Functions of Equal Order and Argument. 521 



and Airy's Integral, viz. in the case of the J»(V) function, 



J n (V)=-(-j 3 | cos J w z + {n-z)i- Yw Vdw 



where p = (n -<s)( - J ', 



or, using De Morgan's expression * for j cos (w z — mw)dw, 



The third and following terms are correctly given in (14). 



z — n 

 When — j- is not very small, the error in (2) becomes 



considerable. From Table II. in the paper f on the 

 "Relation of Airy's Integral to the Be.ssel Functions/* 

 it would appear that the first root p 1 of J n (z) is given by 



p 1= =n + l-1814ni (4) 



From this result, p l for Jiocot*) is 101 1*8, whereas the 

 correct value is 1018*62. It can be shown from other 



,0,7^2 



considerations that when A, is written for (—r^) , p\i$ more 

 accurately represented by 



(nXp 3 (\- 



-) U>) 



n / 



Pi = ^+-T- + 40V 

 = 7i+r841w» + l"017TC~8. 



This gives 1018*51 for the first root of Jiooo(~)- 



The function J»(c) has also been expressed asymptotically J 



in terms of Ji and J i when the argument and the order 

 3—3 ° 



* Airy, Camb. Phil. Trans, vol. viii. pp. 596-599. 

 t See' also Lord Ravleigh, Phil. Mag-. Dec. 1910. 



X Nicholson, Phil. Mag. July 1909, p. 14. For tables of the Ji and J_ * 

 functions, see Dinnik, Archiv der Math. u. Phys. 18 Band, S. 337, 338. 



