Kirkwood. ] 436 [March 15, 1872. 
the value of nyiii, found by equation (3), is 7863//.983, differing from New- 
comb’s value (7864//.935) by less than 1//. 
The corresponding relations between the mean longitudes of the four 
outer planets are sufficiently obvious. Thus, 
3lv — 8ivi — Qivil + Yiviii — ,--aconstant . ; . (4). 
With a slight correction of the elements, it will probably be found that 
x = 135°. 
Again: the equation, 
17nvi — 228nvii + 21inviii — 0, i ‘ Os 
found in the same manner as (1), is believed to be exact. Combining (38) 
and (5), we obtain 
68nvi — B25nvii + 257nviii — 0, ‘ : 7 (6); 
Winv — 844nvi + 587nvii — 0, d 3 oui) 
B25nv — 912nvi + 587nviii — 0, : ; <6). 
These equations indicate that in a cycle of about 11657.2969 Julian 
years, the planets Jupiter, Saturn, Uranus, and Neptune return to the 
same relative mean longitudes. The equations are all satisfied by the 
following values. The received values are given for the convenience of 
: j 1296000 
comparison. In column first 0” — 
I == 11657.2969 « 
RECEIVED 
| THEORETICAL VALUES. | vs var ; | DIFFERENCES. 
| | VALUES. | | 
| 
| 
| Le aE 
| nv == 109256//.719 109256’7.719 | 07.000 
nvi —= n¥— 5879” — 43996.971 |  43996.127 +0/7.844 | 
| vil — mv — 8449” —= 15424.986 | 15424509 | -+0/.477 
viii n¥— 9199” == 7865.083 | '7864.985 |  +0/7.148 
The received value of Jupiter’s mean motion is here assumed to be 
correct. Any change would produce a corresponding variation in the 
remaining values. A revision of the theory of the orbits will, of course, 
result in some slight modifications. I believe, however, that the relations 
expressed by the preceding equations will be found strictly exact. If so, 
it must follow that no three of the four outer planets can ever be in con- 
junction at the same time. 
Buioomineton, Inv., February, 1872. 
aii 
