1894.] on Ellipsoids and Anchor-Rings. 177 



hence the expression (42) is equal to 



The coefficient of sin on the right of (35) is 

 \ {SQ, + WQ;) - h,Q: + A, {SQ, + 2GQ:) 



=h,{Q:-\-Q:)-\Q:+A,{Q:+Q:) 

 = - ^ ^o' + §-' (^^o' + bq:) {q: + q:) + a, (q; + q/) 



by (36 a) and (37 a). 



Equating then the two coefficients of sin 6, we have 



= - ^ Qo' + ^' {ap: + bq:) (q;+ q:) + a, {q: + q/) 



which is the first relation between A and B. 



Again, the coefficient of sin W on the left of (35) is 



.(43), 



= - i (- ^ + 2) {AP: + 5^0 by (36) and (37) 



28Hp 

 and on the right we have 



S 



{A1\' + BQ:)-A,Q,', 



hence 



SB 



IQS ip 



.^ (ap: + bq:) = - ^ {ap: + bq:) - a,q:, 



s 



or 



ap:+bq: = '^ 



asq: 



lesip 



+p:q: 



.(44); 



(43) and (44) determine AB in terms of ^0 and thence ^, ^^^^ . 

 VOL. VIII. PT. III. 13 



