524
ON THE DEVELOPMENT OF THE DISTURBING
[479
Terms of seventh order (concluded) :
f
©
L'-W
fc-i
i
©
L-U
(459)* (i e Y (4 e') rf (459
. . 459)
cos
i
+ 1
- i + 2
+ 4
(460)* (ie) 3 (4 e'Y t/ 2 (460
.. 460)
>>
i
+ 2
— i + 2
+ 3
(461)* (Le) 2 (L e'Y rf (461
.. 461)
5J
i
+ 3
— i + 2
+ 2
(462)* (ie) (L e ')V(462
. . 462)
)>
i
+ 4
-i + 2
+ 1
(463)* (4 e'Y f (463
.. 463)
>5
i
+ 5
-4+2
0
(464)* (L e Y f (464
.. 464)
55
i
0
— 4+4
+ 3
(465)* (L e ) 2 (4 e!) f (465
. . 465)
55
i
+ 1
-4 + 4
+ 2
(466)* (L e) {ie'Yf (466
.. 466)
55
i
+ 2
— 4+4
+ 1
(467)* (L e’) 3 f (467
. . 467)
55
i
+ 3
-4+4
0
(468)* (L e ) rf (468
. . 468)
55
i
0
— 4 + 6
+ 1
(469)* (L e’) rf (469
. . 469)
55
i
+ 1
— 4 + 6
0
Here the several coefficients are ultimately given in terms of the before-mentioned
quantities A\ B 1 , C\ D i , E\ G\ H', L l , S\ T 1 (functions of a, a), and their differential
coefficients in regard to a
1 A.
Ï a da
A\
&c -
as follows:—we have Leverrier, pp. 299—330, a list of functions (1), (2), ...(154) of the
form (1) = \K i , (2) = — 2h 2 K i + Kf + Kf (3) = — 2i 2 K { + Kf + Kf, &c., involving i, h, and
K l and its derived functions Kf, Kf &c. The coefficients of the several cosines are
given by means of the functions in question, thus, first coefficient, above denoted as
(1)*(1 ... 20), is
= (1Y + (2) f a e) + (3)* (*✓).•• + W V e
where (l) i = (l), (2^ = (2) ... writing in the functions (1), (2) ... (10), h = i, and K^A 1 :
(ll) i = (l), (12y = (2), &c., writing h = i and K i = — E i ,
(20^ = (1), writing h = i and K l = — H l ,
and so on for the various component coefficients (1)*, (2) 4 ... (469) f .
But the resulting expressions, for the several integer values i = —10 to +10, are
worked out in the Addition II. {Numerical Tables for the Calculation of the Coefficients
of the Development of the Disturbing Function), pp. 358—383. And this Addition
contains also, indicated by the letters 8 and A respectively, the expressions of the
