length of the pendulum vibrating seconds. 
95 
APPENDIX. 
Since the preceding Paper was written, a very curious and 
important theorem has been discovered by M. Laplace, 
of which Dr. Young has favoured me with a concise 
demonstration, together with some other investigations 
which I shall subjoin in his own words. 
My dear Sir, 
I cannot forbear to congratulate you on the discovery of 
the singular property of your pendulum, which has lately 
been demonstrated by M. Laplace, since it appears to 
remove the only doubt, that could reasonably be entertained, 
of the extreme accuracy of the results of your experiments. 
The correction for the curvature of the rolling surfaces, in 
the case of a simple pendulum, is very easily obtained from 
the geometrical determination of the curve described, although 
M. Laplace’s train of reasoning, from mechanical princi- 
ples, is somewhat too elaborate to be readily followed through 
all the symbols in which it is enveloped : and the same geo- 
metrical considerations appear, at first sight, to be equally 
applicable to the case of compound pendulums in general, 
since the motions of all their effective parts are concentric 
with those of a simple one similarly suspended. But upon 
further reflection, it becomes evident that these motions, 
though concentric, are related to each other in proportions 
somewhat different from those of a similar pendulum vibrat- 
ing on a single point, and it is therefore necessary to deter- 
