OF ARTS AND SCIENCES. 37 



It is uoteworthy that the results over the different lines, with the 

 exception of that at 50°, which is based on only 6 observations, 

 all give values less than that of 1G".61, adopted in the Berlin Jahrbuch, 

 Nautical Almanac, and Connaissauce des Temps, and that the lines of 

 greatest position angle, which by this method would be expected to 

 afford the most accurate results, give the smallest values of the series. 



It appears to me that the method of Professor Rogers is not limited, 

 in its application to the interior planets, to their transits over the sun's 

 disk, or to times when the conditions permit the whole disk to be seen ; 

 but that it may, by an appropriate construction of the plate and 

 arrangement of the observations, be employed at any time when they 

 are near inferior conjunction, and that determinations both before and 

 after conjunction will eliminate any errors peculiar to each elongation. 



Let p be the position angle, counted from an assumed zero, of a 

 line on the plate drawn from some point taken as a centre; the true 

 position angle being p -\- Ap. Let D and 8 be the diameter and 

 declination of the planet ; AS the difference of declination from the 

 centre of the plate when it passes north, and A'S when it passes south 

 of that centre ; and t and t\ the corresponding observed times when 

 the planet's limb in its diurnal motion is tangent to the line. Then 

 in the triangle formed by the planet's centre, the intersection of its 

 path with the line, and the observed point of taugency, the distance be- 

 tween the first two points is, 



\D sec {p -\- A;?) = \D sec p -{- \D tan p sec p Aj» 



where, A/) being small, the terms involving its squares are neglected. 



If we imagine a line drawn from the centre of the plate at the angle 

 p from the true position zero, we have, from the triangle formed by 

 the actual and imaginary lines and the portion of the path of the 

 planet's centre between them, the length of the intercepted path : 



AS A^ secj9 sec {p -\- A/)) = AS Ap sec^jo 



If now we call T the time when the centre of the planet is on this 

 imaginary line when the planet passes north, and T' the time for a 

 corresponding position when the planet passes south of the centre, we 

 have the general equations : 



T=:t -\- ^ \±\D&ecp± \D tan p sec p Ap -f- ^^ soc^p ^p\ 



(1) 



T'=.t' -\- r^ r =F |Z) sec p :p ^D tan p sec^ A/) -|- A'S sec^^ Ajo J 



