ON THE ORBITS OF THE ASTEROIDS. 145 
Representing by 2, 2’, and 2” the angles which the direction of projection of any frag- 
ment makes with the axes of co-ordinates, by « the velocity of projection, and by ¿,, 
20 and & the velocities of the planet in the direction of the three axes relatively to 
the velocity due to a circular orbit, we have 
E = &-+acosi 
n = m + «cos Y 
E = & + cos 2". 
Since by hypothesis cos A has all values at random between +1 and —l, and & and 
% are small compared with e, any small positive value of £ will diminish the absolute 
numerical values of £ for the several fragments for which « cos 4 is negative nearly as 
much as it will increase those values for the fragments for which « cos A is positive, 
and vice versa. A similar remark will of course apply to y and £. On the whole, 
however, the mean values of é, y, and £, taken without regard to their signs, will be 
a b 
2 
increased by a quantity of the order of magnitude of », uo respectively. 
A comparison of (11) and (18) shows that the mean value of the eccentricities of 
the fragments ought to be nearly 4/5 times as great as that of the inclinations; and 
that the mean value of = should be about twice as great as that of the inclinations, or 
a little less than that of the eccentricities. 
From the equations (16), (17), and (18), we easily obtain expressions for £, 7, and 
tin terms of the elements. They are, 
S I A 108. — LT 
To apply these equations rigorously, we should know the values of the eccentricities 
and inclinations of the several fragments immediately after the explosion. But from 
the equation (15) of § 2, it appears that the eccentricities of the fragments must im- 
mediately after the explosion have been quite near their mean values ; moreover, the 
eccentricities and inclinations are subject to but comparatively slight variations, as they 
will very rarely approach either of the limits given on page 138. We shall, therefore, 
use the mean values of those elements for the asteroids whose secular variations are ` 
given in $ 1, and the present values for those which are not there included. 
The mean of the perihelion distances of the outer asteroids givén in the Table in 
$2 is 2.60, and the mean of the aphelion distances of the inner ones is 2.52. The 
mean of these may be taken as the most probable value of a, which we shall therefore 
put equal to 2.56. — | 
VOL, VIII. | E 19; 
