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206 Astronomical Latitudes. For the observation of latitude the Zenith telescope and the Tallcott-Horrebow method seemed to have super- seded all others (see 11.610), but the prismatic astrolabe and the method of equal altitudes have advantages for special purposes which entitle them to consideration. This instrument, which was invented by MM. Claude & Driencourt (17), determines the time and the latitude simultaneously, thus doing the work of the Zenith telescope and the transit instrument at the same time. Those who have used the astrolabe claim that a set of observations (18), which can be made with an instrument of small size in two hours, suffices to determine the time within one-tenth of a second, and the latitude within one second of arc. It is probable, however, that the time observation is liable to be considerably affected by the personal equation of the observer.

The principle of the prismatic astrolabe is that of equal altitudes. The axis of the telescope is horizontal. In front of the object glass an equilateral prism is mounted with its edges horizontal and the face next the object glass vertical ; below, and slightly in front of the prism, there is a bath of mercury. The rays of light from a star at altitude 60 strike the upper face of the prism perpendicularly and are reflected from the opposite face into the telescope ; the rays which fall on the mercury are reflected upwards, strike the lower face of the prism at right angles, are then reflected from the opposite face and enter the telescope parallel to the rays which reached the prism direct; thus two images of the star are formed in the telescope which approach each other as the star approaches the altitude of 60 and then separate again. The observation consists of noting the time at which the images pass each other. Each observation of this kind gives a locus, analogous to a " Sumner " line in navigation, on which the zenith of the station of observation must lie. Three such observa- tions to suitably situated stars should, if everything were perfect, result in three concurrent lines, but will in general produce three lines forming a triangle and care must be used in deciding on the true position of the zenith, which will not necessarily be inside the triangle. If four stars are observed the case is clearer and it is advisable therefore to observe four stars as a minimum ; of these one should be in each quadrant of the heavens; that is to say, one to the N.E., one to the S.E., one to the S.W., and one to the N.VV. of the station at the moment of crossing the circle of 60 altitude. When the observations have been made the chronometer error and the latitude of the place can be deduced from them by means of a simple graphical construction which gives results as accurate as the precision of the observations allows. If certain preliminary computations have been performed beforehand the graphical method can be carried out very quickly and the results obtained in a few minutes after the comple- tion of the observation.

Longitudes. The development of wireless telegraphy (see SUR- VEYING) has removed the chief difficulty of determining the longi- tude and there is now no reason why astronomical longitudes should not be used as freely as astronomical latitudes.

The time signals emitted by the Eiffel tower in Paris are of an accuracy superior to that of any ordinary determination of local time, and they have been picked up without much difficulty (1921) by means of a portable apparatus at a place as far from Paris as Dehra Dun, India, to which the distance along a great circle is about 60. The " Scientific time signals " are sent out from Paris at 23-00 C.M.T. (civil). After certain warning signals, 300 dots are sent at equal intervals, the whole occupying about 293 seconds. These are heard in a telephone in which the clock-beats are also produced. It is also necessary to note the time at which there is a coincidence between a dot and a clock-beat. The precise time of the first and last of the 300 dots is also signalled and a simple computa- tion connects the clock with G.M.T. With good hearing conditions there seems little room for personal equation in this part of the observation. In the determination of the local time there is less certainty; unexplained differences of o-l and even 0-2 occur.

Thus in the Paris Washington arc, 1913-4, the values obtained by the interchange of the American observers were :

h m 8 h m B

5-17-36-549 -0051 and 5-1 7-36-758 =*= -0027

A similar difference was obtained when the French observers were exchanged. Two separate determinations of the Greenwich-Paris

ml m s

arc gave 9-20-977 and 9-20-910. These differences have been the subject of discussion (19), and are receiving a good deal of attention ; it is to be hoped that the source of error will be discovered and that longitudes correct to o-oi sec. will be obtained.

Longitude determinations are required in order to provide Laplace points for the control of the triangulation and also for the investiga- tion of the deflection of the plumb-line. The use of wireless telegraphy will greatly facilitate the former because in the past it has often been difficult to find a place where it was possible to get a connexion with the telegraph wires and also to make satisfactory azimuth observa- tions. By the multiplication of longitude stations knowledge of the deflection of the plumb-line in the prime vertical will be much improved, especially in low latitudes where the effect of such a deflection on the azimuth is so small that the comparison of observed and computed azimuths is incapable of giving trustworthy results.

