and an Asymptotic Theorem in BesseVs Series. 195 

 Bessel's functions, viz. for the boundary condition yjr = 

 -^=r-l-cos k(Vt+y) —^coak(Yt — y) 



+ \/2 cos (f + kVt) [j#r) sin | + J f (*r) sin y 4- . . .J 



- V2 sin f j +*V«)[j«(*r) sin-y + J^(At) sin^- +...]; 



. . . (9a) 

 while for 



— I- =0on the barrier 

 n/r = l cos&(V£+y)+£cos&(V« — y) 



+ \/2cos(|+^ 7 ^J|(^)cos~+j3(^)cosy +...] 



+ v/2 sing + *V«) [j : (*r) cos^ + J 5 (jfer) cos ^+ ...]. 



. . . (9 6) 



§ 3. The first step is to obtain expressions for P, Q, and 

 thereafter for the stationary forms, in which the variables 

 r and are separated. We have 



p=r+y=r(l+ sin 6) = ?j(^+ e~ M% \ where © = « — \ (10) 



and then 



TO + i 



±p : 



m+j m 



i) Jo 



|2m+l 



; ,' , n cos (2p + l)». (11) 



The series changes sign with cos co, i. e. we have the 



o 



positive sign on the left from # = to -^-, and the negative 

 sign from -— to 2tt. Linking (11) with (7 a) we get when 

 the left-hand member of (11) has the positive sign 



P0>)= SP,(r) 0os(2p + 1>, Q(p) = 2, Q p (r) cos (2p + l)co 



p 

 where 



(-l)»|2»2(2,-)*<+* 

 r v~ * .t. =k — : rm. 2«== v : 



n \2n—p 2n+p + l 



P 



K12) 



n _ v (-l)»2>» + 12(2r)»» ^^ 



>J9. 



02 



