to the Vibrations of a Circular Membrane. 57 



If a closer approximation to z 2 is desired, it may be 

 obtained by substituting on the right of (6) 2n for z{ 2 — n 2 in 

 the numerators and neglecting n 2 in the denominators. 

 Thus 



Now, as is easily proved from the ascending series for <J n ', 

 so that finally 



* ,>rf+ *' + Ripi)- • • • (7) 



When n is very great, it will follow from (7) that 

 £i 9 > n 2 -\-3n. However, the approximation is not close, for 

 the ultimate form is * 



z? =n 2 + l -02684 w 4 / 3 . 



As has been mentioned, the sequence formula 



jJ n (z)=J n . 1 (^ + J u+1 {z) ... (8) 



prohibits the simultaneous evanescence of J n _i and J„, or of 

 J M _i and J„+i. The question arises— can Bessel's functions 

 whose orders (supposed integral) differ by more than 2 

 vanish simultaneously? If we change n into n + 1 in (8) 

 and then eliminate J„, we get 



{^lil.ijj^^j.-.+ ^j^, . . (9) 



from which it appears that if J„_i and J ?l+2 vanish simul- 

 taneously, then either J w+ i = 0, which is impossible, or 

 z 2 = 4:n (n + 1). Any common root of J 2l _i and J n+2 must 

 therefore be such that its square is an integer. 



Pursuing the process, we find that if J„_i, J, t +3 have a 

 common root z, then 



(2 ? i + l); 2 = 47i(n + l)(?H 2), 



so that z 2 is rational. And however far we go, we find that 



* Phil. Mag. vol. xx. p. 1003, '19.10, equation (8). 