Gravity. A very important addition to the half-seconds pendulum apparatus designed by Col. R. von Sterneck (see 8.809) of the Aus-

trian survey is the means of measuring and correcting for the move- ment set up in the stand by the swinging pendulum. This movement is named by German observers " das Mitschwingen"; in the Coast and Geodetic Survey of the United States it is called "the flexure." Probably the best English word to describe it is " the sway." Two methods of measuring it are in use. The C. and G.S. measure the actual to-and-fro movement of the top of the stand by means of an interferometer (20). The effects of different amounts of movement on the pendulum's time of oscillation are found by experiment and an empirical formula is used for reducing this time to what it would have been if the stand had been perfectly rigid. The movement is expressed as a fraction of the width of one of the bands or fringes in the interference grating. Using monochromatic sodium light it was found that a movement of p- 1 fringe corresponded to a correction of l7 s -8Xio"' in the time of vibration of the half-seconds pendulum. A movement of o- 1 fringe means a linear displacement of the pendulum support of 0-029/4.

The other method introduced by Prof. Schumann (2Oa) of the Prussian Geodetic Institute is as follows: Two pendulums of equal period are suspended on the stand, with their knife-edges par- allel to each other and in the same horizontal plane. Of these one is set swinging, the other being at rest, and the rate at which the latter acquires an oscillation, from the swaying of the stand induced by the swinging of the former, is a measure of the rigidity of the stand and affords the means of computing a correction to the time of oscillation. With this method the probable error of the correction derived from four observations, two taken before and two after a series of pendu- lum swings, is about o"-5 X io 7 . The amount of the correction where circumstances are favourable may be expected to be about 4O 3 X io 7.

The absence of means of determining the correction for sway was perhaps the principal source of uncertainty in the results of the older series of pendulum observations and the methods which have beer described formed an important advance.

Numerous observations with pendulums for the determination of the force of gravity have been made of recent years in various parts of the world and the effect of applying the theory of isostasy to the computation of the normal value has been extensively tried in America and in India. By normal value is here meant that value which is arrived at by the use of the formula which gives the force of gravity at sea-level in any latitude and the application to that value of the correction for the height of the station above sea-level, and for the attraction of the topography of the whole earth with its isostatic compensation.

The formula deduced by Helmert from the discussion of a large number of observations in different parts of the world was yu =978-030 (i +o-oo53O2sin 2 < o-ooooo7sin 2 2<)

(see 8.801).

For the area of the United States it was found that an equatorial value of 978-039 (9) agreed better with the observed results, and in India the value of 978-041 (21) was found the best.

In 1915 Helmert (3) deduced a formula from the 3,000 stations which had been reduced to the Potsdam system by Borrass in 1912, in which he supposes the equator to be an ellipse so that the longitude L as well as the latitude has to be taken into account. The formula

sin 2 < o-ooooo7sin 2 2< +o-ooooi8cos 2 <#> cos2(L + i7) \ 4 6 j

This formula is applicable to continental stations situated at least loo km. from the edge of the continental shelf, that is from the 110- fathom (2OO-metre) line.

It is assumed that the semi-axes of the equatorial ellipse differ by 230 metres and that the major axis lies in 17 W. longitude. Clarke (4) in 1878 deduced an elliptical equator in which the semi-axes differed by 465 metres, the major axis lying in 8is' W. longitude.

The advantage of using an elliptical equator seems rather ques- tionable; it would appear preferable to regard inequalities of the equatorial radius as deviations of the geoid from the spheroid of reference rather than as fulfilments of a mathematical law, but it is of interest to know that the geoid presents such an irregularity.

For measuring the force of gravity at sea various kinds of appara- tus have been tried. The underlying idea is to balance the weight of a column of mercury against a pressure, which is either constant or measurable, and which is independent of gravity.

Hecker used the pressure of the atmosphere as in an ordinary barometer and measured it by means of boiling-point thermometers. Duffield has tried an aneroid in place of the boiling-point thermom- eter and more lately has balanced the mercury against a constant mass of air, the whole apparatus being closed. The success so far attained is not very great but the last-mentioned device seems ca- pable of improvement and may prove a satisfactory instrument.

The variation of gravity on land has been studied by Baron Roland Eotvos (23) (24) (25) with an instrument of quite a different kind, namely, the Gravity Torsion Balance. This instrument does not measure the force of gravity but the rate of change in the force of gravity at any point and the direction in which that rate is greatest.

The force of gravity 7, in latitude <#>, is given by the formula 7=978-00(1+0-00531 sin 2 <) C.G.S., hence the change in 7 corre- sponding to a change of i" in latitude in latitude 45 is 25-2X10-*,