The correct lunar distance for time of observation :-

Lunar distance at Greenwich for 9 p.m. 48 10 20 for the given day 12 night 46 57 20

Difference in 3 hours 1 12 50

Correct lunar distance at Belfast for time of ob

servation Greenwich lunar distance, from above, at 9

We have now

p.m.

Difference of these lunar distances

1° 12′ 50′′: 0° 39′ 5′′-76::3 hours: 1h. 36 m. 37·08 s.

The fourth term of the proportion is the lapse of Greenwich time beyond 9 p.m. for the corrected lunar distance of the place and time of observation to obtain at Greenwich.

Then 9h. Om. Os. + 1h. 36m. 37·08s. =

h. m. s.

10 36 37.08 Greenwich mean time for corrected lunar distance. 10 13 01 mean time at place for do.

0 23 36.08 difference, or longitude expressed in time.

The longitude is west as the lunar distance is decreasing. The longitude may be expressed in degrees by allowing 15° for each hour of time. The above difference of mean time converted into degrees gives the longitude of Belfast 5° 54′ 2′′ nearly.

27. To find the error of the clock, astronomically (fig. 65).

In lat. 54° 36′ 0′′, on July 29, 1868, at 9h. 46m. 18s. of the sun's lower limb was observed = 44° 47' 4'4. Time of observation from apparent noon 2h. 13m. 42s.

a.m.,

the alt.

71 19 405

= 35 24 0

Sun's semi-diameter 0° 15 47"-5 from Nautical Almanac (1868).

=

Taking the formula for sin or tan, p. 191, we compute from these data, in the triangle A s n, as follows:

= 35 32 28.4 the hour angle from apparent noon.

This, expressed in time of 15° to an hour, is 2h. 22m. 10.2s.

h. m. s.

28. Latitude and longitude of trig. points (fig. 68). In this computation the angle which each reference line makes with a fixed line of direction, such as a meridian, should be known. Let 0 represent this angle, cz the meridian, and a c the given line. Then

These formula give the lat. and long. on rectangular co-ordinates passing through an origin, such as z, and the extremities of the line. If the origin be not z, but another point o, the lat. and long. of the point a will be represented by ox, and oy,. From an origin, such as o, the lat. from the axis oy for each consecutive angular point will be found by deducting from the sum of the previously computed latitudes for 0 in the first and fourth quadrants the sum of like lats. for 0, in the second and third quadrants. [The quadrants in this computation should be reckoned from the origin towards the angular point, right or left-with or contrary to the sun- —as the case may be. In what follows the origin will be considered as on the left side, and the reckoning with the sun.] The corresponding longitude will be found in like manner by taking the difference between the like sums of longitude in the first and second, and in the third and fourth quadrants.

The following example computations, and an inspection of diagram 68, will assist in making clear the manner and arrangement for obtaining the results in a tabulated form :

In the tabulated computation, column 1 contains the given data; 2 contains the logarithms of data and of the lat. and long.; 3 contains the lat. and long.; 4 the reduced lat. and long. to a remote origin o; 5 gives the reduced lat. and long. to rectangular axes passing through extreme angular points; 6 and 7 contain + and products; 8 contains logarithms of factors and products, and 9 the factors whose logs are inserted in column 8. For the purpose of finding the latitude and longitude for map making, or like purposes, columns 1 to 4 inclusive are those required. For the purpose of finding the area of a close, as in traverse surveying, column 4 is unnecessary. Column 9 is not generally introduced in practice, as all the factors are obtainable from columns 3 and 5.

It may be desirable to explain more fully the products in columns 6 and 7. The first entry in columns 6 or 7 is one half the rectangle of the increments or decrements of lat. and long. on the given line, and the second entry is the rectangle of the lesser reduced latitude of the extremities of the line, and the difference of longitude of the extremities. These products should be placed in columns 6 or 7, according as @ shall be less or greater than 180°. The difference of the sums of these columns, for a 'close' traverse will give the area in units of measure. If the unit of measure be a link, 100 of which shall be equal to 66 feet, or 40 × 5 yards, 100,000 square links will be equal to one acre. Hence, by pointing off, in square links, five places of figures to the right, we obtain an expression for the area in acres. The decimal part of this expression may be expressed in roods by multiplying by 'four,' the number of roods in an acre. In like manner the decimal parts of the latter, or roods, may be expressed in square perches by multiplying by 'forty,' the number of square