with R2 being a minimum value is, that R = 0; this therefore is the minimum value; whence, by means of (8), and therefore there always are some values of y and z such that equation (15) is satisfied; and therefore every equation of the form (1) has a root. z If in the general form of the root = 0, the root is a possible one; but if ≈ has a finite value, the root is impossible. 147.] Taking a to be the general symbol for the root of an equation, we may thus prove that the equation is divisible by X- a without a remainder. Let f(x) = 0 be the equation; then, since a is a root, ƒ(a) = 0. Observing now that Taylor's Series does not fail for a function of x, such as we have assumed f(x) to be, and that the (n+1)th derived-function vanishes, because f(x) is algebraical and of n dimensions, by means of equation (14) Art. 119, we have the second member of which equation is, as f(a) = 0, divisible by (x-a). Hence also conversely, if x — a is a factor of f(x), f(a) = 0, or a is a root of the equation. Hence we conclude that if any function of the algebraical form assumed in equation (1), Art. 145, vanishes for a particular value a of the variable, the function has a factor of the form xa, and it is owing to its vanishing that the function vanishes; compare Art. 101. 148.] Hence also it follows, that every equation has as many roots as it has dimensions, and no more. For dividing f(x) in equation (16) by (x − a), the highest power of a that remains is -1, being that which is involved in the last term of it, viz. in (x-a)", whence there results an expression of n-1 dimensions; which again, by virtue of Art. 146, has a root, and therefore is again divisible by a factor of the form x-a; whereby the expression is depressed to one of n-2 dimensions; and if a similar process be continued for n-1 times, we shall finally have an expression of one dimension, which will give the last root, and thereby the equation will have been resolved into n factors. Thus suppose the n roots to be a1, a2 an, then f(x) = (x−α1) (x —α2) .............. (x—ɑ„). ...... (17) Also it is manifest that no other value than one of the n roots can, when substituted for x, make any simple factor, and thereby the whole expression, to vanish; and therefore ƒ(x) has only n roots, some of which however may be equal; and therefore although all the n roots may not be different, yet there can never be fewer than n simple factors. Again, if the coefficients of the several powers of x in f(x) are real, and f(x) has impossible roots, they must enter in pairs: so that, if a; and a, are two impossible roots which are conjugate to each other, .. a; = a + ẞ √ −1 = p {cose + √1 sin 0}, a; = a x2−2 (x−a¡) (x−aj) = (x− a)2 + ß2 = x2-2 px cose + p2, which quadratic expression is essentially positive; and by a similar composition of other conjugate factors we have f(x) = (x —α1) (x — α2) ... {(x—α1)2 + B12} {(x−a2)2 + ß22}..... (18) and therefore f(x) is the product of factors, simple or quadratic*. 149.] On the algebraical relation of f(x) to its derivedfunction. Let f(x) be a function of the form (1), Art. 145, which has * A further inquiry into the possibility and nature of the roots of equations would be out of place in the present Treatise, both because it does not so directly require Infinitesimal Calculus as to be cited in illustration of it, and because it is not elementary enough for our purpose. I would however recommend the advanced student to read "Cours d'Algèbre Supérieure,” par J. A. Serret, Paris, 1849; and "Ouvres d' Abel, Christiania, 1839." K k all its roots real, and therefore all its coefficients real quantities; and let the roots of f(x) be a1, a2 . ...... ...... an, so that ±Pn-1 x + Pn, (19) (20) = (x —α1) (x−α2) ...... (x-an), the upper or lower sign being taken in (19) according as n is odd or even, and the roots being arranged in order of magnitude, viz.: a1 > az az > ...... > ani :. f'(x) = nx1−1 — (n−1) P1 xn−2 + (21) + Pn-1, +(x−α1) (x-A2) ...... (x-an-1). (22) Let x = a1, then observing that all the parts, except the first, of the second member of (22) disappear, and that by virtue of the arrangement of the roots, as indicated by (21), every factor of the first part is positive, it follows that, if x = a1, f'(x) is positive; similarly, if x = a2, f'(x) is negative. There is therefore (Cor. II, Art. 95) some value of x between a1 and a2 which makes f'(x) vanish; a root therefore of f'(x) lies between a and ag. Similarly we have the following results: Let the roots of f'(x) be a1, a2...... an-1, arranged in order of descending magnitude, then they stand to the roots of f(x) in the relation indicated in the following table: whence it appears, that the greatest root of f(x) is greater than the greatest root of f'(x), and the least root of f(x) is less than the least root of f'(x). It is on account of this particular relation of the roots of f'(x) to the roots of f(x) that ƒ'(x) is sometimes called the limiting equation of f(x). Hence also it follows, that if all the roots of an equation be real, all the roots of each of its successive derived-functions will be real also. These results admit of the following geometrical interpretation: Let the curve represented in fig. 23 be that whose equation is y = f(x). As f(x) has n real roots, f(x) and therefore y = 0 at n points corresponding to them; that is, if OA1 = ai, OA2 = α2, the curve cuts the axis of x at A1, A2 .... ................, that is, in n points. As f(x) only∞ when x = ∞, the ordinate is finite for all values of a between a1 and a2, a2 and as and by the last Chapter as f'(x) = 0, and changes sign when f(x) is a maximum or minimum corresponding to these roots of f'(x), and therefore we have maxima or minima ordinates at points intermediate to A1 and A2, A2 and An-1 and An; that is, OB1 = α1, OB2 = a2 x = α1, A2, = ...... ....... ...... .....; A3, Also as n is odd or even will the curve towards the left, when be below or above the axis of x. x= 150.] Hence it appears that, if f(x) has m roots equal to each other, f'(x) has (m—1) roots equal to each of the equal roots of f(x); for if a1 = a2 = = am, then a1 = ɑ2 .. ...... ...... = am-1, as the as are intermediate to the as; which proposition is also thus manifest. Let m roots of f(x) be equal to one another and to a, and let Qr symbolize the product of the other n-m roots, then and f'(x) has m-1 roots equal to each of the m equal roots of f(x). Hence if f(x) has equal roots, they may be determined by the method of finding the greatest common measure of f(x) and f'(x), and f(x) may be depressed by as many dimensions as it has equal roots. The latter proof of this proposition manifestly reaches the case of equal impossible roots which the former may not resolve. 151.] Given an equation f(x) = 0, which has real coefficients; it is required to determine the number of real roots which it contains, and the limits of them. The following process, due to M. Sturm, and now generally known by the name of "Sturm's Theorem," theoretically completes the subject of synthetical expressions, and is one of the greatest modern discoveries in Algebraical Analysis. First let us consider f(x) to have been cleared of equal factors by means of the last Article, so that f(x) has not roots equal to each other; then we may enuntiate the following Theorem: THEOREM.--Let f(x) be a function of x of real coefficients, of which f'(x) is the derived-function; let f(x) be divided by ƒ′(x) in the way of finding the greatest common measure, but with the peculiarity of the sign of a remainder always being changed before it be made a divisor, and let this process of division be continued until it terminates by giving a remainder independent of x and not vanishing; which is always the case when f(x) has no equal factors; and let these successive remainders thus modified be symbolized by so that we have the following system of equations: fn-1(x) being the last factor and independent of x; and not vanishing because by hypothesis f(x) has no equal roots. Let a and ẞ be two numbers of which a is the less (regard being had to its sign); substitute a for x in the series f(x), f'(x), fi(x)................fn-1(x), (24) and write down in the same order the signs of the results; and count the number of sequences of two terms having contrary signs in this series of results; and suppose that A is that number. Substitute in the same series of functions, and count |