the Pendulum vibrating Seconds in the Latitude of London. 427 
But upon further reflection, it becomes evident that these mo- 
tions, though concentric, are related to each other in proportions 
somewhat different from those of a similar pendulum vibrating 
on a single point, and it is therefore necessary to determine the 
modification of the motion produced by this difference of con- 
nexion. The investigation may however be conducted in a me- 
thod much more simple and intelligible to ordinary capacities, 
than that which has been adopted by the celebrated mathemati- 
cian to whom we are indebted for the theorem ; and I am tempted 
to send you an “ appergu ” of the reasoning by which I have sa- 
tisfied myself respecting it. 
** It follows immediately from the general theorem for finding 
the curvature of trochoids of all kinds, (Lectures on Nat.Phil. II. 
p. 559) that the radius of curvature of the path of any point, in 
the rod of a pendulum supported by a cylindrical axis, will ini- ~ 
tially be a third proportional to the distances of the point from 
the centre of the cylinder, and from the surface on which it rolls: 
so that when the cylinder is small, and the pendulum simple, the 
centre of curvature of its path may be considered as situated at 
the distance of the radius 7 below the point of contact: and this 
is obviously the only correction required for such a pendulum as 
that of Borda. But when the weight is divided, or of consider- 
able magnitude, it becomes necessary to calculate the effect of 
the different curvatures of the paths of its different parts, and to 
compare these paths with that of a pendulum A of any given 
length a. Supposing, for the sake of simplicity, the weight of 
each horizontal section to be concentrated in the vertical line, 
and calling the distance of any particle P below the surface of 
the cylinder x, the radius of curvature of its path will be a third 
proportional to x+,7 and a, that is, mats 3; and the inclination of 
the curve at a given distance from the vertical line being always 
directly as the curvature, or inversely as its radius, the force de- 
rived from the weight of P will be to the force, at an equal di- 
a(x+r) 
rv 
stance in the path of A, as a to at or as to 1. Now 
the point of the rolling pendulum confined to the vertical line is 
not the centre of curvature, but initially the surface of the cylin- 
der: so that this must be considered as the point of intersection 
with the vertical line, and as the fulerum of the lever; conse- 
quently the distance of P from the vertical line will be to that of 
the pendulum A, as x to a, and its immediate force will be 
a(r+r) x x 
tr” a’ P= 
*"P; but this force, acting only at the end 
cv 
of a lever x, will have its effect at A again reduced in the ratio 
of 
