EXAMINATION FOR SMITH'S PRIZES.

[It is expected that the numerical calculations will be

the coefficients being the same as those in the expansion of

2. By the observations of Captain Sabine, the length of the seconds' pendulum at Sierra Leone, latitude 8° 29′ 28′′, is 39,01997 inches: that at Drontheim, latitude 60° 25′ 54′′, is 39,17456 inches. Calculate from these data the ellipticity of the Earth.

3. Given the dispersive powers of two kinds of glass; find the relation between the focal lengths of two lenses, whose combination will produce an achromatic object-glass, 1st when the lenses are in contact, 2nd when they are separated by a given interval.

4. Express differences in terms of differential coefficients, and calculate to 7 decimals, the 1st, 2nd, 3rd, and 4th differences of the common logarithm of 1000.

5. Explain why the gnomon of a dial must be parallel to the Earth's axis.

6. A vertical dial for latitude 52° 12′ 43′′ N. faces a point of the horizon 15° west of the south point: find the angle made by the hour-line of 4h P. M. with the vertical.

7. State what is meant by the great inequality of Jupiter and Saturn: and mention the peculiarities in the motions of those planets to which it is owing.

8. Prove that the portion of the sky seen within the primary rainbow, cæteris paribus, is brighter than that seen without it.

9. Explain the transformation of the independent variable; and transform the equation

where x is the independent variable, into one where z is the independent variable, z being: common log. x.

10. The object-glass of an astronomical telescope has 7 feet focal length: its eye-piece consists of two lenses of 1 and 3 inches focal length respectively, separated by an interval of 2 inches: the lens of 3 inches focal length, which is nearest to the object-glass, is distant from it 6 feet 10 inches; find the magnifying power.

11. Explain what is meant by the variation of parameters, and apply it to solve the equation

d+y

dx4

-y=a*.

12. The N.P.D. of Regulus is 77° 11' 40"; his right ascension 9h 59′ 11′′.4: his corrected altitude 33° 17', E. of the meridian; as seen at Cambridge Observatory, whose latitude =52° 12′ 43′′. Find the right ascension of the point of the equator on the meridian at the instant of the observation.

13. In the example above, find the error produced in the result by an error of 11" in the altitude.

14. Explain the precession of the equinoxes, and the nutation of the Earth's axis, without analogical reference to the disturbances produced by the Sun in the Moon's motion: and point out a method of calculating theoretically the coefficients.

15. Prove the process for taking an integral between limits; and find the value of between the limits

1

x = 0, x = 1,

16. In an air-pump, similar in construction to the common water-pump, an interval is left between the piston and the lower valve at the lowest position of the piston; find the density of the air in the receiver after n strokes, and after an infinite number.

17. In finding the sine of half an arc, shew that, when is small, a large error may be expected in applying the forand a small one in using the

mula sin

0

=

cos 2

18. Give the method of determining the variation of the elements of the planetary orbits arising from the mutual at traction of the planets.

19. Find the alteration of gravity at the top of a conical mountain 3 miles high, and 24 miles in diameter: the Earth being supposed a uniform sphere whose diameter=8000 miles.

20. In throwing a quoit, it is found necessary to give it a motion of rotation in its plane, in order that it may preserve its parallelism during its motion. Explain this on mechanical principles.

21. Every point of a fluid is acted on by equal and constant forces directed to two centers: find the figure which it will

assume.

22. The evolute of a curve is similar to the curve itself: find an equation to the curve.

23. Give the necessary theorems for constructing Mercator's chart for an oblate spheroid of small ellipticity.

24. Investigate theoretically the quantity of astronomical refraction, the density of the atmosphere being variable.

*These are only the Problems of one of the Examiners (the Lucasian Professor) for the SMITH'S PRIZES.