349 



It might be supposed that if this infinitesimal nutation were pre- 

 vented by restricting the axis to a circular cone round the polar line, the 

 axis would still, as before, follow a fixed star. But this is not so : the 

 relative curve described by its extremity is a spherical cycloid, and the 

 initial tendency of the axis, when set free, being to move towards the 

 polar line, it follows that when this motion is prevented, it remains at 

 relative rest. 



There are one or two points connected with this problem which it 

 may be interesting to examine into. 



1°. Supposing the axis of the gyroscope fixed so as to be compelled 

 to move with the earth, what force would it exert to break its bonds ? 



Let P be the polar line ; 



XX' two consecutive positions of 

 the axis of the gyroscope ; 



QQ! the axes of the resultant 

 couple of all the motion the gyro- 

 scope has at X and X', then G 



= ^(7V + ^'tt^'sm2^o, the axis of the 

 couple added by the connexions in the 

 time {dt), which changes the position 

 of G from Q to Q', must lie in the plane 

 QQf at right angles to Q, the plane of 

 the couple being the plane OQ, let its 

 moment = JSfdt, 



then 



JSfdt 

 G 



sin QQ' 



- QQ' = XX quam proxime, 



= w sin dt. 



N=^ G . to sin 6q - Cnic sin quam proxime, 



that is, the moment of the couple of constraint {N) = that of couple, 

 which, if acting round the axis to stop the spin, would bring the gyro- 



1 



scope to rest in the time 



or that of a sidereal day divided by 



tf sm ^0 

 27r sin 6q. 



This will serve as a measure of the friction to be overcome before 

 the apparent motion of the axis could take efi'ect. 



2°. In the preceding investigation the resultant of the earth's attrac- 

 tion has been supposed to pass through the centre of the gyroscope, and 

 therefore to exercise no influence on its motion. 



In strict accuracy, of course, this is not so, inasmuch as the earth's 

 attraction upon the different parts is neither uniform in magnitude nor 

 direction. The question arises, what is the error induced by supposing 

 it so ? Assuming the earth a sphere, it is evident that its attraction has 

 no moment either round the axis of figure, or round the vertical through 

 the centre of the gyroscope, 



S. I. A. PEOC. VOL. VIII. 3 A 



