51. Now, because the fluxion of a variable quantity, is the rate of its increase or decrease; and because the maximum or minimum of a quantity neither increases nor decreases, at those points or states; therefore such maximum or minimum has no fluxion, or the fluxion is then equal to nothing. From which we have the following rule. To find the Maximum or Minimum. 52. From the nature of the question or problem, find an algebraical expression for the value, or general state, of the quantity whose maximum or minimum is required; thentake the fluxion of that expression, and put it equal to nothing; from which equation, by dividing by, or leaving out, the fluxional letter and other common quantities, and performing other proper reductions, as in 'common algebra, the value of the unknown quantity will be obtained, determining the point of the maximum or minimum. So, if it be required to find the maximum state of the compound expression 100r5x2c, or the value of when 100r 52c is a maximum. The fluxion of this expression is 100% 10xx = 0; which being made = 0, and divided by 10%, the equation is 10 x=0; and hence 10. That is, the value of x is 10, when the expression 100x - 5.x2±c is the greatest. As is easily tried: for if 10 be substituted for x in that expression, it becomes ±c+500: but if, for æ, there be substituted any other number, whether greater or less than 10, that expression will always be found to be less than 500, which is therefore its greatest possible value, or its maximum. 53. It is evident, that if a maximum or minimum be any way compounded with, or operated on, by a given constant quantity, the result will still be a maximum or minimum. That is, if a maximum or minimum be increased, or decreased, or multiplied, or divided, by a given quantity, or any given power or root of it be taken; the result will still be a maximum or minimum. Thus, if x be a maximum or minimum, then also is x + a, or x or, still a maximum or minimum. Also, the logarithm of the same will be a maximum or a minimum. And therefore, if any proposed maximum or minimum can be made simpler by performing any of these operations, it is better to do so, before the expression is put into fluxions. a, or ax, or or ra, 54. When 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, and 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 xy+ 2y is a minimum. Then we have, 422 1 First, *y = 0, and 8.r 2jxj0, and 2 And hence y or 8x = 16. y = 0, or y = 8x. x = 0, or x = 2. 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. 2x So, if we would find the quantity ar a2 a maximum or minimum; make its fluxion equal to nothing, that is, ax - 2xx = 0, or (a 2x)x = 0; dividing by x, gives Now, if in the fluxica = = 0, or xa at that state. (a-2x)x, the value of x be taken rather less than its true value, a, that fluxion will evidently be positive; but if a 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- being positive before, and gative after the state when its fluxion is = 0, it follows that at this state the expression is not a minimum, but a maximum. Again, taking the expression a3-ax, its fluxion 3x22a.xx=(3x-2a)xx=0; this divided by a gives 3x-2a=0, and r = 3a, its true value when the fluxion of x3 ax2 is 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 (3x - 2a) xx, and this will evidently be negative; and again, taking x a little more than 3a, the value of 3x 2a, or of the fluxion, is as evidently positive. Therefore the fluxion of x3 being negative before that fluxion is = 0, and positive after it, it follows that in this state the quantity 3 a.r2 admits of a minimum, but not of a maximum. 56. SOME EXAMPLES FOR PRACTICE. ax2 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 n 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, r and y, that 2.+xy+ 3y may be a minimum. EXAM. 7. To find the greatest rectangle that can be inscribed 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 r is a mininum. THE METHOD OF TANGENTS; OR, TO DRAW TANGENTS 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 between the ordinate and tangent, in the axis produced, by joining the points T, E, the line TE will be the tangent sought. y 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 Eca similar to the triangle TDE; and therefore 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 y, or of which value substitute for it in the expression DT= j 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 ax2 + xy2 — y3 = 0. Here the fluxion of the equation of the curve is 2axx+x+2xyỷ 3y30; then, by transposition, Qaxx + y2x = 3ÿ'ÿ = 3ŷ'ŷ — 2xyÿ; and hence, by division, x 3y2 - 2xy ў = 2ax + y2 - j 2ax + y2 which is the value of the subtangent TD sought. EXAM. 2. To draw a tangent to a circle; the equation of which is axay; 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 c2 (ax − x2)=a'y2; where a and c are the two axes. EXAM. 5. To draw a tangent to an hyperbola; its equation being c2 (ax + x2) = a2y2; 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 xy= a2; where a denotes the rectangle of the absciss and ordinate answering to the vertex of the curve. OF |