with a concentric core of a different material 



409 



8. The case when /x (6 - a)/a is rational remains to be discussed. 

 Suppose that it is equal to i^jq, a positive fraction in its lowest terms. 

 Then the equation ^ (a) = [cf. § 7 (27)] is satisfied by 

 aa = q-TT, 2q7T, etc. 



as well as by the roots of (28). 



When aa = sqir, it will be seen that ^m {h — a) = sip-rr. 

 Thus, in addition to the terms of (26), 



u^ = sin jLta„ {h — a) sin a^r e'^^^^'^ 

 u^ = sin a„a sin /*«„ (6 — r) e-"!""'^} ' 

 where «„ is a root of (28), we have 



Uj^ = iiK^ cos fSTT sin qsv -e "' «' 



/Jj — r\ _^ gV'^^ 

 Ur, = — K^ cos qsTT sin J)Stt f r^-^ ] ^ "' "'^ 



where s is any positive integer. 



The theorem of § 5 [cf. § 7 (30)] apphes to all the solutions 



sin lutn {b — a) sin a„f| 

 sin a„a sin jtxa„ (6 — ^)j ' 



(35), 



and 



jLt/ig COS -pSTT Sm ^STT 



1 



/^6 — y \ i 

 — K^ cos ^fSTT sm 'psTT I T— 3^ I 



Thus we assume that v^r can be expanded in an infinite series 

 with terms of the type 



An sin fittn {b - a) sin a„r, < r < a\ 

 An sin a^a sin ju,a„ {b - r), a<r<bi' 



and 



ylg /LtZa cos fSTT sin (/stt - , < r < a 



Ag A\ cos g-STT sin ^stt 



< r<b. 



Kb-aJ' 



The coefficient A^ has been found above in (32). The coefficient 

 ^s is given by 



A. 



IJi^K^^ cos^ psTT sin^ 5-577 - dr 



.'0 



+ ^ Zj^ cos2 qsTT sin^ ps7r L _ ) dr 



= v. 



^7i2 cos psTT I r sin qsir - dr 



K;^cosqs7T\ r sin fSTT ir——j dr 



