54. When the expression for a maximum or minimum contains several variable letters or quantities; take the fluxion of it as often as there are variable letters; supposing first one of them only to flow, and the rest to be constant; then another only to flow, an the rest constant; and so on for all of them: then putting each of these fluxions = 0, there will be as many equations as unknown letters, from which these may be all determined. For the fluxion of the expression must be equal to nothing in each of these cases; otherwise the expression might become greater or less, without altering the values of the other letters, which are considered as constant. So, if it be required to find the values of x and y when 4x? – xy + 2y is a minimum. Then'we have, First, 8xi - žy = 0, and 8.x - y = 0, or y = 8.x. Secondly, 2j - xj 0, and 2 - *= 0, or x = 2. And hence yor 8x = 16. 55. To find whether a proposed quantity admits of a Maximum or a Minimum. Every algebraic expression does not admit of a maximum or minimum, properly so called; for it may either increase continually to infinity, or decrease continually to nothing; and in both these cases there is neither a proper maximum nor minimum ; for the true maximum is that finite value to which an expression increases, and after which it decreases again: and the minimum is that finite value to which the expression decreases and after that it increases again. Therefore, when the expression admits of a maximum, its fluxion is positive before the point, and negative after it; but when it admits of a minimum, its fluxion is negative before, and positive after it. Hence then, taking the fluxion of the expression a little before the fluxion is equal to nothing, and again a little after the same; if the former fluxion be positive, and the latter negative, the middle state is a maximum; but if the former fluxion be negative, and the latter positive, the middle state is minimum. So, if we would find the quantity ax q? a maximum or minimum; make its fluxion equal to nothing, that is, ax - 2ri = 0, or (a - 2.x).; = 0; dividing by , gives 2x 0, or x = a at that state. Now, if in the fluxioa (a - 2.x)x, the value of x be taken rather less than its true value, a, that fluxion will evidently be positive; but if x be taken somewhat greater than a the value of a 2.r, and consequently of the fluxion, is as evidently negative. Therefore, the fluxion of ar - x being positive before, and ve. A gative after the state when its flusion is = 0, it follows that at this state the expression is not a minimum, but a maximum. Again, taking the expression 23 — ax?, its fluxion 3x 2arx=(3x – 2a)rx=0; this divided by xx gives 3.x – 2a=0, and .r = 'a, its true value when the fluxion of x3 aris equal to nothing. But now to know whether the given expression be a maximum or a minimum at that time, take r a little less than a in the value of the fluxion (3.x – 2a) xx, and this will evidently be negative; and again, taking x a little more than şa, the value of 3.x 2a, or of the fluxion, is as evidently positive. Therefore the fluxion of 23 - axa being negative before that fluxion is = 0, and positive after it, it follows that in this state the quantity x3 ax? admits of a minimum, but not of a maximum. 56. SOME EXAMPLES FOR PRACTICE. Exam. 1. To divide a line, or any other given quantity a, into two parts, so that their rectangle or product may be the greatest possible. Exam. 2. To divide the given quantity a into two parts such, that the product of the m power of one, by the na power of the other, may be a maximum. Exam. 3. To divide the given quantity a into three parts such, that the continual product of them all may be a maximum. EXAM, 4. To divide the given quantity a into three parts such, that the continual product of the 1st, the square of the 2d, and the cube of the 3d, may be a maximum. Exam. 5. To determine a fraction such, that the difference between its m power and n power shall be the greatest possible. Exam. 6. To divide the number 80 into two such parts, x and y, that 2.2 + xy + 3y2 may be a minimum. Exam. 7. To find the greatest rectangle that can be in, scribed in a given right-angled triangle. Exam. 8. To find the greatest rectangle that can be inscribed in the quadrant of a given circle. EXAM. 9. To find the least right-angled triangle that can circumscribe the quadrant of a given circle. Exam. 10. To find the greatest rectangle inscribed in, and the least isosceles triangle circumscribed about, a given semiellipse. Exam, 11, Exam. 11. To determine the same for a given parabola. Exam. 12. To determine the same for a given hyperbola. Exam. 13. To inscribe the greatest cylinder in a given cone; or to cut