1887. ] ‘in an Infinite Inqud. Bil 
In order to find the period of oscillation or revolution, as the 
case may be, we must express 0 in terms of ¢. 
Casella) f= EF: 
(i) Let the roots be real and equal to p and — qg, where 
q>1l>p> 0. 
Equation (5) may be written 
6? = M? (cos 6 — p) (cos 6 +), 
where i — (R — P). 
_1-—Dcos’¢ 
Let COs 7) = 1+Deos'¢ 5 
where nee ; 
1l+p 
_ 2/Dsin ddd 
Then dé = eens ris 
ty (Ge) Caen) je ee 
(cos 0 — p) (cos 6 + q) Si (ueeinicas.h: (1—k’ sin’), 
where = (q as 1) (1 =P) 
2(p+q) 
; < 2 dd 
Therefore Mdt= Meee Ce) a) (eral Vd — Hsin’ ) = fsin® 6)’ 
therefore o=amlt, 
where T=3M J/(1+4+p) (1 +9). 
Therefore 
1l+p—(l—p)on’ tt 
odie. peak) en’ It’ 
and the period of a complete oscillation is 447/T. 
(ii) Let the roots be imaginary and equal to p + ug. 
Then 6 = M? {(cos 0 — p)?+ q't. 
1-D+(1+D)cos¢ 
1+D+(1—D) cos¢° 
Let cos 9 = 
