1916-17.] Adelphic Integral of Differential Equations. Ill 
and so does not contain terms of the type appropriate to <£ 3 . But when 
s 1 = 2, s 2 = 1, the arbitrary part of the solution of the differential equation 
does contain terms of the type proper to $ 3 , and these must be taken 
account of ; so that we must take the integral of the equation 
/A . 
to be 
dp 1 dp, 
A = - \qf cos Pi + a 2r22 cos (Pi ~ 2)9 
where a is an arbitrary constant. In this way we obtain terms with 
arbitrary coefficients in <J> 3 , </> 5 > • . and these arbitrary coefficients 
must be chosen in such a way as to remove terms with vanishing denomi- 
nators from the subsequently determined part of </>. This principle enables us 
to obtain, in Case II, an adelphic integral free from vanishing denominators. 
§ 8. Study of a particular dynamical system, as an illustration of the 
method o/§ 7. — We shall now illustrate the working of this principle by 
an example. Consider the dynamical system which is specified by the 
Hamiltonian function 
H = 2 q 1 sin 2 ^ + q 2 sin 2 p 2 + — — 1 — — - 
2(1 + 2 qp cos p 1 + q L cos 2 p x + 2 q 2 cos 2 p 2 ) 2 
1 + qf cos p 1 
(13) 
(1 + 2 qf cos p 1 + q 1 cos 2 p x + 'lq 2 cos 2 p 2 f 
If this be expanded in ascending powers of Jq 1 and Jq 2 , we obtain 
H = 2 q x + q 2 + qf-( - f cos p 1 ~I cos 3 pf -f qf(f^ + % 5 - cos 2 p 1 + ff- cos 4 pf) 
+ gp/ 2 { - 3 - 3 cos 2 p 1 - 3 cos 2 p 2 - -| cos (2p 1 + 2 p 2 ) - -| cos (2 p 1 - 2 pf) 
+ qf{ - T °g- -- § cos 2 p 2 - pp- cos 4 p 2 ) + terms of the 5th and higher order 
in -\A?i and q 2 , 
so that in this case s 1 = 2, 6' 2 = 1. 
As explained at the end of § 4, we may assume that the lowest term 
of the adelphic integral is simply q 2 . Then if we write 
0 = q* + As + A + A + • • • 
the equation to determine A is 
3 , 3 A 
A 1 
0, 
so by § 7, 
dpi a p 2 
A = aqfq, cos (p l - 2 p 2 \ 
where a is an arbitrary constant. 
The equation for A now becomes 
if + f = Si? 2~(( 6 + y ) sill2 ^2 + ( 3 + y)sin(2pi + 2 lp 2 ) - (3 + ^ t )sin(2p 1 - 2p,)j 
+ q.f( § sin 2p 2 + § sin 4p 2 ) 
