Total Latitude for any Sta. =

Total Latitude for preceding Sta. + Lat. for preceding Sta. Total Departure for any Sta. =

Total Departure for preceding Sta. + Dep. for preceding Sta. The term "Latitude" is an abbreviation of "Difference of Latitude." The terms "Cosines" and "Sines" are more appropriate when the bearings are kept with no particular reference to the true or magnetic meridian.

By the aid of cross-section paper (if true to scale) we can plot the survey from the notes with only a straight-edge. Thus, e.g., to find the position of Sta. 26 + 50, we read off along the N. and S. base a distance to the north equivalent to 164.5 feet, and along the E. and W. base a distance to the east equivalent to 2382.5 feet; the intersection of the coordinates from these two points gives the position required.

On a long plan, if we have the base-lines drawn straight, and points accurately scaled off along them at, say, every 1000 feet, there is very little chance of making an appreciable error in the plotting of the plan if the notes are correctly worked out. But although this method is undoubtedly the best, unless the notes are well checked, it is very liable to give rise to errors owing to arithmetical mistakes in the notes themselves. But where good work is wanted, and in cases where probably the method of plotting by "chords" or "natural tangents" would otherwise have been used, the method of Latitudes and Departures, well checked, gives far better results, and probably takes no longer than the other ways.

By

56. The only way in which to feel sure that there are no appreciable mistakes in the transit-work is to check the bearing of the alignment every now and again by an observation for azimuth. This should be done, if possible, before starting the survey, or in any case as soon after as possible, and the notes then already taken reduced to their true bearings. taking the magnetic pole as the standard of our bearings, we have no means of applying an accurate check to the work at a later period; but if we start with the vernier at zero, when the telescope is pointing to the true north, we can then check our course at any time on the survey.

Engineers generally fight rather shy of anything in connection with astronomical work; but considering that it is almost as easy to check the alignment by means of a star as by any known point on the Earth's surface,-and usually much more accurate,—it is a great pity that observations for azimuth are not used more frequently than they are. It is so much more satisfactory for the transit-man himself to know if he is doing good work; and considering that the transit-line is usually taken as the basis of all the plans to be afterwards constructed, every possible means of checking the work should be used.

57. The handiest methods of obtaining the true north are the following, one of which is applicable in most northern latitudes about every 6 hours, and can be applied without any knowledge at all of astronomical work:

A. By a Maximum Elongation.-In Fig. 10 let
Z represent the zenith,

W

Z

FIG. 10.

Then the small circle round the pole shows the path and direction of the star's motion, the time taken in making the circuit being nearly 24 hours. Now the radius of this small circle in angular measure is only about equal to 11° (or 21 diameters of the sun), so that the apparent motion of the pole-star in azimuth (i.e., horizontally) will, when due east or west, be nothing at all, and for several minutes together when about east or west the motion will be inappreciable to ordinary railroad transits. Thus if we know about what time the star will be at its east or west elongation,-i.e., due east or due west,--and also the amount in azimuth by which when at those points it will be distant from the pole, we can, by setting the telescope on the star when at either of its elongations and applying the required correction in azimuth, obtain the direction of the true north. The following table shows approximately the times at which the elongations will occur. The amount of the correction in azimuth, which really equals the angle WZP (or EZP), may be found by solving the spherical right-angled triangle WPZ, the angle at W being 90°, the side ZP being equal to 90°-the

WP

"declination" of the star. For Declinations of Stars see Table in Sec. 213. Thus we have

Sin azimuth = cos (dec.) sec (lat.),

PZ being the complement of the latitude of the place of observation. Thus suppose in latitude 50° N., in January 1889, we have the telescope clamped on Polaris at its eastern elongation, the vernier reading 2°.05'; then the sine of the azimuth correction = .0349, which gives a value for the correction of 2°.00, so that the telescope will be pointing due north when the vernier is set to read 0°.05'. (See note D, Appendix.)

TIMES OF ELONGATIONS OF POLARIS.

Month.

1st Day.

11th Day.

21st Day.

Eastern. Western. Eastern. Western. Eastern. Western.

