112 Proceedings of the Royal Society of Edinburgh. [Sess. 
of which the integral is 
<#>4 “ M 2 { “ ( 3 + f) cos 2 P-2 - (2 + y) C0S (%>! + 2 ?■>) + (l + y) cos OPi “ } 
+ q - 2 2 ( - f cos 2 t 2 - fV C09 4p 2 )- 
The equation to determine 0 5 is now 
9^5 , 8< ^5_ 0H l 
+ 
0Pi 0p 2 0p 2 
+ (<£ 3 > H 4 ) + W>4» H s)> 
and we have to choose a so as to annul the terms in sin (yq — 2p 2 ) on the 
right-hand side of this equation. On calculating these terms, we find 
i% 2 sin (Pi ~ 2 Pf 
(from 
V dp 2 
{from (<f> 4 , H s )) - - 4 #(l + sin ( Pl - 2 \p 2 ) 
{from (<f> 3 , H 4 )) + if ±aq^q 2 sin (p, - 2p 2 ). 
The quantity a must therefore satisfy the equation 
o ff _ 4_5 
0 o 
which gives 
1 + 
a 
3a 
Ta - 0, 
- 9 
Substituting this value of a in 0 3 and 0 4 our integral becomes 
Constant = q 2 — ^7, </ 2 cos (p 2 - 2jp 2 ) 
+ 7i2 2 {f cos 2 T 2 + i cos ( 2 Ti + 2 Pf - 1 cos (' 2 Pi ~ 2 lh ) } + c l2 2 ( - ? cos 2 P 2 " T6 cos 4p 2 ) 
+ terms of the 5th and higher orders in s Jq 1 and Jq 2 . . . . (14) 
Now it may be verified by differentiation that the dynamical system 
specified by equation (13) possesses the integral 
Constant = J{ N /2g 2 sin_p 2 + qpj 2q 2 cos p x sinp 2 - 2 s j2q l q 2 sin p ± cos p 2 } 2 
1 +7P cos P] _ / 15 x 
(1 + 2 qf cos p 1 + q ± cos 2 p x + 2q 2 cos 2 p 2 f ‘ 
and this integral is adelphic, as may be shown by completing the solution, 
or more simply by remarking that the integral (15) is a function of the 
variables (V^o Vfe Pi’ P 2 ) which is one-valued and free from singularities 
for a certain range of values, and therefore the infinitesimal transforma- 
tion corresponding to it will also be one-valued and free from singularities, 
and so must transform closed orbits into closed orbits. 
But on expanding this integral (15) in ascending powers of ^/q 1 and 
V c k by the multinomial theorem , we arrive at the series (14). This shows 
that, for the dynamical system we are considering , the series obtained by 
the process of § 7 converges for all real values of pj and p 2 so long as | q 4 1 
