100 
IOWA ACADEMY OF SCIENCE Voi.. XXVIII, 1921 
same period of vibration. Similarly, from the other point of sus- 
pension, O 2 , if y is the distance of Og from C we have two other 
equal periods of vibration if y = or — . 
/Z'2 
We can imagine then that our pendulum (Fig. 20- A) has four 
weightless knife edges, two on each side of the center, so adjusted 
with respect to the masses on the pendulum rod that when it is 
hung in turn from the four, the periods will be the same. The 
possible distances between knife edges, x y, will then be seen 
to be, 
r / I ^ 
h^+Khr + j^, +fe. andiC2 + 
In general these four distances will be dififerent, and only one 
of them corresponds to the length of the equivalent simple pendu- 
lum. In practice there is no difficulty, for one can by rough cal- 
culation determine if the knife edges are in the proper positions. 
Again, the laboratory forms of Kater’s pendulum are usually so 
constructed that one can not easily attain on the actual pendulum 
the wrong positions of the knife edges, but in the case referred to 
in this article, the students had succeeded in doing just this thing. 
It is evident that if there are two values, of the distances from 
Oi to C that yield the same period of vibration, there must be some 
.ar beween them which is the only distance corresponding to a single 
period, and that a maximum or a minimum. Our function of x 
^2 I ^2 
to be studied is — , and a simple examination shows that 
this becomes a minimum for x—±K. Only the plus value of 
K need be considered, since the other value relates to the other 
support. . The relation becomes more understandable on platting 
a typical curve connecting ;ir and T. We have 
T = + 
(6) 
^ gx 
(7) 
- J 
where C is a constant. In figure 21 we have such a graph for the 
special case where C = 1, and K = 2 (not supposed at all to rep- 
resent the facts, but merely to represent the nature of the func- 
tion). On the curve the. point D represents the minimum. It 
is the only value of .v corresponding to the period represented by 
the ordinate at that point. From A we draw the line ABC parallel 
to the X-axis. The two intersections B and C represent the two 
