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Ex. 2. To find which pair of conjugate diameters of an ellipse make the least angle with each other.

Let m, n, be these diameters, P the angle which they form with each other. Then by (Art. 59, Conic Sec.) we have mn sin P-ab, and

ab

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n (a2+b2—n2)1⁄2

—ab (a2+b2—2 n2)

n2 ( a2+b2—n2)}

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o; consequently ab (a+b2—2 μ3)—o.

Now since ab cannot be equal to zero, we must have the factor a2 + b2—2 n2 —o, whence n =

√(a*+b2)=m.

2

Hence the equal conjugate diameters of an ellipse are those which by their intersection form the least angle as required. The

sine of that angle is.

2 ab

a2+b2

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1+tan U

sec U

2 sin U cos Usin 2 U; therefore the angle P is equal to that formed by the two lines drawn from the two extremities of the conjugate axis to either extremity of the transverse axis. Ex. 3. Of all triangles constructed on the same base AB, and having the same perimeter, which has the greatest surface.

Let the semi-perimeter=q, the base AB-a, the side AMx, MB will = 2 q· Therefore calling y the surface, we shall have

-a-x.

Y = √ { q (q—a) (q—x) (a +x−q) }

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and therefore 2 log y=log q+log (q−a)+log (q−x)+log (a+x−q);

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=0.

y q

Hence since cannot-o, we must have the second factor

2

equal to zero, and therefore a+-q-q-x, or 2 q—a—x—MB—1, and consequently the triangle required is isosceles.

From this it follows that among all isoperimetrical triangles, or triangles having the same perimeter, the one which has the greatest surface is equilateral. For if AMB is the triangle required, it is evident that it must have a greater surface than any other isoperimetrical triangle AMB, constructed on the same base AB; therefore AM-MB. In the same manner it may be proved that AM AB.

49. Hitherto we have only considered the maximum or minimum of the function of a single variable x. To find in what cases any function Y of two variables r and y becomes a maximum or a minimum, we may employ the following method.

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Let us suppose that y has already the value adapted to render the function Y a maximum or minimum; we shall then only have to find the proper value of x, that is to say we must take the fluxion of the function Y supposing only to vary, and equate the co-efficient of to zero Pursuing a similar mode of reasoning, we shall find that to obtain y we must take the fluxion of the function Y making only y to vary, and then equate the co-efficient of y to zero. Hence it follows that if Ỷ is represented generally by Pr+Qy, we must have Po, and Q=o, two equations which will give the values of r and y proper to render the function Y a maximum or a minimum.

It is easy to see that this same reasoning applies, whatever be the number of variable quantities of which Y may represent a function. Hence in general to determine those values of the variable quantities which will render the function Y a maximum or minimum, we must take the entire fluxion of Y, and equate to zero the co-efficient of the fluxion of each variable, which will give as many equations as there are unknown quantities.

Ex. I. Let it be required to divide a given number a in three parts, whose product may be a maximum.

Calling x and y two of these parts, the third will be expressed by a-x-y, and we shall have for the product xy (a-x—y). The fluxion of this expression is (a-2x-y) yx + (a—2y—z) xy. Equating separately to zero the co-efficient of x, and that of y, we shall have a-2x — y=o—a—2 y—x. Hence y=x=ta. quently the given number must be divided into three equal parts. Ex. 2. Let it now be required to determine among all isoperime trical triangles, that which has the greatest surface. We have before resolved this problem, but indirectly.

Conse

Let x, y, be two of its sides, 2q the perimeter, then 2q--x-y will De the other side, and the surface will be{q (9-) (q−y) (x+y−q)}· This must be a maximum; if we call it Y, we shall have 2 log Ylog q=log (7—2) + log (q—y) + log (x+y−q). Therefore Ÿ=

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the co-efficient of x and also that of y, we have x+y-q=q-y=q-x;

hence ry: 29-2q——y. Consequently the triangle is equi

3

lateral, as we before found.

Examples for Practice.

1. Of all the squares which may be inscribed in a given square, which is the least?

2. Of all fractions, which is the one that exceeds its mth by the greatest possible quantity?

power

3. Required that number x of which the ath root is a maximum? 4. I am desirous of constructing a cylindrical measure of a given capacity, and of which the internal surface may be a minimum. What must be the ratio between the height of the measure and the diameter of its base?

5. Among all the cylinders which may be inscribed in the same sphere, which is that whose convex surface is a maximum?

6. Which among these cylinders has the greatest solid content? 7. What must be the dimensions of the greatest cylinder that can be inscribed in a given cone?

8. Of all triangles standing on the same base, and inscribed in the same circle, which is the greatest?

9. Which, on the contrary, will be the least of all the triangles may be circumscribed about the same circle?

that

OF VANISHING FRACTIONS.

50. We sometimes meet with algebraic expressions in the form of fractions, which on certain suppositions become equal to

Such

for example, is the quantity.

X-a

when x=a. Now though in

appearance indeterminate, these results are nevertheless susceptible of determinate values; and the following is a method for ascertaining them.

Let

P

Q

be a fraction whose numerator and denominator are functions of x, both of which become zero when x=a. To find the value of this fraction, substitute + in place of x in P and Q,

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and we shall have. ; then making xa in this latter fraction,

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P+P
Q+Q

; and this will be the value of the proposed frac

tion, on the supposition of x=a+x, or of ra, provided however

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Q

x

Ex. 2. Let there be the geometrical progression x, x', x3,...x” of

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Taking the fluxions as directed, we shall find(n+1)x”—1=

n, as is evident.

Ex. 3. Let there be proposed the quantity (2a 3x-x1)—a 3/ a2x

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which becomes when xa. Taking the fluxions and proceeding

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as before, we shall have

a3-2 x3 α Va1x

√(2a3x—x1) 3x
3 4

16

ga, the value

of the proposed quantity.

51. But if it happens that on substituting a instead of x in.

this

fraction also becomes; ; we must treat it in the same manner as

the first, and so on, till we arrive at a value of which one term at least is finite.

Ex. If we take the fluxion of the identical equation x+x2+*"

...I =

n+1

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we shall have x+

— (n+1) x2+1

which becomes

(1-x) 2

; but this

-2 (1-x)

P

when x=1. Therefore

n+1 −x” (n+1)2+n (n+2)x

new expression gives also, on substituting 1 for x; we must therefore take the fluxions of its numerator and denominator separately, and we shall have—na11 (n+1)2+n (n+1) (n+2) 2” which

n

2

on making x=1, gives " (n+1) for the sum of the arithmetical

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Ex. 2. In the quadratrix yatan ; and this expression be

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466

Examples for Practice.

1. Required the value of the fraction

ax2+ac2--2 acx

bx-2 bcx+bca

2. What is the value of the fraction_a*- -b

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when r=0.

when

Note. By these principles we may in each particular case find the indeterminate values of ox∞, and of ∞- ∞. For ox becomes

a

since.

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We also reduce to this same process ∞∞,

a

supposing that the first ∞ arises from, and the second from.

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b

0

by

=∞∞ (because log

as directed, it becomes――1.

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