422 Mr. Bowditch on the motion of a pendulum 
by mathematicians, consequently the motion of the compound pend 
lum may in this case be investigated, as if it were two. simple ones, 
agreeably to the former part of this article. ra 
11. For illustration we shall here compare the path of a sso 
pendulum on a spherical surface with that of the compound pendulum 
on the surface DILN, supposing the body to fall from rest, froma 
point N. — In the first case when the surface DILN is supposed spher- 
ical, (the point C coinciding with G) the body in falling from rest 
from the point N, would, by the force of gravity in the first sae 
describe the small arch Nv, situated in the vertical plane, perpendicu- 
lar to the spherical surface at the point N, which plane would pass 
through the lowest point D of ‘the spherical surface, and the action 
of gravity upon the body, in the successive moments of time, contin- 
uing to operate in the same plane, the paths described would be an 
arch NrD of a great circle, whose projection on the plane of xy 
would’be a right line. Ina similar way, if the surface DILN were 
_ not spherical, the line NR passed over in the first moment of the fall 
would be in the vertical plane perpendicular to that surface at the. . 
point N, and this plane woudd not pass through the lowest point D, €x- 
cept bor =o; and the force of gravity would act in she r 
the successive points of the path described by the body, consequen 
ly that path would be represented by a curve, as NRP, meeting the 
arch DEP in P, the points N and P being on different sides of ‘the 
point D (or rather on different sides of the plane yz), when 4 does not 
much exceed a’, For at the point P the value of y=d/ cosine at is 
0, at being =90°, and as ais greater than a’, at would exceed 90° 
and its cosine would become negative, consequently nial saa a 
tee have a different sign from 6. 
2. That we may more fully understand the nature of the ape 
