412 Dr. T. J. Pa. Bromwich : Examples of 



(y) Problems for a cylinder corresponding to 

 those of (a), (£). 



It is now necessary to use the differential equation 



1 ~b / Bw\ = _1 dw 

 r'drX'dr/ ' at'dt' 



which leads to the use of the function I (qrjc) in place of 

 {sinh (qr/c)l(qr/c)}, where I is the modified Bessel function *. 

 We then obtain for the mean temperature 



5=*f% u *»*Ma>G*, . . . ( i9) 



< r Jo qh(g) v ' 



if the surface-conductivity is treated as infinite ; or 



- _ 2 chl \q) 



U -qM ( q ) + KqL >( q ) at * ' * (20) 

 in general. 



On expanding in powers of q we find that 



V(?)=k( 1 +k 2 +-) 



Thus the expansion of the operator in (20) is 



cA(l + ^+...) /! 1 K\ 



oA(l + i ? 2 +...) + iK2 2 +. _ V8 4 MiJ 9 ^--- 



Hence on comparison with (3) we find 



N =l, Kl ,_g + *-)> . ... (21) 

 In equations (19) and (20) we can take 

 F(p) = |vfe), 



A(^) = I (?) or IofaO + Xtflo'fe), 



(where X = K/cA, as before), because each of these functions 

 can be expressed in positive integral powers of p. 



* Thus I («) = l + |: + _!L +§r |L_ +• . • 



