1894.] 



on Ellipsoids and Anchor-Rings. 



173 



The left-hand side of this without the factor 



dd ^^'^ 12 r~BS ^^^^ ^^^ "^ *^ ^^^^ ^^^ ^^^ 

 „^ (3 cos 6* . . 3 sin^ 6 . . 



R 



2SHp 



IS 



+ 2n—-^ cos nd — n- (trO) sin n6\ 

 (aU) ) 



„, f3 2 cos ^ (0-^)2 - sin^ ^ . . 



= ^^^ I* {^^ '^^ ''^ 



„ sin ^ ^ o / /i\ • /i) 



+ 2n 7—37- cos nU — 7r (a-U) sin nt)\- 



{ad) ) 



^^ (3 8'-(aey . . _ sin6' . ,/ m • J 



= SX„ -{-r — ■ ),,„ -- sm n^ + 2w ,— ^r- cos ?i^ — n^ {a8) sm wt^v 



= -—^^ X\n - (t + *^") (o"^)"* sin ^^^ + 2/i (o-^)^ sin 6 cos ti^ + - ;Sf^ sin nOl 



(ad) 



3^^n 



(o-6')2 |- [| + 7iA (C - cos d) sin ?i6' 



+ 71 (sin w + 1 ^ — sin ?i — 1 d)\ + -r S^ sin ?i^ 



4 



1^)!+? 



(c^y 



X\. 



((76'/|-('| + 7i')Csinw^ 



+ ^ (^ + w^ + 2?i ] sin w + 1 

 1 /3 



+ n^ + 7z' - 2*2] sin 71 - 1 6* + ^ .Sf'-' sin nd 



3 



+ 4 '^'^^ 



sinwc 



(adf 



(adf 





sin ^i^ by (30) 



- I (71" - 2n + ^) /in_a| - ^ 6^Xn I sin »i(9. 



