(155.) The altitude of ẞ Orionis was observed when the chronometer showed 6h 10m 25s, and the altitude of a Hydra was observed when the chronometer showed 7h 17m 353, required the polar angle between the two places observed, the rate of chronometer being 65-3 losing, and the right ascension of ẞ Orionis 5h 7m 15s, and of a Hydræ 9h 20m 85.2. Ans., 3h 5m 32o. Rule XXXIX. Given, the altitudes of two heavenly bodies observed at different times, to find the latitude. 1. Proceed as in (1) and (2) p. 150. 2. Find the polar angle as in Rule 38, p. 155. 3. Find arc 1, as in (4) p. 150. 4. Then proceed to find arcs (2) (3) (4) &c., as in Rule 36, p. 145. EXAMPLE. Sept. 27, 1846, in latitude by account 43° 30′ N., the following altitudes of the stars a Pegasi and a Aquila were observed at different times. the index correction + 1' 10" and height of eye above the sea 20 feet, required the true latitude at the second observation. Latitude by Ivory's rule for double altitude. Let x and y be the place of the sun or a star at the times when its altitudes are taken. Then we have given the polar distances Py, PX, the zenith distances Zy and ZX, and angle xpy to find the colatitude PZ, and thence the latitude. W P M Bisect xy (an arc of a great circle passing through x and y) in м, join PM and z м and draw ZE at right angles to PM. Then PM is at right angles to x y, and z M x is the complement of z M P. We have to compute the following arcs: XM, MP, ZE, EM, and M P EM PE. Then knowing z E and PE in the right-angled triangle ZP E, we can find P z the colatitude. If the great circle drawn through x and y pass when produced between the pole and the zenith, the perpendicular ZE will fall without the triangle PZ M; in this case M P + E M = PE, and P E is formed by adding M P and E M together. We may, however, determine whether the sum or difference is to be taken, by considering that since PZ must always be less than 90°, PE must likewise be less than 90°, and therefore if MP + EM exceeds 90°, we may be sure that PE = MP — EM or that PE is found by taking the difference between M P and E M. The investigation from which the following rule is deduced will be found in the author's volume of "Astronomical Problems and Solutions." Rule XL. (Ivory's Rule.) 1. From the time shown by the chronometer or watch at the second observation (increased if necessary by 12 hours) subtract the time shown at the first observation, divide by 2; the result is the half polar angle in time. 2. To the estimated mean time at the ship at the first observation add the half polar angle; the sum will be the ship mean time at the middle time between the observations. 3. Apply the longitude in time, and thus get a Greenwich date. 4. Take out from the Nautical Almanac the declination for this date, and also the sun's semidiameter in the adjacent column. 5. Correct the observed altitudes for index correction, dip, semidiameter, and parallax and refraction. 6. Correct also the first true altitude for run of ship in the interval, and thus get the true altitudes for the same place. 7. Put the first true altitude under the second true altitude, take their sum and difference, and also the half sum and half difference, call the half sum S. and the half difference D. 8. Under the log. sin. half polar angle put log. cos. declination at the same time take out and put a little to the right, the log. sin. declination. 9. Add together the two logs. first taken out, and call the sum sin. arc 1. 10. At the same opening take out sec. arc 1, and put it under the log. sin. declination; take out also and put down in the same horizontal line the log. cosec. arc 1 and also log. sec. arc 1. 11. Add together log. sin. declination and log. sec. arc 1; the sum will be log. cos. arc 2; the arc corresponding thereto found in the Tables will be arc 2, if the latitude and |