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Journal of Natural History, 2002, 36, 1671–1685

First zoealstages of Epigrapsuspolitus Heller, E. notatus (Heller)and Gecarcoidealalandii H.Milne-Edwards,with remarks on zoeal morphologyof the Macleay (Crustacea: Brachyura)

JOSE´ A.CUESTA†, HUNG-CHANG LIU‡and CHRISTOPH D.SCHUBART†1 †Department ofBiology, Laboratory for CrustaceanResearch, University ofLouisianaat Lafayette,Lafayette, LA 70504-2451,USA; e-mail: [email protected] ‡Department ofLife Science, National Tsing Hua University, Hsinchu, Taiwan

(Accepted10 April 2001 )

Thecrab family Gecarcinidae Macleay, 1838 currently consists of 18speciesthat aregrouped in four genera. Larval data for the Gecarcinidae were only known forspecies of the genera Latreille,1825 and Leach,1814. Inthe present paper, the Ž rstzoeal stage of Gecarcoidealalandii H. Milne- Edwards,1837, Epigrapsuspolitus Heller,1862 and E. notatus Heller,1865 are describedand illustrated. Zoeal morphology of the Gecarcinidae is reviewed, takinginto account all previous descriptions and analysing the relationships betweenthe di Verentgenera of Gecarcinidae based on zoeal morphological characters.A seriesof typical morphological features is proposed for the zoea larvaeof this family di Verentiatingthem from the rest of the Grapsoidea.

Keywords: Larvalmorphology, Gecarcinidae, , Epigrapsus, zoea.

Introduction The grapsoid familyGecarcinidae has acircumtropical distribution, with many species known only from oceanic islands. The familypresently consists of 18species that aredistributed among four genera: Cardisoma Latreille,1825, Gecarcinus Leach, 1814, Gecarcoidea H.Milne Edwards,1837 and Epigrapsus Heller, 1862( Tu¨ rkay, 1970,1973, 1974; Tavares, 1991 ).At this point, wefollow Tavares( 1991)in con- sidering Discoplax A.Milne Edwards,1873 and Johngarthia Tu¨rkay,1970 subgenera of Cardisoma and Gecarcinus,respectively. Crabsincluded inthe Gecarcinidae areoften largein size and commonly referred to as‘ land ’, based on the terrestrial habits shown byadults of most of the

1Presentaddress: Biologie I, Universita¨tRegensburg,D-93040 Regensburg, Germany.

Journalof Natural History ISSN0022-2933 print /ISSN1464-5262 online ©2002Taylor & FrancisLtd http://www.tandf.co.uk /journals DOI:10.1080 /00222930110059673 1672 J. A. Cuesta et al. species of this family.Some species canbe found severalkilometres awayfrom the coast (Gilchrist, 1988).However,no gecarcinidspecies isatrue land , since all of them haveto return to the seafor larvalrelease. All zoealarvae so far known aremarine planktonic, and larvaldevelopment consists of Žveto sixzoeal stages and one megalopalstage. Currently, larvaldata are available for only two generaand sixspecies of Gecarcinidae. Complete larvaldevelopment isknown for Cardisoma( C.)guanhumi Latreille,1825 (Costlow and Bookhout, 1968), Cardisoma( C.)carnifex (Herbst, 1794)(Kannupandi et al., 1980), Gecarcinus(G.) lateralis (Freminville,1835 ) (Willems, 1982)and Cardisoma(Discoplax) hirtipes Dana,1851 (Shokita and Shikatani, 1990).The prezoea of Cardisoma( C.)armatum Herklots, 1852has been described byCannon (1923)and the prezoea and Žrst zoealstage of Gecarcinus (Johngarthia)planatus Stimpson, 1860has been described byErhardt and Niaussat (1968).Previous incomplete descriptions of the Žrst zoealstages of byMoreira (1913)and Gecarcinuslateralis byCabrera ( 1965)havenot been considered in the present study due to the availabilityof newer and more complete descriptions. This paper provides adetailed morphological description of the Žrst zoealstages of Gecarcoidealalandii H.Milne Edwards,1837, Epigrapsuspolitus Heller, 1862and E. notatus (Heller, 1865)based on laboratory-hatched material.For both of these genera,there wereno previous zoealdescriptions. The comparison with descriptions of other gecarcinidgenera gives new insights into possible phylogenetic relationships within this family.

