168 
W. Beebe — The Comet of 1771. 
the four elements ; j, which in the unchanged orbit is taken equal 
to 7T — e, q and T, and since the latitude // is affected only by the 
small changes in the position of the orbit, depending on i and the 
elements which satisfy 6 will nearly satisfy i] also. 
The partial differentials of the four elements with reference to A 
and A", computed from equations of the form 
dx . dv de , dv dt , dv do 1 . . „ „ 
+ - rT7 4- — -4-— - — 4 = — cos n sin (0 — u ) 
de dA ~ dt dA ^ do el A r v ’ 
dA de dA ' dt dA ' dq dA 
dr de , dr dt , dr dq .. 
4 — =- -- 4- - 7 — =4 = cos n cos (0 — u ) 
de el A 1 dt dA dq dA 
and the same for A ' ; give by elimination 
dx 
^—0-23212 
dA 
log 
dx 
~jjT 0'ftl5056 
de 
0-49956 
dA 
log 
d€ 
dx 9 ' 74700 
dt 
1-93140 
dA 
log 
(U 
7 l-«69946 
dA 
dq 
^4 9*58362 
dA 
log 
. 9*32929 
lot 
1 °£ 
Ac., having been computed by the ordinary differential for- 
mulae. 
Then the partial differential coefficients of 6 with reference to 
A and A g are computed for each normal place from equations of 
the form 
dd dO dx dd de dO dt , dd dq 
cos n ~7 = cos n -j—. 4- cos n -L cos n 4- cos n — 
el A dx el A ~ de dA dt el A ' dq dA ’ 
as follows : 
de 
cos n — 
e?A 
<te 
cos n 
dY 
de 
cos n -ji 
dA 
de 
008 n dY 
April 6. — *000055 
4- *000036 
Mav 24, 
— *000027 
— *00000 
14, - *11732 
30, — *21066 
+ *06530 
+ *13701 
June 1, 
9, 
+ *11544 
+ *22364 
- *10734 
— *21520 
Mar 8, — *18644 
4- *13184 
25. 
4 *37886 
— *46916 
16, |— *10443 
+ *08130 
] July 11, 
+ *44275 
— *66823 
The values for April 6 and March 24, which should be zero, furnish 
a check for the correctness of the work to the fifth place. 
From Aa and Ad we compute A0, and also An which will be used 
in another place, from the formulas 
cos n A0 = sin y Ad -|- cos Y cos $ a 
An — cos y Ad — sin y cos dAa, as follows : 
