excess of density required in the deeper parts of the crust, to make good the lightness of the alluvium, will from its position exert a positive attraction on a pendulum off the alluvium on either side, more than sufficient to counteract the negative effect of the lightness of the alluvium itself, for the latter will be nearly in the same horizontal plane as the pendulum and therefore unable to produce any appreciable downward pull, so that there will remain a somewhat greater attraction on stations off the alluvium than if the whole had been homogeneous. These considerations tend to remove the discrepancies between the results of observa- tion and those of calculation in the Gangetic and Sub-Hima- layan region and Sir Sidney Burrard concludes that the Gangetic trough is isostatically compensated.
The depth of compensation is not a quantity which should be regarded as definitely determined. Mr. Bowie finds a good deal of evidence for a smaller depth of about 96 kilometres. In average country the effect on the computed deflections, or values of gravity, of changes in the assumed depth are so small that the evaluation of the depth cannot be very precise.
Base-Line Measurements. The system introduced by E. Jaderin c Stockholm, of measuring with stretched wires, may be said to have s oerseded the older methods and it is improbable that any kind of b r will again be used for this purpose. It is now recognized, more- o^er, that nothing is gained, by measuring a short base with extreme ac:uracy as this accuracy is lost in the process of connecting the base with a side of the primary triangulatipn. The best course is to measure a side of the primary triangulation itself, but it is not often that this can be done, though the use of stretched wires instead of bars makes it possible to carry base measurements over much more uneven ground than of old. Jaderin's original plan was to use two wires having different coefficients of expansion and from the differ- ence between their lengths, as disclosed in the process of measuring, to deduce their temperature. In the United States, where tapes are generally used, uncertainty as to the temperature was much reduced by the expedient of making the measurements by night. The dis- covery of " Invar" by C. E. Guillaume almost entirely removed the need for these special precautions. A wire of the thickness generally used, namely I '65 mm., is not much heated by the rays of the sun and the error made in assuming that its temperature is the same as that of the air is not large. The average coefficient of expansion of invar is about 4X10"' per 1 C., so that an error of 25 C. in the adopted temperature of the wire would produce an error in the measurement of only 1/1,000,000. The methods of making measurements with invar wires have been closely studied by Benoit and Guillaume (12) and their procedure may be confidently followed.
For the standardization of the wires at the observatory, before and after the measurement of the' base, different methods of laying out a length of 24 metres, which is the usual length of the wires, have been employed. An apparatus designed by Sir David Gill for the trigo- nometrical survey of India is fully described in Engineering 1915.
The ultimate standard of length which was supplied with this apparatus is a nickel bar of H section one metre long. Standards for ordinary use are H bars of invar one metre and four metres in length. Invar has been observed to undergo a secular change in length which continues for many years. This constituted a serious drawback, but according to a recent investigation by Guillaume the instability is due to the presence of carbon which gradually forms cementite, Fe 3 C, with the iron. The addition of chromium, which has a greater affinity for carbon than iron has, prevents this, and an invar with ten- fold increased stability has been produced.
Triangulation. For measuring horizontal angles the use of theo- dolites with horizontal circles of more than 12 in. diameter is now unusual. For primary triangulation the use of opaque signals has now been almost entirely abandoned. Luminous signals, i.e. helio- tropes by day and lamps by night, are universally employed and there is a tendency to regard the night as the best time for observing horizontal angles, though for vertical angles it is necessary to choose, the time at which refraction is most steady, namely from about I to 3 P.M. : at this time of day refraction is also a minimum.
In countries where continuous sunshine is rare the night is no doubt preferable to the day, but where sunshine can be counted on the best results for horizontal angles, that is to say those least influenced by lateral refraction, are probably to be obtained from a combination of day and night measures.
Control of Triangulation. As triangulation extends from its initial base errors are generated and controls are required to prevent these errors from accumulating unduly. These are provided by the measurement of additional bases and by introducing Laplace points at suitable intervals along the triangulation. At Laplace point, where the azimuth is observed astronomically, and the longitude deter- mined by telegraph, a check is introduced on the triangulated azimuth precisely similar to that given by an extra base on the triangulated length of the side. The question to be decided is the proper interval at which bases and Laplace points should be intro- duced. When the error in length of side generated in the triangula-
tion is probably two or three times as great as that of a base it will be desirable to introduce a check base. The formulae of de Graaff Hunter (14) give the means of calculating the probable error accumu- lated in the triangulation.
