Page images
PDF
EPUB

other. The numerator may, therefore, be found from the denominator, as well in cases where there are two unknown quantities, as when there is only one, by changing the coefficient of the unknown quantity sought, into the known term or second member, and retaining the accents, which belonged to the coefficients.

The same rule may be applied to equations with three unknown quantities, as we shall see by merely inspecting the values, which result from these equations. With respect to the denominator, it is necessary further to illustrate the method by which it is formed. Now, since in the case of two unknown quantities, the denominator presents all the possible transpositions of the letters a and b, by which the unknown quantities are multiplied, it may be supposed, that when there are three unknown quantities, their denominator will contain all the arrangements of the three letters a, b, c. These arrangements may be formed in the following manner.

We first make the transpositions abba with the two letters a and b, then, after the first term a b, write the third letter c, which gives a bc; making this letter pass through all the places, observing each time to change the sign, and not to derange the order in which a and b respectively stand, we obtain

a b c — a cb + cab.

Proceeding in the same manner with respect to the second term - ba, we find

[blocks in formation]

connecting these products with the preceding, and placing over the second letter one accent, and over the third two, we have

[ocr errors]

- cb'a",

a b c — ac' b + c a' b" -ba'c' + bc' a" a result, which agrees with that presented by the formulas, obtained above.

From this it is obvious, that, in order to form a denominator in the case of four unknown quantities, it is necessary to introduce the letter d into each of the six products,

abc-acbcab-bac+bca-cba,

and to make it occupy successively all the places. The term a b c, for example, will give the four following;

abcd — abdc adb c – d a b c
c + c

If we observe the same method in regard to the five other products, the whole result will be twenty-four terms, in each of

[blocks in formation]

which, the second letter will have one accent, the third two, and the fourth three. The numerators of the unknown quantities u, z, y, and x, are found by the rule already given.*

89. We may employ these formulas for the resolution of numerical equations. In doing this, we must compare the terms of the equations proposed with the corresponding terms of the general equations, given in the preceding articles.

To resolve, for example, the three equations

7x+5y + 2 z = 79,

8 x + 7y+9 z = 122,

x + 4y + 5 z = 55,

compare the terms with those of the equations We have then

it is necessary to

given in art. 86.

a

= 7, b

a' = 8, b'

= 5, C

= 2, d =

79,

= 7, c′ = 9, d' = 122,

a" = 1, b" = 4, c' = 5, d"= 55.

Substituting these values in the general expressions for the unknown quantities x, y, and z, and going through the operations, which are indicated, we find

[blocks in formation]

It is important to remark, that the same expressions may be employed, even when the proposed equations are not, in all their terms, affected with the sign+, as the general equations, from which these expressions are deduced appear to require. If we have, for example,

[blocks in formation]

in comparing the terms of these equations with the corresponding ones in the general equations, we must attend to the signs, and the result will be

[merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small]

We are then to determine by the rules given in art. 31, the sign,

* M. Laplace, in the second part of the Mémoires de l'Académie des Sciences for 1772, p. 294, has demonstrated these rules à priori. See also Annales des Mathématiques pures appliquées, by M. Gergonne, vol. iv, p. 148.

which each term of the general expressions for x, y, and z, ought to have, according to the signs of the factors of which it is composed. Thus we find, for example, that the first term of the common denominator, which is a b'c", becoming + 3 × + 4 × — 6, changes the sign of the product, and gives-72. If we observe the same method with respect to the other terms, both of the numerators and denominators, taking the sum of those, which are positive, and also of those which are negative, we obtain

[blocks in formation]

Equations of the Second Degree, having only one unknown Quantity.

90. HITHERTO I have been employed upon equations of the first degree, or such as involve only the first power of the unknown quantities; but were the question proposed, To find a number, which, multiplied by five times itself, will give a product equal to 125; if we designate this number by x, five times the same will be 5 x, and we shall have

5x2125.

