28] NOTE ON THE CONSTANT OF LUNAR PARALLAX. 225 



of angles which increase in proportion to the time. The value found for 

 this constant was 3422"'325. 



This quantity may also be called very appropriately the mean sine of 

 the parallax, although I do not use the term in the papers referred to. 



The value of the corresponding constant in the expression of the parallax 

 itself is 0"'157 greater than this, or 3422"'48, which may appropriately be 

 called the mean parallax. 



The formula in the Introduction to Haiisen's Lunar Tables does not 

 give the sine of the parallax, but the logarithm of the sine of the parallax, 

 and the constant which Han sen calls C is a quantity such that the constant 

 term in his expression for the logarithm of the sine of the parallax is log sin C. 



Now, it is plain that the constant term in the development of log sin 

 parallax is a different quantity from the logarithm of the constant term of 

 the sine of the parallax, and hence my constant of parallax differs from 



. sin C 



Hansen s quantity - . 



sin I" 



We may readily express the relation between these two constants in 

 the case in which the orbit is supposed to be an undisturbed ellipse. 



In this case, if the reciprocal of the radius vector, which is proportional 

 to the sine of the parallax, be developed in terms of cosines of multiples 

 of the mean anomaly, 



then, being the semi-axis major, 



and e the eccentricity of the orbit, 



the constant term in the development will be - . 



In the same case, the constant term in the development of the logarithm 

 of the reciprocal of the radius vector, expressed in terms of the same 

 form as before, will be 



1 /. 1 



very nearly, instead of log - ; so that if c denote the constant term in the 



C* 



former development, and logc' the constant term in the latter, we shall have 



1 . , 



= 1 e very nearly. 

 4 J J 



A. 



29 



