1894.] on Ellipsoids and Anchor-Rings. 163 



,1 p A A„ A,— A 



thereiore — - = 



n -(n + l) 2n + l 



ipa(A^+A,) 



(n+l)(2n + l)f- 



by (10), 



and therefore = ipoA, ^^^^ 



(71 + 1) \{2n + 1) 27^ + ipa\ 



Thus both the unknown constants are determined and <& and O 

 are completely known ; for example, 



^ {^n + l)ipaAo . 



(n + 1) {(2n + 1) o- + 4!7ripa] «* 



Again, the whole potential inside the sphere is 



(A+A) e^P^ r-X Y - (^^+^)''^o e^pt (VX r 

 {A, + A,) e \^J r„ - ^2^^ _^ 1) ^ _^ ^^-^^ ^ [J ^n 



= ^oCos:,^e^<i^*-x)QV^, 

 where ^ = tan~' - 



(271+1)0-' 



shewing a field diminished in the ratio cos % : 1, and lagging behind 

 by a time equal to 2ir\x of a complete period. This agrees with 

 the result given by Mr Larmor {Phil. Mag. 1884). 



For free currents of the type e^vty^ we must put A^^O in (12), 

 and we then have 



{2n + 1) o- + ^iripa = 0, 



(2w + l)o- 

 or ip = — ~ — — , 



giving a value for the modulus of decay which is well known. 



The results given by Prof. C. Niven {Phil. Tynans. 1881) for 

 the law of decay of such currents can also be found from our 

 equations. 



(4) Let the current sheet be an infinite right circular cylinder 

 of radius a, the specific resistance being constant at every point of 

 its surface. 



Here ds"^ = dr^ + r^dd"^ + dz^ 



with cylindrical co-ordinates and equation (3) is 



12—2 



