72. Substituting this value of C in the equation (a), we

obtain

(velocity)2 = λ(a2 + b2 — p2),

= λ (semi-conjugate diameter)2.

73. This might have been obtained without reference to (b); thus:

If v1, v2, be the velocities at the extremities of a and b,

2h2C

.. by equating these we get

λ= λ,

a2 + b2

or h2C = λ(a2 + b2);

.. (velocity)2 = h2 C − λr2 = λ (a2 + b2 — r2), as before.

74. The axes of the ellipse are determined by the conditions

The first of these might have been got at once from the results of the last two Articles.

75. For the determination of the period, i. e. the time required to describe the whole ellipse, we have

which is independent of the form of the ellipse, and is therefore the same for all ellipses described under accelerations subject to the same law as above, and of the same magnitude at all equal distances. (NEWTON, Prop. 10.)

76. (ij) Let the acceleration vary as the inverse square of the distance from the fixed point.

MOTION IN A CONIC SECTION ABOUT THE FOCUS. 41

This is the equation to a conic section whose focus is at the pole,

2h2

the angle vector of the apse being B, the latus rectum and

Ah2
λ

the eccentricity ; and therefore it is an ellipse, parabola, or

hyperbola, according as

Ah2
λ

is<= or > 1.

77. This might have been obtained, as in Art. 52, by putting

2

A2

(du) + u2 = 4a sin2 (0 − §) + {}3 + 4 cos (0 – B)}"

h2

2

79. The constants A, B, and h, are determined by the initial circumstances of the motion, viz. by the following equations:

80. If a be the semiaxis major of the conic section, and e the eccentricity,

which is independent of the initial direction of motion;

.. if v1, v, be the velocities at the extremities of the major axis,

2λ

(1 + e) − ho C (1 + e)2 = 21 (1 − e) – h°C . (1 − e)" ;

4e,

82. In the two preceding Articles the conic section has been treated as if it were an ellipse: if e were > 1, i. e. if it were an hyperbola, we should have a (e-1) = semi-latus rectum; the only effect this has is to change the sign of a: then we shall have

If e were = 1, i. e. if the curve were a parabola, we should have