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



C. n<l. 



We may put n = cos rj^. Then 



^^_ . p. i/cZ+&sin2^ (26 



= ^\ll 



^ l/(cos ^ — COS 7)) (cos t;^ — cos 0) 



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

 tions = 7) (^minimum deviation from ZZ ' ) and 6 = rj^ (tJiax- 

 imum deviation from ZZ'), 7iev er passing through ZZ\ 

 The period of a complete oscillation is 2t, where 



/'^ 

 \/d + b^m'^e d0 

 l/(cos^ — COST?) (cos 77^ — cos^) 



Case II. m = — 1. 



Formula (15) now becomes 



\_a_ Vd + b sin^d dO 



dt = ±y2A d . \ 



cos Q V n — cos^ 



where 



267./„ 



"^12 



(19) n= 1 + 



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



It should be observed that if ^0' = ^ the case A) alone is 

 possible. In fact, if ^0 = ^» since 7?j = — 1, we have 



d {I,— CI, cos e,y _ ^ 



d + b si"^2^i^ ~ ^'2 + Ch) ^ 



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

 and, therefore, n < 1. 



