ON THE ORBITS OF THE ASTEROIDS. 129 
functions of the mean distances only of the asteroids, it will be convenient to tabulate 
them for different values of that element; and by means of such tables, the equations 
required can be obtained for any asteroid whatever, of which both the eccentricity and 
inclination are small, with very little labor. In the computation I use the following 
values of the masses of the disturbing planets. 
Venus sgoov; Earth cade; Mars sggbes7> Jupiter Tos 
Saturn syr Uranus ` un, Neptune 44yg. 
The direct action of Mercury is neglected, it being entirely insensible. 
Making use of these values of the masses, and of the usual values of the mean 
distances; calling (0,1), [0,1], the coefficients between an asteroid and Venus, (0,2) 
[0,2], between an asteroid and the Earth, and so on, we have the following values of 
those coefficients in seconds, the unit of time being 3651 days. 
den (0,1) (0,2) (03) | (04) (0,5) (0,6) (0,7) 
2.2 0.102 0.268 0.184 31.076 1.156 0.022 0.007 
2.4 0.073 0.183 | 0.104 38.359 1.345 0.025 0.008 
9.6 0.053 0.131 0.066 47.372 1.550 0.028 0.009 
2.8 0.040 0.096 0.044 58.727 1.775 0.032 0.010 
3.0 0.031 0.073 0.032 73.315 2.021 0.036 0.011 
3.2 0.025 0.057 0.023 92,495 2.291 0.039 0.012 
a pu] | rg | tos] | t4 | ros | tos] | wa 
2.2 0.041 0.148 0.148 16.036 0.331 0:003 0/001 
2.4 0.027 0.093 0.078 21.485 0.419 0.004 0.001 
2.6 0.018 0.062 0.046 28.587 0.523 0.005 0.001 
2.8 0.013 0.042 0.029 37.930 0.644 0.006 0.001 
3.0 0.009 0.030 0.019 50.381 0.784 0.007 0.001 
3.2 0.007 0.022 0.013 67.277 0.947 0.008 0.002 
We next require the numerical values of the quantities represented in (4) and (5) 
by N and M. These are obtained by the simultaneous integration, for the larger 
planets, of the systems of equations (1) and (2), a process so laborious, that only three 
or four astronomers have ever attempted to carry it through. It seems to have been 
done most completely by Le Verrier, before the discovery of Neptune, as he then gave 
the differential coefficients of the several quantities with respect to the masses; and he 
| has since taken into account the action of that planet, but has not given these differ- 
ential coefficients, nor has he considered in what way the action of Neptune might be 
modified by that of the planets inside of Jupiter. Still, it is quite possible, that the 
inaccuracies thus introduced are no greater than those which proceed from the neglect 
of terms of the third order, and we shall therefore use throughout the values given by 
VOL. VIII. 17 
