^ 



J. G. Barnard on the Gyroscope. 53 



v^cf^zrzsm 6 sin <jp(7^// — cos qxiO 

 Vydt:=.^m 6 cos qprf^'+sin g)c//9 J- (2.)* 



v^dti^d (jp ^cos ddyj 



The general equations (1.) are susceptible of integration only 

 in a few particular cases. Among tliese cases is that we con- 



its axis of figure. 



Let tlie solid A BCD 



2/ 



solid, 



whicli Oz^ is the axis of figure. It will be, of course, a princi- 

 pal axis, and any two rectangular axes in the plane, tbrougli 

 perpendicular to it, will likewise be principal. By way of de- 

 termining tbem, let Ox^ be vsupposed to pierce tlie surface in 

 some arbitrarily assumed E point in this plane. Let G be the 

 center of gravity (gravity being the sole accelerating force). 



(1.) reduce to 



and 



Cdv. = 



Advy —{ C— A) Vz Vx dtz^y a Mg dt 

 Adv^-[-{C—A)vyV:,df=^yhMgdt 



in Avhich the distance OG of the point of support from the cen- 

 ter of gravity is represented by j^, g is the force of gravity, M 



the mass^ and a and b stand for the cosines x^ Oz and y^ Oz and 

 of which tlic values are (p. 52) 



a 



sin 6 sill <p, J=: —sin 6 cos ?) 



y 



The first equation (3) gives by integration v^^rx^ n being an 

 arbitrary constant; it indicates that the rotation about the axis 

 of figure remains always constant. 



Multiplying the two last equations (3) by Vy and v^ respect- 

 ively and adding the products, we get 



A (vy d Vy -{-Vx d Vx ) r= J' Mg {a Vy — h v^ ) dL 



From the values of a and h above, and from those Vx and v 

 (equations 2) it is easy to find 



{aVy — 6i'a:)rf/:^— sin 0d6:^d .CQ% d\ 



substituting this value and integrating and calling h the arbitrary 

 constant 



A{vy^-\-Vx^)=1yMg cos d^h (a) 



* To avoid the introduction of numerous quantities foreign to our particular in- 

 vestigation and a tedious analysis, I have departed from Poisson and substituted the 

 above simple nlethod oi getting equations (2.), which ib an inptructive illustration 

 of the principles of the composition of rotary motions. 



f See Bartletfs Mech. Equations f225) and (118) for the values of L^ M. iV, : 

 in the case we consider the extraneous force P (of eq. 118) is 9; the co-ordinates 

 ar',y' of its point of application G (referred to the axes Oxy, Oy^, Oz^,) are zero 

 and z^=^OG=j\ cosines of o, |3 and 7 a^e a, b and c: hence Ly=^0, My^'^aMg^ 



