R: 



698 Dr. J. W. Nicholson on the Asymptotic 



subject. The results obtained by Lorenz may be sum- 

 marized as follows : — 



WHting J +i (-')= ©-</> •••(!) 



J (~~) = (-)"(^)W> • • • (2) 



in all cases in which n-\~\ is less than ~, which is large; 

 then, if z—n — \ is of higher order than z$, 



'(r^=ii)* < 3 > 



^, = (.- 2 -^+l 2 )*-^ r +(» + i) S in-'^, . . (4) 



The formulae deduced by Lorenz for higher values of n, 

 more closely equal to, or greater than z, are not relevant to 

 the present purpose. 



When z and n are only moderately large, these forms cease 

 to be good approximations, and in this paper it is proposed 

 to generalize them, and to carry the calculation to higher 

 orders. 



Making, with Lorenz, the substitutions (1) and (2) in the 

 relation 



j'jWj'j=(-)».— , 



it appears, after some reduction, that 



dz -R {b) 



But for extremely great values of z, in comparison with 

 n, Lommel's* ordinary formula yields 



_ T17T 



ThUS ^-T-ffe- 1 )* •■■:(«) 



in the more general case when n is of order z. 



The differential equation for the functions J must there- 

 fore be satisfied by the form 



/=(?)W£ «> 



* E. g,, vide Whittaker, Modem Analysis, 1902, p. 294. 



