I' 



52 Mr. A. R. McLeod on the 



Take f(r/c) = 1. Then, since 



£Jo(&?)^=HJ o (A0//3n 2 , 







ff{Jo(&) P^=(A 2 + H 2 ){J (A)} 2 /2/3. 2 , 

 Jo 

 we have 



_ ^ 2HJ o (fl.r/<0 



,S(/3» 2 +H 2 )J (&)" • • • ; ^ iU > 



Write 



i(ft.*+H»)Jd(ft) 



X (r,0 = 0, t<0. 

 Consider, for £>0, 



*(r, t) = 1 + H ^ - 1) 2 ^ + H2) Jo0 g, } . 



We see that 



i/r(c, *)=!, £>0, 



and we define 



^(c,f)=0, KO. 



Further, as r->c, the ratio of yfr(r, t) — l to ^t ^ +% — 1 



tends to unity. 



Now by the usual method *, the function 



reduces to the value </>(£) when r = c. Performing the 

 differentiation, this becomes 



c 2 V r/^ (/?J-t-H 2 )J (/3„) Jo 



Hence, by inspection, the solution corresponding to %, 

 which is the one we require for condition (6), is the second 

 of the terms on the right of (11), and the solution we 

 require is 



« = 5 08,» + H»){J o0 S.)}' Jo f F(cf ) *° m)d * 



2am ^ MUf3 n r/c)e-*W* C' (T)eaVi , T!c , dT 



!> • (11) 



* Phil. Mag. Jan. 1919, or Weber, Partielle Differ eniialgleichung en y 

 Band ii. 



