Lunar Observations. 25$ 



we obtain the one part of Mendoza y Rio's approximate 

 formula, to which the other is analogous. 



By separating in Lions's method the moon's parallax from 

 the refraction, we obtain the thence proceeding correction of 



, j- . ti T sm €'* »!*• sin O's alt.n , . . 



the distance = nor. par. ^ — .— w — , which 



1 L tan g Jf sin D J 



is identical in Elford's, Thomson's, and Lynn's tables, and is 

 found graphically by Kelly and Norie in his linear tables, and 

 by Thomson with his lunar scale ; and it is only the manner 

 of allowing for the refraction and sun's parallax wherein the 

 above authors differ, all of whom, however, with the exception 

 of Lynn, have erred in using in the calculation the apparent 

 altitude in lieu of the altitude corrected for refraction. On the 

 same account the usual tables giving the moon's correction 

 by inspection are erroneously computed. By approximate 

 methods are not to be understood methods that admit of less 

 accuracy, but such ones that approach the truth by a series 

 of which the last terms vanish, or, which answers the same 

 purpose, by successively substituting in the calculation terms 

 found by a former approximation. The approximative me- 

 thods have, particularly to seamen, that advantage above the 

 direct ones, that the trigonometrical calculation need only to be 

 executed to the nearest minute; and as the corrections of the 

 distance never can exceed those of the altitudes, and as their 

 amount, as well as sign, may nearly be estimated from the 

 places which the observed bodies occupy in the heavens, 

 essential errors may easily be avoided; alterations may be 

 made in the refraction and parallax without rendering it ne- 

 cessary to resume the calculation ; and, moreover, they afford 

 an easy mode of reducing this refraction to the points of the 

 limbs brought into contact when observing their distance, as 

 shall be shown by the following lines : we will designate by 



D, the apparent distance of centres; 



d 9 the distance of the points of contact or observed distance 

 of limbs ; 



H, the greater, h the less, apparent altitudes of centres ; 



H', the greater, k' the less, apparent altitudes of the points 

 of contact ; 



r, the semidiameter parallel to the horizon (augmented if 

 the moon's) of the upper body ; 



g— 7T, the correction of the upper altitude; 



g— 7r', the correction of the lesser altitude; 



0, the true distance of the limbs, to which the equatorial ho- 

 rizontal semidiameters must be added to obtain the true di- 

 stance of the centres. 



Let A be the distance of the middle of the apparent di- 

 stance of the centres, and A' the distance of the middle of the 



2 I 2 



