101 
1916-17.] Adelphic Integral of Differential Equations. 
The differential equations will be taken in the Hamiltonian form 
where 
dq x 0H dq 2 0H dp x 0H dp., 0H 
dt dp x ’ dt dp 2 ’ dt dq x ’ dt dq 2 
H(g l3 q 2 , p x , p 2 ) = Constant . 
(1) 
( 2 ) 
is the integral of energy. 
In general — at any rate in the problems of practical importance — it is 
possible * to choose the generalised co-ordinates in such a way that H can 
be expanded as an infinite series proceeding in powers of y/q 1 and y' q 2 , and 
in trigonometric functions of multiples of p x and p 2 : that is to say, in terms 
of the type 
qp m qp n cos (ip x +jp 2 ) 
where m and n are integers (positive or zero) and i and j are integers 
(positive or negative or zero): moreover, if we call the “order” 
of a term, the terms of lowest order are linear in q x and q 2 and free 
from p x and p 2 , so that they may be written (s^ + s 2 q 2 ), where s x 
and s 0 are constants. In most cases we find also the condition that 
m — \i\ is zero or an even integer, and n — \j\ is also zero or an even 
integer. 
The Hamiltonian function H may therefore be expanded in the 
form 
H = + s 2 q 2 + Hg 4- H 4 + H 5 + ... . . . (3) 
where H y denotes the terms of order r, so we may write 
H b = cos p x + U 2 cos 3 p x ) -1- q x q<p{ U 3 cos p 2 4- U 4 cos (2^ +p 2 ) + U 5 cos (2 p 1 -p 2 )} 
+ ^i% 2 {^ 6 C0S i°i + U 7 cos(2p 2 4-pd 4- U 8 cos(2^> 2 -p x ) } + q 2 *{U 9 coap 2 4- U 10 cos3p 2 }, 
and 
H 4 = g 1 2 (X 1 + X 2 cos 2 p x + X 3 cos 4 p x ) 
+ ?1%H X 4 C0S (Pi +Pz) + X 5 C0S (Pi -P 2 ) + X 6 C0S ( 3 Pl +P 2 ) + X 7 C0S ( 3 Pl -P 2 )} 
+ q x q 2 {X 8 4 - X 9 cos 2 p x + X 10 cos 2 p 2 + X n cos (2 p ± + 2 \p 2 ) 4- X 12 cos (2 p ± - k 2p 2 )} 
+ ?1%2 5 { X 13 C0S (Pi + P%) + X 14 C0S iPl -P 2 I + X 15 C0S (Pi + 3 P%) + X 16 C0S (Pi ~ 3 £ 2 )} 
4- q^{ X 17 + X 18 cos 2 p 2 4 - X 19 cos 4 p 2 }, 
the coefficients U 1 , U 2 , . . . U 10 , X 4 , X 2 , . . . X 19 being constants. 
It will appear that it is necessary to distinguish three cases : in each 
case an adelphic integral exists and will be determined, but the form of the 
adelphic integral is different in each of the three cases. 
* Cf., e.g ., Analytical Dynamics , §§ 184-6. 
