DOUBL E B U B B L E S 5 9

[M ] F r a n k M o r g a n , Soap bubbles in R

2

and in surfaces, p r e p r i n t ( 1 9 9 2 ) .

[Ta ] J e a n Ta yl or , The structure of singularities in soap-bubble-like and soap-film-like

minimal surfaces, A n n a l s o f M a t h . , 1 0 3 ( 1 9 7 6 ) , 4 8 9 - 5 3 9 .

Receive d A u g u s t 1 2 , 1 9 9 1 . T h i s r e s e a r c h w a s part iall y s u p p o r t e d b y t h e Nat ion a l S c i -

enc e F o u n d a t i o n R e s e a r c h E x p e r i e n c e s f o r U n d e r g r a d u a t e s P r o g r a m , t h e F o r d F o u n -

dation , t h e N e w E n g l a n d C o n s o r t i u m f o r U n d e r g r a d u a t e S c i e n c e Ed uc at i o n , t h e S h e l l

Foundation , G . T . E . , a n d t h e B r o n f m a n S c i e n c e C e n t e r a t W i l l i a m s C o l l e g e .

c/ o F R A N K M O R G A N

WILLIAM S C O L L E G E

WILLIAMSTOWN , M A 0 1 2 6 7