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



Then, by formula (13) 



(15) dt = ±: 



la i/d + bsin^S dd 



A l/ (cos ^ — m)(n — cos^) 

 from which we see at once that 



m ^ cos ^ ^ n , 



and, therefore, m < 1. If ?w = 1 and 6^ not = 0, the problem 



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

 compatible. If rw = 1 and ^o = 0» t^^n the axis NN' of the 

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

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



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

 equations 



r ?i = "^'o + (<^'o + ®) cos ^0 



{IQ)\ I, = Cl^ cos e^-\- {d-^h 8in2 6^) {<^\ + «) 



[ ?3 = a (97 + ((^ + 6 sin2 6,) {<(>', + <oy 



Case I. -1< m < 1. 



We will put m = cos -q and consider separately the cases of 

 n >, = and < 1. 

 A. n > 1. 



Equation (15) becomes 



~ l/d + b sin" dd 



dt 



---li 



l/{ cos — cos v) {n — cos ) 



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

 amplitude of these oscillations being 2r], If we put 