In the first place a quantity M =(/+/)OT /i for each series
\ S
of the triangulation under discussion is to be computed. This quantity measures the precision of the series and enters into the determination of the .probable error. In it m = VSA 2 / 3 n is Ferrero's error of mean square of a single observed angle, / is a factor ranging from o to 1/6 depending on the type of figures of which the series consists, and / is the average length of side in miles.
Then P.E. in seconds of azimuth of terminal side of a series
P.E. in 7th place of log. of terminal side of a series
in which S is the length of the series in units of 100 miles. The summation is for different series for which values of M differ. If one straight series only is considered the above quantities become:
33-2MVL
where L is the length of the series in units of 100 miles. It may be pointed out that there is no symbolic difference between azimuth error expressed in radians and error in Napierian log. side, and with these units the same description applies to one or other.
If we take T&J as the probable error of a measured base it will be desirable to introduce a check base as soon as the probable error of the length of a side of the triangulation amounts to three times this quantity ; this stage is reached when
33-2MVL=3Xio 7 X log. (i+io- 8 ).
In first-rate triangulation the value of M will be about O-2, using this value L = 384 m.
If AA= astronomic-geodetic azimuth, Laplace's equation be- comes
AA.cosec Xo AA cosec X = (L L ) 1ST
where (L Lo) is the computed longitude difference and T is the difference between the local times at the two ends of the triangula- tion.
This equation serves to determine AA whence the true geodetic azimuth follows. The P.E. of an azimuth observation, 6A, may be estimated at about o"-2 in high-class work, that of T at about o"-O3, or 5L=o"-45, hence in latitude 45 the P.E. in azimuth determined from a Laplace equation is V(8A 2 o+5A 2 +8L 2 sin 2 X) =o"-4, which in radians is roughly 2Xlo 6 . This is twice as large as the value which was adopted as the P.E. of a base and shows that the precision of azimuth and longitude observations must be increased if they are to be brought up to the standard already reached in base-line measurements; that is to say, it indicates that in the case of triangu- lation in which M is as small as 0-2 it would not be justifiable to attempt to control the azimuths by means of frequent Laplace points unless the observations at the latter can be improved.
Determinations of Height. The precision of spirit levelling is so great as to justify the recognition of the lack of parallelism of the various level surfaces, each of which is approximately spheroidal. It is nowadays customary in levelling of high precision to apply to the observed differences of height the correction (i5) (16) for the convergence of these surfaces, that is to say the orthometric correc- tion, and to publish the orthometric heights.
As regards differences of height found by triangulation much improvement is called for. Refraction has always been a source of great uncertainty, and it has perhaps been looked on as more intract- able than it really is. Further research is required. Few field obser- vations provide suitable material for investigating the question, owing to lack of information as regards (a) plumb-line deflection, without which it is impossible to reduce the observations to the reference spheroid, and (b) rate of change of the density of the air with height, on which the refraction depends. A consideration (5) of the ordinary physical laws leads to a formula which represents very well the refraction usually met with, when these are not burdened with error due to neglect of plumb-line deflection.
Observed vertical angles are referred to the local geoidal vertical ; when reciprocal observations have been made at two points A and AI, if EI, wi, 81 are respectively the angle of elevation, the refraction and the deflection at AI towards A 2 and with changed suffixes for A 2 , then
where c is the angle between the verticals at AI and As. It has been customary to neglect 5j and 62 and to assume wi=u2 = Q, whence the equation becomes
fl = Ei+E 2 +c=o.
As S t +5 2 may easily exceed the error of observation and as o>i and uj are appreciably different, values of refraction so deduced are of little value. There is no reason why this state should continue. Properly reduced vertical angles will give good values of heights above the selected spheroid and the differences of these heights from the heights obtained by spirit levelling will reveal the separation of the geoid from this spheroid.
