Functions of high order to the Whispering Gallery. 107 



Thus (32) becomes 



J n '(ha) pk . , 



T ) 7 \ — — ~r smh ft, ... . (34) 



the right-hand member being real and negative. Of this a 

 solution can always be found in which ha = n very nearly. 

 For * J n (z) increases with z from zero until z = n + *8065^, 

 when JV(~) = 0, and then decreases until it vanishes when 

 z = n + 1*8558/2*. Between these limits fore, J»'/J n assumes 

 all possible negative values. Substituting n for /*a on the 

 right in (34), we get 



-^sinh/3, or -^tanh/5, . . . . (35) 



while cosh /3 = p. The approximate real value of ha is thus 

 ?z simply, while that of ha is njpu. 



These results, though stated for aerial vibrations, have as 

 in all such (two-dimensional) cases a wider application, for 

 example to electrical vibrations, whether the electric force 

 be in or perpendicular to the plane of r, 0. For ordinary 

 gases, of which the compressibility is the same, 



pja = h 2 /P =p?. 



Hitherto we have neglected the small* imaginary part of 

 D B 7D B . By (18), (20), when z is real, 



Mf) = _ sinh/5 |£±g = - sinh/3.(l+^) (36) 



approximately, with cosh ft = n/z. We have now to determine 

 what small imaginary additions must be made to ha, ha in 

 order to satisfy the complete equation. 



Let us assume ha = x-\-iy, where x and y are real, and y 

 is small. Then approximately 



JVQg + ty) ^ J» ; (a-) + »yJ»"(fl) 



Jn(x + iy) J»(#)H-^/J„'(#) ' 



and 



J.» = -;J.'W-(l-J)j^*). 



Since the approximate value of x is w, J,/' is small compared 

 with J?* or jy, and we may take 



j^+y) a J^r 1H y W\ . . (37) 



Jn(* + ty) Jn(^) L J»00 J 



* See paper quoted on p. 100 and correction. 



