ERRATA. On page 75, fourth line from bottom, insert the coefficient 2 before "sin" &c., in both numerator and denominator. On page 157, line 17, for "Put A, = x = [(x + 1) — 1]' = (x + 1)," read, Because a = [(x + 1) — 1] [(x + 1) — 1] = (x + 1)', &c. ; and on same page, line 19, for "Then" read Therefore. On page 158, lines 8 and 9 from bottom should read '4 = 127 [(2 + c)2 + (2 − 12π 2 + c ) 2 − ( 2 − c ) 3 ]. Additional formulæ in Finite Differences. A Demonstration of the Binomial Theorem for Negative Exponents. A New Theory of the Rule of Signs.... The Potential Function... Turrell, Isaac H.... Scheffer, Prot. J. The use of Imaginary Quantities in the Integral Calculus.. The Peaucellier Machine and other Linkages.. Trowbridge, Prof. David. 125 Zielinski, Prof. A.. 77 VOL. 11. JANUARY. 1875. ON THE MAXIMA AND MINIMA OF ALGEBRAIC POLYNOMIALS. No. 1. BY PROF. DAVID TROWBRIDGE, WATERBURGH, N. Y. 1. Let u= A1x” + Aqx”−1 + Az”—2 + ... + An-, = 4x . . . . . (1) The expression or is read "function of a". Suppose we wish to find a value of x such that if we substitute it in the polynomial (1), this polynomial, or u, will be a maximum, or the greatest possible; or a minimum, or the least possible, according to the conditions of the problem. To discover such values of x,-for there may be several maxima or minima of different orders, we shall proceed as follows: Suppose x to be increased or decreased by a small quantity h, so that we have, if we accent the u for this case, u' = 4(x+ h), or u' = Let each of these be developed according to the ascending powers of h, and let the result be denoted as follows: In these equations u = 4x, and if we transpose this term we shall have h2 u' — u = 4'x.h + "x.1.2+ h2 u' — u = — f'x.h +"'x.1.2-""'x. 1.2.3 1.2.3+ In these equations 'x, '', '"'x, &c., are finite quantities; and if we take h very small, 'x.h will be of the 1st order of small quantities, or infinitesimals, and 'x × (h2 ÷ 1.2), of the 2nd order, &c., so that "x× (h2÷1.2) is infinitely small as compared with 'x.h, and 4'"'x×(h3÷÷1.2.3) is infinitely small as compared with "xX(h2 ÷ 1.2), &c. (See Prof. Ficklin's Complete Algebra, p. 129.) Now if u, or its equal the polynomial, has its greatest value, or is a maximum, then u', which is found by increasing or diminishing x by a quantity h, no matter how small, is less than u; and if u is the least possible, or a minimum, then u' is greater than u. We hence see that the infinitesimal value of h, which we have supposed, is all that we need consider in this demonstration. We can easily see, then, that when h is thus small, the term 'x.h will be greater than all the other terms; for the number of terms in the second members of (5) and (6), is limited, so that their number cannot compensate for their smallness. In the case of a maximum u'-u is negative; and in the case of a minimum u'-u is positive. But because 'x.h is greater than all the other terms in the second members of (5) and (6), the sign of this term will determine the sign of the second member; and from (5) we see that when h is positive, supposing 'r positive, the second member is positive, and hence u'-u is positive for a maximum; and from (6) we see that u'— u is negative if h is negative; that is, the second member, or u'-u changes sign with h. But if u is a maximum, whether x be increased or diminished, u'- u is negative. To satisfy all these conditions we must make 'x=0. For a minimum we must have in all cases, u'-u a positive quantity, and in order to secure this, we must also have 'x=0. The term "x(h2÷1.2), which is greater than all the remaining terms, has the same sign whether h be plus or minus. For a maximum we see further, then, that ''x must be negative; and for a minimum, "x must be positive. Our conditions, then, are, for a maximum, for a minimum. 'x0, and "a negative, and Now 'x is called the first derived polynomial, or function, and 'x, the second derived polynomial, &c. (See Ficklin's Com. Algebra, p. 400, where he uses fa instead of x.) To find what value of x will render u a maximum, put the first derived polynomial equal to 0, and find the value of x from the resulting equation. Substitute these values, in succession, in the second derived polynomial, and all the values that give a negative result indicate so many maxima; and all that give a positive result, indicate so many minima. If any of the values of a render "a = 0, then, in order that such values of x may produce maximum or minimum values of u, we must also have "x = 0, for 4'"'xX(h3÷1.2.3) changes sign with h; and to determine whether we have a maximum or a minimum, we must substitute the values of x in "'"'x, which will be negative for a maximum and positive |