102 
IOWA ACADEMY OF SCIENCE Voi.. XXVIII, 1921 
and 
mx 
^ ~ M 
( 10 ) 
Let Ko be the radius of gyration of M about Co, and ko that of m 
about Oi- Then the moment of inertia about Cx is given by 
/q = {M + m)k^ — + Mx'^ + + mx^ (11) 
Substituting for x' from (10) we have 
Let 
then 
MKo^ + mko^ 4 " m^x^/M + mx^ 
X2= r (12) 
M + m ^ ^ 
M = aM (13) 
Ko^ + ako^ . „ / , ^ N 
= 1 + 0 + 
or 
X2 = B + o,r2 (15) 
where 
„ Xo2 + ako^ 
B = 1 + 0 ~ 
Substituting (15) in (8) we have 
r. = 2ir J5_±_2£i±+! = 2ir jB_±_(l±_5)£i (17) 
^ gx ^ gx 
Again for every value of there are twp values for x, except 
at the minimum point. Determining from (17) the value of x 
to make a minimum, we find 
X 
=v 
B 
1 + a 
This readily reduces to the former value .r = iC, if w = O (and 
hence a= 0). 
The form of (17) is seen to be the same as that of (1), and so 
here too we have the possibility in the most general case of four 
positions of the knife edges which would yield the same period of 
vibration. 
In conclusion then we can state the following : The equality of 
periods, when a pendulum is suspended in turn from two knife 
edges, is a necessary condition that the length between the knife 
edges may be equal to the equivalent simple pendulum, but it can 
not be said to be a sufficient condition. 
State University Oe Iowa. 
