a Horizontal Layer of Fluid. 545 



Let us now suppose that a-\-da is the equivalent radius, 

 so that J '(ka + kda) = 0, that is the radius of the exact circle 

 which corresponds to the value of k appropriate to the 

 approximate circle. Then 



J '(z)+kdaJ n (z) = Q y 

 and 



Again, if a+da f be the radius of the true circle which has 

 the same area as the approximate circle 



da'=~Z(*J>+/3S), .... (67) 

 and 



"-*--*^#& • • • <«> 



where z is the first root (after zero) of J '(z)=*0, viz. 3*832. 



The question with which we are mainly concerned is the 

 sign of da' — da for the various values of n. When n=l, 

 Ji( z ) = — JYOO = 0, so that da = da', a result which was to be 

 expected, since the terms in a 1? /3i represent approximately 

 a displacement merely of the circle, without alteration of size 

 or shape. We will now examine the sign of J„/J n ' when 

 w = 2, and 3. 



For this purpose we may employ the sequence equations 



Jn+l "■"»*"" Jn-lj J« — 2"»-l — "2«J»+1» 



which allow J„ and J n f to be expressed in terms of J x and J , 

 of which the former is here zero. We find 



J 2 = — J , J 3 — — 4:Z Oq, 



Ji / = Jo? "2 =22;" J , 



Thus 



Ji n J 2 z 

 J/ ' J 2 /_ 2 J 



J 4 =(l-24z- 2 )J ; 

 J 3 ' = (12- 2 -l)J . 



J 3 4z 



J/ ~* 2 -12' 



whence on introduction of the actual value of z y viz. 3*832, 

 we see that J 2 /J2 / is negative, and that J3/J3' is positive. 



When n>z, it is a general proposition that J n [z) and 

 J n '(z) are both positive*. Hence for n = 4 and onwards, 

 Jn/Jn' is positive when £ = 3832. We thus arrive at the 



* See, for example, Theory of Sound, § 210. 

 Phil. Mag. S. 6. Vol. 32. No. 192." Dec. 1916. 2 P 