Materials andmethods First stagezoea larvae were obtained from two hatches of Gecarcoidealalandii , nine hatches of Epigrapsuspolitus and two hatches of E. notatus.Ovigerous crabs of Gecarcoidealalandii werecollected in Hsiang-Chiao-Wan,Pingtung Province, Taiwan( 30May 1997 and 3August 1999by H.-C.L.). One ovigerous Epigrapsus politus wasfrom Pingtung, Taiwan( 28August 1999by H.-C.L.), another one from Hsiang-Chiao-Wan,Taiwan ( 2September 1999by H.-C.L.), and seven werecollected from MalalayangBeach, Sulawesi, Indonesia (23January 2000 by C.D.S.). Two ovigerous crabs of E. notatus werecollected from Hengchun and Hualien, Taiwan (2and 30September 1999by H.-C.L.). Ovigerous crabs weremaintained in con- tainers with natural seawater until hatching. Freshlyhatched larvaewith anactive natatory behaviour wereŽ xedin 70%ethanol. Appendages weredissected under a Wild MZ8binocular microscope, and drawingswere made using anOlympus BH-2 microscope equipped with Nomarski interference contrast and attached camera lucida.All measurements weremade byan ocular micrometer. Drawingswere based on 10larvae, and measurements on 20–30 larvae. One hatch of G. lalandii, three hatches of E. politus (one from Taiwan,and two from Indonesia) and two hatches of E. notatus weremeasured. No signiŽcant di Verences in morphology and morpho- metry werefound between hatches of the same species. In the caseof E. politus, all drawingsare based on larvaefrom one hatch (female 4)from Sulawesi.The following measurements weremade: rostro-dorsal length (rdl )wasmeasured from the tip of the rostral spine to the tip of the dorsal spine; carapacelength (cl )from the base of the rostrum to the posterior margin;carapace width (cw) asthe distance between the tips of the lateralspines. The long natatory setae on the distal exopod segments of the Žrst and second maxillipedsare truncated in Žgures 2and 5.Descriptions Larvaldevelopment of Gecarcinidae 1673 and Žgures arearranged according to the standard proposed byClark et al. (1998). Maternal crabs of Gecarcoidealalandii , Epigrapsuspolitus , and E. notatus were deposited atthe National TaiwanMuseum, Taipei( TMCD3276– 3278 ).Samples of larvaeof the three species weredeposited atthe United States National Museum of Natural History, Washington, DCunder the accession number USNM 310337, 310336and 310335respectively.

Results Descriptions

Gecarcoidealalandii H.Milne-Edwards, 1837 Zoea I (Žgures 1A–C, 2A–D, 3A–C )

Dimensions. rdl: 0.54Ô 0.02mm; cl:0.33 Ô 0.01mm; cw:0.37 Ô 0.01 mm. Cephalothorax (Žgure 1A). Globose, smooth and without tubercles, with an anterodorsal protuberance asa carina.Dorsal spine short and slightlycurved. Rostral and lateralspines short and straight. One pair of posterodorsal setae. Anterodorsal region, posterior and ventralmargin without setae. Eyessessile. Antennule (Žgure 1B).Uniramous. Endopod absent. Exopod unsegmented with three aesthetascs (two long and one thin and short)and two setae. Antenna (Žgure 1C). Well-developed protopod almost reachingthe tip of rostral spine and bearingtwo rows of well-developed spines. Exopod elongated, reaching the middle of the protopod and bearinga group of three minute subterminal spinules, one long and one short terminal setae, and Žveshort terminal spines. Mandible.Endopod palp absent. Maxillule (Žgure 2A). Coxalendite with sixplumodenticulate setae. Basialendite with Žvesetae (two cuspidate and three plumodenticulate). Endopod two-segmented with one plumodenticulate seta on the proximal segment and one medial,two subterminal and two terminal plumodenticulate setae on the distal segment. Exopod absent. Maxilla (Žgure 2B).Coxal and basialendites bilobed with 5 1 4plumodenticulate setae. Endopod unsegmented, bilobed with one long plumodenticulate and one short simple seta on inner lobe and two long plumodenticulate setae on outer lobe. Scaphognathite with four plumose marginalsetae and one long setose posterior process. First maxilliped (Žgure 2C).Coxawith one seta. Basiswith 10medial setae arranged2, 2,3, 3.Endopod Žve-segmented with 2,2,1, 2,5(one subterminal 1 four terminal )setae. Exopod two-segmented, with four long terminal plumose natatory setae on the distal segment. Dorsal part of basis and exopod covered with minute spinules. Secondmaxilliped (Žgure 2D). Coxawithout setae. Basiswith four medialsetae arranged1, 1,1, 1. Endopod three-segmented with 1,1, 6 (three subterminal 1 three terminal )setae. Exopod two-segmented, with four long terminal plumose natatory setae on distal segment. Dorsal part of basis and exopod covered with minute spinules. Thirdmaxilliped . Absent. Pereiopods. Absent. 1674 J. A. Cuesta et al.

