ON THE ORBITS OF THE ASTEROIDS. 133 
elements is less than 11^. The uncertainty in the present mean values of the elements 
is such, that no advantage will result from any attempt to carry the results to more 
than four places of decimals; which are, moreover, quite sufficient for our present 
purposes. The following are the assumed values of the longitude of perihelion, lon- 
gitude of node, eccentricities, and inclinations, for January 1, 1850, of such of the 
asteroids fulfilling the above conditions as I have to the present time (March, 1860) 
been able to obtain accurate elements. 
Name. Symbol, m 3 e i 
o o o I 
Ceres 149 12.5 80 49.0 0.0803 10 36.5 
Vesta 250 23.1 103 19.5 0.0902 7 81 
Astrea. 134 35.6 141 24.8 0.1900 5 19.6 
Flora 32 57:2 110 18.3 0.1567 5 53.2 
Metis 70 51.2 68 30.0 0.1284 5 35.9 
Hygea 227 44.0 287 39.7 0.1005 3 47.2 
Parthenope 315 57.1 125 2.6 0.0988 4 36.9 
Irene 179 15.0 86 38.8 0.1652 9 74 
Psyche 12 24.1 150 32.0 0.1363 3 3.9 
Thetis 259 12.8 125 26.5 0.1268 5 35.6 
Fortuna 30 8.7 211 24.9 0.1578 1 32.2 
Massilia 98 16.0 206 40.2 0.1438 0 41.0 
Lutetia 326 58.4 80 26.8 0.1620 5 53 
Themis 137 37.6 36 14.3 0.1177 0 49.1 
Proserpina 235 4.2 45 53.2 0.0875 3 35.7 
Euterpe 87 25.4 93 36.8 0.1730 1 35.5 
Bellona 122 16.1 144 43.3 0.1544 9 22.5 
Amphitrite 56 52.4 356 26.8 0.0725 6 8.0 
Urania 30 47.5 308 13.0 0.1268 2 6.0 
Pomona 194 12.0 220 48.9 0.0824 5 29.0 
Circe 149 48.3 184 47.3 0.1083 5 26.5 
Fides 65 56.1 8 10.7 0.1749 8 T3 
Leda 100 99.5 296 27.8 0.1556 6 58.5 
Laetitia. 1 48.0 157 20.5 0.1110 10 20.8 
Harmonia 1 63 93 29.5 0.0462 4 15.8 
From these elements, and from the preceding tables, we obtain the following expres- 
sions for h, l, p, and q, in terms of the time, for the above asteroids. 
Ceres. (1) 
h = .0020 sin (0) + .0286 sin (1) — .0148 sin (2) + .1102 sin (158° 51’ -++ 58".58 1) 
1 = .0020 cos (0) + .0286 cos (1) — .0148 cos (2) + .1102 cos ues DU + ein 
p= M sin y +.0012 sin [1] —.0103 sin [2] +.0001 sin [5] —.0015 sin [7] + .1652 sin er ee t) 
M cos y + .0012 cos [1] —.0103 cos [2] + .0001 cos [5] — .0015 cos [7] + .1652 cos (80? 11.— 58,58 t) 
| Vesta. (4) 
h = .0018 sin (0) + .0254 sin (1) — .0188 sin (2) — .0001 sin (4) + .0002 sin (5) + -0003 sin (6) 
+ .1048 sin (231° 29' + 38".56 1) | 
= .0018 cos (0) + .0254 cos (1) — .0188 cos (2) — .0001 cos (4) + .0002 cos (5) + .0003 cos (6) 
| + .1048 cos (281° 29' + 38".56 1) 
q= 
