Moots ofBessel Function J n (x) and its derivate J/(V). 191 



we find, on substituting the value of ft in terms of «, 



4ti 2 -1 1127i 4 -lf>2n 2 -h31 /1Q . 



p p = n<x-\ ^ Q o 3 ..., . {16) 



rF one*. oo4?ra 



•x- ^ c (4» — l + 2n)ir , r „ 



or writing \ tor n<x = ^-^ — and m tor 4w , 



„ m-1 4(m-l)(7ro— 31) /1/1X 



^ = X --^T- 3(8X)» -'• ' (U) 



the same result as McMahon's as far as the third term. 



Roots of J n (V). 



(a) When n is positive and larger than 4, the second and 

 higher roots of J n (x) can be found with considerable accuracy 

 from the formulae 



p p =n sec<£, 

 where 



and „(tan^-^)=^=^ (16) 



When p=l, ,=000222...; p>l, , = _^__. 



From (16), tan ^= ^J" ^ + <£„ and as before 



tan^/3-i-A-..., 



where R __ (4p — 1 + 2w)7r 



^ 4^ * 



Substituting this value of tan <j> in (15) we get 



n(tan <£ — $) 



tan^> = 7 tTl""--'j where 7 = *+^, 



and , 1 7 83 



sec <p = <y — 



2 7 24 7 3 240 7 5 * 



