52 Mr Basset, On the Motion of a Ring [Jan. 31, 
2/Ddd 
Then os Tea eT DSHS? 
and 
{(cos 0—p)’+q"} {1+.D+(1—D)cos¢}={1—D—p(+D)?+@(14+D)? 
+2coso[1—p)+q°— Did +p) +@3]+ [1+ D—p (1 —D)P 
+ q° (1 — D)’] cos’ ¢. 
D= (Oe a 
(1+ py +9" 
the coefficient of cos will vanish; substituting this value of D, 
we obtain 
Hence, if 
dd 2 1 dé 
V{(cos0—p)?+q7}  {((L+p*+@')*— 4p}? VL — Hsin’ $) 
where B= : ji “is io eet 1 
| Ga Rd apa 
Hence ¢=amnlt, 
where T=M\1+p4+¢y- ap} ; 
and we finally obtain 
tans = / Goer 1—en Lt 
2° \/ (L+p)?+q?°14 en It’ 
and the time of a complete revolution is 4K/l’. 
Case Lik PSs. 
In this case both roots are real, and one root is always negative 
and numerically greater than unity. 
(i) Let the roots be p and —qg, where g>1>p>0. The 
transformation is the same as in Case I. sub-case (i). 
(i) Let the roots be — p and —q, where g>1>p>0. 
Then 6° = M* (cos @ + p) (cos 8 + q), 
2 
(2 = aap 
where AUT ss APR (P — R). 
In this case we employ the same transformation, but must put 
eae 
ae —p’ 
LG CEC) 
U = 
Meg Digey 2(¢- p) 
