From the true horary motions of the moon in right ascension and declination subtract these variations; the remainders will be the apparent horary motions of the moon in right ascension and declination, and the former, multiplied by the cosine of the moon's apparent declination, gives her apparent motion on her parallel of declination: let this last be represented by p, and her apparent motion in declination by 9:

The differences between the true right ascensions and declinations of the moon and star found as above for the time T' being taken; if to these be applied, by addition or subtraction, the values of a and 8 respectively, and the first of the two results expressed in seconds of a degree be multiplied by the cosine of the moon's apparent declination, there will be obtained the apparent distances between the star and the centre of the moon, on the parallel of declination and on a horary circle passing through the star: let these be represented by m and n.

Then, if the estimated difference between Greenwich and the station, in longitude, were correct, we should have, as in the process for a solar eclipse, m2 + n2=s2. But t (in decimals of an hour) denoting the error, there must be substituted in the equation, m +pt for m and n + qt for n; and from the equation thence arising the value of t may be found. Finally, the difference between T't and T will express (in time) the required longitude.

Ex. March 15. 1840. The emersion of a Leonis from the moon was observed, by mean time, at

Estimated longitude from Greenwich, in time,

Greenwich mean time of the observation

8 ho. 17' 4" (T)

ho.

3

20 4 (T)

9 58 43.85

41.13

9 59 24.98 7 50 52.95

8

1 21.25

Moon's r. asc. at T
R. asc. mid-heaven

With the elements above found we have, as in art. 558., the moon's true altitude at the station, equal to 43° 54'.

On putting in numbers, with the above elements, the formula for tan. a (art. 161.) we obtain, in angle, a=19′ 40′′.5, and in time, 1' 18".7. Again, on putting in numbers, the formula for tan. 8 (art. 162.) neglecting cos.a, we have S=36′ 46′′.7. Also, on putting in numbers the formula

Now putting t (in decimals of an hour) for the error in the estimated difference in the longitude of the station, the equation m2 + n2 = s2 becomes

(739.31 + 1339.8 t)2 + (590.4 + 789.28 t)2 = (945.9)2 from which we obtain t (in hours)=-0.00098 or (in seconds of time) — 3′′.5 nearly; and therefore the difference, in lon

gitude, between Greenwich and Sandhurst is, in time, 2′56′′.5 nearly, subject to the uncertainty respecting the precise instant of the emersion.

Instead of solving this quadratic equation it may be more convenient to obtain the error in the estimated longitude by the following process:

Moon's appar. hor. mot. in r. asc. (in time), =1′ 31′′.51 prop. log. 2.0720 is to 1 hour

as appar. dist. in r. asc. (in time) between the moon's centre and
the star at the time T',:
= 50".5

to the interval, in time, between T' and the instant of conjunction in right ascension

0.4771

2.3301

2.8072

33' 7" 0.7352

P

Now s being the place of the star and м that of the moon at the instant T', let PS, PM represent hour circles passing through s and м, and let fall MN perpendicularly on PS: then SN= 590.4, and MN= 739.31; and by plane trigonometry, the angle MSN will be found to be 51° 23'. But this may be considered as equal to the value which the angle would have if the values of SN and MN were in accordance with the equality of SM to the apparent semidiameter of the moon; therefore, in the spherical triangle PSM

S

N

sin. PM (the moon's appar. pol. dist. at r'=77° 25′ 12′′) co. ar. 0.0105543 is to sin. PSM (suppt. of 51° 23′)

as SM (=945′′.9, using the angle in seconds for its sine)

is to sPM = 757′′.25 in angle (=50′′.45 in time) Then, moon's appt. hor. mot. in r. asc. in time

is to 1 hour

as 50".45

9.8928395

- 2.9758452

2.8792390

is to the interval in time between the instant of conjunction in r. asc. and the corrected time of emersion (=33′ 4′′)

- 0.7359

The difference between this interval and 33' 7", found above, is the error in the assumed Greenwich time of emersion, or in the estimated difference in longitude (in time); and the corrected difference in longitude is therefore equal to 2' 57" nearly.

