REMARK. This problem shows how much the introduction of an unknown auxiliary often facilitates the determination of the principal unknown quantities. There are other problems of the same kind, which lead to equations of a degree supe rior to the second, and yet they may be resolved by the aid of equations of the first and second degrees, by introducing unknown auxiliaries.

3. Given the sum of two numbers equal to a, and the sum of their cubes equal to c, to find the numbers

By the conditions

x + y = a

x3 + y3 = c.

and

Putting xs + 2, and y = s — 2, we have

Jx3 = s3 + 3s2z + 3sz2 + 23

a = 28,

4. The sum of the squares of two numbers is expressed by a, and the difference of their squares by b: what are the numbers ?

Ans.

5. What three numbers are they, which, multiplied two and two, and each product divided by the third number, give the quotients, a, b, c?

Ans. √ab, ac, √bc.

256

5

6. The sum of two numbers is 8, and the sum of their cubes is 152: what are the numbers ? Ans. 3 and 5.

7. Find two numbers, whose difference added to the difference of their squares is 150, and whose sum added to the sum of their squares, is 330. Ans. 9 and 15.

8. There are two numbers whose difference is 15, and half their product is equal to the cube of the lesser number: what are the numbers ? Ans. 3 and 18.

9. What two numbers are those whose sum multiplied by the greater, is equal to 77; and whose difference, multiplied by the lesser, is equal to 12?

11. It is required to divide the number 24 into two such parts, that their product may be equal to 35 times their differAns. 10 and 14.

ence.

12. What two numbers are they, whose product is 255, and the sum of whose squares is 514? Ans. 15 and 17.

13. There is a number expressed by two digits, which, when divided by the sum of the digits, gives a quotient greater by 2 than the first digit; but if the digits be inverted, and the resulting number be divided by a number greater by 1 than the sum of the digits, the quotient will exceed the former quotient by 2: what is the number? Ans. 24.

14. A regiment, in garrison, consisting of a certain number of companies, receives orders to send 216 men on duty, each com pany to furnish an equal number. Before the order was executed, three of the companies were sent on another service, and it was then found that each company that remained would have to send 12 men additional, in order to make up the complement, 216. How many companies were in the regiment, and what number of men did each of the remaining companies send?

Ans. 9 companies: each that remained sent 36 men.

15. Find three numbers such, that their sum shall be 14, the sum of their squares equal to 84, and the product of the first and third equal to the square of the second.

Ans. 2, 4 and 8.

16. It is required to find a number, expressed by three digits, such, that the sum of the squares of the digits shall be 104; the square of the middle digit to exceed twice the product of the other two by 4; and if 594 be subtracted from the number, the remainder will be expressed by the same figures, but with the extreme digits reversed. Ans. 862.

24.

17. A person has three kinds of goods which togetner cost $2305 A pound of each article costs as many dollars as there are pounds in that article: he has one-third more of the second than of the first, and 31 times as much of the third as of the second: How many pounds has he of each article?

Ans. 15 of the 1st, 20 of the 2d, 70 of the 3d.

18. Two merchants each sold the same kind of stuff: the second sold 3 yards more of it than the first, and together, they received 35 dollars. The first said to the second, "I would have received 24 dollars for your stuff.” The other replied, "And I would have received 12 dollars for yours." How many yards did each of them sell?

19. A widow possessed 13000 dollars, which she divided into two parts, and placed them at interest, in such a manner, that the incomes from them were equal. If she had put out the first portion at the same rate as the second, she would have drawn for this part 360 dollars interest; and if she had placed the second out at the same rate as the first, she would have drawn for it 490 dollars interest. What were the two rates of interest? Ans. 7 and 6 per cent.

CHAPTER VII.

FORMATION OF POWERS-BINOMIAL THEOREM-EXTRACTION OF ROOTS OF ANY DEGREE-OF RADICALS.

In

128. THE solution of equations of the second degree supposes the process for extracting the square root to be known. like manner, the solution of equations of the third, fourth, &c., degrees, requires that we should know how to extract the third, fourth, &c., roots of any numerical or algebraic quantity.

The power of a number can be obtained by the rules for inultiplication, and this power is subject to a certain law of formation, which it is necessary to know, in order to deduce the root from the power.

Now, the law of formation of the square of a numerical or algebraic quantity, is deduced from the expression for the square of a binomial (Art. 47); so likewise, the law of a power of any degree, is deduced from the expression for the same power of a binomial. We shall therefore first determine the law for the formation of any power of a binomial.

129. By taking the binomial

+ a several times, as a factor,

the following results are obtained, by the rule for multiplication:

(x + a) = x + α,

(x+a)2 = x2 + 2ax + a2,

(x + a)3 = x3 + 3ax2 + 3a2x + a3,

(2 + α)1 = x1 + 4ax3 + 6a2x2 + 4a3x + aa,

(x + a)5 = x5 + 5axa + 10a2x3 + 10a3x2 + 5aax + a3.

By examining these powers of x+a, we readily discover the law according to which the exponents of the powers of a de

crease, and those of the powers of a increase, in the successive terms. It is not, however, so easy to discover a law for the formation of the co-efficients. Newton discovered one, by means of which a binomial may be raised to any power, without performing the multiplications. He did not, however, explain the course of reasoning which led him to the discovery; but the law has since been demonstrated in a rigorous manner. Of all the known demonstrations of it, the most elementary is that which is founded upon the theory of combinations. However, as the demonstration is rather complicated, we will, in order to simplify it, begin by demonstrating some propositions relative to permutations and combinations, on which the demonstration of the binomial theorem depends.

Of Permutations, Arrangements and Combinations.

130. Let it be proposed to determine the whole number of ways in which several letters, a, b, c, d, &c., can be written, one after the other. The result corresponding to each change in the position of any one of these letters, is called a permutation.

Thus, the two letters a and b furnish the two permutations, ab and ba.

cab

ach

In like manner, the three letters, a, b, C, furnish six permutations.

abc

cba

bca

bac

PERMUTATIONS, are the results obtained by writing a certain number of letters one after the other, in every possible order, in such a manner that all the letters shall enter into each result, and each letter enter but once.

To determine the number of permutations of which n letters are susceptible.

Two letters, a and b, evidently give two permutations.

ab

ba