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 &> 

 must be now replaced by — w, 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 ^'^ = ^'o = and i/r'^ 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 



j/d + bam^Sde 



l/{cosO — cos t; ) ( cos 7^^ — cos 6) 



If -t/r'^ > 0, i. e. the rotation of the torus appears from left 

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

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

 axis of the earth (i. e. its northward direction), then 77j = 

 0Q < 7). If, on the contrary, yjr'f, < 0, ^. e. the direction of 

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

 ■»7 = ^5 > 7)^. Hence, the axis of the torus oscillates between the 

 positions 6 = 6^^ and 6 = rj or rj^y never passing through ZZ ' 

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

 a complete oscillation is 2tj, where 



^if 



y'd + bsiD'e de 

 (27) -- ' • 



\/{C09,6 — COS*;) (cOSt;^ — 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- 



