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2 NOTE ON BESSEL'S FUNCTIONS AS APPLIED
increasing without limit the potential energy of the displacemei every point of the small sector to be cut off. We may imagine springs to be introduced whose stiffness is gradually increased, without limit. During this process every frequency originally fin increase*, finally by an amount proportional to d$\ and, as we knov root can become finite. Thus before and after the change the fir correspond each to each, and every member of the latter series ex< corresponding member of the former.
As /3 continues to diminish this process goes on until when ft res n again becomes integral and equal to 2. We infer that every fin.il Jz exceeds the corresponding finite root of Jlt In like manner ev< root of 1/3 exceeds the corresponding root of J2, and so onf.
I was led to consider this question by a remark of Gray and Mai " It seems probable that between every pair of successive real ro< there is exactly one real root of Jn+1. It does not appear that this strictly proved; there must in any case be an odd number of rool interval/1 The property just established seems to allow the prc completed.
As regards the latter part of the statement, it may be consider a consequence of the well-known relation
When Jn vanishes, Jn+l has the opposite sign to «/,/, both these q being finite§; But at consecutive roots of Jn, Jn' must assume oppoa and so therefore must Jn+i- Accordingly the number of roots of Jn interval must be odd.
The theorem required then follows readily. For the first root must lie between the first and second roots of Jn. We have pro it exceeds the first root. If it also exceeded the second root, the would be destitute of roots, contrary to what we have just seen, manner the second root of Jn+1 lies between the second and. third Jn, and so on. The roots of Jn+1 separate those of Jn ||.
* Loc. cit. §§ 88, 92 a.
t [1915. Similar arguments may be applied to tesseral spherical harmonics, prop cos s<p, where (/> denotes longitude, of fixed order n and continuously variable s.]
J Bessel's Functions, 1895, p. 50.
§ If Jn, Jn+1 could vanish together, the sequence formula, (8) below, would require succeeding order vanish also. This of course is impossible, if only because when u ii lowest root of </„ is of order of magnitude n.
|| I have since found in Whittaker's Modern Analysis, § 152, another proof of this p: attributed to Gegenbauer (1897).is Volume, p. 275].