Fig. 1. Gecarcoidealalandii H.MilneEdwards, 1837, Zoea I. (A)Cephalothorax,lateral view;( B)antennule;(C )antenna.Scale bars: ( A)0.1mm, ( B,C)0.05mm.

Abdomen (Žgure 3A,B). Fiveabdominal somites. Somites 2–3with one pair of dorsolateral processes. Somites 3–5with conspicuous posterolateral processes. Somites 2–5 with one pairof posterodorsal setae. Pleopods absent. Telson (Žgure 3A–C ).Bifurcated with three pairs of stout spinulate setae on posterior margin.Along the distal part of each furcal arm two rows of minute spines. Twolateral spines on outer marginof eachfurcal arm. Larvaldevelopment of Gecarcinidae 1675

Fig. 2. Gecarcoidealalandii H.MilneEdwards, 1837, Zoea I. (A)Maxillule;( B)maxilla; (C)Žrstmaxilliped; ( D)secondmaxilliped. Scale bars: ( A,B)0.05mm, (C, D)0.1mm.

Epigrapsuspolitus Heller, 1862 Zoea I (Žgures 4A–C, 5A–D, 6A–C ) Dimensions. rdl: 0.42Ô 0.02mm; cl:0.24 Ô 0.01mm; cw:0.33 Ô 0.01 mm. Cephalothorax (Žgure 4A). Globose, smooth and without tubercles, with an anterodorsal protuberance asa carina.Dorsal spine short and slightlycurved. 1676 J. A. Cuesta et al.

Fig. 3. Gecarcoidealalandii H.MilneEdwards, 1837, Zoea I. (A)Abdomen,dorsal view; (B)abdomenlateral view; (C )telsondetail. Scale bars: ( A, B)0.1mm, (C )0.05mm.

Rostral and lateralspines short and straight. One pair of posterodorsal setae. Anterodorsal region, posterior and ventralmargin without setae. Eyessessile. Antennule (Žgure 4B).Uniramous. Endopod absent. Exopod unsegmented with three aesthetascs (two long and one short),and two setae. Antenna (Žgure 4C).Well-developed protopod reachingthe tip of rostral spine and bearingtwo unequal rows of well-developed spines. Exopod elongated, bearing Larvaldevelopment of Gecarcinidae 1677

Fig. 4. Epigrapsuspolitus Heller,1862, Zoea I fromSulawesi, Indonesia. ( A)Cephalothorax, lateralview; ( B)antennule;(C )antenna.Scale bars: ( A)0.1mm, ( B,C)0.05mm. agroup of four minute subterminal spinules, one long and one short terminal setae, and three short terminal spines. Mandible.Endopod palp absent. Maxillule (Žgure 5A). Coxalendite with sixplumodenticulate setae. Basialendite with Žvesetae (two cuspidate and three plumodenticulate). Endopod two-segmented with one plumodenticulate seta on the proximal segment and one medial,two subterminal and two terminal plumodenticulate setae on the distal segment. Exopod absent. 1678 J. A. Cuesta et al.

Fig. 5. Epigrapsuspolitus Heller,1862, Zoea I fromSulawesi, Indonesia. ( A)Maxillule;( B) maxilla;(C )Žrstmaxilliped; ( D)secondmaxilliped. Scale bars: ( A,B)0.05mm, (C, D)0.1mm.

Maxilla (Žgure 5B).Coxal and basialendites bilobed with 5 1 4plumodenticulate setae. Endopod unsegmented, bilobed, with one long plumodenticulate and one short simple seta on inner lobe and two long plumodenticulate setae on outer lobe. Scaphognathite with four plumose marginalsetae and one long setose posterior process. Larvaldevelopment of Gecarcinidae 1679

Fig. 6. Epigrapsuspolitus Heller,1862, Zoea I fromSulawesi, Indonesia. ( A)Abdomen, dorsalview; ( B)abdomenlateral view; (C )telsondetail. Scale bars: ( A, B)0.1mm, (C ) 0.05 mm.

