ya, we shall have, at the same time, y = a. Hence it follows, that this equation must be made up of terms involving only even powers of the unknown quantity; for its first member must be the product of a certain number of factors of the second degree of the form y2 — a2 = (y—a) (y +α) (184); it will therefore itself be exhibited under the form ́y2n +p y2n−2+ q y3n-1..... + ty2 + u = 0. y2n−4... If we put y2=%, this becomes ལ"+p2"-1+༡2"-2. +tz+u=0; and as the unknown quantity is the square of y, its values will be the squares of the differences between the roots of the pro-, posed equation. It may be observed, that as the differences between the real roots of the proposed equation are necessarily real, their squares will be positive, and consequently the equation in will have only positive roots, if the proposed equation admits of those only, which are real. Let there be, for example, the equation Suppressing the terms a3-7a+7, which, from their identity with the proposed equation, become nothing when united, and dividing the remainder by y, we have 3a2+3ay + y2 — 7 = 0; eliminating a by means of this equation and the equation 209. The substitution of a + y in the place of x in the equation xm−2 x+Pxm-1+Q x2-2... . . + U=0 (204), is sometimes resorted to also in order to make one of the terms of this cquation to disappear. We then arrange the result with reference to the powers of y, which takes the place of the unknown quantity x, and consider a as a second unknown quan tity, which is determined by putting equal to zero the coefficient of the term we wish to cancel; in this way we obtain ym + maym−1 + 1.2 m (m—1) + Pym−1+ (m—1) Pa ym-2.. + ym-2. ..... Q ym-2 If the term we would suppress be the second, or that which involves ym-1, we make ma + P=0, from which we deduce Substituting this value in the result, there remain P m only the terms involving y", ym-2, ym-3, &c. Hence it follows, that we make the second term of an equation to disappear, by substituting for the unknown quantity in this equation a new unknown quantity, united with the coefficient of the second term taken with the sign contrary to that originally belonging to it, and divided by the exponent of the first term. Let there be, for example, the equation x3+6x-3x+4=0; x=y—=y—2; we have by the rule substituting this value, the equation becomes y3 — 6y2+12y- 8 which may be reduced to 3y+6 0, +4 y315y+26= 0, in which the term involving y2 does not appear. We may cause the third term, or that involving ym-2, to disappear by putting equal to zero the sum of the quantities, by which it is multiplied, that is, by forming the equation Pursuing this method, we shall readily perceive, that the fourth term will be made to vanish by means of an equation of the third degree, and so on to the last, which can be made to disappear only by means of the equation ..... am+Pam-1+ Q am-2 perfectly similar to the equation proposed. + U=0, It is not difficult to discover the reason of this similarity. By making the last term of the equation in y equal to zero, we suppose, that one of the values of this unknown quantity is zero; and if we admit this supposition with respect to the equation xy+a, it follows that x= a; that is, the quantity a, in this case, is necessarily one of the values of x. 210. We have sometimes occasion to resolve equations into factors of the second and higher degrees. I cannot here explain in detail the several processes, which may be employed for this purpose; one example only will be given. Let there be the equation x3 24 x3 + 12 x2 11x+7=0, in which it is required to determine the factors of the third degree; I shall represent one of these factors by x3 +p x2 + qx+r, the coefficients p, q and r being indeterminate. They must be such, that the first member of the proposed equation will be exactly divisible by the factor x3 +p x2 + q x+r, independently of any particular value of x; but in making an actual division, we meet with a remainder (p3 2 p q - 24p+r—12) x2 an expression, which must be reduced to nothing independently of x, when we substitute for the letters P, q, and r, the values that answer to the conditions of the question. We have then p3-2pq-24p+r—12=0 p3 q- pr— q2-24 q+11=0 p2r-qr-24r-7=0. These three equations furnish us with the means of determining the unknown quantities p, q, and r; and it is to a resolution of these that the proposed question is reduced. Of the resolution of numerical equations by approximation. 211. HAVING Completed the investigation of commensurable divisors, we must have recourse to the methods of finding roots by approximation, which depend on the following principle ; When we arrive at two quantities which substituted in the place of the unknown quantity in an equation, lead to two results with contrary signs, we may infer, that one of the roots of the proposed equation lies between these two quantities, and is consequently real. Let there be, for example, the equation x3 — 13 x2 + 7 x -1=0; if we substitute successively 2 and 20 in the place of x, in the first member, instead of being reduced to zero, this member becomes, in the former case, equal to 31, and in the latter, to +2939; we may therefore conclude, that this equation has a real root between 2 and 20, that is, greater than two and less than 20. As there will be frequent occasion to express this relation, I shall employ the signs and which algebraists have adopted to denote the inequality of two magnitudes, placing the greater of two quantities opposite the opening of the lines, and the less against the point of meeting. Thus I shall write x2, to denote, that x is greater than 2, 20, to denote, that x is less than 20. Now in order to prove what has been laid down above, we may reason in the following manner. Bringing together the posi-. tive terms of the proposed equation, and also those, which are negative, we have x3+7x-(13x2+1), a quantity, which will be negative, if we suppose x = 2, because upon this supposition, x3 +7 x 13 x2 + 1, and which becomes positive, when we make x 20, because in this case x37x 13 x2 + 1. Moreover it is evident, that the quantities x+7x and 13x2 + 1, each increase, as greater and greater values are assigned to x, and that, by taking values, which approach each other very nearly, we may make the increments of the proposed quantities as small as we please. But since the first of the above quantities, which was originally less than the second, becomes greater, it is evident, that it increases more rapidly than the other, in consequence of which its deficiency is made up, and it y=a, we shall have, at the same time, y=a. Hence it follows, that this equation must be made up of terms involving only even powers of the unknown quantity; for its first member must be the product of a certain number of factors of the second degree of the form y2 — α2 = (y — α) (y+α) (184); (૫ a) it will therefore itself be exhibited under the form ̈ ́y2n +p y2n¬2+q y2n−4 If we put y2=%, this becomes z"+pz"¬1+q zπ-2, and as the unknown quantity is the square of y, its values will be the squares of the differences between the roots of the proposed equation. It may be observed, that as the differences between the real roots of the proposed equation are necessarily real, their squares will be positive, and consequently the equation in z will have only positive roots, if the proposed equation admits of those only, which are real. Let there be, for example, the equation Suppressing the terms a3-7a+7, which, from their identity with the proposed equation, become nothing when united, and dividing the remainder by y, we have eliminating a by means of this equation and the equation 209. The substitution of a + y in the place of x in the equation 2m+Pxm-1+Q xm−2. . . . . + U=0 (204), is sometimes resorted to also in order to make one of the terms of this equation to disappear. We then arrange the result with reference to the powers of y, which takes the place of the unknown quantity x, and consider a as a second unknown quan |