115 
1916-17.] Adelphic Integral of Differential Equations. 
where a, /3, y are arbitrary constants, since these terms satisfy the 
differential equation and are of the type proper to <^ 4 . It should be noticed 
that these terms are not now superfluous, as they were in the general case 
studied in § 4 ; for in the general case the addition of such terms to <^> 4 would 
merely be equivalent to adding on an arbitrary quadratic function of the 
integral of energy and the adelphic integral : but in our present case the 
adelphic integral does not begin with terms linear in q x and q v and therefore 
a quadratic function of it does not account for terms like those in (18). 
The arbitrary constants in (18) are to be determined in such a way as to 
make terms with vanishing denominators disappear from the higher-order 
terms of (p . Thus, writing now 
2 cos (Ti +p 2 ) - cos ( 3 Pi + P 2 )} + a< li 
and substituting in the differential equation satisfied by <p 5 which is 
^-2^ = (^,H 3 ) .... 
dp, dp 2 
(19)' 
we find that on the right-hand side of (19) the terms involving sin (2 'p 1 +p 2 )' 
(which would lead to vanishing denominators on integration) are 
- 3 2iW u i 2 sin ( 2 Pi +P 2 ) “ 4a 2iW U 4 sin ( 2 Pi + P 2 ) 
and these will collectively vanish provided 
TT 2 
— _ 3 U 1 
u, 
In this way, by repeated application of the principle of § 7, we are able to 
remove all terms with vanishing denominators and obtain an adelphic 
integral free from them. 
§ 11. Completion of the integration of the dynamical system in Cases 
II and III . — Having now in §§ 7-10 overcome the difficulty caused by the 
presence of terms with vanishing divisors in the ade]phic integral in Cases 
II and III, we can use this integral in order to integrate the dynamical 
system completely, just as was done for Case I in § 6. We thus obtain 
expansions for the co-ordinates in terms of the time in all cases : but these 
expansions are completely different in form, according as the dynamical 
system falls under Case I, II, or III. This result supplies the underlying 
explanation of Poincare’s theorem that the series of Celestial Mechanics 
cannot converge uniformly over any continuous range of values of the 
constants : for the series to which he was referring were of the kind which 
we have classified under Case I, and we have seen that when the constants 
s v s 2 are continuously varied, these series must be replaced by the series. 
