320 Astronomical and Nautical Collections. 



This method of correcting the lunar distances from the effect of 



the earth's spheroidal figure is derived from La Lande's method of 



computing the parallaxes on the spheroidal hypothesis, in which 



the latitude on the sphere, and the horizontal parallax augmented 



by a small quantity, are employed. The true place being deduced 



from the apparent in this manner, the true right ascension is at 



, : . , , . xl . p sin. a cos. D dec. , , 



once obtained ; but the equation £ (where p is 



cos. lat. 



the horizontal parallax, and a the angle of the vertical with the 

 radius of the earth for the given place,) must be added to the cal- 

 culated true co- declination or distance from the elevated pole, to 

 have it correct. Therefore when the augmented horizontal parallax 

 is employed in computing the true lunar distances, the result must 

 be increased by this equation, multiplied by the cosine of the angle 

 at the moon in the spherical triangle formed by the moon, sun, or 

 star, and elevated pole. Let A denote this angle, the correction of 



,. . . p sin. a cos. D dec. cos. A p sin. a cos D dec. 



distance is *- - = u — 



cos. lat. cos. lat. 



2 p sin, a cos. > dec, sin.' \A The ^ rf ^ ^ rf 



cos. lat. 

 this expression, being the quantity to be added, is contained in 

 Table XII., and the second part to be subtracted is expressed thus, 

 S denoting the half sum of the three sides of the spherical triangle. 

 "2 p sin. a cos. D dec. sin. (S— J co-dec.) sin. (&— D dist. from ©or-^-) 

 cos. lat. cos. D dec. sin. dist. 



The logarithms of — ^ — Ll_^ are contained in Table XIII., to 

 cos. lat. 



which the logarithms of the other parts of the expression being 

 added, as in the practical rule, (expunging cos. D dec. from both 

 numerator and denominator,) the sum is the logarithm of the se- 

 cond part to be subtracted. In Tables XII. and XIII. the mean 

 value of the moon's horizontal parallax 57' is assumed as sufficiently 

 correct for computing the equation in every case. The augmenta- 

 tion of the horizontal parallax is contained in Table XL 



Besides the corrections taken notice of in this paper, a third ought 

 to be considered, namely, the contraction of the semi-diameters of 