Although the hour-angles from which the above times are calculated vary year by year and in different latitudes, they may be considered to be sufficiently correct between the years 1890 and 1900, and between latitudes 25° and 65° N. Where extreme accuracy is wanted, the time of observation may be calculated as in note D, Appendix. The above times increase by about 4 minutes every 10 years. But as these elongations occur only at intervals of 12 hours, more or less, it is well to have some other means of obtaining the true north, which can be used when the above method is inapplicable. The two following are similar to one another in principle, but occur about 12 hours apart, and from 5 to 7 hours from the time of the elongations given above.

B. In Fig. 11 let P be the pole and S the Pole-star, and let A

represent Alioth (e Ursa Majoris), and C represent the star "Gamma" (y) Cassiopeia. The arrows and dotted lines show the paths and the directions of the

The

motion of the three stars.
positions of the stars in the figure
are those which they would oc-
cupy about the time of the western
elongation of Polaris; but since the
complete circuit occupies about
24 hours, we see that in about 6
hours will be about vertically
under S. When this occurs (i.e.,
when S and C are in the same
vertical plane), clamp the tele-

FIG. 11.

scope on Polaris, and wait through an interval of time which

2 is to be found from the interval of 29 minutes 30 seconds for

Jan. 1, 1889, by applying for any later date an annual correc 77 tion of 19 seconds. After the lapse of this interval Polaris will be due north.

C. The third method consists in making use of Alioth in a similar manner to that in which we have just made use of y Cassiopeia. But in this case, when Alioth is vertically below Polaris, Polaris will be nearly at its upper "culmination" (or "transit," as its passage across the meridian is called), but this makes no difference in the mode of procedure. The interval to wait when using Alioth was, on Jan. 1, 1889, about 27 minutes, and increases annually by 17 seconds. To calculate the above intervals, see note E, Appendix; but for ordinary work the figures given above are sufficiently correct as far north as 70°, and as far south as A or Care visible at their lower culminations. The altitude at which C or A will be above the horizon when due north equals about

so that observations B and D cannot practically be used farther south than latitude 35° N. If, however, the instrument has a reflecting eye-piece, if either observation B or C is needed farther south than these limits, A and C can be used at their upper culminations, which will take place near the zenith; the intervals of time and modes of procedure will be the same as for the lower culminations.

To obtain the azimuth of Polaris at any time see Sec. 202.

There can be no difficulty about finding these stars if it is remembered that the altitude of the pole-star is about equal to the latitude of the place; that the "pointers" pp, Fig. 11, point towards it; that A and C are each about 30° from the pole-star; C, A, and S being all three more or less in a straight line.

The remarks made in Sec. 45 regarding the vertical axis, etc., should be carefully attended to. The times at which observations B and C will occur can be found near enough by noticing the positions of the stars themselves.

In observation A the instrument should be "reversed" on the star at the elongation. In observations B and C, where the star's motion in azimuth is comparatively rapid, observe, say, 2 minutes before the star is due north, and then again 2 minutes after its transit: the mean result should then be taken. An error of about 2 minutes in time in observations B and C causes an error in azimuth of about 1'. The verticality of the two stars should be also tested by a reversal of the instrument. 58. In checking the line by an azimuth observation as already described, it must be borne in mind that the convergence of the meridians needs a very important correction in the bearings relating to other points east or west of the place where the observation is taken. This may be best shown by means of Fig. 12.

Let ONEF represent a sector of the northern hemisphere, and let A be the point on the earth's surface at which the survey was started, a continuous "straight" line being run

B

which had at A a bearing due west. After we have traversed a difference of longitude which is represented by the angle EOF (or the spherical angle N) and have arrived at C, we shall be considerably south of the point A, our line having taken the course AC in the figure: so that, if at C we take an observation for azimuth, we shall find our line to have a bearing considerably south of east; and similarly all straight lines run from A, either towards the east or west, have a tendency to run to the south; similarly in the southern hemisphere they would have a tendency to run to the north. Thus in order to run a line from A to a point B, keep

F

FIG. 12.