28 Trans. Acad. Sci. of St. Louis. 



C. n<l. 



We may put n = cos v^. Then 



—1 



^^_ . ,^ l/rf+& 8102 6/ dd 



^ l/(cos^ — cost;) (cos77j — cos 6) 



and we see that the axis of the torus oscillates between the posi- 

 tions = 7) [minimum deviation from ZZ ' ) and 6 = Vi (max- 

 imum deviation from ZZ'), never passing through ZZ'. 

 The period af a complete oscillation is 2t^ where 



/^V 

 Vd + b siu^e dd 

 l/(cos^ — C08 77) (cost;, — cos (9) 



Case II. m = — 1. 



Formula (15) now becomes 



==fc\2A 



Vd + bsin^d dO 



dt = :ty2A e / 



cos rt V n — cos^ 



where 



26V. 



(19) 71= 1 + 



Hence, two cases are possible: ^) n < 1 and B) n > 1. 



It should be observed that if 6^ = the case A) alone is 

 possible. In fact, if 0^= 0, since m = —1, we have 



d + b sm^^o 



from which it is clear that l^ and l^ must have different signs> 

 and, therefore, n < 1. 



