THE ROLLING MOTION OF A CYLINDER. 
551 
Now let it be observed that in this function a>(3 so long as a is less than g, since 
-\-t^ > — , or 2«^cos^i+A^> — 
and F + «^+^^>2aAcos — 
l/F . a h\ k^ + P ^ 
!+« 1 — « a— cosfl 
1+?=?’’ -13-^=?“. 
Then when ^=0, sec ?>= 1 and ^=0. 
1/A:® A fl!^ , 
When let (p=^i 
also 
cosS] — /3 
fk + l^ 2 
\ 2gh r 
l/A® a h' 
\aJ 
\ 
{A® + /®)a)® ffcu® ’ 
. l-« 
) 
(9\ 
^®+(ffl-A)® 
?=-Tr5 
■ (1 -cos «,) + (^)»* *■ + 
, (F + /^)a)® + 2^Avers9, 
•■• {F+(fl-A)2}w2 . 
Now 
/’®i /a— cos 3\^ , . /u— cosS\i dQ j 
'^1 /«— cos 
(5.) 
And since 
a— COS S 
c3r5^=? ‘p’ 
2cos0 — (« + jS) cos^ip — 
(a— /3) cos^<p4-§^’ 
COS ^ 
cos^ <P + 9^ 
COS 
« cos^ <p + (8^^ 
cos^ '? + 9^ 
( 6 .) 
. 2 (cos^ <p + 9^)^ — (« cos^ (p + /3<7^)^ 
(cos^<p + 9^)^ 
(cos® <P + 9^ — cos® <p — /39 ®)(cos® (p + 9® + a cos® (p + /3g'®) 
(cos® <p + g'®)® 
{(1 — |3)g® + (l— at) cos® (?}{(! +/3)g® + (1 + a) cos® <p} 
(9®+cos®<p)® 
_(i p;9 .sin 9- (^ 2_j. cos2<p)2’ 
1 . (fl® + n® cos® (S)^ 
•■. sm <l=y( 1 -<3’)»sin p . cos" f 
4 B 2 
(7.) 
