The Hon. J. W. Stmtt on BesseFs Functions. 337 



ting the functions under any simple form when n is very great. 

 Both the ascending and descending series fail in such a case if z 

 is comparable with n. Light is thrown on the question by a 

 consideration of the differential equation (1) itselfc The ascend- 

 ing series is founded on the supposition that the coefficient mul- 



/ n^\ 

 tiplying y, really ( 1 -A, can be approximately represented 



n^ 

 by —-^•, if the term 1 be absolutely neglected, the solution is 



y Oiz'^. 



On the other hand, the descending series has its foundation in a 

 substitution of \ for n, when the exact solution becomes 



y ocJ^(2') Qc5^~^sin^. 

 When, however, n is great and z comparable with it, neither 



term in 1 ^, over a large range of z, can be treated as rela- 



z 



tively unimportant. For a considerable range in the neighbour- 

 hood of n the differential equation approximates to 



^ + 1^=0 



dz^ z dz ' 



of which the solution would be 



2/ = A + Blog5'. 



The question is worthy of the attention of analysts ; for BesseFs 

 functions with large values of n (1000 or more) would have phy- 

 sical applications, for instance, to the problem of the rainbow. 



In virtue of the integral formulae, a combination of the partial 

 solutions (25) can be found to correspond to any initial values 



The problem becomes only a little more complicated if we 

 suppose the cylinder closed sX z — and z=il, and discard the 

 restriction that the motion shall be independent of z. The par- 

 ticular solution is 



^ = cos(A-fl;/ + e)cosA3yjcos(?z6> + a)J„j T/c^-^^^jV [,(26) 



p being an integer. For a rigid cylmder of radius unity we have 



«^=;>^^Vk^ (37) 



where K denotes the values corresponding tojo = 0, being those 

 given in the Table. To every value of K corresponds a series 

 of values of «:, found by ascribing to p in succession the values 

 Phil. Mag. S. 4. Vol. 44. No. 294. Nov. 1872. Z 



