VOL. LXI.J PHILOSOPHICAL TRANSACTIONS. ISQ 



sibly differing from the arches themselves. Then a denoting hl or hm, and b 

 the sum of gh and hi, the time sought for the greater chord may be considered 



equal to [— — (— )^] X — " — X the time of the moon's passing from g to h, or 



from H to I. The time for the lesser chord will be [ 1- (— )*] X -iLf.!li ^ j^g 



time above named; and in the last place, the time of the moon's passage between 

 H and K equal to [ + ( — )''] X — - — X the time specified. 



This calculation Dr. P. deduced from Sir Isaac Newton's differential method; 



and in the last case — or + ( Y X &c. is to be taken, as k shall fall within 



the greater or lesser of the arches gh, hi : but for the most part the term may 

 be wholly omitted. 



If this method be applied to the occultation of a star, the distances cd, ce,' 

 cp must be the parallaxes in longitude computed according to the first of the 

 preceding propositions, united with the respective distances of the moon from 

 the star in longitude, contracted in the proportion of the cosines of the moon's 

 latitudes, or at least of the star's latitude to the radius. Also the moon's appa- 

 rent latitudes must, for the most part, be corrected by the 3d corollary of the 

 3d prop, and the apparent diameters, if the correction could amount to any sen- 

 sible quantity, by the 4th corollary. 



The proposition mentioned above, for estimating the distance of the true con- 

 junction from the visible, is this. Fig. 8, in any circle, whose diameter is ab, 

 let the arch ac measure twice the complement of the declination of any point 

 in the ecliptic cd; in like manner measure twice the complement of the latitude, 

 and, AD, BD being drawn, let de be the versed sine of the distance in right as- 

 cension, of that point of the ecliptic from the meridian, taken to a radius equal 

 to the perpendicular let fall from c on the chord ad ; then be will be the sine of 

 the distance of the point assumed in the ecliptic from the horizon, to a radius 

 equal to the diameter of the circle. Therefore, if the diameter of the circle be 

 the measure, on any scale of equal parts of the moon's horizontal parallax, and 

 the point taken in the ecliptic be 90° distant from the moon's apparent longitude; 

 the right ascension and declination of this point being first taken from tables of 

 right ascension and declination, be, found as above, will be the measure of the 

 parallax in longitude, as assigned in the corol. to prop. 1 , and if the point as- 

 sumed in the ecliptic be 90° distant from the moon's true place, be will approach 

 near enough to that parallax for the purpose intended. 



After the same manner may the parallax in longitude be found for any other 

 time assumed. Also if the arch ac be taken equal to twice the complement of 

 the obliquity of the ecliptic, that is, bc equal to twice that obliquity, be will 

 be nearly equal to the parallax in latitude, provided de be taken equal to the 



