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 
quantities C. &c. by different formulae; and the fundamental number differed 
IT 
by a few units in the last place of decimals. The numbers admitted of com¬ 
parison at several intermediate points before arriving at the fina] 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 meanlcmg. 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 ) (10) 
■j L . e'. cos (brd) -{- L . e ^ e . cos f4 rd r») -f- L . e ^ e 2 . cos (3 rd -j- 2 ary 
rn) , ri2) 
+ L .d 2 d. cos (2 rd -f- 3 vj) -{- L . e e 4 . cos (rd -j- 4 r») 
+ L be 5 , cos (5 w) -j- M \ d z f 2 . cos (3 rd + 2 6) 
(10) , (11) 
+ M d 2 ef 2 . cos (2nr' -f?»+2 Q) -f-M . d e 2 / 2 . cos (rd -f- 2 rz -f- 2 6) 
(12) (10) 
+ M . e 3 / 2 . cos (3 ar -j- 2 6) + >.' . d . cos (rd + 4 &) 
,(n) •) 
+ N . e/ 4 . cos (rs -f 4 0) cos {13 {jit -f- i) — 8 (nt + z)} 
~h ^ L . d 0 . sin (5 id) -j- L .d^e. sin (4 id -f- rz) -f- L . e 2 . sin (3rd 2 ~j 
(11) (12) 
-j- L .d 2 e j . sin (2 w' + 3 -f- L . d d. sin far 1 ’ + 4 ) 
MDCCCXXXII. 
p 