First maxilliped (Žgure 5C).Coxawith one seta. Basiswith 10medial setae arranged2, 2,3, 3.Endopod Žve-segmented with 2,2,1, 2,5(one subterminal 1 four terminal )setae. Exopod two-segmented, with four long terminal plumose natatory setae on the distal segment. Dorsal part of basis and exopod covered with minute spinules. Secondmaxilliped (Žgure 5D). Coxawithout setae. Basiswith four medialsetae arranged1, 1,1, 1. Endopod three-segmented with 1,1, 6 (three subterminal 1 three terminal )setae. Exopod two-segmented, with four long terminal plumose natatory 1680 J. A. Cuesta et al. setae on the distal segment. Dorsal part of basis and exopod covered with minute spinules. Thirdmaxilliped . Absent. Pereiopods. Absent. Abdomen (Žgure 6A,B). Fiveabdominal somites. Somites 2–5with one pair of dorsolateral processes and one pairof posterodorsal setae. Somites 3–5with long posterolateral processes. Pleopods absent. Telson (Žgure 6A–C ).Bifurcated with three pairs of stout spinulate setae on posterior margin.Along the distal part of each furcal arm two rows of minute spines. One dorsal and one lateralspine on outer marginof eachfurcal arm.

Epigrapsusnotatus (Heller, 1865) Zoea I (Žgure 7A,B) Dimensions. rdl: 0.49Ô 0.03mm; cl:0.26 Ô 0.02mm; cw:0.38 Ô 0.01 mm. The Žrst zoealstage of E. notatus isverysimilar to that of E. politus; diVerences wereonly found in the size and abdominal morphology. Wetherefore do not repeat here the complete description, and the appendages arenot illustrated, except for the abdominal characters. Abdomen (Žgure 7A,B). Fiveabdominal somites. Somites 2–5with one pair of dorsolateral processes and one pairof posterodorsal setae. Somites 3–5with long posterolateral processes, those of somites 4–5 longer and more acute than in E. politus.Pleopods absent.

Discussion The genera Gecarcoidea and Epigrapsus areboth restricted to the Indo-West PaciŽc region. The distribution of Gecarcoidealalandii includes the Red Sea (Holthuis, 1977),South-East Asia (de Man, 1929)and the western PaciŽc (Daiand Yang,1991 )(see Ng,1998 ).The distributions of Epigrapsuspolitus and E. notatus haverecently been revised byNg et al.(1998,2000 ).However, E. politus had not been previously reported from Taiwan( Liuand Schubart, unpublished)and Sulawesi,localities from where ovigerous females werecollected for the present study. Previously, E. politus wasonly known from Tahiti,Tuamotu, CarolineIslands, Japan,Bertrand Island (Papua NewGuinea), Christmas Islands and northern Sumatra (see Ng et al., 2000). The larvalmorphology of the Gecarcinidae is poorly documented. For only six species and two generawas there knowledge of larvaldata (see Introduction). Furthermore, some of the larvaldescriptions present incorrect setal counts or lack detail,as shown bya recent comparison of zoealarvae of Cardisomacarnifex from Taiwanwith the ones described byKannupandi et al.(1980)from India (table 1) and further exempliŽed bythe study of Erhardt and Niaussat (1968)on larvaeof Gecarcinusplanatus .With the present description of the Žrst zoealstage of Gecarcoidealalandii , Epigrapsuspolitus and E. notatus,larvaldata become available for allfour generaof the Gecarcinidae and allowŽ rst intergeneric comparisons. Morphological characters of allGecarcinidae zoeaI that havebeen described are compiled in table 1.The following observations canbe made: allŽ rst zoeae show the same setation pattern on the basis of Žrst maxilliped( 2,2, 3, 3 )and second maxillipedendopod (1,1, 6 ),and also the same type of antennal and telson Larvaldevelopment of Gecarcinidae 1681

