let yn; then the inclination of the image of the given object to the major axis is had from

and therefore the inclination to the given axis is known, viz.,

The portion of the image due to the given object is the arc of the conic section intercepted by the straight lines drawn from the extremities of the given object at right-angles to it.

827. SINCE the day is given, the sun's place in the ecliptic is known, and therefore, by the tables, its declination; which suppose D. Also let L denote the given latitude of the spectator. Hence the meridian altitude of the sun is

+ or

a = 90° – L±D

being used according as the given time is in summer or winter. This is the subtended by the tower; consequently its altitude is

828.

ha x tan. a = a x cot. (LF D).

Once for all let us investigate a general formula for

the aberratic curve.

Let p and p denote the radius vector and upon the tangent of the given orbit ; p′ and p' the corresponding ones to the aberratic Also let be the described by ; ' the described by . Then since the angular velocity of p = the angular velocity é'. of g' (Woodhouse's Ast. 2d edit., p. 304,) we have (see 443)

curve.

These two equations will give the equation to the aberratic

L being the principal parameter of the parabolic orbit;

But if a be the distance of the centre of polar co-ordinates from the centre of a circle, its equation is

P =

2r

and consequently when that pole is in the circumference

p=

2r

Hence it appears that the aberrative curve is a circle whose radius is

2c L

Ex. 2. Let the orbit be an ellipse whose equation is

so that the aberratic curve is a circle whose radius is

ac

b9

and the distance of whose centre from the centre of co-ordinates is

Ex. 3. Let the orbit be the logarithmic spiral whose equa

Therefore the aberratic curve is also a logarithmic spiral.

829.

Let a, a' be the altitudes when the sun is due east and at six o'clock, L the latitude of the place, and D the declination of the sun; then from the two right-angled A whose sides are

830.

Let a, b denote the semi-axis of the planet's orbit;

then its area is

παι

Hence if a circle be taken whose radius is

r = √(ab)

its area will be equal to that of the orbit; and if, with an uniform motion, a body be supposed to go through the whole circumfer ence in the periodic time of the planet, the so described by the radius in an hour, will measure the mean horary motion (6) of the planet in its orbit.

Again if do denote the true described by the radius vector in the same hour, by Kepler's law of the equable description of areas, we have

which is the true horary motion of the planet in its orbit.

Let the planet's orbit Np (Fig. 106.) and ecliptic Nm intersect in the node N, then make

and describe the arcs PM, pm, Pr 1 NM and pm. Now since Pp is very small compared with the whole extent of the orbit, Ppr may be considered rectilinear, and we have

which gives the horary motion required.

In like manner the horary motion in latitude is found to be