ON THE ORBITS OF THE ASTEROIDS, 139 
not possible so long as the equations (12) of the preceding section will give nearly 
true values of the elements by attributing any values whatever to the angles (1), (2), 
(3), Ee In order that any conclusion that one may draw respecting this question may 
rest on as broad a foundation as possible, we shall now consider the same question 
with respect to other asteroids. 
In the paper referred to, it is shown that an intersection of the orbits is not possible 
unless their elements and secular coefficients are such as to fulfil the condition 
(Eh — k hy + (11 — k1)? > (k — k}, 
K' representing the semi-parameter of the outer orbit, and & that of the other. Substi- 
tuting for h and / their expressions in the preceding section, and putting B + bt = N, 
B + bt = N', this inequality will take the form 
— 2 kk AA! cos (N' — N) — 2 5, k A’ (k's, — K e) cos (IN! — (n) ) 
+2 zx, E A’ (k's, — k e'n) cos (N — n) + 2 En X, (Ke, — ke'm) (Ke, — k e'n) cos ( (m) — (n)) (13) 
> (E— by — z (Ke — key — p? A3 — BAY, 
If A and A’ are both small, the eccentricities of the orbits will likewise be small, and 
k and E will be subject to only very slight variations. If we suppose k and E con- 
stant, the maximum value of the first member of the above inequality will be, after 
transposing the last three terms, 
$E A+ E A! 4 z (k's — ke’)? 
and by finding the value of this expression, using the mean values of the parameters 
E and k, and so taking the doubtful signs that the value of this expression shall be 
the greatest possible, we may at once find whether an intersection of the orbits is 
possible, by observing whether the condition K A+ k A 4 £ (Ke — ke’) > KE—k 
is fulfilled. 
If A and A' are both large, the first member of the above inequality can attain its 
greatest magnitude only when we have very nearly N! — N — 180°. If this condition 
is fulfilled, the eccentricity of the one orbit will be at its maximum when that of the 
other is at its minimum. The second member of (13) will attain its least magnitude 
when the eccentricity of the outer planet is at its maximum, and that of the inner one 
at its minimum; that is when we have 
(0) =(1) = T = (5) = (6) = (1) = a (14) 
(2) = (4) = N= 180 — IN 
But, since Az — kz’ has in general the same sign as e, it is evident from an inspection 
of (13), that these conditions will also make its first member attain its least magnitude ; 
