Substituting these values in the expression for z, and reducing all the terms to the same denominator, we have, (a) d (b' a" ―a'b'') + d' (a b" — b a")—d" (a b' — ba') c (b'a'' — a'b'') + c' (a b” — ba′′) — c'' (a b′ — ba') If we had made the terms containing x and z to disappear, we should have had y; the letters m and n would have depended upon the equations am+a'n=a" cm + cn=c", and proceeding as before, we should have found d (c'a" a' c') + d' (a c"-ca")-d" (a c'-ca') y = b (c' a'' — a' c") + b′ (a c" — c a") — b′′ (a c' — c a') * Lastly, by assuming the equations we make the terms multiplied by y and z to disappear; and we have d (c'b'b'c') + d' (b c" — c b')—d" (bc'-cb') x= a (c'v'' — b' c'') + a' (bc" — c b'') — a" (bc' — c b')' These values being developed in such a manner, as to make the terms alternately positive and negative, if we change, at the same time, the signs of the numerator and denominator, in the first and third, we shall give them the following forms; a b' d' a d'b'+da' b' ལ = y= x = a b' c" -a c'b" +ca'b' a b' c' d b' c" -a c'b" + c a' b" - d c′ b'' + c d′ b′′ ba' d' + b d' a" +bc' a" da' c"+dc' a" ba'c' + bc' a" - b d'c' + b c' d" a c'b" + ca'b' —b a' c' + b c' a" 87. Let there be the four equations a x+by+c z+du=e a b' - ba' ba' d" a b'. ba' a b' ba a b if we multiply the first by m, the second by n, the third by p, and from the sum of their products subtract the fourth, we shall *have The preceding equations, which must give m, n, and p, may be resolved by means of the formulas found for the case of three unknown quantities. This method will appear very simple and convenient; but the nature of the results obtained above will furnish us with a rule for finding them without any calculation. 88. To begin with the most simple case, we take an equation with one unknown quantity, a x = b; from this we find x= b a in which the numerator is the whole known term b, and the denominator the coefficient a of the unknown quantity. The denominator in this case is composed also of the letters a, a', b, b', by which the unknown quantities are multiplied. We first write a by the side of b, which gives a b; we then change the order of a and b, and obtain ba; prefixing to this the sign we have a b ba; lastly we place an accent over the last letter in each term, and the expression becomes a b'-ba' for the denominator. From this expression we may find the numerator. To obtain that for x, we have only to change each a into c, and each b into c for that of y, putting an accent over the last letter as before; in this way we find c b' - - bd for the one, and a c-c a' for the 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 tranpositions 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 ab-ba with the two letters a and b, then after the first term a b, write the third letter c, which gives a b c; 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 abc-acb+cab. Proceeding in the same manner with respect to the second term -ba, we find -bac+bea-cba; connecting these products with the preceding, and placing over the second letter one accent, and over the third two, we have ab'c"-acb" +ca'b"-ba'c"+bc' a"-cb' a", a result, which agrecs 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-acb+cab-bac+bca-cba, and to make it occupy successively all the places. The term abc, for example, will give the four following; abcd-abde+adbc-dabc. If we observe the same method in regard to the five other products, the whole result will be twenty four terms, in each of which, the second letter will have one accent, the third two, and the fourth three. The numerators of the unknown quantities u, , 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%= 79 x+4y+5x= 55, compare the terms with those of the equaWe have then = 5, c = 2, d = 79 it is necessary to tions given in art. 86. 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 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, 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 a 20 + 3, b =➡ 9, c = + 8, d = + 41 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 de Mathématiques pures appliquées, by M. Gergonne, vol. iv. p. 148. which each term of the general expressions, y and z ought to have, according to the signs of the factors which it is composed. Thus we find, for example, that the first term of the common denominator, which is a b' c", becoming + 3x+4x-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 Equations of the second degree, having only one unknown quantity. 2 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 We cannot here obtain the value of the unknown quantity a 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 equations of the second degree. 920486A |