294 



A,(p) = l (^=1.2.3...) 

 A,(p)=p + 2 (pr= 1.2.3.4. .) 



A^ip) = ^{p'+p-4) (p = dA.b...) 



A^i'^ = ^ (p-4) ip' +P-Q) (/^==5.6.7...) 



A/.P) = ^^{p-h) {p-6) ip'^p-S) {p=7.S.d...) 



A,(p) = l-(p-ö)(p-7){p-S)(p^+p-\0) (/.=9.10.1l..) 

 5! 



.(15) 



where the law of succession is evident. With these values the equa- 

 tions (13) and (14) give the required solution. 



4. To generalise the preceding results we will proceed to examine 

 the more general integral equation. 



f{,v)=J<f{^)K{x-i3)dii (16) 







assuming that the functions /(.n) and K{,c) may be expanded in series 

 of Bessel's functions 



ƒ(.^•) = cj,{.v) + cj,{.c) -r cj,{^) + ... 

 K{w) = aj„{.v) + aj,{.v) + aj^{,v) + ... 

 which is the case if these functions are finite and continuous 

 from to X. 

 If now 



the second member reduces to 



22 





Thus, comparing the two members, we find 



Cj = 2a„6„ 



C2 — 2a,6„ + 2a„6i 



<^8 = ^ («2— «0) ^„ + 2«,6, + 2a„&, 



c, = 2 {a— a,) 6, + 2 {a— a,) ^ + 2a,/,, + 2«„6, 



etc. 



which give 



