vert this into mean solar time at Greenwich; with this mean solar time and the hourly ephemeris of the Nautical Almanac, compute the moon's right ascension and N. P. D. by simple proportion of the hourly change. Then proceed to find the hour angle, exactly as if it were a star, using the geocentric zenith distance of centre, the colatitude of the place, and the N. P. D. just computed. Apply the hour angle to the right ascension, and thus obtain the computed sidereal time; obtaining two computed sidereal times from the two assumptions of longitude, you will find (as in other cases) what correction must be applied to the smaller longitude, in order to make the computed sidereal time agree with sidereal time at the observations of the moon.

17. (ồ.) Longitude by occultations of stars by the moon. Suppose the disappearance of a star behind the moon, or the reappearance of a star from behind the moon, has been observed, and the chronometer time noted (the calculation is precisely the same for disappearance or reappearance). Correct the chronometer for its error, and thus the true sidereal time at the place is found.

18. Assume two values of longitude, one greater and one less than the reputed values, and by applying these to the sidereal time, form the sidereal times at Greenwich on the two assumptions, and convert them into mean solar times at Greenwich. With these mean solar times compute (by the hourly ephemeris) the right ascension and N. P. D. of the moon's centre on each assumption; also the equatorial horizontal parallax and the semidiameter, and from the equatorial horizontal parallax obtain the true horizontal parallax as in 5.

19. The latitude of station to be used in the following computations is the geocentric latitude, which will be found generally by diminishing the astronomical latitude by the angle of the centre, and which in the boundary latitudes, viz. 45° to 48°, will be found by diminishing the astronomical latitude by 11' 25".

20. Take from the Nautical Almanac, section occultations-elements, the right ascension and N. P. D. of the star whose occultation has been observed. From the right ascension and time find the hour angle. Put 0 for the hour angle and for the N. P. D. Then determine a new right ascension, 1, and a new N. P. D., d', by the following equations :

21. These equations are to be solved by successive substitution. Two substitutions will usually be sufficient. Thus-first assume to be the same as d, and from the first equation determine . Use this in the two other equations, and you will get 1st No., 2nd No., and S', very nearly. Use this new value of in the first equation and you will get 0-0′ much more accurately then by means of the other two, & can be got still more accurately; and so on. again if you think fit.

22. With this new hour angle, ', and the sidereal time, determine a new right ascension; then calculate the following quantity:

Computed semidiameter of moon—

{

+ sin sin N. P. D. of moon's centre, new R. A.-R. A. of moon's centre

This is to be computed on the two assumptions for longitude (which have given two right ascensions and N. P. D. of moon's centre).

23. The computed semidiameter ought to agree with the Nautical Almanac semidiameter found in 18. If neither of the two assumptions of longitude makes it do so, one of them must be altered by a proportional part of their difference, found in the same manner as in other cases where two longitudes are tried.

24. (ε.) Longitude by eclipses of Jupiter's satellites—a very rough method. The observation is merely to note the last instant, or the first instant (according as it is disappearance or reappearance) of the satellite. The computation is merely to correct this for chronometer error, so as to obtain sidereal time, and to compare this with the Greenwich sidereal time given in the Nautical Almanac.

25. In regard to the effects of errors of observation, the following remarks should be borne in mind :

A certain error of time in taking a lunar distance produces that same error in the deduced longitude. An error in the measure of one second produces about two seconds of time in the longitude.

An error of one second of time in a lunar transit produces about thirty seconds' error in the longitude.

An error of one second of time in a lunar zenith distance will produce at least thirty seconds of time error in longitude, sometimes considerably more. An error of one second in zenith distance produces at least two seconds of time in longitude, sometimes considerably more.

An error of one second of time in an occultation produces one second of time in the longitude.

The same in the observations of eclipses of Jupiter's satellites.

In illustration of the part (6) transits of the moon, the following observa

tions and calculations, made at the station of Lake Memphramagog, in Canada, are given.

The transits of the stars and the moon's bright limb, are the mean of the five wires without any correction (which is unnecessary) for azimuthal error. LAKE MEMPHRAMAGOG, 1845.

Calculations for Longitudes from observed Lunar Transits.

L

-

L 2

Values of (12) — (125) &c., assuming the Longitudes at 4h. 48m. and

H. M. S.

H. M. S.

R. A.

= 19 0 37.05

+ bx (1)

R.A. 19 0 37.05

= +0 12 55.90 = 2.889354 + 0 12 5778=2.890859

In addition to the foregoing methods, it may be useful to lay down the process of determining the difference of longitude between two places by a chronometer. Suppose A and B to be these two places. At A find the error and rate of the chronometer, and then transport it to B. At the latter station take a time observation with the same chronometer; the time at B being known, and that at A being deduced from the in

*From the Corps Papers and Memoirs on Military Subjects of the Royal and East India Company's Engineers, vol. I., pp. 311, 318.

dication of the chronometer, the difference between these two elements is the difference of the longitude required.

In practice when a chronometer is carried over-land, it cannot be relied upon in furnishing a very accurate difference of longitude between two places, as its rate becomes liable to irregular variations from the jolting, attendant on its transport. But the uncertainty which arises from the employment of one chronometer may often be got rid of by the use of several, when the mean of all the results may be assumed as the true difference of longitude, between the two stations of observation.

In exploring new countries, or in accompanying armies on a foreign expedition, the time, necessary for making longitude observations, cannot be spared; in such cases the difference of longitude between two places may be determined by means of a route survey combined with Azimuth observations in the following manner:

It ought to be premised that the object of such a survey is not so much to lay down the road, as to fix with accuracy the positions of distant places; with this view the stations selected along the route, should be as few as possible, and not less than one mile apart. The line of the road may be followed whenever stations can be fixed thereon fulfilling these conditions. But as this is a circumstance which is not always obtainable in practice, the road sometimes deviating from a straight course, and passing through towns which obstruct a distant view in front; the trace of the route in such cases may be carried out of the direction of the road, so as to pass clear of the obstructing towns, which, if required, may be connected with the trace aforesaid by offsets, or by subsidiary routes executed with different or inferior instruments.

There ought to be at least three perambulators for executing the linear measurement, their errors being previously ascertained by rolling them over a distance fixed by a trigonometrical operation. Two perambulators would be insufficient, for in