( 689 ) 
dL; _ OR’ dG; _ OR; 
dy DR a = Raga 
The functions ,,Y,,W,.w, and yy, therefore contain only 
sines of linear combinations of the angles /, /', w, w'. Further we have 
Or’ OF, 2 OZD 
and therefore the equation wy, = 0 is a necessary consequence of w, = 0 
and w, == 9. The four remaining mutually independent equations 
| eae a, 0 oe Le 
correspond to what has in the beginning of this paper been called 
the “conditions of symmetry”. If we put 
L=(, —2l,),=ea v= ( — 21), =e 
Je (x, ry Xs), EE 8 0, = (x, ik Xs) — B, 
(the subscript O indicating the unperturbed value, or the average 
value over a complete period in the periodic solution), then the 
equations (4) are satisfied, if each of the angles 
eae aes ed 
is either 0° or 180°. 
These conditions being satisfied, we can, in the differential coeffi- 
cients of R’ (after the differentation) replace the angles J, !, w, w' by 
a;'a', B, B'. 
The developments of the functions p,, ... W‚ in powers of 3; are 
ò’R' On 07 R’ terms of 
», =F |B dl, +8, 01,01, de cls has 01,09, AG higher orders 
and similar formulae for w,,w,,w, and w,. Then we find easily 
or 
Pr B ST: + terms of higher order, 
OL, 
and similarly w, and w,. These equations give 8, = B, = 8, = 0, 
in other words the mean motions in anomaly (n;) are not changed. 
Finally we have 
T 
ME dt OR OR 
IE tam om 
WED OR 
Y,, = — Trae OI, 
The equations W,=w,,—0 are thus found to represent the 
conditions (1), since 
