318 Astronomical and Nautical Collections, 



Therefore 



2 cos. 60° - 2 sin. a sin. 60° = 1 — 1-73206 sin. a o 



cos. M + wsin. rs'm.M - cos. £ + m sin. r' sin. S _ ,* , 



— — x — ( l + m 



cos. M cos. £ 



sin. r tang, ikf) x (1 *f- m sin. r' tang. »S) = (1 + m sin. 57") x 



(1 + wi sin. 57') = 1 + 2m sin. 57" = 1 + m sin. 114". 



Therefore - 1*73206 a = m x 114", and a = »X 



-1-73206 

 ?» X — 66". 



Therefore the auxiliary angle, so far as it depends on refraction, 

 is equal to the mean refraction 66 seconds taken negatively, and 

 corrected for the given state of the atmosphere. Hence the cor- 

 rection of the auxiliary angle is evident. 



Although this investigation involves assumptions not quite cor- 

 rect, yet the result will be sufficiently exact for the purposes of 

 navigation. 



Tables XI., XII., and XIII., are designed to give the corrections 

 of lunar distances, arising from the spheroidal figure of the earth. 

 No instructions for their use are given by the author, but as they 

 are much better adapted to show these corrections than any others 

 yet published, and as they are equally applicable to every method 

 of computing the true distances, the following directions may be of 

 service. 



The moon's horizontal parallax employed in calculating the true 

 distance, must be increased by the number taken from Table XI. 

 The calculated true distance is to be augmented by the number 

 taken from Table XII., and diminished by the number obtained 

 from Table XIII., and the annexed rule, viz. : — From the half sum 

 of the moon's distance from the elevated pole, the sun or star's dis- 

 tance from the same, and the distance of the moon from the sun or 

 star, subtract the moon's distances from the pole and the sun or 

 star respectively. Add the logarithmic sines of the two remainders, 

 the logarithmic cosecant of the moon's distance from the sun or 

 star, and the logarithm taken from Table XIII., the sum is the 

 logarithm of the number of seconds to be subtracted from the true 

 distance. The result is the true distance on the hypothesis of the 



