J. G\ Bai^nard on the Gyroscope. 55 



EE 



This section may be called the equator. E being some fixed 

 point in the equator (through -which the principal axis Ox^ 



•n^s;sf«^^. the ancrle od is the an^e EON. 



the 



!* ^is the ascending node of the equator — that is, the point 

 -•hich ^in its axial rotation rises above the horizontal plane, 

 angle 9 must increase from iV^ towards E — that is, dtp (in 

 equation 5) must be positive and (as the second term of its value 

 is usually very small compared to the first) the angular velocity 

 n must be positive. That being the case the value of d(p will be 

 exactly that due to the constant axial rotation ndt^ augmented 

 by the term cos i9cZv', which is the projection on the plane of the 

 equator of the angular motion d^ of the node. This term is an 

 increment to ndt when it is positive, and the reverse when it is 

 negative. In the first case, the motion of the node is considered 

 retrograde — in the second, direct. 



The first member of the second equation (4) being essentially 

 positive, the difference cos ^— cos a must be al^vays positive — 

 that is, the axis of figure O2, can never rise above its initial an- 



gle of elevation a. As a consequence -y- [in first equation (4)] 



must 



K 



in the direction in which v is laid off positively, and the motion 

 "will be direct or retroerrade. with reference to the axial rotation. 



positive 



as 



of figure is above or below the horizontal plane. In either case 

 the motion of the node in its own horizontal plane is always 

 progressive in the same direction. If the rotation n Avere re- 



d, so would also be the motion 



If 



■ -;— must also be 

 ' dt 



equation (4) reduces at once to the equation of the compound 

 ^ pendulum, as it should. Eliminating -77 between the two equa- 



dt 



tions (4) we get 



Bm^,d- — z= ^ fsm* 6^ -.--^ — (cos <?— cos a)] (cos 5— cos a). 



dt^ A '- 2AMYg^ /j v j 



The length of the sirai")le pendulum which would make its 

 oscillations in the same time as the body (if the rotary velocity 



A 



n were zero) is -^^.^ If we call this X and make for simplicity 



^ My ^ 



* The length of the simple peadulum is (see Bartletfa MecL, p. 252) A= 



A 



The moment of inertia A=z3f[kj^^ +/^); hence jj- = x. 



