IN THE MOTIONS OF THE EARTH AND VENUS. 
105 
had the advantage however of comparing the calculated values several times 
with the values which I calculated nearly four years ago. At that time I 
developed the principal fraction in a different manner, and I expressed the 
(t) 
quantities C 5 &c. by different formulae ; and the fundamental number differed 
by a few units in the last place of decimals. The numbers admitted of com- 
parison at several intermediate points before arriving at the final results ; and 
one small error was discovered in the old calculations, and one in the new 
ones. Upon the whole, I am certain that there is no error of importance in 
these numbers ; and I think it highly probable that there is no error, except 
such as inevitably arise from the rejection of figures beyond a certain place 
of decimals. It is impossible to assert that the last figure preserved is correct, 
or even the last but one ; but 1 do not think that the last but two is wrong. 
Section 14. 
Numerical calculation of the long inequality in the epoch, depending on 
(13 X mean long. Earth — 8 X mean long. Venus). 
50. The most convenient form in which the expression of (29) can be put is 
the following. 
f (8) (9) (io) 
j L . e' 5 . cos (5z/) -{- L . e' 4 e . cos (4 zd + *r) -j- L .e' 3 f 2 .cos (3w' + 2w) 
(n) (12) 
-f- L . e' 2 e 3 . cos (2 & -j- 3 ot) -f- L . e e 4 . cos (&' -f- 4 zj) 
(13) (9) 
+ L . e 5 . cos (5 zu) + M . e' 3 f 2 . cos (3 w' -f- 2 0) 
(10) (11) 
+ M e' 2 ef 2 . cos (2®' + zn + 2 &) + M . e' e 2 f 2 . cos (z&' -j- 2 zs -{- 2 6) 
(12) (10) 
+ M . e 3 /* 2 . cos (3 tu -f- 2 &) + N . e 1 f 4 . cos ( zs' -j- 4 6) 
(ii) -) 
+ N . e/ 4 . cos (©■ + 4 ff) ^ cos {13 ( n't + s') - 8 (n t + s)} 
+ ^ L . e' 5 . sin (5 zu') + L . e' 4 e . sin (4 zb zb) -j- l} . e ’ 3 e 2 . sin (3 zb' -J- 2 tv) 
(11) (12) 
+ L .e 2 e i . sin (2 zs' -f- 3 zs) -f- L . e' e 4 . sin (zb 1 + 4 zb) 
MDCCCXXXII. 
P 