Fig. 7. Epigrapsusnotatus (Heller,1865 ),ZoeaI. (A)Abdomen,dorsal view; ( B)abdomen, lateralview. Scale bar 5 0.1 mm. morphology (telson with the current exception of G.lateralis ).The setation pattern of the endopod of the maxillule( 1,1 1 4)and basis of second maxilliped( 1,1, 1, 1)arenot shown in the table 1because they areshared within the Gecarcinidae as wellas with allother Grapsoidea. Zoealarvae of the familyGecarcinidae displaya combination of characters that uniŽes them and allowsthem to be distinguished from the rest of the grapsoid families (, Plagusiidae,Sesarmidae, Varunidae, sensu Schubart et al.,2000a).The combination of antennal and telson morphology, and setation of the second maxillipedendopod (1,1, 6)is not present in anyother familyof grapsoids. This seems to reect apossible monophyletic origin 1682 J. A. Cuesta et al. , 8 o h 0 t 6 n t 9 9 o s g i 9 1 r t n 1 e , e a f l t , r e i u a r n d n o p i a o e h e t A , r p c k a r p o e n e o k t i e g p o n o r h i y n r B e t S 2 n f , p y y y A . 8 e o d l d d d n d i f 9 a n d t R u u u o n 1 o a t t t a s n t a l s s s , r s a e e s a w a t t t d t T a a p r o n n n m i t t i l e e e . e e s s t k r l h s s s a . e e s l t o p e e i e t - e u u o r r r h a a o C C C S P P W P d d w i t o n i n o c d t 3 3 3 3 5 5 3 3 p r , l b – – – – – – – – ? a l c 2 2 2 2 2 2 2 2 D a ; A e s u e q G s e s y e l d i c ‡ 2 o o 6 6 6 6 6 6 6 6 p m r p , , , , , , , , p d a p o f 1 1 1 1 1 1 1 1 x n x l , , , , , , , , e E e a M 1 1 1 1 1 1 1 1 h r t e o t t n a i § l s 3 3 3 3 3 3 3 3 r o d 1 e s , , , , , , , , s f e i r 3 3 3 3 3 3 3 3 e p s d o r , , , , , , , , x a u d l 2 2 2 2 2 2 2 2 b c , M A , , , , , , , , n p 2 2 2 2 2 2 2 2 i l e p s D e y i t ; c d . † l p e a s o 3 3 3 3 2 2 2 2 x d p n e p s , , , , , , , , n n n M o 2 2 2 2 2 2 2 2 i f e E t d p o s n n s e t e A r , g . o a p a s h t n d e s s e n n n l i l p e A A A A A A A A E t a y a p t e s n ; n i o n r A z e e m t r t m e s u t r o o Ž d e n t e b v o n u a s p Ž B l A A A A A A A o y w , e t h o o d t T t i n b k o w A f w ; a t o c l s d r s a e e d 6 e u e g n p ? ? ? 5 – 5 5 6 f r i o a a l 5 u t l Z i t o s e t a x a e a t s f e r 5 m s . l e 5 1 3 ) ) f a l , 2 8 5 e 0 c 4 a . 8 p i r 8 1 8 ) 9 1 n x g 1 , i 9 7 0 e , o B , 1 1 l 8 l M e m s l ( l i 9 t o , e r . l ) ; t . i v e 1 o h p n l 5 a s t l e ( n l y p a s o r 6 i l b k t 2 i . r i t e e r r 8 l t t l 6 x o a p m e e 1 a e a i d p 8 ´ e a r t L , i r n t H 1 H 7 M r a m r i e ( i m d a i 3 , F e p h s r l i n 8 , , m m e l s x l e a s i d 1 r u u l e e ) l d x e l t f p n p h , e x a i e a H n s a u n r z i a n M ( i i n l e d a p n r i H m i p s t r : p r u s - u n a d s s , s a l a o a c g u . l a n n u n a l c t r a w t n s e a ) ) ) ) a l o u i K i . . . . i t d l t a a q t t 2 l o o u n D G E C C C e e a K i 5 p n ( ( ( ( i o ( n a e 8 v a s s s e n 3 u a a a a t 1 n e h i u u u d l s , t r i i , s s i m m m m o n e 4 3 b i o a p p o o o o s u w w , , c c b M s s s s n a a e t i i i i r r i 3 1 r r C a . a A † ‡ § d d d d a a c g g h c r r r r i i c c e D H t r a a a a e e i p p p u S C C C C E E G G f w Larvaldevelopment of Gecarcinidae 1683 of the Gecarcinidae, which wasalso suggested for the genera Cardisoma and Gecarcinus based on mtDNA sequence data(Schubart et al.,2000a,2000b ). Within the Gecarcinidae, the only important di Verence is the setation of the maxillarendopod. According to this, two major groups canbe distinguished within the Gecarcinidae: Epigrapsus, Gecarcinus and Gecarcoidea on one hand (with a2, 2setation), and the genus Cardisoma on the other hand (with 2,3 ).The setation of the maxillarendopod has alreadybeen noted byRice (1980)asof important taxonomic value,and recently it has been used to