the greatest cylinder out of a given cone. Exam. 14. To determine the dimensions of a rectangular cistern, capable of containing a given quantity a of water, so as to be lined with lead at the least possible expense. Exam. 15. Required the dimensions of a cylindrical tankard, to hold one quart of ale measure, that can be made of the least possible quantity of silver, of a given thickness. Exam. 16. To cut the greatest parabola from a given cone. Exam. 17. To cut the greatest ellipse from a given cone. Exam. 18. To find the value of r when x' is a mininum. THE METHOD OF TANGENTS; OR, TO DRAW TAN GENTS TO CURVES, 57. The Method of Tangents, is a method of determining the quantity of the tangent and subtangent of any algebraic curve; the equation of the curve being given. Or, vice versa, the nature of the curve, from the tangent given. If ae be any curve, and E be any point in it, to which it is required to draw a tangent TE. Draw the ordinate Ed: then if we can determine the subtangent TD, limited be A* DAT tween the ordinate and tangent, in the axis produced, by joining the points T, E, the line Te will be the tangent sought. 58. Let dae be another ordinate, indefinitely near to DE, meeting the curve, or tangent produced in e; and let Ea be parallel to the axis AD. Then is the elementary triangle Ees similar to the triangle TDE; and therefore therefore ca : aE :: ED : DT. ea : aE :: flux. ED: flux. AD. yos That is, j :: :: y: :y =DT; y which is therefore the general value of the subtangent sought; where x is the absciss AD, and y the ordinate DE. Hence we have this general rule. GENERAL RULE. 59. By means of the given equation of the curve, when put into fluxions, find the value of either * or j, or of gas which value substitute for it in the expression DT = ģ and, when reduced to its simplest terms, it will be the value of the subtangent sought. EXAMPLES. Exam. 1. Let the proposed curve be that which is defined, or expressed, by the equation ax + xy2 — 93 = 0. Here the fluxion of the equation of the curve is 2ax: + y2 + + 2xyj – 3yž j = 0; then, by transposition, 2ax: + y** = 3y’j — 2xyj; and hence, by division, a. – 3y3 -- 2xy ; у 2ax + y j 2ax + 92 which is the value of the subtangent to sought. EXAM. 2. To draw a tangent to a circle;, the equation of which is ax x2 = y2; where x is the absciss, y the ordinate, and a the diameter. Exam. 3. To draw a tangent to a parabola ; its equation being ax = y*; where a denotes the parameter of the axis. EXAM. 4. To draw a tangent to an ellipse; its equation being c4 (ax — xo) = a’yo; where a and c are the two axeś. ExAM. 5. To draw a tangent to an hyperbola; its equation being c (ax + **) = a*y*; where a and c are the two axes. Exam. 6. To draw a tangent to the hyperbola referred to the asymptote as an axis ; its equation being wy = a*; where a’ denotes the rectangle of the absciss and ordinate answering to the vertex of the curve. OF OF RECTIFICATIONS; OR, TO FIND THE LENGTHS OF CURVE LINES. z 60. RECTIFICATION, is the finding the length of a curve line, or finding a right line equal to a proposed curve. By art. 10 it appears, that the elementary triangle Eae, formed by the increments of the absciss, ordinate, and curve, is a right-angled triangle, of which the increment of the curve is the hypothenuse; and therefore the square of the latter is equal to the sum of the squares of the two former; that is, Eé = Ba2 + aé. Or, substituțing, for the increments, their proportional fluxions, it is żż = is + ýý, or ż = š? + j?; where z denotes any curve line Aė, x its absciss AD, and y its ordinate de. Hence this rule. RULE. 61. From the given equation of the curve put into fluxions, find the yalue of $? or j, which value substitute instead of it in the equation ż=V x2 + ją; then the fluents, being taken, will give the value of z, or the length of the curve, in terms of the absciss or ordinate. EXAMPLES. Exam. I. To find the length of the arc of a circle, in terms both of the sine, versed sine, tangent, and secant. The equation of the circle may be expressed in terms of the radius, and either the sine, or the versed sine, or tangent, or secant, &c, of an arc. Let therefore the radius of the circle be ca or ce = r, the versed sine AD (of the arc af.)= x, the right sine DĖ = y, the tangent te =t, and the secant CT =s; then, by the nature of the circle, there arise these equations, viz. g2 42 s2 - 2 yz = 2rx gol. goh +ť s2 Then, by means of the fluxions of these equations, with the general fluxional equation z = ? + j?, are obtained the following fluxional forms, for the fluxion of the curve; the fuent of any one of which will be the curve itself; viz. r* rý gai 2rx god ✓ - 2 VOL. II. Y Hei |