Induced Stability. 235 
except for r=0 and 1, for which 
-B.g + j^ft-O, (0)' 
and 
-*J$+*) + ±%*(SB.+*d=fi%* . . (1)' 
The series is evidently convergent. Since S=^ = fi 2 is small 
compared with n 2 , 
approximately if r>l. 
Also Bi =— -„B , and therefore B 2 = o^ - 2^0- 
1 <m 2 u ' z 2frnr 
B 2 is therefore negligible in (1)', and finally 
B = 
2gW 
i 2 a 2 h 
As before for the stability of the " free " motion n 2 must 
exceed //. 2 /2a, which is equal to 2gk*[a 2 h : if, therefore, n is 
in the vicinity of this limit B , the mean inclination of the 
pendulum, is large compared with /3, the inclination of the 
path of the pivot. Since 2g/an' 2 is small, being less than 
00 
ah/k 2 , it is clear that B is large compared with S I B r | ; i. <?., 
1 
the forced oscillation about the inclined position is compara- 
tively small. This fact is very evident experimentally. 
As n increases B decreases to the limit ft, and the rod 
approaches the line of vibration. 
2. A case of steady motion for which the stability equation 
is of the type 
x + \x = 
is furnished by a solid of revolution rolling on a horizontal 
plane. Consider the rectilinear motion of such a body sym- 
metrical about the middle plane normal to its axis. The 
small oscillations are determined by 
(A + m 2 yx+{Cp*(C + Mb 2 )/A-{b--p)Mg}x=0, 
where p is the angular velocity, b the radius of the mid- 
section, p the other principal radius at any point on it, and 
the other constants are in the usual notation. 
When, therefore, p 2 <g rWrrnvfpY tne motion is unstable. 
