56 J. G, Barnard on the Gyroscope. 



o A^ "n~~ *^^ above equation becomes 



d6^ 2 G 

 sin^O—^ =-f [sin ^0^2p (cos^-cosa)] (cos(9-cosa) (6) 



and tlie first equation (4) becomes 



sin 



2 ^1^— 2^ J-| (cos 5- cos a). (7.) 



Equation (6) would, if integrated, give tlie value of d in terms 

 of the time ; that is, the inclination which the axis of figure 

 makes at any moment with the vertical ; while eq, (7) (after sub- 

 stituting the ascertained value of 0) woidd give the value of V 

 and hence determines the progressive movement of the body 

 about the vertical Oz. 



These equations in the above general form, have not been 

 integrated ;^' nevertheless thej furnish the means of obtaining all 

 that we desire with regard to gyroscopic motion, and in particu- 

 lar that self-sustaining pOwer, Avhich it is the particular object 

 of our analysis to explain. 



In the first place, from eq. (6), by putting — equal to zero, we 



u C 



t 



can obtain the maximum and minimum values of 6. This diff. 

 co-efficient is zero, when the factor cos 6— cos cc—0^ that is, when 

 6=a'^ and this is ^ maximum^ for it has just been shown from 

 equations (4) that cannot exceed a. It will be zero also and 

 a minimum ^\ when 



sin^ (9—2(52 (cos ^- cos a)—0 

 or cos ^= «.^2^v'l-j-2^2"cos«+5* (8.) 



(The positive sign of the radical alone applies to the case, since 

 the negative one would make a greater angle than «.) 



It is clear that (« being given) the value of 6 depends on ^ 

 alone, and that it can never become zero unless ^ is zero ; and 

 as long as the impressed rotary velocity n is not itself zero (how- 

 ever minute it may be), § will have a finite value. 



Thus, however minute may be the velocity of rotation, it is 



suflicient to prevent the axis of rotation from falling to a vertical 

 position. 



The self-sustaining power of the gyroscope when very great 

 velocities are given is hut an extreme case of this law. For, if ^ 

 is very great, the small quantify 1 — cos ^ a maybe subtracted 

 from^ the quantity under the radical (eq. 8) without sensibly 

 altering its value, which would cause that eq. to become 



cos n: COS a* 



* The integration may be effected by tlie use of elliptic functions : but the pro- 

 cess is of no interest in this discus&iion. 



f It is easy to show that this value of $ belonga to an actual minimum ; but it is 



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