the fourth, the sixth, &c. The business then is to establish a rule, by which any power of a + b, however high, may be determined without the necessity of calculating all the preceding ones.

303. Now, if from the powers which we have already determined we take away the numbers that precede each term, which are called the coefficients, we observe in all the terms a singular order; first, we see the first term a of the root raised to the power which is required; in the following terms the powers of a diminish continually by unity, and the powers of b increase in the same proportion; so that the sum of the exponents of a and of b is always the same, and always equal to the exponent of the power required; and, lastly, we find the term b by itself raised to the same power. If, therefore, the tenth power of a+b were required, we are certain that the terms, without their coefficients, would succeed each other in the following order; a1o, a b, aa b2, a7 b3, a® ba, a§ b3, aa b®, a3 b2, aa b3, abo, b10.

304. It remains, therefore, to show how we are to determine the coefficients which belong to those terms, or the numbers by which they are to be multiplied. Now, with respect to the first term, its coefficient is always unity; and with regard to the second, its coefficient is constantly the exponent of the power; but with regard to the other terms, it is not so easy to observe any order in their coefficients. However, if we continue those coefficients, we shall not fail to discover a law, by which we may advance as far as we please. This the following table will show.

X.

1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, &c.

1, 6, 15, 20, 15, 6, 1

1, 7, 21, 35, 35, 21, 7, 1

1, 8, 28, 56, 70, 56, 28, 8, 1

1, 9, 36, 84, 126, 126, 84, 36, 9, 1

6 4

We see then, that the tenth power of a + b will be a1o + 10ab45 a bb + 120 a' b3 +210 a ba + 252 as b3 + 210 a1 b + 120 a3 b7 + 45 a ab3 + 10 a bo + b1o.

305. With regard to the coefficients, it must be observed, that for each power their sum must be equal to the number 2 raised to the same power. Let a = 1 and b = 1, each term, without the coefficients, will be = 1; consequently, the value of the power will be simply the sum of the coefficients; this sum, in the preceding example, is 1024, and accordingly

(11)1o 21° 1024.

It is the same with respect to other powers; we have for the

I. 1+1=2 = 21,

II. 1+2+1 = 4 = 22,

III. 1 + 3 + 3+1=8=23,

IV. 1+4 +6+4+1= 16 = 24,

V. 15+ 10 + 10 + 5 + 1 = 32 = 25,

VI. 1 + 6 + 15 + 20 + 15 +6+1= 64 = 2®, VII. 17+ 21+ 35 + 35 + 21 + 7 + 1 = 128 = 27, &c.

306. Another necessary remark, with regard to the coefficients, is, that they increase from the beginning to the middle, and then decrease in the same order. In the even powers, the greatest coefficient is exactly in the middle; but in the odd powers, two coefficients, equal and greater than the others, are found in the middle, belonging to the mean terms.

The order of the coefficients deserves particular attention; for it is in this order that we discover the means of determining them for any power whatever, without calculating all the preceding powers. We shall explain this method, reserving the demonstration however for the next chapter.

307. In order to find the coefficients of any power proposed, the seventh, for example, let us write the following fractions, one after the other;

In this arrangement we perceive that the numerators begin by the exponent of the power required, and that they diminish successively by unity; while the denominators follow in the natural order of the numbers, 1, 2, 3, 4, &c. Now, the first coefficient being always 1, the first fraction gives the second coefficient. The product of the two first fractions, multiplied together, represents the third coefficient. The product of the three first fractions represents the fourth coefficient, and so on.

Eul. Alg.

13

So that the first coefficient = 1; the second

third

=

= } = 7; the

= 35; the fifth

X=21; the fourth X X = 7 1 X X = 21; 1 3 = the seventh = 21 X = 7; the eighth 7 X = 1.

= X X X 35; the sixth X X

2

=

=

303. So that we have, for the second power, the two fractions 1 ; whence it follows, that the first coefficient = 1; the second == 2; and the third = 2 × 1 = 1.

The third power furnishes the fractions,,; wherefore the first coefficient = 1; the second == 3; the third=3x=3; the fourth 1.

We have for the fourth power, the fractions,,,; consequently the first coefficient = 1; the second = 4; the third X = 6; the fourth XX=4; and the fifth × × × 1 = 1.

