338 The Hon. J. W. Strutt on BesseVs Functmis. 



1, 2, 3^ . . . The purely axial vibrations correspond to a zero 

 value of K. A similar analysis would apply on Euler and La- 

 grange^s hypothesis if the ends or the side of the cylinder were 

 opeU; though in the latter case the result would be no approxi- 

 mation to the truth. At the plane ends we may legitimately 

 take as an approximation (p — 0, provided that the radius of the 

 cylinder bears but a small ratio to the wave-length of the vibration 

 under consideration. (See a paper "On Resonance/-' Phil. 

 Trans. 1871.) 



In the problem^ partially considered by Fourier {Theorie Ana- 

 lytique de la Chaleur, Chap. VI.), of a cylinder of uniformly con- 

 ducting material heated arbitrarily and then allowed to cool by 

 radiation, the cosine factor containing the time would be replaced 

 by an exponential, the coefficient of t being negative, and (18) 

 would have to be used as boundary condition, but otherwise 

 there would be little change. 



Hitherto we have had to do with integral values of n only ; 

 but if instead of the complete cylinder we take the sector cut off 

 by 6 — 0, = ^j fractional values will be introduced. The ele- 

 mentary solution for the acoustical problem in two dimensions 

 then becomes 



^ = cos {Kat 4- e) cos nO . J,, {kt) , ... (28) 

 where 



TT 



«= (m= integer) X 3 (29) 



Hence, as might have been foreseen, if B be an aliquot part of tt 

 (or TT itself), the complete solution requires only integral values 

 of ?z; but otherwise functions of fractional orders must be intro- 

 duced. An interesting example is when /S=27r, corresponding 

 to a cylinder with a rigid partition from the centre to the cir- 

 cumference. When m is even, n becomes integral, and we have 

 motions which might take place without the partition, and there- 

 fore presenting nothing peculiar for consideration; but if m is 

 odd, n assumes the form (integer +^),in which case, as we have 

 seen, J,j is expressible in finite terms. Thus if m = l, 7z = J, and 



0=cos (/<:«^ + 6)cos-Ji(«:r) GCCOs(A:fl^f+e)cos^.(/^r)~2sinA:r.(30) 



The admissible values oIk are those which render tan k equal to 

 2k. The first value of k is 1*1655, giving a much lower tone 

 than any of which the complete cylinder is capable. 

 If instead of (14) we were to take 



6 =e~^ cos 710 J n {r), 1 



