Page:Encyclopædia Britannica, Ninth Edition, v. 10.djvu/178

Rh 166 To detern1i11e the direction of the meridian, it is well to erect two marks at nearly equal angular distances on either side of the north meridian line, so that the pole star crosses the vertical of each mark a short time before and after attaining its greatest eastern and western azimuths. If now the instrument, perfectly levelled, is adjusted to have its centre wire on one of the marks, then when ele- vated to the star, the star will traverse the wire, and its exact position in the ﬁeld at any moment can be measured by the micrometer wire. Alternate observations of the star and the terrestrial mark, combined with careful level readings and reversals of the instrument, will enable one, even with only one mark, to determine the direction of the meridian in the course of an hour with a probable error of less than a second. The second mark enables one to c0111- plete the station 1nore rapidly, and gives a check 11po11 the work. As a11 instance, at Findlay Seat, in latitude 57° 35', the resulting azimuths of the two marks were 177° 45' 37"'29 =1: 0"'20 and 182° 17'15"'61 =1: 0"'13, while the angle between the two marks directly measured by a theodolite was found to be 4° 31' 37"‘-13 i 0"'23. We now come to the consideration of the determination of time with the transit instrument. Let fig. 3 repre- sent the sphere stereographi- cally projected on the plane of the horizon,—-ns being the meridian, we the prime verti- cal, Z, P the zenith and the pole. Let 2) be the point in which the production of the axis of the instrument meets the celestial sphere, S the posi- tion of a star when observed on a wire whose distance from the collimation centre is c. Leta Fig. 3. be the azimuthal deviation, namely, the angle 2:-Zp, 6 the level error so that Zp = 90° — 6. Let also the hour angle corresponding to p be 90° —n, and the declination of the same = m, the star’s declination being 8, and the latitude 96. Then to ﬁnd the hour angle ZPS = -r of the star when observed, in the triangles pPS, pPZ We have, since — Sin c=sin m sin 6+ cos m cos 6 sin (n— 1-), Sin m=sin 1) sin <p— cos b cos (1: sin a, Cos m sin n=sin b cos 4) + cos b sin «p sin a. And these equations solve the problem, however large be the errors of the instrument. Supposing, as usual, (1, b, m, n to be small, we have at once 7 = n +c sec 5 +722 tan 8, which is the correction to the observed time of transit. Or, eliminating m and 72. by means of the second and third equations, and putting 2 for the zenith distance of the star, at for the observed time of transit-, the corrected time is (1, sin ::+b cos :+c t+ cos 6 Another very convenient forn1 for stars near the zenith is this- -r=b sec <p+c sec 6+7: (tan 6-tan (,5). Suppose that in commencing to observe at a station the error of the chronometer is not known; then having se- cured for the instrument a very solid foundation, removed as far as possible level and collimation errors, and placed it by estimation nearly in the meridian, let two stars dif- fering considerably in declination be observed—-—the in- strument not being reversed between them. From these two stars, neither of which should be a close circumpolar star, a good approximation to the chronometer error can be obtained ; thus let :1, :2 be the apparent clock errors given by these stars, if 8,, 82 be their declinations the real error 1s GEODESY tan ¢p— tan 6, 6- 6l+(61_ 62 1:11:61 — tau 6._, ' Of course this is still only approximative, but it will enable the observer (who by the help of a table of natural tangents can compute a in a few minutes) to ﬁnd the meridian by placing at the proper time, which he now knows approxi- mately, the centre wire of his instrmnent 011 the ﬁrst star that passes—not near the zenith. The transit instrument is always reversed at least once in the course of an evening's observing, the level being frequently read and recorded. It is necessary in most instruments to add a correction for the difference in size of the pivots. The transit instrument is also used in the prime vertical for the determination of latitudes. I11 the preceding ﬁgure let (1 be the point in which the northern extremity of the axis of the instrument produced meets the celestial sphere. Let nZq be the azimuthal deviation=a, and I; being the level error, Zq = 90° — b ,' let also nPq == 7 and Pq = mp. Let S’ be the position of a star when observed on a wire whose distance from the collimation centre is c, positive when to the south, and let It be the observed hour angle of the star, viz., Zl’b". Then the triangles gPS', ql’Z give '— Sin c = sin 6 cos tl/- cos 6 sin up cos (h+7_, Cos dz = sin 1) sin <p+ cos b cos <p cos (1, Sin mp sin 7 = cos b sin (1.. N ow when (L and b are very small, we see from the last two equations that it: 43 — b, a=-r sin tb, and if we calcu- late cf)’ by the formula cot 43’: cot 5 cos /1, the first equa- tion leads us to this result— , (’LSll1Z+l)COS’C+C "°_ cos: ’ the correction for instrumental error being very similar to that applied to the observed time of transit in the case of meridian observations. When a is 11ot very smalland 2 is s111all, the formulae required are more complicated. The method of determining latitude by transits in the prime vertical has the disadvantage of being a somewhat slow process, and of requiring a very precise knowledge of the time, a disadvantage from which the zenith telescope is free. In principle this instrument is based on the proposi- tion that when the meridian zenith distances of two stars- at their upper culn1inations—one being to the north and thu. other to the south of the zenith—are equal, the latitude is the mean of their declinations; or, if the zenith distance of a star culminating to the south of the zenith be Z, its de- clination being 8, and that. of another culminating to thc north with zenith distance Z’ and declination 8', then clearly the latitude is ~l(5+5') + .‘_.('/. — Z’). Now the zenith telescope does away with the divided circle, and sub- stitutes the measurement micrometrically of the quantity Z’ - Z. The instrument (fig. 4) is supported on a strong tripod, ﬁtted with levelling screws 5 to this tripod is ﬁxed the azi- muth circle and a long vertical steel axis. Fitting on this axis is a hollow axis which carries on its upper end a sho"t transverse horizontal axis. This latter carries the telescope, which, supported at the centre of its length, is free to rotate in a vertical plane. The telescope is thus mounted excentrically with respect to the vertical axis around which it revolves. An extremely sensitive level is attached to the telescope, which latter carries a micrometer in its eye- piece, with a screw of long range for measuring differences of zenith distance. For this instrument stars are selected in pairs, passing north and south of the zenith, culminating within a few minutes of time and within about twenty minutes (angular) of zenith distance of each other. When a pair of stars is to be observed, the telescope is set to the mean of the zenith distances a11d in the plane of the
 * 2PS= 90 +7 — 72,