32 Trans. Acad. Sci. of St. Louis. 



We may now immediately apply the formulas derived for 

 the polytrope of Sire to the case of a gyroscope on the sur- 

 face of the earth. The metallic ring ( C^) of the polytrope 

 will, in the following discussion, be replaced by the meridian 

 of the place of observation. 



We will assume that ZZ ' is parallel to the axis of the 

 earth. Referring to the figures in the text it is clear that (o 

 must be now replaced by — «, since the positive axis of rota- 

 tion of the earth is directed southward. 



We will consider the motion only under Foucault's condi- 

 tions, i. e. when 6\ = <^\ = and -v/r'j, is very great. Then 

 only one among the cases discussed for the polytrope pre- 

 sents itself, namely the case /C, as can be readily seen. 

 Therefore, 



( 26 ) dt 



= ±\1 



i/ d -\- h ^m^ d ae 



;/(COS^ — C08?7)(COS7?^ — COS 6) 



If -v/r'j) > 0, i. e. the rotation of the torus appears from left 

 to right to an observer standing along the axis ON^ with his 

 feet at O, this axis forming an acute angle with the negative 

 axis of the earth (i. e. its northward direction), then 17^ = 

 6q < 77. If, on the contrary, yjr'^ < 0, ^. e. the direction of 

 rotation of the torus is opposite to the one just described, then 

 7? = ^5 > 77j. Hence, the axis of the torus oscillates between the 

 positions 6 =^ 6q and 6 •= rj or ij^^ never passing through ZZ ' 

 (which is parallel to the axis of the earth). The period of 

 a complete oscillation is 2r^f where 



4ij\ 



l/d + b sin2 e dO 

 (27) '-- ' ■ 



( cos 6 — cos r}) [ COS t]^ — COS 6) 



Of the three cases of motion when the nutation is == 0, only 

 one is possible in Foucault's gyroscope, namely the one deter- 



