26 Trans. Acad. Sci. of St. Louis. 



Then, by formula (13) 



la i/d-hbsm'^0 dO 



^ ^ ^^ l/ (cos ^ — m) {n — cos^) 



from which we see at once that 



m ^ cos 6 ^ n i 



and, therefore, m ^1. If m = 1 and 0^ not = 0, the problem 



admits of no solution. In fact, the initial conditions are in- 

 compatible. If m = 1 and 6q = 0, then the axis NN' of the 

 torus will remain fixed relatively to the meridian circle (C^). 

 Thus, we are reduced to considering the case when m < 1. 



The constants of integration l^, l^, l^ are determined by the 

 equations 



r ?i = ^\ + (<^'o + ^) cos ^0 



(16) J l^= CI, cos d,+ (d + bsm^e,)(cj,', + <o) 



[ l, = a e\^ + {d + b sin2 9,) (f + a>y 



Case I. — 1 < w < 1 . 



We will put m = cos t; and consider separately the cases of 

 w >, = and < 1. 

 A. 71 > 1. 



Equation (15) becomes 



~ ^d + b sin2 e dd 



dt 



= ±\ll 



l/(cos 6 — cos?;) (n — cos^) 



and shows that the axis of the torus oscillates about ZZ\ the 

 amplitude of these oscillations being 2r). If we put 



