Suppose that about paft four P. M. on the 26th Nov. 1809, in lat. 54° 25'S. long. by account 10° E. fix obfervations were made, the mean of which were taken at 4hs. 44m. and the altitude was 27° 42′ 35′′ the error of the inftrument, 24", to be added, the eye of the observer 21 feet above the furface of the sea, required the true time? H. M. " 4 44 obf. alt. L. L. 27 42 35 zen. dift. 62 6 49 + 24 co. lat 35 35 co-fec. o 23516 16 14 27 42 59 4 40's fe.di. 16 Dip 62 649 н. м. Co. lat. 35 35 35° 30' fine 9 91069 71 oin time 4 44 On the fame evening the following obfervations were made of the distance of the star Regulus from the moon's fartheft limb, lon. by account as before, and the error of the inftruments by which the moon's altitude and diftance were taken was 7' 30' 25" to be added; the true longitude is required? Here I have given one method of finding the longitude, illuftrated by a fufficient number of examples, all of which are reduced to the year 1809, in order that the reader, or teacher, may have fufficient time to furnish himself with a N. A. for that year, which is now printed. But as many would wish to have fome other method of reducing the diftance, that, by comparing them together, they may not only have the advantage of proving their calculations, but alfo of making choice of which they prefer to work by; the fecond method I fhall prefent the Reader with, is chiefly deduced from that invented by Mr. Witchell, late Matter of the Royal Academy at Portsmouth, as it is fhort, and requires but four places of figures in the logarithms, befides the index; the preparations in both methods being exactly the fame. RULE. Firft. Add the fun, or ftar's and moon's apparent altitudes together, half the fum; fubtract the lefs from the greater, and half the difference; then add together, the co tang. of half the fum, the tang. of half the difference, and the co-tang, of balf the apparent distance; their fum (rejecting 20 in the index) will be the log. tang. of an angle, which call A. Secondly. When the fun or ftar's altitude is greater than the moon's, take the difference between angle A, and half the apparent distance; but if lefs, take their fum. Then add together the co-tang. of this fum or difference, the co-tang. of fun or ftar's apparent altitude, and the prop. log. of the correction of the fun or ftar's altitude; their fum (rejecting 20 in the index) will be the prop. log. of the first correction. Thirdly. If the fum of ang'e A and half the diftance was taken in the last article, take now their difference, but if their difference, now take their fum; then add together the co-tang. of the fum, or difference, the co-tang. of the moon's apparent altitude, and the prop. log. of the correction of the moon's apparent altitude; their fum (rejecting 20 in the index) will be the proportional logarithm of the fecond correction. Fourthly. When the angle A is lefs than half the apparent diftance, the firft correction must be added to, and the fecond fubtracted from, the apparent diftance; but when the angle A is greateft, their fum must be added to the apparent distance, when the fun or ftar's altitude is less than the moon's; but when the moon's altitude is leaft, their fum must be fubtracted to give the corrected distance. Fifthly. In Table XXVI. look for the corrected dift. in the top column, and the correction of moon's alt. in the left-hand fide column; take out the number of feconds that fland under the former and oppofite to the latter. Look again in the fame Table for the corrected diftance in top column, and the fecond correction in the left-hand fide column; take out the number of feconds that stand under the former and oppofite the latter, the difference be 2 G 2 twean tween thefe two numbers will be the third correction, which muft be added to the corrected distance, if less than 90°, but fubtracted from it, if more than 90°; the fum, or difference, will be the true distance. To illuftrate this laft method of reducing the apparent distance to the true distance, I fhall take the apparent altitudes and distances as they stand in the firft examples, worked by the former method. EXAMPLE I. See Example I. p. 231. Given, the apparent distance of the fun and moon's centres, 105° 10' 0", the fun's apparent altitude 12° 51', that of the moon 43° 32', and horizontal parallax at reduced time 57' 49". Required the true distance of their centres by Mr. Witchell's method? M. S. O's refrac. 4 6 D's hor. par. at red. ti. 57 49 P.L. O 4932 O's parallax 9 D's ap. alt. 42 32 Sec. O 1397 Sum 56 23 1 0 Half fum 28 11 Co-tang. 10 2710 Diff. 30 41 o Half diff. 15 20 Tang. 94381 Ap. dift. 105 10 o Half dift. 52 35 Co-tang. 9 8837 +38 ift. cor. 105 13 8 Arc A 21 23 Tang. 95928 23 33 Sum 73 58 Co-tang. 9 4584 O'sap.alt.12 51 Co tang. 10 6418 16587 3 EXAMPLE II. See Example p. 232. Given, the apparent distance of the fun and moon's centres 68° 42' 11", the fun's apparent altitude 32° 0′ 12′′, apparent altitude of the moon 24° 0' 10", the fun's correction 1' 23", the moon's correction 51' 30". What is the true diftance of their centres by Mr. Witchell's method? fun's O's ap. alt 32° 0 12" D's ap. alt. 24 Ο 10 Sum 56 0 22 Half fum 28° 0' 11" Co-tang. o 2743 Diff. 8 O 2 Half diff. 4 0 I Tang. 8 8447 Ap. dift. 68 42 11 Half dift. 34 21 5 Co-tang. o 1653 EXAMPLE III. See Example p. 233. Given, the apparent distance of the moon's centre from the ftar Regulus 31° 18' 30", the apparent altitude of the ftar 20° 10' 55", that of the moon 31° 18′ 30′′, the star's correction 2′ 34", that of the moon's correction 48′ 33′′. What is the true diftance of their centres by Mr. Witchell's method? |