*, (a+b)=a+6ab+15ab2+20a3b3+15a2b*+6ab3+bo. The powers of a-b, found in the same manner, are as fol lows": (a—b)'=a-b, (a—b)2=a2-2ab +b2, (a—b)3=a3—3a2b+3ab2 —b3, (a-b)=a-4ab+6a'b' -4ab3 +b', (a-b)=a3-5a*b+10a3b2-10a2b3+5ab* -bo, (a—b)®=a®—6ab+15a*b2—20a3b3+15a3b*—6ab3+b”. In On comparing the powers of a+b with those of a−b, we perceive that they only differ in the signs of certain terms. the powers of a+b, all the terms are positive. In the powers of a-b, the terms containing the odd powers of b have the sign-, while the even powers retain the sign +. The reason of this is obvious; for, since -b is the only negative term of the root, the terms of the power can only be rendered nega tive by b. A term which contains the factor -b an even number of tines, will therefore be positive; if it contain it an odd number of times, it must be negative. Hence it appears that it is only necessary to seek for a method of obtaining the powers of a+b; for these will become the powers of a-b by simply changing the signs of the alternate terms. (263.) If we consider the exponents of the preceding powers, we shall find that they follow a very simple law. Thus, In the first term of each power, a is raised to the required power of the binomial; and in the following terms, the exponents of a continually decrease by unity to 0; while the exponents of b increase by unity from 0 up to the required power of the binomial. It is obvious that this will always be the case, to whatever extent the involution may be carried. Also, the sum of the exponents of a and b in any term is equal to the exponent of the power required. Thus, in the second power, the sum of the exponents of a and b in each term is 2; in the third power it is 3; in the fourth power, 4, &c. We hence infer, that for the seventh power the terms, without the coefficients, must be a', aob, a3b3, a1b3, a3b1, a2b3, abo, b'; (264.) It remains to determine the coefficients which belong to these terms; and in order to discover the law of their formation, let us take the coefficients already found by themselves. The coefficients of the 1st power are 1 1 The numbers in this table are identical with those in the table of combinations on page 214. For example, the coefficients of the fifth power denote the number of combinations of five letters taken one and one, two and two, &c.; the coefficients of the sixth power denote the number of combinations of six letters taken one and one, two and two, &c. The reason of this will appear if we observe the law of the product of several binomial factors, x+a, x+b, x+c, x+d, &c. we obtain x2+(a+b+c)x2 + (ab + ac + bc)x + abc = 2d product. Multiplying by x + d, we obtain x+(a+b+c+d)x2+(ab+ac+ad+be+bd+ cd)x2+(abc+abd+acd+bcd)x+abcd=3d product. We observe that in each of these products the coefficient of x in the first term is unity; the coefficient of the second term is the sum of the second terms of the binomial factors; the coefficient of the third term is the sum of all their products taken two ana two; the coefficient of the fourth term is the sum of all their products taken three and three, &c. It is easily seen that if we multiply the last product by a new.factor, x+e, the same law of the coefficients will be preserved. Hence the law is general. If now, in the preceding binomial factors, we suppose a, b c. d, &c., to be all equal to each other, the product becomes (x+a) (x+b) (x+c) (x+d)... . . The coefficient of the second term of the product, or a+b+ c+d....., becomes a+a+a+a. that is, a taken as many times as there are letters a, b, c, d, and is, consequently. equal to na. The coefficient of the third term, or ab+ac, &c., reduces to a2+a2+a2 . . . . ., or a2 repeated as many times as there are different combinations of n letters taken two and two; that is, The coefficient of the fourth term reduces tc a' repeatca as many times as there are different combinations of n letters n(n-1) (n-2) taken three and three; that is, -a3, and so on. 1.2.3 Thus we find that the nth power of x+a may be expressed which is called the BINOMIAL FORMULA, and is generally ascribed to Sir Isaac Newton. So important was it regarded, that it was engraved on his monument in Westminster Abbey as one of his greatest discoveries. On comparing the different terms of this development, we perceive that any coefficient may be derived from the preceding one by the following rule: If the coefficient of any term be multiplied by the exponent of x in that term, and divided by the exponent of a increased by one, it will give the coefficient of the succeeding term. Thus, the fifth power of x+a is x+5ax+10a'x' +10a'x'+5a*x+a.. If the coefficient 5 of the second term be multiplied by 4, the exponent of x in that term, and divided by 2, which is the exponent of a increased by one, we obtain 10, the coefficient of the third term. So, also, if 10, the coefficient of the fourth term, be multiplied by 2, the exponent of x, and divided by 4, the exponent of a increased by one, we obtain 5, the coefficient of the fifth term; and so of the others. The coefficients of the sixth power will also be found as follows: 6X5 15X4 20×3 15X2 6x1 1, 6, 1, 6, 15, 20, 15, 6, 1. that is, 7X6 21×5 35X4 35X3 21×2 7X1 1, 7, |