This is an equation of the second degree, because it contains x2, or the second power of the unknown quantity. If we free this second power from its coefficient 5, we obtain

[blocks in formation]

We cannot here obtain the value of the unknown quantity x, as in art. 11, and the question amounts simply to this, to find a number which, multiplied by itself, will give 25. It is obvious that this number is 5; but it seldom happens that the solution is so easy; hence arises this new numerical question; to find a number, which, multiplied by itself, will give a product equal to a proposed number; or, which is the same thing, from the second power of a number, to retrace our steps to the number from which it is derived, and which is called the square root. I shall proceed, in the first place, to resolve this question, as it is involved in the determination of the unknown quantities, in all tions of the second degree.

equa

91. The method employed in finding or extracting the roots of numbers, supposes the second power of such, as are expressed by only one figure to be known. See the nine primitive numbers with their second powers written under them respectively. 2 3 4 5 6 7 8 9

1

1

4 9 16 25 36 49 64 81.

It is evident from this table, that the second power of a number expressed by one figure, contains only two figures; 10, which is the least number expressed by two figures, has for its square a number composed of three, 100. In order to resolve the second power of a number consisting of two figures, we must attend to the method by which it is formed; for this purpose we must inquire, how each part of the number 47, for example, is employed in the production of the square of this number.

We may resolve 47 into 40 + 7, or into 4 tens and 7 units; if we represent the tens of the proposed number by a, and the units by b, the second power will be expressed by

(a + b) (a + b) = a2 + 2 a b + b2;

that is, it is made up of three parts, namely, the square of the tens, twice the product of the tens multiplied by the units, and the square of the units. In the example we have taken, a = 4 tens or 40 units, and b = 7; we have then

a2 = 1600 2ab= 560

[ocr errors][merged small]

Total, a2 + 2 a b + b2 = 2209 47 X 47.

Now in order to return, by a reverse pro ess, from the number 2209 to its root, we may observe, that the square of the tens, 1600, has no figure, which denotes a rank inferior to hundreds, and that it is the greatest square, which the 22 hundreds, comprehended in 2209, contain; for 22 lies between 16 and 25, that is, between the square of 4 and that of 5, as 47 falls between 4 tens or 40, and 5 tens or 50.

We find, therefore, upon examination, that the greatest square contained in 22 is 16, the root of which 4 expresses the number of tens in the root of 2209; subtracting 16 hundreds, or 1600 from 2209, the remainder 609 contains double the product of the tens by the units, 560, and the square of the units 49. But as double the product of the tens by the units has no figure inferior

to tens, it must be found in the two first figures 60 of the remainder 609, which contain also the tens, arising from the square of the units. Now, if we divide 60 by double of the tens 8, and neglect the remainder, we have a quotient 7 equal to the units sought. If we multiply 8 by 7, we have double the product of the tens by the units, 560; subtracting this from the whole remainder 609, we obtain a difference 49, which must be, and in fact is, the square of the units.

This process may be exhibited thus ;

[blocks in formation]

We write the proposed number in the manner of a dividend, and assign for the root the usual place of the divisor. We then separate the units and tens by a comma, and employ only the two first figures on the left, which contain the square of the tens found in the root. We seek the greatest square 16, contained in these two figures, put the root 4 in its assigned place, and subtract 16 from 22. To the remainder we bring down the two other figures, 09, of the proposed number, separating the last, which does not enter into double the product of the tens by the units, and divide the remainder on the left by 8, double the tens in the root, which gives for the quotient the units 7. In order to collect into one expression the two last parts of the square contained in 609, we write 7 by the side of 8, which gives 87, equal to double the tens plus the units, or 2 a+b; this multiplied by 7 or b, reproduces 6092ab+b2, or double the product of the tens by the units, plus the square of the units. This being subtracted leaves no remainder, and the operation shows, that 47 is the square root of 2209.

If it were required to extract the square root of 324; the operation would be as follows;

3,24 18

1

22,4 | 28
224

000

« PreviousContinue »