107 
1916-17.] Aclelpliic Integral of Differential Equations. 
equating* the Hamiltonian function to a constant) and the integral expressed 
by equation (5). But it is known * that if, in any conservative holonomic 
dynamical system with two degrees of freedom, we know one integral in 
addition to the integral of energy, the system can be completely integrated, 
i.e. we can find expressions for the co-ordinates and momenta (q v q 2 , p v p 2 ) 
in terms of the time and 4 arbitrary constants of integration. We shall 
now perform this process, which incidentally will show that the integral (5) 
is the adelphic integral of the system. 
If we add the integral of energy to the integral (5), and divide 
throughout by 2 s v we obtain 
l x = q x + qp filJj COS^q -f — U 2 COS 3pj 1 
vSj 6q J 
{ 9 9 A 
+ » --" - - U 4 cos (2y t +y 2 ) + J U 5 cos (2/q - p 2 ) - 
V ^-iS x + Sr, ZiS x + &\> J 
+ 
I" 1 6 C0S Pl + 7 ,V. U 7 C0S ( 2 ^2 +Pl) + U 8 C0S ( 2 ^2 “ 
+ terms of the 4th and higher orders, 
where denotes an arbitrary constant. 
Similarly by subtracting the integral (5) from the integral of energy, 
and dividing by s 2 , we obtain 
h = r h + Mo-I-Us C0S P2 + ~ — u 4 C0S ( 2 Pl + P 2 ) - 
\&o ~ ^9 
U, cos 
2 s 1 -s. 2 5 
(2jPi. ~p 2 ) } 
r 2 
+ tfr&l — tw-D 7 cos (2 p. 2 +p ± ) - 
s 1 - 2 Si 
U 8 cos (2 \p 2 -p 1 
+ q£ ! ' ' cosp 2 + - U 10 cos 3p, \ + terms of the 4th and higher orders, 
l So So J 
So 
where l 2 represents a second arbitrary constant. 
It is an easy matter to obtain q x and q 2 from these equations in terms 
of (l v l 2 ,p v p 2 ) by successive approximation : in fact, the first approximation 
gives q l = l v q 2 = l 2 , and the second approximation gives 
q x — l l — Ipi — U 1 cos p A + — U 2 cos 3 p x j 
v Si Sn J 
1 
r 9 9 
- cos ( 2pi +p ^ + 2s -8 1X5 cos (' 2pi ~ ^>j 
(Itt 1 TT 1 
l ik\ — U 6 cos lh + U 7 cos (2 p 2 + Pl ) + - \ U 8 cos (2 p 2 ~iq)| 
v. 6*2 + aSo oj — ZjSo ) 
terms of the 4th and higher order in Jl x and Jh, 
* Of., e.g ., Analytical Dynamics , § 121 . 
