474 MR. 8. S. HOUGH ON THE OSCILLATIONS OF A 
The 1st condition requires that y? should not lie between a? and £?. 
Now 
yi — 8 (0? + B) 7? + 5028" = (7? — 3a?) (y* — 88") — 40%? 
= (3a — y’) (88? — y’) — 402°. 
The 1st form shows that condition (2) is certainly satisfied if 
ye > ba” and also.) >) 582 
The 2nd shows that it is satisfied if y? < a and < f”. 
Lastly, 
{yt — 8 (02 + B’) 7° + 5028}? — 16258" (9° — 02) (7° — B) 
= (y? foe a”) (y? aan B’) fy! — & (a? + B?) y + 9a°B?} + 4 (a? pt 8’)? y4 
(y? — a2) (9° — B) {(y? — 502) (y* — 5B") — 16028" + 4 (02 — BY 4 
=, or 
(a? — y*) (B® — y?) {(5a? -- y*) (58? —y?) — 160787} + 4 (a? — 6)? 
Hence condition (3) will certainly be satisfied if y? > 9a? and also > 98%, or if 
y? < @ and also < B’. 
Thus the roots of (6) will both be real and positive if 
y<a andalso <8, 
or if 
y>3a andalso > 3p. 
These conditions are sufficient, but not necessary, to ensure stability; the neces- 
sary conditions are given by the inequalities (1), (2), (8). 
The analytical conditions here discussed are approximately realised in the case of a 
liquid gyrostat (vide ‘ Nature,’ vol. 15, p. 297) mounted on gimbals in such a way that 
the centre of gravity is held at rest. The inertia of the gimbal-rings will be unim- 
portant when the rotation is rapid, and, if we may also neglect the mertia of the case 
compared with that of the fluid, the gyrostat will be stable when set rotating about 
its least axis; it will also be stable when set rotating about its greatest axis when 
this axis is, at least, three times as great as either of the others. It will, however, 
certainly be unstable when set rotating about its mean axis. 
§ 3. Approximate Solution of the Period Equation. 
Let us for the future suppose that the cavity which contains fluid is approximately 
spherical, so that ~— » Bea 
Y 

ave small quantities. 
en 
