238 Bessel Functions of Equal Order and Argument. 



integral are given in (7) and (16). The following expressions 

 were obtained by Graf from Schlomilch's integral : 

 When x<l, 



J a (ax)- 

 "When x=l, 



When x>\, 



•a(y- tanhy) 



-y/ 2ttu tanh y 



, where cosh 7= -. 



r (*) T(i) /6\t 



7ra*2*3i 27r^/3 



m- 



2cos a(tanS — S) — -r 



JJotx)^- , -— r , where cos 8=.-. 



K J V^TratanS * 



Graf proceeded to throw the last expression into the form 



and found that ft n , the nth. root of J a (#), is represented 

 approximately by 



overlooking the simple formula 



/3 n =«sec</>, 



where 



,. . ., (4n-l)7r 

 (tan <£-<£) = ^—^ 



and which is especially useful in calculating the roots of 

 the functions . J a (a?) . of high order. This result is given 

 in detail in the Phil. Mag. July 1916, where, as in the paper 

 on "The Bessel Functions of equal order and argument/' 

 the integral 



C 6*- 1 cos 0d6 = T(x) cos^ 



was employed for values of x outside the limits 0<x<l. 



Perhaps, in this case also, the correct result may be due 

 to the errors being mutually destructive. In the expression 



1 



( 14) for tan e, r is equal to 



the paper. 



n tan <b 



, not n tan cf> as stated in 



I am, Gentlemen, 



Yours faithfully, 



John R. Airey. 



