44 Proceedings of the Royal Irish Academy. 



II. Tor cylindrical sheath between planes],between radii i2, R' : 

 Here 



v= Fy + a + ^logr + ^yr2 + y2;(?-s)+2(^„7o(wr) + -ff„Z'(jWr)sinw2, 



where a, /? are determined so that 



a + /? log r + lyr^ 



shall vanish for 



r = R, r^E. 



This gives 



a + ^logi2 + ^yi2- = 0, 

 a + ^logi2'-l-iyi?'2=0, 



whence 



III. For external cylinder between plates extending from R' toco : 



z 



V = V j + J^o(^^) sin ws. 

 We have then, proceeding in the usual method, the following equations. 

 A\a TlnR') = A,, YlnR') + B,K,{nR') + ^yjnH, (47) 

 A„Y,{nR') = A^Y^nR') + Br,K^{nR') + 4y^/w% (48) 

 ^„ YlnR) + B,,KlnR) + 8y/w'? = R\,Ko{nR), (49) 

 ^„ ri(wi?) + B,,K,{nR) - 4yB/nH = B\,K,{nR). (50) 



n = ^~ , s having all positive values. 



It will be noted here that the y terms correspond only to s odd. 

 Hence for s even we shall have 



A\, = A,, = B,, = = 0. 



We shall therefore understand now, throughout, s to be odd. 

 The former pair of equations now give 



B,, = - nR'Yi{nR')Sy/nH + ^BynR! YlnR')nH ; (51) 



the latter pair 



A,, = nRKlnR)^yBlnH + nRK^{nR) • (52) 



