g6 Capt. Kater's experiments for determining the 
mine the modification of the motion produced by this difference 
of connexion. The investigation may however be conducted 
in a method much more simple and intelligible to ordinary 
capacities, than that which has been adopted by the celebrated 
mathematician to whom we are indebted for the theorem ; 
and I am tempted to send you an “ apergu” of the reasoning 
by which I have satisfied myself respecting it. 
It follows immediately from the general theorem for find- 
ing 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 cylindri- 
cal axis, will initially 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 r 
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 considerable magni- 
tude, it becomes necessary to calculate the effect of the dif- 
ferent 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+ r and x, that is, ~L ; and 
the inclination of the curve at a given distance from the ver- 
tical line being always directly as the curvature, or inversely 
as its radius, the force derived from the weight of P will be 
