105 
1916-17.] Adelphic Integral of Differential Equations. 
gained by doing this, and we may therefore omit these arbitrary terms 
in 0 4 , 0 6 , <{> 8 , . . . 
§ 5. An example of the integral found in % 4, with remarks on its 
convergence . — As an example, consider the dynamical system which is 
specified by the Hamiltonian function 
H = 2 sin 2 ^ + q 2 sin 2 p 2 - 
1 + 3.2 iqf cos pj 
3(1 + 2lqf cos p x + 2%q x cos 2 p 1 + 2 q 2 cos 2 p. 2 )i ; 
or expanding, 
H = 2^q x + q 2 + 2^ 1 #( - cos^ - J cos 3 p-f 
+ 2 ~iqfq 2 { ~ 2 cos p 1 - cos (p 1 + 2 \p 2 ) - cos (y 1 - 2 p 2 )} + . . . . 
( 6 ) 
The corresponding integral, obtained by substituting in formula (5) is 
Constant = cf)=2iq 1 - q 2 + 2iqf{ - cos^ - cos "dpf 
+ 2~^qfqf - 2 cos + (1 - v^) 2 cos (p x + 2pf + (1 + \/2) 2 cos (p x - 2y 9 )} + . . . . 
(7) 
Now it may be verified readily by differentiation that this dynamical system 
possesses the integral 
Constant = (qf sin p 2 + 2iqfqf sinjo 2 cos p 1 - 2hqfqf sin p 1 cos y 2 ) 2 
1 + 2h/pcosp 1 
(1 4- 2 cosyj + 2 iq x cos 2 p ± + 2 q 2 cos 2 p 2 )l ’ 
which when expanded takes the form 
Constant = q 2 + 2 _ i( 1 - \Z2)qfq 2 cos (p 1 + 2pf 
— 24(1 + -\/2)qfq 2 cos (jp ± - 2/> 2 ) + (8) 
It is evident, on comparing the series, that the series (7) is what would 
be obtained by subtracting twice the series (8) from the series (6), which 
represents the integral of energy. This shows that for the particular 
dynamical system we are considering, the <£- series (5) is identical with the 
expansion, formed b 3 ^ ordinary algebraic and trigonometric processes under 
conditions which ensure convergence, of a known integral : and the 
convergence of the series (5), for sufficiently small values of Jq 1 and Jq 2 , 
is thereby established for this particular system. 
It is by considering particular dynamical systems such as this, in which 
the convergence of the series can be proved, that I have formed the opinion 
that the series (5) is in general convergent, for sufficiently small values of 
q 1 and q 2 , so long as the ratio sjs 2 is an irrational number. A general 
proof of its convergence would probably be very difficult, and I have not 
as yet succeeded in obtaining one. But the following considerations may 
be adduced in support of the opinion of convergence. 
Since the ratio sjs 2 is an irrational number, none of the denominators 
(s t +s 2 ), (s x — .s 2 ), (26—KsA (2s x — s 2 ), (s-l -f- 2s 2 ), (3s 1 + s 2 ), . . . can vanish, and 
