1887. | in an Infinite Liquid. 59 
Differentiating (8) with respect to t, and using (9) and (10), we 
obtain 
3 ye | . Gk 
Ape cos a— COn+(_—p) Z eos a — —3°=0... (11), 
where Z=IK' + €, cosa. 
In order that steady motion may be possible, we must have 
C°O? > 4ZA cos a ee Z cosa — a| tia (12). 
il 
R -3) R 
Hence, if R> P steady motion will always be possible, but if 
P> fh, steady motion will be impossible unless the condition (12) 
is satisfied. 
If x, y, 2 be the co-ordinates of O, we have 
&=(ucos 8+ wsin 8) cos v= 2 (a- p) eosa— ith sin a cos pf, 
y = (ucos 6 + w sin 6) sin = 1% (5-5) nie sina sin pt, 
Ji Mowe. 
ati en a COsne Gc Osi2 
2=weos@ usin 0 = Z ( =n — ae 
whence the centre of inertia describes the helix 
Ly (S Le So 
ia z {2 iz >) cos a — zp sina sin yt, 
y=-2 12 (5 pp) 00s a = 5th sin a 0s pt 
iby sin’@  cosa\ ¢, cosa) 
z= J Z ( P + R ) mt t. 
This last result may be at once obtained from the fact that the 
impulse of the motion must consist of wrench about a fixed axis. 
To examine the stability differentiate (8) with respect to t, and 
we obtain vhs 
Ad +f (0) = 
_ Hence the motion will be stable or unstable according as /"' (a) 
18 positive or negative. 
Now 
Ze ae ee ah 26,5. 0) CO 
#(=5 (p-%) sin? 9 + R + Fang (F + CO (cos a — cos 4)} 
cos 8 
Tana {G + CO, (cos a — cos 6)}’; 
