1894.] 



on Ellipsoids and Anchor-Rings. 



175 



These are 



2^ 2 (''' ^ l) ^^^^" ~ ^"-^ ~ ^"+'^ ^ "'' ^^«-' " ^"^'-^ ~ 2 '^'^« 



+ {^>„ ('S^Q. + 2Ca;) - Z>„_,Q;_, - 6„.,QUJ = . . .(35 a), 

 where n may have any positive integral vahie except ni — 1, m, or 

 m + 1 ; in the first and third cases we have instead of zero on the 

 right -i4oQ„/, and in the second case 



If w = 1 there is a solution given by 



\= af:+bq: 



,j.„ = -S(AP^ + BQJ] 

 for all values of w > 1, A and B being constants, or more simply 



tt] 

 S 



and when n = 1 



/., = - >sf (^p, + BQ^) + ap: + bq:] 



h=-B 



.(36), 



.(36 a), 



or 



...(37), 

 .(37 a). 



a,= A'^ + ^^iAP.'+BQ:, 



h,=-B'^ + ^(AP: + BQy 



For these values of X, /jL, satisfy equations (30) by reason of the 

 relations 



p'.-^ + p\.» = ^gp: + sp) 



{Phil Trans. 1881, p. 646) and they will also satisfy (35) if A and 

 B have suitable values. To prove this it will be sufficient to sub- 

 stitute PJ for X and — SP for /*, as the relations between the Q 

 terms are always similar to those between the P^ terms. The left- 

 hand side of (35) then becomes 

 R , 



(38) 



2SSp 



R 



iSip 



2^.. +|)^(2C'P„-P„.,-P„J 



-n8{P„_,-P,,J-ls-\ 

 (n^ + ^) (26T„ - P„_, - P,^J + 2C'P„ 

 + {2n - 1) P„_, - {2n + 1) P„,, -1- 3,SP; 



sin nd 



sin >i^. 



