102 Proceedings of the Royal Society of Edinburgh. [Sess. 
Case I. The ratio sjs 2 is an irrational number. 
Case IT. The ratio sjs 2 is a rational number, say equal to m/n (where 
m and n are integers and the fraction m/n is in its lowest terms) and 
terms in cos ( np 1 — mp 2 ) are absent from H 3 . 
Case III. The ratio sjs 2 is a rational number, say equal to m/n, and 
terms in cos (' np 1 —mp 2 ) are present in H 3 . 
We shall now determine the adelphic integral in each of these cases 
in turn. 
§ 4. Determination of the adelphic integral in Case I . — Let us then 
first suppose that the Hamiltonian function is expanded as in § 3, and 
that the ratio sjs 2 is an irrational number. We shall now show how 
to set up formally a series which, if it converges, is an integral of the 
system. 
If <f>(q v q 9 , p v p 2 ) = Constant is an integral, we must have (from the 
equations of motion) 
dcf> 3H 00 0H 00 0H 00 0H 
0^1 dp 1 + dq 2 dp 2 0p x dq x dp 2 dq 2 ~ 
an equation which we may write (0, H) = 0. 
Let us see if this equation can be satisfied formally by a series proceeding 
in ascending powers of Jq 1 and Jq 2 and trigonometric functions of p x and p> 2 
(like the series for H), whose terms of lowest order are {s 1 q 1 — s 2 q 2 ) : so that 
we may write 
0 = ^ 1-^2 + 03 + 04 + 05 + • • • 
where 0 r denotes the terms which are of degree r in Jq 1 and Jq 2 . 
Substituting in equation (4), and equating to zero the terms of lowest 
order, we have 
+ £> 2 
?03 = s 
dp 2 1 dp 1 
0Ho 
o o 
On — • 
“0^2 
This evidently implies that to any term A cos (mp 1 + np 2 ) in H 3 , there 
corresponds a term 
spn 
■s a n 
A cos (mp l + np 2 ) in 0 3 : so the value of 0 3 
spn + s 2 n 
may be written down at once. Having thus determined 0 3 , we equate to 
zero the terms in equation (4) which are of order 4 in q x and V c h '• f his 
gives the equation 
0 0 4j . 804 3H 
+ 5 
Sc 
0H 
dpi dp 2 d Pl dp. 
- + ( 03 » H 8 ). 
As the quantities on the right-hand side are all known, we can solve 
this equation for 0 4 in the same way as the preceding equation was 
