APPENDIX TO PART V. A VERY portable and accurate instrument for the measurement of altitudes of the heavenly bodies is THE CIRCLE OF REFLECTION.* This instrument is constructed on the same principles as the sextant, the only dif ference being that the circle, as its name imports, has a limb which is a complete circumference, measuring on the doubling principle of optics already explained 720° instead of 360°. By having two verniers 180° apart, the instrument corrects its own eccentricity, and by having three, as in Troughton's construction, for error of figure and division, to some extent also. By reversing the face of the instrument the angle may be taken on what is called the off arc, that is, on the other side of the zero. This, by taking the mean, corrects for index error, so that with three verniers six readings may be obtained for one angle.. Dolland's circle has an inner movable circle, with a vernier upon it, in consequence of which it admits of the repeating process explained for the theodolite, at p. 242. The horizon glass and telescope are attached firmly to the inner movable circle, which has a clamp screw and screw of slow motion. The index glass is attached to an arm which moves freely around the centre, and is unconnected with the inner circle, telescope and horizon glass. This arm has a vernier and screw for clamping it to the outer circle, and a screw of slow motion. The repeating process is conducted as follows: Place the zero of the vernier of the inner circle clamped accurately at 720° on the outer, and move the free index carrying the other vernier forward until the two objects are brought in contact, as in observing with the sextant. Leaving this index fast to the limb, unclamp the inner circle, which carries the telescope and horizon glass, and move it forward also, not merely by the same amount, which would bring back the horizon glass to parallelism again with the index glass, but move it twice the angular distance necessary for this purpose; the horizon glass will now be inclined to the index glass, just as much as when the free index was first moved forward to produce the contact of the objects, but the inclination will be the other way. The contact may again be produced between the objects by the tangent screw of the inner circle, the telescope being Angles in any plane may be observed with the Circle of Reflection as with the Sextant. presented to the other object if the face of the circle be continued one way, but to the same object if the face of the circle be reversed. The reading now of the vernier attached to the inner circle would be twice the angle between the objects. If now the free index be moved forward a distance equal to the angle between the objects, the horizon and index glasses will be parallel again; and if it be moved forward still further by the same amount, the glasses will be inclined to each other, exactly as at first, and the contact of the objects may be made with the screw of slow motion attached to the free index, the face of the circle being as at first. Again, the inner circle is to be moved forward as before, over twice the angle between the objects, the face of the circle being inverted, the telescope directed always to one object, and the contact made with the screw of slow motion attached to the inner circle. The process above described is to be repeated until the vernier of the inner circle approaches near, or is a little past the 720 point again. The reading at which it stands, divided by twice the number of times that the inner circle has been moved forward, will give the angle subtended by the two objects corrected for all errors of division, centering and observation. If the angle between the objects be changing, as is often the case with celestial objects, the times of each contact of the object should be noted. The sum of the times, divided by the number of contacts, will be the mean time corresponding to the mean angle obtained, as already described. The Sextant and Reflecting Circle are used for taking altitudes on land, by the aid of a basin of mereury, called an artificial horizon. The telescope is presented to the reflected image of the sun or other heavenly body, seen in the mercury, and the angle between this and the sun in the heavens measured by moving the index forward. This angle will be double the altitude of the sun. The following example of an observation of the altitude of the sun for time will illustrate the mode of using this instrument. Observation with Repeating and Reflecting Circle, and Box Chronometer, of the sun's lower limb. The mean of the angles being 41° 52' 48' 5, and of the times, 24 31m 18.3, the former is the apparent altitude at the instant expressed by the latter, with no error of excentricity or graduation, and a probable error of observation that which would have been obtained without the repeating process. VARIOUS MODES OF DETERMINING LATITUDE. LATITUDE BY A SINGLE ALTITUDE. The data for this problem are the declination, the altitude, and the hour angle. We have thus three elements of the triangle SPZ (p. 296), viz., sp, sz, and the angle P, known, to calculate a fourth. This method requires an accurate knowledge of the time. If the latitude be known nearly, which it generally is, by means of the dead reckoning, it may be