1002 Lord Rayleigh on the 



the first or lowest root (after zero) which we may call : x . 

 In this case J„ (:) remains throughout of one sign. For the 

 aerial vibrations, in which we are especially interested, the 

 boundary condition, representing that r=B behaves as a 

 fixed wall, is that J n '(AR)=0. We will suppose that k and 

 R are so related that h\\ is equal to the first root (V) of this 

 equation. The character of the vibrations as a function of 

 /• thus depends upon that of J„ (z), where » is very large 

 and z less than:! ore,'. And we know that in general. 

 n being integral. 



i c* 



J (r) — 1 <•«)- ( : sin (o — tiro) d(0. 



(2) 



Moreover, the well known series in ascending powers ot z 

 -h.»w- thai in the neighbourhood of the origin J„ (:) is very 

 small, the lowest power occurring being :". 



The tendency, when n is moderately high, may he recog- 

 nized in MeissoPs tables*, from which the following is 



extracted : — 



--. 



Jm 

 -0-0031 



J,, (--). 



+0-.: 







J,, (--)• 



24 



16 



+0-0668 



+0*0079 



23 



+0-0340 



0-2381 



15 



00346 



0-0031 



22 



el. mi 



02105 



14 



0-0158 



0-0010 



21 



0-2316 



0-1621 



13 



00063 



00003 



20 



0-25U 



01 106 



12 



00022 



o-uooi 



10 



2235 



0-0675 



11 



00006 



00000 



18 



01706 



00369 



10 



0-000-2 





17 



01138 



0-0 1 so 



o 



00000 





From the second column we see that the first root of 

 Ji8 { z ) = ® occurs when £=23*3. The function is a maximum 

 in the neighbourhood of £ = 20, and sinks to insignificance 



when z is less than 14, being thus in a physical sense limited 

 to a somewhat narrow range within ^=23'3. 



The above applies to the membrane problem. In the case 

 of aerial waves the third column shows that J 2i (z) is a 

 maximum when z= 23*3, so that J 2 / (23*3) = 0. This then 

 is the value of /cR, or z/. It appears that the important 

 part of the range is from 23*3 to about 16. 



The course of the function J n (z) when n and z are both 

 large and nearly equal has recently been discussed by Dr. 

 Nicholson f. Under these circumstances the important part 



* Gray and Matthews' Bessel'a Functions. 



t Phil. Mag. xvi. p. 271 (1908) ; xviii. p. (1000). 



