54 /. G. Barnai^d on the Gyroscope. 



Multiplying the two last equations (3), respectively, by h and a 

 and adding and reducing by the value just found of c?.cos d and 

 of Vz, we get 



Differentiating the values of a and h and referring to equations 

 (2) it may readily be verified (putting for Vz its value n) that 



dhz=z[v^ cos d-^anjdt ' 



da:=z[hn — Vy cos O^dt ^ 



and multiplying the first by Av^^ and the second by Av^;^ and 



adding 



A {Vy db'\-Va;da)=^An(bvx -^avy)d /= — And, cos 6, f 



Adding this to equation (6), we get 



Ad. {bvy-^av:^)'^Cnd. cos dzizO, the integral of which is 



A {bVy-f-aVj:)-]-Cn cos 6z=zl (Z being an arbitrary constant), (c) 



Eeferring to equations (2) it will be found by performing the 



operations indicated, that : 



dy^^ d6^ 



dip 



Substituting these values in equations (a) and (c), we get 



Cn cos d- A sm^d~=:zl 



dt 



^^ ' ■^jj^U2 M g Y COS e^h 



If, at the origin of motion, the axis of figure is simply de- 

 viated from a vertical position by an arbitrary angle «, in the 

 plane of xz^ and an arbitrary velocity n is imparted about this 

 axis alone; then v^ and t;^ will, at that instant, be zero, 6=a^ 

 and the substitution of these values in equations (a) and (c) will 

 determine the values of the constants I and A. 



7^z= — 2 Mff Y cos oc 



which substituted in the above equations, malce them 



. dip Cn 



sin2 d"—=:--~r (cos a— cos «) 



dt A ^ * 



8m2 e ~~ 4-r-;- = — -i^ (cos (?— cos 



(substitut 



b 



d(p=:ndt + cosOdtp (5.) 



will, (if integrated) determine the three angles % 6 and ^ in 

 terms of the time L They are therefore the differential equa- 

 tions of motion of the gyroscope. 



