324 Proceedings of the Royal Irish Academy. 
"We have now to deduce the periodic time, and also the epoch, 
when the star last passed through periastron. AVe have the for- 
mulae — 
A value of 6 "being substituted in the last of these equations, v be- 
comes known, whence u is known by the second equation ; and by 
substitution in the first, we have a relation between n and e, when 
the known value of t is introduced. 
At the date 1802, the value of ^, read off from the interpolating 
curve, is 96-5, and at 1872 -2, the value of 0 is 24*-19. 
From these we deduce 
w = - 6<'-012, 
€= 1816*405, 
whence the periodic time is, 
It remains to determine the semi-axis major of the true ellipse. 
In the determination of this element alone do we employ the observed 
distances. 
Observed measures of the distances from ^fos. 4 to 22 are given 
in Table lY. 
n {t - e) =^ u - e sin u. 
tan {v - K) cos 7 = tan {0 - O). 
69-88 years. 
