DIFFERENTIAL EQUATIONS OF ASTRONOMICAL INTEREST. 
14 I 
the value of q , in certain cases, is a particular case of Poincare’s theorem, ‘ Mhth. 
Nouv.,’ I., § 79, p 219. The phenomenon presents itself, however, as a consequence 
of the use of elliptic functions in Tisserand’s theory of the small planets; see 
Tisserand, ‘ Mec. Cel.,’ IV., p. 426 (or ‘ Bull. Astr.,’ IV.). 
§ 5. A very important question in regard to the differential equation under 
discussion is whether q is real or not, since upon this depends the conventional 
stability of the secondary oscillation determined by the differential equation. We 
have remarked above (§ 3) that when n is not an integer, and Jc x A, k 2 \ 2 , ... are small 
enough to render the series there obtained convergent, the value of q is necessarily 
real. The cases in which n is an integer and k 2 — 0 = k s = ... have been discussed 
by Tisserand, ‘ Bull. Astr.,’ IX., 1892, who obtains the result that the motion is 
unstable for n — 1 or n = 2, that is for the equations 
( ~- + [l + 4A& 1 w 1 ] x — 0, < ~ + [4 + 4A^ 1 ^ 1 ] x = 0, 
when A is small enough, but stable for greater integer values of n. The formula for 
q 3 , given in the earlier part of § 4 preceding, shows that for cases in which 
the motion is stable provided 
n 2 = 1 + 4/qA 
(hJkS a > 1 , 
the values of c/ta and shot, being then both real. It shows further that it is stable for 
h l = ±&! = positive 
provided A be small enough. The critical equation is thus 
pp- + x [l + 4 k Y (1 + 1 (\) + &k 2 w 2 + ... J = 0, 
the other sign of k v being obtainable by changing t into 
A 
§ 6. We proceed now to the case when n = 2. 
If in the equation 
72 
~ +x[m 2 +4A (hi + hWi) +4A 2 (h 2 -\-k 2 w 2 ) + ...] = 0, 
in which m is an integer, we put 
t = 2 it, £ = e T , 
U = £e imT+?T 
— imxj , V = ^e~ imT+qT +imx 
