1911] TO THE VIBRATIONS OF A CIRCULAR MEMBRANE 5
Pursuing the process, we find that if Jn-i, Jn±s have a common root z, then
(2?i 4-1) z* = 4n (n + 1) (n + 2),
so that z*- is rational. And however far we go, we find that the simultaneous evanescence of two Bessel's functions requires that the common root be such that z- satisfies an algebraic equation whose coefficients are integers, the degree of the equation rising with the difference in order of the functions. If, as seems probable, a root of a Bessel's function cannot satisfy an integral algebraic equation, it would follow that no two Bessel's functions have a common root. The question seems worthy of the attention of mathematicians.J_ /y "*2 1_ « "~~2 _l