326 Vn. DYNAMICS OF ROTATING BODIES. 



not variable), which give by the use of the table of direction cosines, 



the values of the rotations of the axes 



p = SI sin # sin op, 

 QCD\ f) CL drp 



T = i sin # cos op, 



and for the angular momenta, o being again the velocity of spinning, 



H x = A& sin # sin op, 



o r" i"\\ ~TT A i 4~\ d ($)\ 



H z = Co. 

 Inserting in equations 29), 84, the constraint producing the couple Z, 



4 f~\ * tt CD y^f / f~* (t QP\ 



yl 5i sin v cos op -T -j- C o ( 5i cos v" ~^^ I 



- J. te cos 0- - ~\ 1 sin ^ cos op = L, 



d 2 cp 



260) A -=TJ + ^15i 2 sin 2 ^ sin op cos op C&& sin ^ sin op = 0, 



C-~ -\- A ( 5i cos ^ - ~p ] 5i sin -O 1 sin op 



A& (1 cos & -=?] sin ^ sin op = 0. 



V / 



The last equation again shows that o is constant, while from the 

 second, neglecting i 2 we have 



261) -v-^ -j~ -j" &> sin -& sin op = 0. 



The first equation determines the constraint L. The gyroscope again 

 performs oscillations about the meridian, with the period 



A 



which is greater the greater the latitude, being infinite at the poles. The 

 gyroscope in this mounting therefore constitutes a dynamical compass. 

 It is to be noticed in both cases that the equilibrium is stable 

 for # = or qp = if o is positive, and for & = it, cp = it if o is 

 negative, in other words the gyroscope tends to set its axis as nearly 

 as possible parallel with the earth's axis, so that its direction of 

 rotation shall correspond with that of the earth. This was clearly 

 stated by Foucault, although he employed no mathematics. 



