1072 CHARLIER, ON PERIODIC ORBITS. 



The minür raass jx being always smaller than 0.04, if a 

 periodic orbit shall exist, it is evident from (27) that the angle 

 d deviates but little from + 30^ 



Hence one axis of the ellipse is nearly directed to the 

 greater mass (in^), the other being rectangular to this line. 



For the axes a and h of the ellipse we ohtain the values 



d-9\ I d'^^V d-Q 



^^' + ^\ cos- ^ + [v'^ + ^rnr] sin2 d + sin 2ß -^-^ 



da?} \ ob-} da ob 



_^|s,n-.+(.^+^)cos-.- - 



which expressions through (26) may be written 



(28) 





dadb ' 



-,cos2«= v= + ^cos= «-„'-+ 3p-|sin=« 



The exact values of the half-axes may hence be easily 

 found. In consideration of the small value of f.i for the periodic 

 Orbits it is raeanwhile preferable to expand the quantities in 

 powers of i^i. We obtain now to the first degree of f.i: 



cos 2(9 = h + I i-i 

 cos- ö = I + I /.t 

 sin- 6 = 1 — I /,< . 

 Further is (exactly) 



^ = 1 + 1^* 



As to V- we must distinguish between two cases correspond- 

 ing to the two difFerent roots of (13*). 

 1) V- = 6.75 ß . 



We then have 



r^ = 144^* (ß-^ + ßl) 

 a2 = 16(/r- + /y^) 

 b^- = a8i.i{ßl + ßl) . 



