94 
Dr. Young’s remarks on the reduction 
4. Euler’s formula for the rolling pendulum. 
I beg leave to observe, in conclusion, with regard to Mr. 
Laplace’s theorem for the length of the convertible pendulum 
rolling on equal cylinders, that its perfect accuracy may 
readily be inferred, without any limitation of the form of the 
pendulum, or of the magnitude of the cylinders, from the 
general and elegant investigation of Euler, which also 
affords us the proper correction for the arc of vibration. This 
admirable mathematician has demonstrated, in the sixth volume 
of the Nova Acta Petropolitana, for 1788, p. 145, that if we 
put k for the radius of gyration with respect to the centre of 
gravity, a for the distance of the centre of gravity from the 
centre of the cylinder, c for the radius of the cylinder, It for 
i a — C Y> and b for the sine of half of any very small arc 
of semi vibration, we shall have, for the time of a complete 
oscillation, ■ * b — -j- ~- b - b S h] ^+^ ac ) and ultimately, if b = o, 
only? which, for a simple pendulum, of the length a , 
k and c both vanishing, becomes and for any other 
len S th ’ conse< l uentl y- makin g 7755 = ftir we have 
jL 
V7 = — — , and al = hh — p -}- a* — 2ac + c-. Now if we 
find another value of a, which will fulfil the conditions of the 
equation, all the other quantities concerned remaining unal- 
tered, and add the two values together, we shall have the 
distance of the centres of the two cylinders correspond- 
ing to the length / of the equivalent pendulum ; but since 
ct — [l + 2(7 ) a = — It — -c 2 , we have a — \l — c = + . . ., 
and a-=^\l so that the sum of the two values of 
