Methods of correcting Lunar Observations. 367 

 reduced for the moon in the same ratio of cos. A, the primitive 

 refraction being very nearly expressed by q tang, b =z q — '—, 



so that the correction may be called a cos. A — '—;, and if we 

 •' ^ cos. b 



— , , , _ , , cos. A sin. b ^ 



find the angle E, such that ; =. tang. E, the cor- 



cos. 



rection becomes q tang. E, which is the refraction appropriate 

 to the zenith distance E, being always somewhat more correct 

 than taking the refraction simply proportional to the tangent, 

 and of course abundantly accurate for the present purpose. It 

 15 also obvious, from an inspection of the figure, that the dis- 

 tance of the same radius from the solar line will determine the 

 effect of solar refraction in the same manner. 



The correction for the obliquity of the true distance, com- 

 pared with the apparent, is found in the same manner as in the 

 Preface to Dr. Maskelyne's British Mariner's Guide. Since the 

 cosine of the hypotenuse of a triangle h is equal to the rect- 

 angle of the cosines of the legs d and k, we have cos. h rr cos. 

 d cos. k, or since k is very small, and cos. k zz 1 — ^ A*, cos. d 



— cos. h z= ^ k' cos. d ; and the small differences of cosines, 

 divided by the sine, being as the differences of the arcs, h — d 



cos. d ^ k" 



— A A' . " , = - — ;. Now k iz p sin. "6 sin. A, and ¥ — p* 

 ^ sm. d ta. d ^ ^ 



sin.'fc (1 — cos. 'A) = p' (sin. 6 + sin. b cos. A) (sin. b — sin. 



b cos. A), whence the construction is easily derived. 



The process requires, in this form, only 12 references to 

 tables of four places, besides the refractions, which are taken 

 out with the same ease as the direct refractions in the ordinary 

 methods of computation. 



B. A similar computation may, however, be performed by 

 1 1 references to tables only, besides the refractions, and with- 

 out having occasion for a table of natural sines. 



1. Call the observed distance d, the altitudes m and s, and 

 the half sum of these three angles k. 



2. To the proportional logarithm of the horizontal parallax, 

 add the logarithmic secant of the moon's altitude m, the sum 

 will be the proportional logarithm of the parallax in altitude. 



