The answer to the following question, proposed in the Smith's Prize Examination, 1871, is sent in compliance with a request from one of the Editors:

“In a compound pendulum consisting of masses m, m′ attached to strings of length l, l′, in which of course the most general small motion in one plane consists of two harmonic vibrations superposed, if the upper mass m be very large compared with the under mass m′, it is clear that one of the two periodic times {that corresponding to the mode of vibration in which m is nearly at rest) must be very nearly the same as in a simple pendulum of length l′, and the other very nearly the same as in a simple pendulum of length l. By α continuous variation of l′, the former may be made to pass continuously from less to greater than the latter, and therefore for some value of l′ nearly equal to l the two must be equal. But when a system is in stable equilibrium (as is clearly the case here) the equation the roots of which give the times of vibration cannot have equal roots, for that would imply the transitional condition between stable and unstable.

“Point out precisely the fallacy which leads to the above contradiction.”