284
ASTRONOMY.
one of the morning of Jan. 11, 1813, according to commoni eckoning.
Eiainjylo 3. 'Required-the t-ue time of the apparent and realrising ot Algol, (a variable star commonly of the second magni-tude, in the head of Medina,) at the Cape of Good Hope, onDec. 1, 1S10 i the star’s right ascension at that time being 2h.Stint. 52s. its declination 40° 13' 9 " north, the Sun ’s right ascensionat apparent noon, in the meridian of the 11 oval Observatory atGreenwich , lhh. 27m. 52,9s. his daily variation in right ascension4m. 19is. the latitude of the C,.pe 34" 29' syuth, and its longitude18° 23' 15" east from Greenwich .
Wc must begin with reducing the Sun ’s right ascension for theapparent noon of the meridian of Greenwich , to apparent noonin tiie meridian of the Gape, as follows: 360°, the Sun ’s apparentambit-round the Earth in 24 hours, is to 4m. 19)*. t he variation of theSun ’s right ascension in 24jh<nirs; so is 18° 23' l .V E. the longitude ofthe Cape, to 13~s. the difference of the Sun ’s right ascen-ion atapparent noon in tiie two meridians. 'This last found quantitysubtracted from Kill. 27m. a2,9*. because tiie place h to the cast,the Sun ’s right ascension being alwavs progressive, the remainder,i(ih. 27m. 3p,6s. will be the right ascension required. We mustnext proceed in fading the apparent time of the star’s passing themeridian, as instructed in pp. 2S 1,282; thus:
Eirst Approximation. The Sun ’s right ascension at noon,in the meridian of the Cape, 161i. 27m. 39,0s. subtracted from thatof the star, 2h. 56m. 52s. the remainder, lOh. 29m. 12,4s. is thefirst approximate time of the star’s passing the meridian.
Second Approximation. Subtract two minutes from thetime found by the first approximation, because the star’s passageover the meridian is after 9 hours,and the remainder, 10b. 27m.12,4s. is the second approximation of Algol’s passing the meridianof the Cape,
Third Approximation. 24 hours : 4<n. 191s. the Sun ’sdaily variation in right ascension, : : 10b. 27m. 12,4s. the time ofthe star’s meridian-passage, as obtained by the second approxima-tion : lm. 53,027s. This last quantity subtracted from the firstapproximate time of tiie star’s passing the meridian, leaves lOh.27m. 19,37s. for the correct time of the star’s southing.
We must, in tiie third place, find the ascensional difference, as
follows:
Is to the tangent of the star’s declination, 40° 13' 9".9271850As the tangent of the latitude of the place, 34° 29'.9,8368636
Is to the sine of the ascensional difference, 35° 30' 32"....9,7640486
The ascensional difference turned into time, at the rate of 15°per hour, gives 2h. 22m. 2s.; this subtracted from 6 hours, be-cause the declination of the star, and the latitude of tiie place,are of a different name, i. e. the one is north, and the other south,the remainder, 3h. 37m. 58s. is the correct semidiurnal arc ofAlgol at the Cape. If 3h. 37m. 58s. the star’s semidiurnal arc, besubtracted from lOh 27m. 19,37s. the apparent time of the star’spassing the meridian, the remainder, 6ii. 49m 21,37s. will be theapparent time of the star’s real rising at the Cape at the proposedtime; but if 3h. 37m. 58s. be added to lOh. 27m. 19,37s. the sum,14h. 5m. 17,37s. will be the true time of its setting required.
Lastlv, To find the quantity which refraction accelerates therisinsr, and retards the setting, there is given the latitude of theplace, 34° 29', the declination of Algol, 40° 13'9", and die hori-zontal refraction, 33' 30", which reduced to time is equal to 2m.14s. With these we find the required quantity as follows: (See-
the preceding column.)
Logarithm-cosine of (l -{- (1) 74° 42' 9”.9,4213257
Logarithm-cosine of (d — /) 5° 44' 9".9,9978211
Sum of the above logarithms.2)9,4191468
Half the sum.9,7095734
Which subtract from the logarithm of I34s.2,1271048
And the remainder is the log. of 4m. 21,53s.2,4175314
If this last-found quantity be subtracted from Gli. 49ni. 21,37s.became the refraction is greater than the parallax, the latter in thiscase being insensible, the remainder, 6h. 44m. 59,84s. will be thetrue time of Algol’s apparent rising at the Cape, and 4m. 2),53s.
added to 14b. 5m. 17,37s. will give 14h. 9m. 39,9s. the apparenttime of the star’s real immediately subsequentsetting. Q. K. E.
