4 



J. G, Barnard on the Gyroscope, 65 



and whose direction is always normal to tlic plane of motion of 

 tlieaxis; we onglit, introducing these forces, and making the 

 axial rotation n zero, in general equations (3), to be able to de- 

 duce therefrom the identical equations (4) which express the mo- 

 tion of the gyroscope. 



This I have done ; but as it is only a verification of what has 

 previously been said, I omit in the text the introduction of the 



somewhat difficult analysis."^' 



i ntegr ati 



(pz=int-{- 



which, with the values of 2i and ^p already obtained, determines 

 completely the position of the body at any instant of time. 



Knowing now not only the exact nature of the motion of the 

 gyroscope, but the direction and intensity of the forces which 



h - 



* To introduce these forces in eq. (3) I observ*e, first, that as both are applied at 

 G (in the axis Oz^) the moment Z^ is still zero and the Jirst eq. becomes, as before, 



Cdv^ =z or v^ = const. 



And as we disregard the Impressed axial rotation, we mate this constant (or v^^ ) 

 zero. 



On 

 Tlie deflecting force — r> v, (taken with contrary sign to the counteracting force 



Cn <f 9 Cn d^l , 

 just obtained) resolves itself into two components — j> -jr and j^ -3- sm 9, the 



first in a vertical, the second in a horizontal plane, and both normal to the axia of 

 figure. 



The first is opposed to gravity, whoso component normal to the axis of figure, 



is g sin 9. 



Hence we have the two component forces (in the directions above indicated), 



Cn d^ I Cnd^ . ^ 



These moments witli reference to the axes of y x ^''^^ ^ i ^^^ ^^ 



Cn d^\ Cn S 



sm ViM [s-:^-^j sia e - cos <F7Jf ^ ^ ' and 



Cn d^ \ Cn rf3 



Hence equations (3) (making v^ zero, and putting for M^ and iV^ the above values, 

 and recollecting the values of a and 6, (p. 53) become 



d"4/ (fa 



Adv^ = a^Mgdt — aCn -^ dl— Cn cos ^'ludt 



Adv^ =- hiMgdt^bCn -^ dt— Cn ^^^V^ dt 



Multiplying the equations severally by Vy and v^., adding and reducmg (as on 

 p. 53) we get 



A{vifdvy-\- Vj^dv^ )= 71/jri .cos 9 — Cn -jr d . cos 6— Cn <7v ( Vy cos ff-f-^^jr s^" ^) 

 But VyCos(^-\-Vj^m\(i> will be found equal to sm 9 -^ (by substituting the values 



SECOND SERIESj VOL, XXIV, KO. TO. JULY, 1857. 



9 



