154 
PROF. H. F. BAKER OK CERTAIN LINEAR 
The value found for x is of the form 
* = T[l + (H, H~') + (AT, AT 2 , A 2 ) + (at, at 2 , X'V AT 1 ) 
+ (AT, AT 4 , AT, AT 2 , A 4 ) + ...], 
or, say, 
x = A* 9 [v 0 + A£m 1 + \£~ 1 w _ 1 +A 2 £ 2 , w 2 i-X 2 £ - 2 w_ 2 +...], 
where every one of u 0 , u u u_ u u 2 , r_ 2 , ... is a power series in X 2 with real coefficients, 
not generally vanishing with X 2 . And similarly for y. 
§ 10. The interesting case of the preceding solution is that corresponding to the 
value of <r given by 
cr = |-[(l+«r) i — (1— m)*], = ~ + ...j. 
The quantity 
7 — 6q- 2 
K * Pq (7(l-2cr 2 )(l-4(7 2 
is then equal to 
bn (7 + % 5 -m 2 ) 
approximately, and k 2 X 2 is of the order m\ 2 . When m 2 cc x, this is of the order m° or 
X 5/2 ; when m oc x 3 , it is of the order m bl3 or X 5 . Thus a very few terms of the preceding 
solutions would seem to be sufficient for practical cases. 
PART II. 
§11. A large part of the interest of Poincare’s ‘ Methodes Nouvelles de la 
Mecanique Celeste ’ depends on his criticism of the convergence of the series used 
by astronomers, particularly those series in which the time enters only under 
trigonometrical signs. In t. II., p. 277, he refers to a linear differential equation 
4p+Ti+V') = o, 
in which A, for our purposes, may be supposed to have a form 
A = 4a cos ht + ±b cos kt, 
in which a, b are small. When h , k are commensurable the equation has periodic 
coefficients, and Poincare makes the convergence of the series expressing the solution 
depend on this circumstance (‘ Meth. Nouv.,’ t. I., p. 66). Considering the case in 
which h and k are incommensurable, and so A n °t periodic, and supposing a, b to 
have common a small factor ju, he obtains formal solutions of the differential equation 
in sines and cosines, and says “ les series . . ., qu’on peut ordonner suivant les 
puissances de /x, ne sont plus convergentes ” (‘Meth Nouv.,’ t. II., pp. 277, 278). On 