A process similar to that which has just been exhibited might have been employed, in finding the longitude by the eclipse of the sun, instead of solving the equation for t in that example.

384. In order to determine the longitude of a station by Jupiter's satellites, the observer, having ascertained the mean time at the place and thus found the error of his watch, has

only to ascertain with his telescope the instant when the immersion or emersion of the satellite takes place. Supposing either of these to be accurately observed, the difference between the time at the station (by the watch) and the mean time for Greenwich, in the Nautical Almanac, is evidently equal to the required difference in longitude from Greenwich, in time. However simple the observation may appear to be, some practice is required (since. the satellite, particularly if it be any but the first, loses or acquires its light gradually) to enable the observer to be certain of the exact moment when the immersion or emersion takes place. In preparing for such an observation, it is necessary that an approximate knowledge of the longitude of the station should be had, in order to ascertain nearly the instant when the phenomenon will take place; otherwise the opportunity may be lost for want of time to make the proper dispositions. The traveller in a region distant from Greenwich, in longitude, should take every convenient opportunity of applying his approximate difference of longitude, in time, to the times of the phenomena as they are given in the Nautical Almanac, in order that he may know what are the particular eclipses which will be visible at his station. It is understood that the motion of a ship would render the observation of an eclipse of a satellite impracticable at sea; it is right, however, to remark that the difficulty is said to have been recently overcome, but it may have been in favourable circumstances which do not often occur.

Ex. September 5. 1843, at Sandhurst there was observed an emersion of Jupiter's first satellite at 10ho.19′ 50′′ by the watch. Error of the watch (too slow)

Mean time of the emersion
Do. at Greenwich (Naut. Alm.)

Difference of longitude, in time

1 25

CHAP. XVIII.

GEODESY.

METHOD OF CONDUCTING A GEODETICAL SURVEY. MEASUREMENT OF A BASE. -FORMULÆ FOR VERIFYING THE OBSERVED ANGLES AND COMPUTING THE SIDES OF THE TRIANGLES. -MANNER OF DETERMINING THE POSITION, AND COMPUTING THE LENGTH OF A GEODETICAL ARC.- PROPOSITIONS RELATING TO THE VALUES OF TERRESTRIAL ARCS ON THE SUPPOSITION THAT THE EARTH IS A SPHEROID OF REVOLUTION. THE EMPLOYMENT OF PENDULUMS TO DETERMINE THE FIGURE OF THE EARTH.-INSTRUMENTS USED IN FINDING THE ELEMENTS OF TERRESTRIAL MAGNETISM. FORMULE FOR COMPUTING THOSE ELEMENTS AND THEIR VARIATIONS.

385. THE figure presented by the visible disk of each of the principal planets, which is that of an ellipse whose minor axis is coincident with that of the axis of rotation, sufficiently indicates that the form of the earth must, since like those planets it has a rotation on an axis, be such as to present a similar appearance to a spectator supposed to be situated at a great distance from its surface; that is, its figure must be, either accurately or nearly, that of a solid of revolution whose shorter axis joins the north and south poles. If the materials of which the earth is composed were originally homogeneous and in a fluid state, and there had been no external attractions by which the effects arising from the rotation on its axis might be deranged, the figure which the earth would have assumed is well known to be that of a geometrical spheroid; but as no such homogeneity nor freedom from external attractions can with the least probability be supposed to have at any time existed, it ought to be inferred that a deviation from perfect regularity must unavoidably have place in the figure of the earth.

386. The results, however, which have been obtained from the effects of the earth's figure on the moon's motions, from geodetical measurements and from the experiments made to determine the intensity of gravity at different parts of the earth's surface, place beyond doubt the fact that the figure of the earth does not differ considerably from that of a sphere, and also that it is very nearly identical with that of a spheroid: therefore, while measuring only small portions of the earth's surface, its figure may without producing any sensible error