64 /. G. Barnard on the Gyroscope. 



To determine the force of g' ; multiply tlie first of preceding 

 equations by h^ and the second by a, and add the two, and add 

 likewise A\vydh-\~Va;da)=—AndQ0&6 (seep. 54) and we shall 



get 



A d {b Vy -\-a V2;)-{-Cnd cos 6z=i'^ Mg' (^a'b'-a 6') d L 



By referring to the values of a, a', 5, Z>', and performing the 

 operations indicated and making cos y^=o^ sin V'^l, the above 

 becomes, 



Ad{bvy^av:c)'\-Cnd QOB Oz=:-^Mg' sin ddU 



But the value of {bVy-\-aVa;) (p, 54) becomes zero when 777=0. 



__ . Cnd cos d Cn dO ^ 



Hence g — ^— — — _ * 



dt 



yMsmOdt y M dt 



dO . 



The second factor y- is tbe angular velocity with which the axis 



of rotation is moving. 



Hence calling v^ that angular velocity, the value of the defied- 

 ingforce^ g' may be written (irrespective of signs), 



that is, it is directly proportional to the axial rotation n, and to t 



the angulojT velocity of the axis of that rotation. By putting for . 



<7, Mh- (in which Ic is the distance from the axis at which the 

 mass M^ if concentrated, would have the moment of inertia^ (7,) 

 the above takes the simple form 



r 



In the case we have been considering above, in which g' is sup- 

 posed to counteract the deflecting force of axial rotation, the angu- 

 lar velocity v^ , or — y- (equation g) is equal to -^ (cos ^ cos of). 



But in the case of the free motion of the gyroscope, this de- 

 flecting force combines with gravity to produce the observed 

 movements of the axis of figure. 



If, therefore^ we disregard the axial rotation and consider the 

 body simply as fixed at the point 0, and acted upon, at the cen- 

 ter of gravity, by two forces — one of gravitv, constant in inten- 

 sity and direction--the other, the deflecting force due to an axial 



C 



rotation ?2, whose variable intensity is represented bj— ^^^*^j 



* The effect of gravity Is to diminish S and the increment dO is negative in the 

 case we are considering. Hence the negative sign to the value of a^ indicating that 

 the force is in the direction of the poiltive axis of y, as it should, since the tendency 

 of the node is to move in the reverse direction. 



