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165. The normal of a curve touches its evolute. Determine the whole length of the evolute of an ellipse.

166. Investigate the differential coefficient of a surface of revolution. What is the surface generated by the revolution about the axis of x of the curve whose equations are

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of the hyperbola referred to its focus as pole; and thence determine the position of its asymptote.

168. If x, y be rectangular coordinates of any point in a curve, and r its distance from the origin, prove that the coordinates of the corresponding point in the evolute are ́d2 (r2)

đ (2)

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169. Having given cx (by) = ay (cz)

= bz (ax), find the maximum value of each of these expressions.

170. The area of a polygon of a given number of sides, circumscribing a given oval figure, will be the least possible when each side is bisected in the point of contact.

171. Determine the multiple point in the curve whose equation is

ay3 - 2ax2y x4 =

0.

Find also the points where it is parallel to the coordinate axes.

172. If z = f(x, y), where x and y are both functions of

dz

two other variables r and 0, express and in terms of r,

dz
dx

dy'

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173. Apply Lagrange's theorem to find x" from the equation x2+2ax + 1 = 0.

174. What are the analytical characteristics of a point of

osculation, and of a cusp? Ex. y = a vers-1

175. Trace the curve whose equation is

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a

3

+ √2αx - x2.

(1)2 + (3)

3

=

1;

determine its asymptote, points of inflexion, and the angles at which it cuts the axes.

176. Eliminate by differentiation the constant from the equation

√ T= x2 + √T — y2 = a (x − y).

H

1821

1822

SECTION IX. .

QUESTIONS IN INTEGRAL CALCULUS.

1. FIND the area of a curve in which the abscissa
cos2 0
the ordinate =
>
T

0, and between the values of = 0, and 0=90°.

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dx

axx3 dx.

dx

ʼn being an even number; x2√a + bx + cx2

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5. How much of the Earth's surface may be seen by a person raised ʼn radii above it?

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7. Find the area included between any two radii of a spiral, where the angle contained between them is the measure of their ratio.

8. Find the integral of

xm dx (log x)" from x = 0 to x = 1.

9. Shew that the content of a sphere

the content of the

greatest cone that can be inscribed in it :: 33: 23.

10. The equation to a curve is y determine the values of the abscissa

=

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9x2+24x + 16;

when the ordinate is a

maximum, and when a minimum; and find the area included

between those ordinates.

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15. If there be taken the evolute of a logarithmic spiral, the evolute of that evolute, and so on ad infinitum, find the sum of the arcs of all the successive evolutes.

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1823

17. Find the integrals of

xdx
✓ (a + x1)'

and of v2dx, where

v = log x.

18. Trace the curve whose equation is y = sec x, draw a tangent to it, and find its area.

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21. The sum of the squares of the coefficients of the expan

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23. In a parabola find the area included between the curve, its evolute, and its radius of curvature.

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26. ACB is a quadrant of a circle whose centre is C, CA, CB its radii, AD, BE equal arcs, DE the chord of the arc DE; shew that the solid generated by the revolution of the circular segment DE, about either radius is equal to twice the sphere whose diameter is √2 sin DE.

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