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80. Define conjugate diameters, and prove that in an ellipse there can be only one pair which are at right angles to each other.

81. Determine the chord of curvature through the focus at any point of an hyperbola.

82. In any conic section, SG « SP, G being the foot of the normal, and if GL be perpendicular to SP, PL = } latus rectum.

83. Assuming the equations to the tangent of an ellipse, and to the perpendicular upon it from the focus, find the locus of their intersection; and account for the presence, in the result, of the factor which is rejected.

84. Shew that the curve y= bx + has the origin of the coordinates for its centre ; trace it, and find the magnitude of its axes.

85. If a right cone be cut by a plane, and a, b denote the distances from the vertex of those points of the curve of intersection which lie in a plane through the axis perpendicular to the cutting plane, prove that the distance between the foci = a F b according as the section is an ellipse or hyperbola.

86. Two conjugate diameters are produced to intersect the same directrix of an ellipse, and from the point of intersection of each one a perpendicular is drawn on the other ; prove that these perpendiculars will cut one another in the nearer focus. 87. In an ellipse shew that

PSH PHS 1

tan 2

2 Ite 88. Investigate the polar equation to an ellipse, the focus being the pole ; and find the position of the distance which is half the sum of the greatest and least distances.

89. If E be the point of intersection of SP and CD in an ellipse, shew that PE = AC.

90. Investigate those sections of an oblique cone which are circular.

91. In the ellipse PV . VG : QV2 :: CP2 : CD2.

е

tan

1835

92. Shew that Ay+ By + Cx + D = 0 is the equation to a parabola, and determine the position of the vertex and magnitude of the latus rectum.

93. The tangent at any point of a parabola will meet the directrix and latus rectum in two points equally distant from the focus.

94. From any point Q in the line BQ, which is perpendicular to the axis CAB of a parabola, whose vertex is A, QP is drawn parallel to the axis to meet the curve in P; shew that if CA be taken equal to AB, the locus of the intersections of AQ and CP is a parabola.

95. The length of the perpendicular upon the tangent from the centre of an ellipse is equal to av I - e2 (cos p)?, where o is the inclination of the tangent to the axis major.

96. Find the magnitude and position of the axes of the curve whose equation is 3x2 + 2xy + 3y2

167 + 23 = 0. 97. a, b being the axes of an ellipse, and a', b' being conju- 1836 gate diameters respectively inclined to them (viz. d' to a, and b' to b) at angles a, ß; shew that

a'2 – 02 cos (a + B)

a? 62 cos (a B) 98. Of two conjugate diameters of an hyperbola' one only meets the curve. If one be drawn through a given point of the curve, find where the other meets the conjugate hyperbola.

99. If PQ be a chord of a parabola, normal at P, and T the point of intersection of the tangents at P and Q; prove that PT is bisected by the directrix.

100. If two chords of a conic section be drawn, of which one bisects the other, and the straight lines joining their extremities be produced to intersect, the line joining the two points of intersection shall be parallel to the chord bisected.

101. If a right cylinder of circular base be cut by a plane, the section will be an ellipse whose eccentricity is equal to the cosine of the inclination of the cutting plane to the axis of the cylinder

[graphic]

form y?

102. From the general equation to a curve of the second order deduce the equation to that diameter, which bisects all chords parallel to one of the coordinates axes; and state the successive transformations by which it may be reduced to the

= mx + naa. 103. If a paraboloid of revolution be cut by a plane parallel to its axis, the section is the same as the generating parabola.

104. The chord of curvature, through the focus, at any point P of a parabola = 4SP.

105. In an ellipse, SY. HZ = BC?. Prove this, and deduce the relation between SY and SP.

106. If the tangent at any point of an hyperbola be produced to meet the asymptotes, the area of the triangle cut off is constant.

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2. Of all triangles upon equal bases and with equal vertical angles, the isosceles has the greatest perimeter.

3. Investigate a differential expression 'for the radius of curvature of a curve, referred to rectangular coordinates.

4. Investigate the differential expression for the subtangent of a curve; and mention the analytical characters of a double, triple, and conjugate point. 5. If a be the arc of a circle whose radius is unity, a', a", a“,

the arcs of its successive involutes, then

a + d + a" + a"' + .... in infinitum 1. 6. Investigate the differential expression for the length of

= ea

a curve.

7. Trace the curve whose equation is

24
a? x2 + a'y

0,
and find its points of contrary flexure.
8. Trace the curve, the equation to which is

xy + ay + bx 9. Trace the curve, the equation to which is

у?

ab - 0. Draw a tangent to it at any point, and determine the angle at which the curve cuts the axis.

C.

1822

ax

sec X

10. Shew how the true value of a fraction may be found, the numerator and denominator of which both vanish upon assigning a particular value to the variable quantity, and find the value of tan x + sec X

1
when x =

- 0. 1 + tan x 11. Find the radius of curvature at any point of the catenary.

12. Prove Newton's fourth Lemma (Sect. 1.), and by means of it find the content of an oblate spheroid. 13. Differentiate

4ax + x2 + x
(1) log

✓ 40x + x2
ex ve + x
(2)

e-De- x

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14. Required the radius of curvature of the curve whose equation is

a + y, and determine the coordinates of the y

X' centre.

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a

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Sv

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dx

dx
find also
VI

X2 and fdx (a? + b2 – 2ab cos x)" from x = 0 to x = 180°; and determine the relation between x and

y

when
dạy
dy2

dy:
A

= 0. d.x2

dx3

+ B dx2

16. Given the magnitude of a spherical surface, find the radius of the sphere so that the corresponding spherical segment may be the greatest possible. 17. Differentiate

(1) log (2x + 1 + 2V1 + it + x2):

(2) cos 0 + sec 0; and find the values of alog x 1

log tan x (1)

when x =
1: (2)

when x = 0.
log 2

log tanz

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