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1911] TO THE VIBRATIONS OF A CTBCULAE MEMBRANE 3 The physical argument may easily be extended to show in like manner that all the finite roots of /,/ (2) increase continually with n. For this purpose it is only necessary to alter the boundary condition at r — 1 so as to make dw/dr = 0 instead of w= 0. The only difference in (1) is that z^ now denotes a root of Jn' (z) = 0. Mechanically the membrane is fixed as before along 6 = 0, & = ft, but all points on the circular boundary are free to slide transversely. The required conclusion follows by the same argument as was applied to Jn. It is also true that there must be at least one root of J'n+l between any two consecutive roots of Jn't but this is not so easily proved as for the original functions. If we differentiate (2) with respect to z and then eliminate Jn between the equation so obtained and the general differential equation, viz. "« = 0, .....................(3) # \ & / we find In (4) we suppose that z is a root of Jn', so that Jnf = 0. The argument then proceeds as before if we can assume that z2 — 'ii- and z* — n(n + I) are both positive. Passing over this question for the moment, we notice that Jn' and J'n+i have opposite signs, and that both functions are finite. In fact if Jn" and Jn could vanish together, so also by (3) would Jn, and again by (2) Jn+1; and this we have already seen to be impossible. At consecutive roots of Jn, Jn" must have opposite signs, and therefore also J'n+i. Accordingly there must be at least one root of J'n^ between consecutive roots of Jn'. It follows as before that the roots of J'n+i separate those of Jn'. It remains to prove that z* necessarily exceeds n(n 4-1). That zli exceeds n2 is well known*, but this does not suffice. We can obtain what we require from a formula given, in Theory of Sound, 2nd ed. § 339. If the finite roots taken in order be zi} zz,... za..., we may write log Jn (z) — const. + (% — !) log z + 2, log (1 — z*/*:*), the summation including all finite values of zs\. or on differentiation with respect to z 7700 =.....T'^zf-f This holds for all values of z. If we put z — n, we get * Riemann's Partielle D'iff'erentialgleichungen; Theory of Sound, § 210. 1owest root of </„ is of order of magnitude n.