These values for stars near the pole may differ siderably from the first.

very con

335. Problem I. To reduce the right ascension and declination of a star from the epoch 1800 + t to 1800+t', the proper motion being known.

First. Suppose the proper motion given in reference to the mean equator of 1800+t, the solution is as follows:

Add to the right ascension for 1800+t the effect of proper motion for the interval (t't), viz., μ(t' — t); similarly add to the declination μ'(t − t). With these values of the right ascension and declination the precession is computed as before by formulæ (542).

Second. The proper motion being given for the mean equator of 1800 + t'.

add to the results

Reduce the star's place to 1800+t' by formulæ (542), and (tt) and '(t' - t) respectively. 336. Problem II. Having given the proper motion in right ascension and declination, referred to the mean equator of 1800t, to derive the values in reference to the equator of 1800 t'.

Equations (539), giving the values of a' and ' in terms of a and d, are as follows:

The proper motion which changes the position of the star itself produces no change in the quantities z, z', 9, 9', or 0, as these quantities merely serve to fix the positions of the

2' +9) cos

z'+9) sin

+ sin d' sin 9; (553) + sin d' cos 0.

reference planes. Therefore, proper motion alone being considered, these quantities will be constants, a, a', d, d' being variable.

Differentiating the first two of (552) on this hypothesis, we have

cos d' cos (a'

z' + 9') da'

sin d' sin (a' — 2' + 9') d8'

= cos & cos (a + z + 9) da sin ở sin (a +2t9) ườ; z' + 9') da' sin d' cos (a'

2' + 9') d8'

= cos o sin (a++9) cos 6da-sin d cos (+z+9) cos Odd—cos ô sin îďồ.

Multiply the first of these by cos (a' - ' + 9'), the second by sin (a' — ' + 9'), subtracting and reducing by (552) and (553); then multiply the first by sin (a' -- ' + 9'), the second by cos (a' — z' + 9'), add, and reduce. We find

da, dô, da', and do' have been changed to 4a, 46, etc. These equations solve the problem above enunciated with all necessary precision; 4a, 48, etc., being so small that it is unnecessary to consider terms of the higher orders. They may be used for the entire proper motion between the two dates and tor for the annual proper motion.

337. Problem III. The proper motion being given in reference to the mean equator of 1800 + t' to derive the values of 4a and 48 in reference to the mean equator of 1800+t.

Differentiating equations (553) and reducing by (552) and (553) in a manner similar to that explained above, we have

Example.

In the example Art. 332 we have found by applying the precession to the catalogue place of Polaris the mean position for 1900.0, as follows:

These values are referred to the mean equator of 1900. If we wish to reduce them to the equator of 1825 we employ formulæ (555). From the values of (a + ≈ + 9) and 0, Art. 332, we find

The above treatment of the problem is due to Bessel.

* This is, of course, not an observed place, but it answers equally well for illustrating the method.

+4a' being given in time and 46' in arc.

Proper Motion on the Arc of a Great Circle.

338. Let p = the annual motion on the arc of a great circle; x = the angle which this great circle forms

with the hour-circle of the star.
When the star is on the meridian,

X will be measured from the
north towards the east.

In the figure P is the pole, S and S' the first and second positions of the star respectively.

SS' = p;

PSS' = X;

S'A= 4a cos = p sin X;

SA = 46 p cos x; p2 = 48° + 4a2 cos2 d.

Expansion into Series.

S

A

S FIG. 70.

(556)

339. The foregoing problem of reducing the mean place of a star from one epoch to another is treated in a very convenient and elegant manner by expansion into series in terms of the time.

If we let a, and

α

δ

=

=

the right ascension and declination. for any time T,

a and the right ascension and declination. for any time T+t,

the changes in a and d are functions of these two independ

ent variables, and [da], [db], e

etc., are the total differential

coefficients with respect to both precession and proper

motion.

If we write da, do to indicate a variation due to precession, and dua, dud to indicate changes due to proper motion, we have