189-4.] on Ellipsoids and Anchor- Rings. 169 



The modulus of decay, viz. — 1/ip for free currents of the 

 type considered, is found as before by putting A =0 in equa- 

 tion (24). Thus 



{2n + 1) ^ {n (n ■hl)a'-q {W + F)| - ipl/o'E^F^ = 0, 



(2n + l)K{n(n + l)a^-q{h' + F)] 



When $ = C^ we have E^ (p) = Jp'^ — F, and we easily find 

 by substituting this vahie in the equation satisfied by E^ (p) 

 that 



Thus for free currents 



. K {2a? - h') 



" 1 "i« E:J{p^-h^){p^-F) EJ,c 



-^ , a a 



or smce -c-.. = 



ip = 



K{a^-V¥) 



'^iralfc^ \ - 



a C"^ dp 



[c J a E:J{p''-K'){p''-]^) he- 



K ja' + b') 

 ab^iN-iir] ' 



dp 



where N = ^irahc 



(p^-Ji^)Hp'-k^f' 

 and the modulus of decay is 



1 1 Cib^ /Tir i ^ 



- ^ = . — j., . (N - 47r), 



ip K a. + 0^ 



and this will be found to agree with Prof. Lamb's result (Phil. 

 Trans. 1887), when we take into account the difference between 

 his p and our k. 



(6) We proceed now to the case of the anchor-ring. The 

 notation used is that of Dr W. M. Hicks in his paper on Toroidal 

 Functions, Phil. Trans. 1881, except that we use o- and 6 in- 

 stead of u and v respectively, and we have 



ds'' = T-^4 (da-'' + dd^ + sinh^ a■d^l/'), 

 (o-o) 



