VOL. LIV.] PHILOSOPHICAL TRANSACTIONS. ~ iS5 



the two corrections of parallax and refraction duly, according to the rules, to 

 the observed distance of the moon from the star, you will have the true and 

 correct distance of the moon from the star, cleared both of refraction and parallax. 

 A Rule. For computing a 2d, but smaller correction than the first, necessary to 

 be applied to the observations of the distance of the moon from a star on account 

 of parallax. 



Call the principal effect of parallax, found by the preceding rule, the parallax 

 in distance; and find the parallax answering to the moon's altitude. Then to 

 the constant logarithm 0.g4 1 add the logarithm of the sum of the parallax in alti- 

 tude and the parallax in distance, the logarithm of the difference of the same pa- 

 rallaxes, and the co-tangent of the observed distance of the moon from the star 

 (corrected for refraction, and the principal effect of parallax), the sum, abating 

 13 from the index, is the logarithm of the number of seconds required, being 

 the 2d correction of parallax ; and is always to be added to the distance of the 

 moon from the star, first corrected for refraction, and the principal effect of pa- 

 rallax found above, in order to obtain the true distance; unless the distance ex- 

 ceed 90°, in which case it is to be subtracted. 



^ concise rule to find the distance of the moon from a zodiacal star, very nearly; 

 the difference of the longitudes of the moon and star, and the latitudes of both 

 being given. 



To the cosine of the difference of the longitudes add the cosine of the dif- 

 ference of the latitudes, ifbothof the same denomination , or sum, if of contrary 

 denomination; the sum of the two logarithms, abating 10 from the index, is the 

 cosine of the approximate distance. This gives the true distance of the moon 

 from the sun, being then nothing more than the common rule for finding the 

 hypothenuse of a right angled spherical triangle from the two sides given. But 

 in the case of a zodiacal star, apply the following correction to the approximate 

 distance thus found. 



To the constant logarithm 5.3144 add the sine of the moon's latitude, the sine 

 of the star's latitude, the versed sine of the difference of longitude, and the co- 

 secant of the approximate distance; the sum of these 5 logarithms, abating 40 

 from the index, is the logarithm of a number of seconds, which subtracted from 

 the approximate distance before found, if the latitudes of the moon and star are 

 of the same denomination, or added to it, if they are of different denominations, 

 gives the true distance of the moon from the star. 



This rule, though only an approximation, is so very exact, that even if the 

 latitude of the moon was 5°, and that of the star 1 5", the error would be only 

 10"; and if the latitude of the moon be 5°, and that of the star 10°, the error 

 is only 4"^ ; and if the latitudes be less, it will be less in proportion as the squares 

 of the sines of the latitudes decrease. 



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