A532 $, Bowdiich on the motion of a pendulum 
will be as in Fig. 12, both these curves being described according to 
the order of the letters ed/g. When n=3m (which corresponds to 3m 
vibrations of the pendulum 7+’) the arch 27C will be 180°, the curve 
will become a parabola (as in Fig. 13), and the pendulum will vibrate 
backwards and forwards in the archCSA. When n={m, the arch 
2nC will be 225°, the pendulum will then again revolve in the curve 
described in Fig. 12, but in a contrary direction, the motion being ac- 
cording to the order of the letters gfedcba ; for at the point e where 
d d a 
the curve touches the line CK, we have = =o and changes sign 
by making 2nC successively equal to 135° and 225°, as evidently ap- 
pears by the equations (C). When n=$m, the arch 2nC will be 270°, 
~ and the curve as in Fig. 11: When n=Zm, the arch 2nC will be $15 
and the curve as in Fig. 10, both these curves being also described in 
a contrary direction to that when m=1 and2. When n=m, which cor- 
responds to m vibrations of a simple pendulum of the length r+’, oF 
2m-+1 vibrations of the simple pendulum of the length r, the arch 2nC 
will be 360°, the curve described will be as in Fig. 9, at the commence- 
ment of the motion, and the cycle of motions will be complete, pr0- 
vided m be a whole number, and the body will again hegin to describe 
the same curves as in the former period. The remarks, made at the 
end of § 15 for the case of m not being a whole number, apply without 
modification to this. 
”“_, consequently 5r 
= I 
If we put gu —, we shall have m= a 
must exceed r by a very small quantity to render m positive and great. | 
In this case a will become €qual to 4. cosine (2a’t— es and in the 
- | 
_. Successive revolutions, mentioned in the first part of this article, it will 
become b, b cos. (—2C), &e. equal respectively to b, 6 cos. 2C, U 
_ Sos. 4C, &c. as in the case of = =2+—. Consequently the curves 
