552 
THE TIME OF ROLLING OF A CYLINDER. 
Now 
M M d cos S d cos <p sin <p c? cos 6 
d cos $ d cos d<^ sin 9 d cos <p' ’ ' ° ‘ ‘ 
Also by equation (6.), 
c?cos9 2a(§’^+ cos^ <p) cos f — 2(« cos® <P + /35^) cos ip 2(«— cos ip 
(g'®+ cos® <p)® 
( 8 .) 
d cos (p 
by equations (7.) and (8.), 
d& 2(« — (S)g® g'®+cos®ip 
cos ^5 2(a — | 3 )g 
(5® + cos® f ) 
cos p 
2 ^\2 » 
( 1 -/ 3 ®)^? (g®+j 9 ® cos® p)2 (^®+cos®p)® (g'®+ COS® p)(g'®+j 9 ® COS® p)*’ 
/ce—cosS\^dS 2(x — ^)q^ 
[ 1 1 
ycos 9 — / 3 / d<^ (i_|32ji ^ 
[ (g® + COS® p) {(f +j0® cos® p) 2 / 
2 («— / 3 )g® 1 
1 
1 
1 
L (g® + 1 — sin® p) (g® +^®— g)® sin' 
2(«-/3)g® 1 
1 
(l-/3®)l(g,2 + g®)i(l+g®)| 
(1 
2(«-^)g® 
If 
and 
. , ^ ^ 
(1 — /3®)2(j(j® + g'®)2(I +22)\(1 —n sin® p)(l — c® sin® p)'^J 
1 1-/3 
'H-§® «— / 3 ’ 
1-/3 
1 + « 
2 1 +/3 _(1 +«)(!— / 3 ) 
^ p® + 9®~l+« 1 — « 2(« — /3) ’ 
( 9 .) 
( 10 .) 
fi / a — COS 9 \ 2 * C?9 
2(«-/3)?® 
c?p 
0 \ cos 
— /3J (1 _ 132^2(^2 _j_ ^2 j (1 — n sin® p)(l — c® sin® pj’i 
Ilf — nc<X> ^ 
- (1 - | 32 )i(/ + /) 4(1 + 9®) '^^7 ’ 
(II.) 
where !!( — ncipj) is that elliptic function of the third order whose parameter is —n and 
modulus c. 
1 
Now 
1 
a, /I — «\'li V 2(«— /3) 
{p^ + qy J fl±_ , _ 
/I — 
_q^ _ /«-l\ 
/l — u\ \«~/ 3 /' 
* I cannot find that this function has before been integrated, except in the case in which 9 is exceedingly 
small. 
