# The Pamphlet Collection of Sir Robert Stout: Volume 50

# University of New Zealand. — Mathematics and Mathematical Physics. — Paper e (3). Astronomy

## University of New Zealand.

## Mathematics and Mathematical Physics.

##
Paper *e* (3). Astronomy.

"Every body that by a radius drawn to the centre of another body, however moved, describes areas about that centre proportional to the times, is urged by a force compounded out of the centripetal force tending to that other body, and of all the accelerative force by which that other body is impelled."

Discuss also the truth or falsity of this proposition if motion take place in a resisting medium.

2. Prove the following equations for a right-angled spherical triangle:—

3. Prove that any two confocal solid ellipsoids of equal mass produce equal attraction through all space external to both of them.

page 2Extend the proof of the preceding so that it shall include a proof of the proposition for any two shells of equal mass (of homogeneous density), each shell being bounded by two con-focal ellipsoidal surfaces.

4. Explain the method of deducing the Sun's distance from that of the Moon (supposed known), by observation of the angle of elongation of the Moon when the latter is dichotomized. State why the method is liable to introduce considerable errors of observation.

Show that any small error in observing the elongation will necessarily produce such a large error in the result as to render the method utterly unreliable.

5. The inner satellite of Mars is unique in having its "day" longer than its "month." Discuss the conditions of equilibrium of such a case.

6. The elementary expression for the Moon's longitude is

Show that this is correct to the first order of approximation, and explain how the arbitrary constant of integration introduced during the process of integration has been eliminated.

7. The differential equation of the moon's radius vector may be written in the following form:—

and in the solution the first approximate value of *u* is:—

show that the terms which follow in the higher approximations contain *θ* as a factor of the coefficient of the longitude-function.

Why are such terms inadmissible and incompatible with the known facts of the Moon's distance? How comes it that such terms appear in our mathematical expressions, and how can they be got rid of?

8. Show how to determine the latitude of any place from observation of the time that elapses between the rising of two known stars.