distinguish between grapsoid families (Cuesta, 1999).This character would thus support the recognition of two distinct taxaby separating Cardisoma with both of its subgenera ( Cardisoma and Discoplax)from the rest of the Gecarcinidae. The assumption that the Gecarcinidae is aheterogeneous group wasalready mentioned byWillems (1982),but new dataon adult and megalopalmorphology, aswell as molecular evidence willbe necessary to establish the taxonomic status of these two groups of generawithin the Grapsoidea. Within the genera Gecarcoidea , Gecarcinus and Epigrapsus the only diVerences arein abdominal morphology. The two species of Epigrapsus havedorsolateral processes on somites 2–5, whereas Gecarcoidea and Gecarcinus only on somites 2–3. Gecarcoidealalandii possesses conspicuous posterolateral processes on somites 3–5, similarto those of the Grapsidae s. str. (see Cuesta et al.,1998;Cuesta and Schubart, 1999)and characterized bythe lobulate shape of the tip versus the triangularshape of the tip of these processes in Epigrapsuspolitus , E. notatus and Gecarcinuslateralis . Despite the fact that larvaeof Gecarcinus(Johngarthia) planatus havebeen men- tioned and illustrated in the literature (Erhardt and Niaussat, 1968),larvalcharacters of this species could not be included in our comparison because aproper description of the larvaeis lacking,and the drawingsand photographs arepoor and do not allowdistinction of morphological characters. Within the Grapsoidea, the gecarcinidgroup comprising Epigrapsus, Gecarcinus and Gecarcoidea shows greatsimilarity with the Varunidaebased on zoealmorpho- logy.The features shared between these two groups arethe setation pattern of endopod of maxilla( 2,2 )and the presence of lateralcephalothoracic spines. On the other hand, these groups canbe clearlydi Verentiated bytelson and antennal morphology and the setation of the endopod of second maxilliped( 1,1,6versus 0, 1,6 ).The genus Cardisoma shares antennal and telson morphology, the number of abdominal dorsolateral processes and the setation pattern of the endopod of the maxilla( 2,3 )with Sesarmidae. However,the presence of lateralspines on the cephalothorax, and the setation of the endopod of the second maxilliped( 1,1, 6 versus 0,1, 6 )areclearly distinguishing characters. The setation of the basis of the Žrst maxilliped( 2,2, 3,3 )is shared byall Gecarcinidae, Sesarmidae and Varunidae (see table 1;Schubart and Cuesta,1998; Cuesta et al.,2000)and distinguishes them from the Plagusiidaeand Grapsidae s. str. (2,2,2, 2).The description and compar- ison of the larvaeof Gecarcinidae show that these larvaecarry a mosaic of characters that arealso present in other grapsoid families (especiallySesarmidae and Varunidae) and conŽrms the conclusion of Schubart et al.(2000a)that this familyshould be included within the Grapsoidea, atthe same taxonomic levelas the other grapsoid families.

Acknowledgements Thanks aredue to Pierre No¨el,Joseph Poupin and Isabel Fernandez for their help in obtaining acopy of the publication byErhardt and Niaussat (1968),aswell 1684 J. A. Cuesta et al. asto Colin McLayand P.J.Haywardfor the corrections and suggestions to the manuscript. Materialfrom Sulawesiwas collected byC.D.S. while funded through RaZes Research Fellowship thanks to Peter Ng.Some of the dissections and drawingswere made atthe laboratory of DarrylFelder, atthe University of Louisianaat Lafayette, partially funded by‘ Subprograma General de Perfeccionamiento de Doctores en elExtranjero, Ministerio de Educacio´nyCultura’, Spain and by‘ U.S. Department of Energy’(grant DE-FG02-97ER12220to D.L. Felder). This is the contribution 87of the Laboratory of Research of the University of Louisianaat Lafayette.

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