The Hon. J. W. Strutt on BessePs Functions. 339 



we should find by an application of Greenes theorem to the space 

 bounded by 5' = 0j z= oo^ r=r, ^=0^ 6 = a: — 



r{Jn-3'rr.-rn.Jm\={m'-n')Cjn^J.n.y^ • (32) 



Jo 

 In a similar manner, by use of 



another useful formula may be arrived at. But it is perhaps 

 simpler to proceed from the differential equation (1) itself. 



Now 



j^Vr . r-+^ [j". + I r^ + (l - ^)k } =0.' 



Thus 



I V"^+^J„^r = wr'^ J„ —r''''+'J'n + {n^—m^) 1 V-'J^^r (34) 



• — a formula of reduction, by means of which, if m be even, the 

 integral on the left may be evaluated. 

 li m = n, we have 



i 



r'*+'J„^r = wr^J„— r'^+iJ'^=r^+^4+i by (9), 



a formula given by Lommel^ p. 20, 

 If ?i=0. 



Jq J q 



^0 



Thus 



rdr5^{r) = -rJ'o(r) =rJ,{r), 







I: 



r^^r Jo(r) = 2r^ Jq - r^ J'o + 4r J'^, 



If r is a root of J'Q(r) =0, the three integrals become simply 



0, 2r^3,(r), 4r^(r^-8)Jo(r). 

 Z2 



