little more than 3a, the value of 3x

=

2a, or of the fluxion,

is as evidently positive. Therefore the fluxion of x3 — ax2 being negative before that fluxion is 0, and positive after it, it follows that in this state the quantity 3 ara admits of a minimum, but not of a maximum.

SOME EXAMPLES FOR PRACTICE.

EXAM. 1. Of all triangles, ACB, constructed on the same base AB,and having the same perimeter, to determine that whose area or surface is the greatest.

Let p denote the semiperimeter, b the base AB, x the side AC, then BC will = 2p — b — x. Therefore putting s for the surface, we have by rule 3 for the area of triangles (pa. 408, vol. i.)

s2=p (pb) (p − x) (b + x − p).

Expressing this equation logarithmically, we have, 2 log. s=log. p + log. (p—b) + log. (p—x) + log. (b+x−p) which (art. 93) is to be a max. or when put into fluxions equal to zero or nothing.

Now, here it is evident, since s must be a max. that can

2

not = 0; consequently the second factor must that is,

Cor. Hence it follows that of all isoperimetrical triangles, the one which has the greatest surface is equilateral. A truth, indeed, which may be readily shown by a direct investigation.

EXAM. 2. Amongst all parallelopipedons of given magni. tude, whose planes are respectively perpendicular to one another, to determine that which has the least surface.

Let x, y, and z, be the measures of the three edges of the required parallelopipedon. Then, since the magnitude is given,

we have xyz = a, a given magnitude;'
and 2xy + 2xz +2yz = a minimum.

Here, substituting for z, and dividing by 2, there results

Therefore, adopting the principle of art. 94,

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." × sin."ya max.

In logs. m log. sin x + n log. sin. y = a max.
mi cos. x my cos. y
+

=0.

Hence, (art. 93)

sin. z
mi cos. x

sin. y

But y =- i .

Hence m cot. x = n cot y, or m tan. y = n tan. x.

n tan. y

. and

m-n tan. x-tan. y

(See equa. 9 and 10, p. 394, vol. i.)

:

sin. (x+y)

sin. (x—y)

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 MN 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.

R B

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 differ. ence between its m power and n power shall be the greatest possible.

EXAM. 10. To divide the number 80 into two such parts, ≈ and that 2x2 + xy + 3y may be a minimum. VOL. II.

y,

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 x 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 be

E

α

y

tween the ordinate and tangent, in T ADd C

the axis produced, by joining the points T, E, the line TE will be the

tangent sought.

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 xea 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 put into fluxions, find the value of either or ý, or of

which value substitute for it in the expression DT ==

yi

y'

ў

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 2axi + y2'i + 2xyÿ — 3y3ý = 0; then, by transposition, 2axi+yi 3y'y-2xyy; and hence, by division,

; consequently

which is the value of the subtangent TD sought.

EXAM. 2. To draw a tangent to a circle; the equation of which is ax xy; 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 c(ax-x)= 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) = a3y3 ; where a and c are the two

axes.

ง