113 
1916-17.] Adelphic Integral of Differential Equations. 
and | q 2 1 are inferior to certain fixed quantities , and that the series repre- 
sents the adelphic integral of the dynamical system. 
§ 9. Determination of the adelphic integral in Case III. — The principle 
for the removal of vanishing divisors from the adelphic integral, which 
was explained in § 7 and illustrated in § 8, is not sufficient for the purpose 
if the Hamiltonian function contains, among its third-order terms, a 
term in cos ( s 2 pi — sppf ) : for this term gives rise to a vanishing divisor 
in <t> 3 , and the arbitrary terms which are used in order to annul terms 
with vanishing divisors do not come into operation early enough to 
remove vanishing divisors from <£ 3 . 
In this “ Case III ” we must make use of another principle (concurrently 
with the principle of § 7) which may be explained thus: Suppose that an 
integral of a system of differential equations in variables (q v q 2 ,Pv P 2 ) 
the form 
ffiv Pv To) + = 
P 
where y is the arbitrary constant and /x is a definite constant formed of 
quantities occurring in the differential equations. The integral in this 
form ceases to have a meaning when /x tends to zero. But we may derive 
from it an integral which has a meaning when /x->0, by merely supposing 
first that jj. is different from zero, and multiplying the equation throughout 
by fi, so that it becomes 
pffiv To P 2 ) + Pj Pv To) = py 
and then making /x->0 ; the equation becomes 
9{<lv Pv T 2 ) = 
where c denotes Lt^o (/ay). This is the desired form of the integral when 
/x is zero. 
Our case is not so simple as this, since the vanishing divisor occurs not 
only in the inverse first power, but in an infinite series containing all 
the inverse powers. The method we follow, which will be illustrated in 
the next article, is really equivalent to using the principle of § 7 in order 
to remove all inverse powers of the small divisor except the first, and 
then using the principle of this article in order to remove this inverse 
first power. 
§ 10. Example of the principle of § 9. — We shall now show by consider- 
ing a particular dynamical system how the principle just mentioned is 
applied in order to obtain an adelphic integral free from vanishing divisors 
in " Case III.” 
Consider the dynamical system whose Hamiltonian function is 
H = 2 , i- 2 2 , 2 + ^ fu i C0S Ti + M2 iU 4 C0S ( 2 Ti+T 2 ) • • ( 16 ) 
VOL. XXXVII. 8 
