1921-22.] 
and 
so that 
A Fourier-Bessel Expansion. 
T(k) = J/(K5)G„(Ka) - G/(k 6) J„(Ka) , 
f ^/(^)Q(Ks)dx = As^S'(Ks)T(Ks), . 
Ja Jt 
91 
• (3) 
• (4) 
^ S'{ks)T(ks) 
Validity of the Expansion . — In order to establish the validity of the 
expansion, consider the integral 
fm^i, 
where C is a path above the ^-axis from E to A, A is ^ = M, E is f — M, 
M lies between Ky and , 
and 
S(0 
P(0 = JnmG^n'iCa) - Gnm^n'iCa) . 
It should be noted that P(^), E(C), S(4) are uniform functions 
of I, P(^) being odd and the others even. The first terms of their 
asymptotic expansions are — 
cos {^(b-a)}, 
1 
sin {t,{x — a)}. 
1 
sin {^{r — a)}, 
1 
Cj(ah) CJ(xa) 
sin {((b — a)} respectively, and these are the 
CJ(ra) (J{ab) 
same for all values of amp The first term of the asymptotic expansion 
of F(C) is thus 
h 
0 ( 0 = - 
J{xr) 
cot {^{h - a)} sin [i^{x - a)} sin {^(r - a)}. 
