.Expansions of Bessel Functions. 229 



only, and include three cases : (1) z and n very large, but 

 z—n not o£ low order in comparison : (2) z and n very 

 large, and n — z not of low order ; and (3) z and n nearly 

 equal. The limits of validity were left very doubtful in each 

 case, and especially in (3), there were many points in the 

 investigation which cannot bear examination. In a paper 

 by the author*, the defect in this case was indicated, and 

 expansions deduced when n and z do not differ by an amount 

 of higher order than zk, whether n be greater or less than z. 

 These results are general, and hold for all large real values 

 of n. They were subsequently f applied to the calculation 

 of a table for the function J„ (z) in this case, based upon 

 Airy'sJ tabulation of a type of integral occurring in physical 

 optics. 



The Bessel functions of nearly equal argument and order 

 may be reduced to an approximate dependence on this and 

 an associated integral, and thence also to Bessel functions 

 of small argument and fractional order J, whose tabulation 

 is readily effected. In another paper §, the special case of 

 restricted order of Lorenz has been further investigated when 

 the order is less than the argument, and a type of expansion 

 obtained which can be used to a degree of accuracy determined 

 only by its convergence. 



The consideration of corresponding expansions for the 

 remaining cases of large real argument or order is the object 

 of this paper. A scheme is developed which will furnish 

 the approximate values of the functions in all cases in which 

 one or both of the magnitudes n and s is large, and both are 

 real. The order is not restricted in any other way. Some 

 interesting analytical results appear in the course of the 

 work, and a general theory is indicated, applicable to all 

 solutions of differential equations of the second order 

 which can be expressed in series whose general term is 

 known. 



The Associated Equation of the Third Order. 



If (?/!, f/ 2 ) are two independent solutions of a differential 

 equation of the second order with invariant 1, so that 



(£+1) <*.*)=<>, (i) 



* Phil. Mag. Aug. 1908. 



f Phil. Mag. July 1909. 



I Airy, Cainb. Phil. Trans, vi. p. 379 ; viii. p. 595. 



§ Phil. Mac. Dec. 1007. 



