THE TIME OF OSCILLATION OF A PENDULUM HAVING A CYLINDRICAL AXIS. 553 
/A® a h\ 
2(a— 1) \ah^ a) 
2(«-/3)^2 
•• (1-/32)V + ?')^(1 + 9") >/2(«-/3) 
k^ + {a-hY 
a h 
ah~^ Ji a 
■2 cos fij + 
gh 
( 12 .) 
by equations 1 1 and 4, 
sj .i(F+P)(l + y)’ 
.n(-«c?,), (13.) 
where (9.) (2.) (3.) 
1 o 
a— /3 \(k^ a h\ . 
and (lO.) (2.) (3.) 
2^A " 
-=a- 
fZ. ii , + 2 
2ft vers 6, H w' 
9 
(14.) 
(i+(.)(i-/3) i 
f 1 / h a\ 
[2\«A'^ft'^ h) 
1 + 1 ] 
[{ 
(3 + 2 I 
vers 9 .+ »<= 
l-flA 
1 
1 
(N 
! 
(A2 + Z3) 
|f+ (ft + A)®||vers + 
2^A 
2(*‘ + l')(l+^) 
(15.) 
The value of n(— wc<Pi) being determinable by known methods (Legendre, Fonc- 
tions Elliptiques, vol. i. chap, xxiii.), the time of rolling is given by equation 13. 
In the case in which the rolling motion is not continuous but oscillatory, we have 
TT 
ft;=:0; and therefore (equation 5.) ^ 1 = 2 ; n(~wc<pi) becomes therefore in this case a 
complete function. 
To express the value of this complete elliptic function of the third order in terms 
of functions of the first and second orders, let 
sin^%l/=-2=T-^: 
^ 1 -f (St 
2ah 
Then* 
-ha k^ + {a + A)^ 
n(— I) = F(4) + (Tz 5 !^{f( 4) ■E(c4)-E(cf) .F(c+) 
Representing therefore the time of a semi-oscillation by 
(16.) 
t,= 
k^ + (a — k)^ 
a/ g/i(k^ + P) 
F(4)+TrSW{F(<|)E(c^)-E(o0F(c^)}}. 
where (15.) 
c"= 
A:® + (ft + A) 
2(k^ + r 
vers 
( 17 .) 
( 18 .) 
* Legendre, Calcul des Fonctions Elliptiques, vol. i. chap, xxiii. Art. 116. 
