144 ON THE ORBITS OF THE ASTEROIDS. 
in the direction of the axes of X and Z respectively; and by y the velocity in the 
direction of the axis of Y relatively to that of a planet moving in a circular orbit at 
the distance a. E y, and £ will then be equal to the velocities of projection of the 
fragment in the directions of the corresponding axes, plus the velocities of the planet 
in the same direction over and above those due to a circular orbit. | 
The only elements which we can determine are the eccentricity, inclination, and the 
amount by which the mean distance differs from a, In determining these, we may, in 
a rough approximation like the present, neglect all quantities of the second order with 
respect to the velocities of projection E y, and £. Represent by da the difference be- 
tween a, and a, the latter being the mean distance of the fragment after its projection, 
by v, the velocity of the planet in a circular orbit, and by v the actual velocity of the 
fragment after projection. We then have 
1 21 a, 
ico osi P= HHH 
a 
We then obtain by suitable reductions, neglecting quantities of the second order, 
and observing that v — e DEE 
SCH (16) 
k here representing the Gaussian constant. 
Representing for the present by p the parameter of the orbit, we have 
pateo 
DG SE ay Y 
P Ze ELE 
2 at 
f= Dee aE d ug) 
But by the conditions of circular motion 
e—1— 
— B+ 3a ci — EL 0, 
Wherefore 
gë (20 lie 4208 SHE ERC, 
Developing the expressions within the parenthesis to quantities of the second order 
3 
inclusive, and observing that x == B , we find 
on Hi, BEIGE YS i 
For the EEN we easily find , 
e EE (18) 
If the eccentricity of the planet before the explosion were small, the mean values 
of &, y, and ¢ would be very nearly the mean velocity of projection of the fragments. 