accurately found as follows. From (7) of Art. 72 we have which substituted in the formula deduced at Art. 82, written thus (1) cos acos b cos c + sin b sin c cos a (2) gives by (8) of Art. 70, cos acos (bc) — 2 sin b sin c sin2 A (3) substituting in (3) the sides and angles of zrs, calling the polar dist. 7, the zenith dist. 5, and the colatitude A, that equation becomes But if ' denote the meridian zenith distance of the heavenly body S' = (TX) (4) (5) The use of formula (6) is facilitated by Table XXIII., commencing p. 154 of Bowditch's Navigator, entitled log. rising, in which log. 2 sin2P is calculated for every value of the hour angle P. To this it is only necessary to add the logarithms of the sines of the polar distance and approximate colatitude, and we have the difference between the observed and meridian altitudes, or the correction to be applied to the former to obtain the latter, from which the latitude is calculated as at p. 280. A formula may be derived from either of the forms (B) of Art. 86. Applied after clearing of fractions to the present triangle it becomes sinP sin sin λ= sin(3+λ —7) sin § (3 + T − d) or from (5) above sin2 P sin a sin λ=sin(+ 3') sin ↓ (3—5') sin()—>') = sin? P sin sin λ cosec † (3+5) (7) which will serve to determine -', the correction for the zenith distance, if in the second member the value of ' be used, which would be given by the approximate value of the latitude. CIRCUM-MERIDIAN ALTitudes. If the heavenly body be near the meridian, P and -' will be very small, and writing the small arcs in place of their sines, to which they are sensibly equal (7) becomes (considering (=' in the second member) in which the value of the first member is expressed in terms of the radius as unity. To express it in seconds of arc, its value, as well as that of P must be divided by the sine of 1", which may be regarded as the length of 1", in terms of radius as unity. (8) thus becomes, striking out at the same time the common factor P2 1 2 sin 1" If p - in which is expressed in seconds of arc, as is also P the hour angle. denote the hour angle in time, 15p must be substituted for P, and (9) becomes The value of -' for the same station and star being proportional to p2, if it were calculated for a value of p=1", it might be found for any other value of p, by multiplying this by p2 in minutes of time. Table XXXII. of Bowditch's Navigator contains values of -' for p= 1m for all latitudes and all declinations less than 240, so that entering this table with the declination of the star and proximate latitude of the station as arguments, the number taken from the table multiplied by the square of the hour angle, in minutes and decimals of time, which is given in Tab. XXXIII. of Bowditch, for every value of p up to 13", will produce the correction to be applied to the observed altitude, to obtain the meridian altitude. -' may be calculated more accurately by means of a table (Tab. XXXVI.), P2 adapted to formula (9) above, in which is equal to the versed sine of P, as may be seen by referring to (2) on p. 94, from which, using only two terms on account of the smallness of P, we have Table XXXVI. gives vers P and the arithmetical complement of the logarithm of the sin 1" is 9.3144. N. B. The correction -' is subtractive from the observed zen. dist. to obtain ' the merid. zen. dist. or it is additive to the observed altitude to obtain meridian altitude. If several altitudes or zenith distances were observed near the meridian, and each reduced to the meridian by the above formula, the mean of the latitudes thence derived would be the true latitude more nearly in proportion to the number of observations. But the mean of the values for ', obtained from (10), since for each the quantities entering into (10) are the same except p2, may be obtained by taking the mean of the values of p2, and multiplying this by the constant factors. And if the mean of the values of -' be subtracted from the mean of the values of, the same latitude will result as if the latitude were calculated for each observed zenith distance, and the mean of all the latitudes taken, but with much less computation. EXAMPLE. Observed circum-meridian altitudes of O's lower limb, at head of Upper Mistigougiche Lake, latitude about 480, July 24th, 1841, with Repeating and Reflecting Circle, Artificial Horizon, and Box Chronometer. Vernier Reading 19° 26' 20'' <-' for p=1" in lat. 48°, dec. 190, Table XXXII. 2''•6 mean 2 .05 19 <-' for mean of hour angles=product 5-3 The times in the first column above are the instants of contact of the images as observed with the reflecting circle, according to the method described at p. 300. The numbers in the 2d column are obtained by subtracting those of the first from 114 12m 48 1, the time at which the sun is on the meridian. The numbers in the 3d column are the squares of those in the 2d, expressed in minutes and tenths. The mean of the values of p2 is 2.05, which, multiplied by 26, the value of ¿ —¿' lat. 48° and dec. 19°, gives 5'3, the correction required, by which the mean of the observed altitudes is to be increased, to produce the meridian altitude. The fourth column is for another mode of computation, the numbers in it being This of course depends on the equation of time, and the error of the chronometer. |