8 Lord Rayleigh on the Correction to the 



It is particularly to be noticed that although (29) is an over- 

 estimate, it vanishes when a tends to zero. 



The next step in the approximation is the inclusion of H^ 

 corresponding to the first root h x of (j>(Jib) — 0. For a given 

 k, B has only one term, expressed by (13) when we write 

 /&!, Hi for h, H. In (16) when we expand (A + B) 2 , we 

 obtain three series of which the first involving A 2 is that 

 already dealt with. It does not depend upon Hp Constant 

 factors being omitted, the second series depends upon 



^ J n 2 (/ca) x£q\ 



k^-k^J^kb)' ' ' 

 and the third upon 



s kJ Q 2 {ka) / 31 n 



(V-^?Ji 2 W 



the summations including all admissible values of k. In (24) 

 we have under 2 merely the single term corresponding to 

 Hi, />i. The sum of (16) and (24) is a quadratic expression 

 in Hi, and is to be made a minimum by variation of that 

 quantity. 



The application of this process to the case of a very small 

 leads to a rather curious result. It is known (' Theory of 

 Sound/ § 213 a) that k r 2 and h^ are then nearly equal, so 

 that the first terms of (30) and (31) are relatively large, and 

 require a special evaluation. For this purpose we must 

 revert to (10) in which, since ha is small, 



Y (ha) = log ha J'o(Aa) + 2J 2 (/ia), . . . (32) 



so that nearly enough 



JoW=(^^JoW = ^ = gg, 



k ~ ll=: ''bJ^kbJ\ogka (33) 



and 



Thus, when a is small enough, the first terms of (30) and 

 (31) dominate the others, and we may take simply 



f 3 fh - _ h lo £ k ' a (ZA\ 



W- 2 *i«Yo0fci&)J 1 (* 1 6)' ' * ' W 