Example 4. Required the apparent time oftlie true and apparentrising and setting ot Sirius, at Cape Horn , on Jan. 29. 1811 ; theSun’s rigid ascension at noon in the meridian of Greenwich being20h. 44m. 48,7s. his daily difference in right ascension 4m. 7s. thestar’s declination 16° 29' 2'S. and right ascension Oh. 37m. 31,3s.;the latitude of Cape Horn 55° 5b' 30”S, and its longitude 67°26' 0" west from Greenwich .
We must begin with reducing the Sun ’s right ascension for theapparent noon oftlie meridian of Greenwich to apparent noon inthe meridian of Cape Horn , as follows: 360°: 247 seconds oftime, the daily difference of the Sun ’s right ascension, : : 67° 20',the longitude of Cape Horn to the west of Greenwich , : 46,26s.the diiiennce oftlie Sun ’s right ascension at aoparent noon in thetwo meridians. This last-found quantity added to 20h. 44in.48,7s. because the place is to the w est, the sum, 20h. 45m. 34,96s.will be the r'-ght ascension required. We must next find the ap-parent tuna ot the star’s passing the meridian of Cape Horn , asfollow s:
Eirst Ai-troximation. The Sun ’s right ascension at noon inthe meridian of Cape Horn , 20h.45m.’34,!,6s. subtracted from thatof the star, Oh. 3im. 31,3s. the remainder, 9h. 51m. 56,34s. willbe the first approximate time of the star’s southing. '
Second Approximation. Subtract two minutes from thetime found by the first approximation, because (lie star’s passageover the meridian is after 9 hours, and the remainder, 9h. 49m.56,34s. will be the second approximate time of the star’ssouthing.
Third Approximation. Twenty-four hours: 247s. the Sun ’sdaily variation in right ascension,:: 9h. 49m. 56,34s. : lm. 41,2s.'I’his last quantity subtracted from tiie first approximate time ofthe star's southing, leaves 9h. 50m. 15,14*. for the correct time ofthe star's passing the meridian. Eor the ascensional difference usethe following proportion:
Radi us ....... i.10,000000O
Is to tangent oftlie star’s declination, 16° 29' 2". 9,4711562
As the tangent of the latitude of the place, 55° 53' 30".. 10,1706039
Is to the sine of the ascensional difference, 25° 59' 41'.. 9,6417601
The ascensional difference turned into time, at the rate of 15°per hour, gives Ih. 43m. 59s. and added to 6 hours, because thelatitude of the place and the declination of the star are of the samename, the sum, 7h. 43m. 59s. is the star’s semidiurnal arc at CapeHorn . If 7h. 43m. 59s. the star’s semidiurnal arc, be subtractedfrom 9h. 50m. 15,14s. tiie time of the star’s southing, the remain-der, 2h. 6m. 16,14s. will be the apparent time of Sirius’s realrising; but if 7h. 43m. 59s. be added to 9h. 50m. 15,14s. thesum) 17h. 34m. 14,14s. will be the apparent time of the star’sreal setting.
To find the quantity w hich refraction accelerates the rising and re-retards the setting, enter the Table (page 282,) with the declinationand latitude of the place, and after making the necessary proportions4m. 43s. is found for the effect of retraction if the declinationhad been north, but it being south in the present case, 4s. musthe subtracted from it according to the direction given in thelast column of the Table ; so the whole effect w ill he 4m. 39s.This quantity subtracted from 2h. 6m. 16,14s. gives 2h. 11m.37,14s. for the correct time of the star’s apparent rising, andadded to 17h. 34m. 14,14*. gives 17h. 38m. 53,14s. for the cor-rect time of the star’s apparent setting. Thus it appears that theacceleration of the rising and retardation of the setting of Sirius,on account of refraction is 4m. 39s. so that the star appears abovethe horizon of Cape Horn 9m. 18s. longer than the double of itsreal semidiurnal arc.
(93.) In order to facilitate the calculation oftlie risings and settingsof the planets at London , we have thought proper to subjoin thefollowing New Table of Ascensional Differences and SemidiurnalArcs for the latitude of London (51° 30') and for every 10 minutes ofdeclination as far as 33 degrees, which is somew hat more than (lieutmost limit of any of the planets. All the ascensional differenceshave been carefully calculated from the proportion radius: tangentof declination ; : tangent of latitude : sine of the ascensional dif-ference ; ami the truth of the calculations ascertained by the me-thod of differences. The effect of refraction lias also been care-fully applied to the semidiurnal arcs.