DIFFERENTIAL EQUATIONS OF ASTRONOMICAL INTEREST. 
143 
The case discussed by Tisserand is that in which k 2 = k 3 = ... = 0 . Then 
CL = 
and the quantity A in the formula for W, or x, is found to be -J-( — 2±\/— 5). 
When m — 3, for the equation 
d 2 x 
~df 
+ (9 + 8A& cos 2 1) x = 0, 
we find A = 0, and 
k 2 \ 2 
q = - 
269 
12 64.27.5 
&V+..., 
Ui- + W — “ + i) + — 
The question of the reality of q, in cases where Jc 2 = 0 = k 3 = ..., is discussed by 
Poincare, ‘ Meth. Nouv.,’ II. (1893), p. 243, and by Callandreau, £ Ann. Observ.,’ 
Paris, XXII. (1896), p. 23. So far the results are :— 
(1) For the equation at the bottom of p. 135 (§3) q is real when n 2 is not an 
integer, provided the series obtained converges. 
(2) This condition does not however include, for instance, the case when n 2 is near 
to unity. For q is imaginary, for the equation 
dt 2 
+ [n 2 + 8k, cos 2t + ...] x = 0, 
if (n 2 — l) 2 < (4&j) 2 . It is real if (n 2 — l) 2 > (4^) 2 , and real if n 2 — 1 is positive and 
equal to ±4^. This has been proved here. 
(3) q may be real when n is just greater than 2, when k„ k 2 , ... are small enough. 
This has been proved here. 
(4) q is real when n is any integer greater than 2, if k 2 = k 3 = ... = 0, but 
imaginary when n — 1 or n = 2. This result is given by Tisserand and 
Callandreau, as above.* 
[Oc£o&er 30, 1915 .—It may be worth adding, in connexion with the numerical 
results given in § 6, that the equation 
+ c sin t . x — 0, 
dt 2 
in which c is small, is solved by 
x = e^U, 
* See the note at the conclusion of § 21 (p. 184). 