309. This rule evidently renders it unnecessary for us to find the preceding coefficients, and enables us to discover immediately the coefficients which belong to any power. Thus, for the tenth power, we write the fractions,,, 1, 8, 6, 4, 3, &, ', by means of which we find

6 5

= 10,

=

10 X 45, = 45 X 120,

=

= 120 × 1 = 210,

the third

the fourth

the fifth

the sixth

310. We may also write these fractions as they are, without computing their value; and in this way it is easy to express any power of a+b, however high. Thus, the hundredth power of a + b, will be

whence the law of the succeeding terms may be easily deduced.

CHAPTER XI.

Of the Transposition of the Letters, on which the Demonstration of the preceding Rule is founded.

311. Ir we trace back the origin of the coefficients which we have been considering, we shall find, that each term is presented, as many times as it is possible to transpose the letters, of which that term consists; or, to express the same thing differently, the coefficient of each term is equal to the number of transpositions that the letters admit, of which that term is composed. In the second power, for example, the term ab is taken twice, that is to say, its coefficient is 2; and in fact we may change the order of the letters which compose that term twice, since we may write a b and ba; the term a a, on the contrary, is found only once, because the order of the letters can undergo no change or transposition. In the third power of a+b, the term a a b may be written in three different ways, a ab, aba, baa; thus the coefficient is 3. Likewise, in the fourth power, the term a3 b or a a ab, admits of four different arrangements, a aab, aaba, abaa, baaa; therefore its coefficient is 4. The term a abb admits of six transpositions, a abb, abba, baba, abab, bba a, b a ab, and its coefficient is 6. It is the same in all cases.

312. In fact, if we consider that the fourth power, for example, of any root consisting of more than two terms, as (a + b + c + d)a, is found by multiplying the four factors, I. a + b + c + d ; II. a+b+c+d; III. a + b + c + d ; IV. a+b+c+d; we may easily see, that each letter of the first factor must be multiplied by each letter of the second, then by each letter of the third, and lastly, by each letter of the fourth.

Each term must therefore not only be composed of four letters, but also present itself, or enter into the sum, as many times as those letters can be differently arranged with respect to each other, whence arises its coefficient.

313. It is therefore of great importance to know, in how many different ways a given number of letters may be arranged. And, in this inquiry, we must particularly consider, whether the letters in question are the same, or different. When they are the same, there can be no transposition of them, and for this reason the simple powers, as a2, α3, α, &c.,'all have unity for the coefficient.

314. Let us first suppose all the letters different; and beginning with the simplest case of two letters, or a b, we immediately discover that two transpositions may take place, namely, a b and b a.

If we have three letters a b c, to consider, we observe that each of the three may take the first place, while the two others will admit of two transpositions. For if a is the first letter, we have two arrangements, abc, a cb; if b is in the first place, we have the arrangements bac, bea; lastly, if c occupies the first place, we have also two arrangements, namely, ca b, c b a. And consequently the whole number of arrangements is 3 × 2 = 6.

If there are four letters, a b c d, each may occupy the first place; and in each case the three others may form six different arrangements, as we have just seen. The whole number of transpositions is therefore 4 X 6 = 24 = 4 × 3 × 2 x 1.

If there are five letters, a b c d e, each of the five must be the first, and the four others will admit of twenty-four transpositions; so that the whole number of transpositions will be

5 X 24 120 5 X 4 X 3 X 2 X 1.

315. Consequently, however great the number of letters may be, it is evident, provided they are all different, that we may easily determine the number of transpositions, and that we may make use of the following table":

Number of Letters.

I.

II.

III.

IV.

V.

VI.

VII.

VIII.

IX.

8 X 7 X 6 X 5 X 4 × 3 × 2 × 1 = 40320.

9 X 8 X 7 X 6 X 5 X 4 X 3 X 2 X 1 = 362880.

X. 10 X 9 X 8 X 7 X 6 X5 X 4 X3 X 2 X 1

3628800.

316. But, as we have intimated, the numbers in this table can be made use of only when all the letters are different; for if two or more of them are alike, the number of transpositions becomes much less; and if all the letters are the same, we have only one arrangement. We shall now see how the numbers in the table are to be diminished, according to the number of letters that are alike.