428 Mr. Bowditch on the motion of a pendulum 
consequence of the increase of the arch c—a’.¢ by 180°. After this 
the value of GR (Fig. 7) independent of its sign decreases, and finally 
becomes 0, at the end of 2m vibrations, when the body will proceed to- 
wards the point K (Fig. 2), nearly in the right line IK, and when ar- 
rived at K the arch z—c’. ¢ will be 360°, and the pendulum will be at 
rest, as at the commencement of the motion, and it will begin again to 
describe the same curves as inthe former period. In Fig. 6, the curve 
described in a quarter of a cycle of revolutions when m=1@ is marked. 
What has been said for the case c=o, c =o and ma whole number, will 
apply, with but little modification, to other values of those soni so 
it will not be nesessary to examine this class of motions more partic 
larly. 
17. When =, =2, or 3r=r’, the equation (H) becomes = 
¢. cosine (2. Ax =>) —sine c. sine QA+ 5 =), which by substituting 
the values of cosine (2A. =) and sine (2A. =) becomes 
== (4 —1), cosine c—2, 2 MW sine. Grey ws 
which is in general a curve of the fourth order, but becomes of : 
second when sine e=o; that is, when e=o, or 180° &c. 
When c=o the equation (I) becomes =- — —1i or (ban). 
=yy, which is the equation of a parabola, whose axis is fie | 
y, and parameter“. Hence if we take r'=3r, and let the pendulum , 
fall from rest from the point K (Pl. 3, Fig, 9) of the parallel ui ¢ 
gram, it will describe the parabolic are KND, whose axis is NS,Ve™ 4 
tex N; the parts KN, DN, being exactly similar and equal. 
the arch af is 180°, a’t will be 90° and the body will be at N, as is evi: ee ) 
deat by the equations (F). When at=360°, a't=190°, and i pis 
Pe Oe Se TeES 6 Mea een Pe 
