1 Here, substituting for z, and dividing by 2, there results EXAM. 3. Divide a given arc a into two such parts, that the mth power of the sine of one part, multiplied into the nth power of the sine of the other part, shall be a maximum. Let x and y be the parts: then x + y = A, and sin." X sin."y a max. Ꮖ Hence m cot. an cot y, or m tan. y = n tan. x. and ... n tan. y m-n tan. x-tan. y (See equa. 9 and 10, p. 394, vol. i.) : Hence x and y become known and the same principle is evidently applicable to three or more arcs, making together a given arc. EXAM. 4. To find the longest straight pole that can be put up a chimney, whose height RM = a, from the floor to the mantel, and depth мN b, from front to back, are given. Here the longest pole that can be put up the chimney is, in fact, the shortest line PMO, which can be drawn through м, and terminated by BA and BC. EXAM. 5. 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. 6. 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. 7. To divide the given quantity a into three parts such, that the continual product of them all may be a maxi mum. EXAM. 8. 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. 9. To determine a fraction such, that the difference between its m power and n power shall be the greatest possible. EXAM. 10. To divide the number 80 into two such parts, ≈ and y, that 2x2xy + 3y2 may be a minimum. VOL. II. 47 EXAM. 11. To find the greatest rectangle that can be inscribed in a given right-angled triangle. EXAM. 12. To find the greatest rectangle that can be inscribed in the quadrant of a given circle. EXAM. 13. To find the least right-angled triangle that can circumscribe the quadrant of a given circle. EXAM. 14. To find the greatest rectangle inscribed in, and the least isosceles triangle circumscribed about, a given semi-ellipse. EXAM. 15. To determine the same for a given parabola. EXAM. 16. To determine the same for a given hyperbola. EXAM. 17. To inscribe the greatest cylinder in a given cone; or to cut the greatest cylinder out of a given cone. EXAM. 18. 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. 19. 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. 20. The cut the greatest parabola from a given cone. EXAM. 21. To cut the greatest ellipse from a given cone. EXAM. 22. To find the value of x when a is a minimum. THE METHOD OF TANGENTS; OR OF DRAWING TANGENTS TO CURVES. 96. 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. E a У T ADd C 97. 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 rea similar to the triangle TDE; and 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. 98. By means of the given equation of the curve, when i put into fluxions, find the value of either & or y, or of which value substitute for it in the expression DT = yi 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 + xy-y3 = 0. Here the fluxion of the equation of the curve is 2axi + y2i + 2xyý — 3y23ý = 0; then, by transposition, Qaxi+y'i=3y'ÿ -2xyy; and hence, by division, which is the value of the subtangent TD sought. EXAM. 2. To draw a tangent to a circle; the equation of which is ax-x2= y; where x is the absciss, y the ordinate, and a the diameter. EXAM. 3. To draw a tangent to a parabola; its equation being px = y2; where p denotes the parameter of the axis. EXAM. 4. To draw a tangent to an ellipse; its equation being c2(ax x2) = a2y2; where a and c are the two axes. EXAM. 5. To draw a tangent to an hyperbola; its equa tion being c2 (ax + x2) = a1y2; 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 a2 denotes the rectangle of the absciss and ordinate answering to the vertex of the curve. By slight and obvious extensions of the same principles, tangents may be drawn to spirals, and asymptotes may be drawn to such curves as admit of them. OF RECTIFICATIONS; OR, TO FIND THE a 99. 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 الا A Da C of the squares of the two former; that is, Ee2= Ea2 + ae2. Or, substituting, for the increments, their proportional fluxions, it is żż = iî + ÿÿ, or ż= √(x2+ÿ3) ; where z denotes any curve line AE, x its absciss AD, and y its ordinate Hence this rule. DE. જી RULE. 100. From the given equation of the curve put into fluxions, find the value of 2 or y3, which value substitute instead of it in the equation (+); then the fluents, ż = being taken, will give the value of z, or the length of the curve, in terms of the absciss or ordinate. EXAMPLES. EXAM. 1. 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 AE)=x, the right sine DEy, the tangent TE=t, and the secant CTS; then, by the nature of the circle, there arise these equations, viz. |