( 686 ) 
kind are not arbitrary, but must be determined from the equations 
(1). When the value of x is the same for both cases, the two solu- 
tions are entirely equivalent. 
In order now to investigate these solutions according to the theory 
of PormcarÉ, we must write down the conditions of periodicity 
T 
dE; 
— a = 0 
a (5 
0 
where for E; we must take successively each of the elements of 
the system. If further 8; be the small correction to be applied to 
the value of B; (for f=0) in the unperturbed orbit, in order to 
retain the periodicity in the perturbed orbit, then the stability of the 
solution depends on the roots of the equation 
0 0 Ou, 
ie + 28 bil a) Paes Co wel eli re v = 
dp, DB, 2, 
Ow Ow, Ow \ 
AE 5 aen an Per ef = =S 
” a ee OB, ©) 
dum i) Be Ou, 
2 eae 1 — 
08, 03, 03, a . | 
If we put s—e*! (or, approximately, 1—s——aT), then the 
condition that the orbit shall be stable, is that all the values of a 
are real and negative (with the exception of one or more, which 
may be identically zero). 
I will introduce the elements 
(Ord 1 EA a te 
of which the meaning is 
L7=mpy a OH; = L;V1 — e? 
1; = mean anomaly 
zt; — longitude of pericentre. 
Supposing the units to be so chosen that the constant of Gauss 
and the mass of the central body are unity, the equations of motion 
are : 
dh; dF CUT; OF 
KE Ee Te 
dl; OF da; OF 
dt ae OL; dt En Ò 
FSE h 
Pane m,* m,° m,° 
, replies. aes BS ame 0 
