66 J. G. Barmard on the Gyi^oscope. 



produce it, it is not difficult to understand wliy sucli a motion 

 takes place, 



Fig. 1 represents the body as supported by a point within its 

 mass ; but the analysis applies to any position, in the axis of 

 figure, within or without ; and figs. 3 and 4 represent the more 

 familiar circumstances under which the phenomenon is ex- 

 hibited. 



Let the revolving body be supposed (fig. 3, vertical projection), 

 for simplicity of projection ^ an exact sphere^ supported by a 

 point in the axis prolonged, at 0, which has an initial elevation 

 a greater than 90°. Fig. 4 represents the projection on the hor- 

 izontal plane xy] the initial position of the axis of figure (being 

 in the plane of xz) is projected in Ox. 



Oxj Oy^ Oz, are the three (fixed in space) co-ordinate axes, to 

 which the body^s position is referred. 



In this position, an initial and high velocity n is supposed to 

 be given about the axis of figure Oz^^ so that the visible por- 

 tions move in the direction of the arrows &, &', and the body is 

 left subject to whatever motion about its point of support 0, 

 gravity may impress upon it. Had it no axial rotation, it would 

 immediately fall and vibrate according to the known laws of the 

 pendulum. Instead of which, while the axis maintains (appar- 

 ently) its elevation «, it moves slowly around the vertical Oz, re- 

 ceding from the observer, or from the position ON^^ towards 01^. 



It is self-evident that the first tendency (and as I have likewise 

 proved, the first effect) of gravity is to cause the axis Oz^ to de- 

 scend vertically, and to generate vertical angular velocity. But 

 with this angular velocity, the deflecting force proportional to 

 that velocity and normal to its direction, is generated, which 

 pushes aside the descending axis from its vertical path. — But as 

 the direction of motion changes, so docs the direction of this 

 force — always preserving its perpendicularity. It finally acquires 



of vy and v^) ; hence the two last terras destroy each other, and the above equation 

 becomes identical Trith equation (a) from which the 2d eq, (4) is deduced. 



Multiplying the 1st equation (i) by cos^p and the second by sin<p and adding", 



"vre get, 



-4(cos <pdvy -f" sin <^dv^) = - Cnd 9. 



By differentiating the values of Vy and v^, performing the multiplications, and 

 substituting for dqt its value, coa 9 d^r, (proceeding from the 3d equation (2) when 

 Vj = 0) the above becomes 



. , d^^ d^ d9\ ^ rf9 



Multiplying both members by sm^dt, and integrating, the above becomes 



. ,d^ Cn 



sma 9 — r- = -r- cos 9 + Z ; 



Of A 



the same aa the 1st equation (4) when the value of the constant I is determined. 



