SHELMERDINE, SUSAN C., Hermes and the Tortoise: A Prelude to Cult , Greek, Roman and 
Byzantine Studies, 25:3 (1984) p.201 

Hermes and the Tortoise: 
A Prelude to Cult 

Susan C. Shelmerdine 

THE HOMERIC HYMN to Hermes, one of the four longer poems of 
the corpus, tells the story of Hermes' conception, birth, and 
inauguration into the Olympic pantheon after his first two mis­

chief-filled days of life. Almost immediately after his birth, the infant 
springs up from his cradle, intent on stealing the cattle of Apollo 
(22).1 On the threshold of his mother's cave, however, he meets a 
tortoise, whom he greets politely and quickly dispatches. Then, using 
her shell, he invents the lyre, and composes a song to its music. The 
episode is described in some detail (30-61), as if the poet, like his 
young subject, had been diverted from his main purpose by this 
chance encounter with the animal. But the narration then returns to 
the god's original goal, as Hermes places the new toy in his cradle 
and sets off once again after Apollo's cattle. This time he steals the 
animals and sacrifices two of them to the twelve gods before return­
ing home to await Apollo's discovery and anger. The remainder of 
the hymn deals with the inevitable confrontation between Apollo and 
his younger brother, and the final resolution of their conflict. 

The myth of Hermes' trickery was a popular one, and several 
versions existed in antiquity. But the other accounts of which we hear 
either omit the tortoise scene altogether and focus entirely on the 
theft of the cattle, or place the cattle theft and sacrifice before the 
invention of the lyre. Thus the Hesiodic version describes the adult 
Hermes' theft of the cattle and his subsequent encounter with a man 

1 We learn later, however, that he has a greater purpose in mind. At 167-75 he tells 
his mother why he has stolen Apollo's cattle: 

, . , ov8e ()WL(J'L 
vwi,' J-UT' a()aVClTounv a&VpTjTOt Kat aAtO'TOt 
atiTov rfi8€ Il-EVOVW; av€~oJ-U()', ~ a-V K€A€V€t~. 
/3EAnpov T/,.wTa 1TaVTa J-UT' alJavaTOtS' oapl1;,Hv 
1TAOV(J'LOV acjWELDV 1TOAVA.,]i.DV 7j KaTO: 8W,.w 
aVT{XfJ EV T,EpOEVn lJaaO'U'EJ-UV' aJ.I4X Be nil-TiS' 
Kayw TTiS' OO'LTjS' Em/3.,]U'o,.wt -qS' 1TEp 'A1TOUWV, 
Ei BE KE J.,L7J &ffnO't 1Ta~p EJ.,LOS', -q TOt eYWYE 
1TELP.,]O'W, Bvva,.wt, cjJTjATjTEWV opx.alLO~ ElVat, 

201 



SHELMERDINE, SUSAN C., Hermes and the Tortoise: A Prelude to Cult , Greek, Roman and 
Byzantine Studies, 25:3 (1984) p.201 

202 HERMES AND THE TORTOISE 

named Battos, whom he turns to stone for informing on him (fr.256 
M.-W.). Alcaeus' account apparently told of the god's theft, not only 
of Apollo's cattle, but also of his bow and arrows (fr.a.2 L.-P.). Both 
Sophocles and Apollodorus include the episode of the lyre in their 
versions,2 but treat the invention as a secondary feat of the god, 
describing it only after they have dealt with the cattle theft, and then 
almost as an afterthought. Both these authors tie the two episodes 
together logically, with the slaughtered cattle providing a hide and 
strings for the lyre, whose creation remains a subordinate point. 

In contrast, the unique treatment of this episode and its placement 
in the narrative of the hymn suggest that, for this poet, the invention 
of the lyre, not the cattle episode, is of primary interest. Hermes' theft 
and sacrifice of the cattle are important in establishing the character of 
the new god, and, more specifically, in precipitating his conflict with 
Apollo. But it is the lyre, and the music and song this instrument 
makes possible, that have the power to bring about a reconciliation of 
the two rivals.3 And it is the power of music (and song) which ulti­
mately wins for Hermes the place among the gods that is his goal from 
the start. This is the theme of the hymn. So the poet, I would argue, 
has quite deliberately orchestrated his treatment of the lyre's creation 
to emphasize the importance of the music which results. 

The priority of this scene in the narrative serves two purposes. 
First, by allowing the encounter with the tortoise initially to intrude 
on a quite different scheme in Hermes' mind, the poet keeps our 
attention on the (necessary) opposition between Hermes and Apollo, 
while at the same time showing the younger god for what he is: a 
child, easily distracted from one venture to another.4 This is impor­
tant, for the subject of the hymn is precisely a celebration of Hermes' 
'coming of age' and initiation as a god. At the same time, the hymnist 
places the scene first in order to give it the prominence it deserves as 

2 Soph. Ichneutae fr.314.312ff Radt; Apollod. Bibl. 3.10.2. 
3 Apollo's speech upon hearing the lyre for the first time makes clear that he is taken 

both with the sound of the instrument and with the song itself. So e.g. 441-47: 

~E Tt~ a9avaTwv T,E 9V'T/TWV all9pW7rwv 
8Wpov ayaoov EWKE Kai Eq,pafTE 9Emnll aOt~hiv; 
fJav,uxfT;:rjV yap rrjvBE IIET,q,aTOV OfTfTav aK01JW, 
Tjv OV 11'W 11'OTE c/rqJ.U OOT,/-UllaL OWE "TLII' allBpWv, 
OWE "TLV' a6avaTwv oi 'OA.VIJ,11'Ul BW,uxT' exovrrt, 
VOfTcf>t fTEfJEv cfn1Aijra aw~ Kai Mawoo~ viE. 
TIS TEXV"IJ, TIS IJ,Owa aWIJxallEWV /-UAEBWIIWV ... 

4 Note however that in the swift mind of the infant god, this discovery of the tortoise 
(and her subsequent metamorphosis) becomes almost instantly another way of out­
doing Apollo. 



SHELMERDINE, SUSAN C., Hermes and the Tortoise: A Prelude to Cult , Greek, Roman and 
Byzantine Studies, 25:3 (1984) p.201 

SUSAN C. SHELMERDINE 203 

the origin of the central theme, the power of music. To underscore 
the essential role this invention plays, moreover, the poet links the 
earlier scene thematically with the cattle sacrifice (which sets up a 
need for the newly created lyre) and presents the killing of the tor­
toise and its aftermath as a sacred act in its own right. A closer exami­
nation of the episode reveals its connection with song and sacrifice. 

The standard elements of a hymn are the identification of the 
subject, the invocation, the greeting (Xa'iPE), the naming of the deity 
(or subject of the song) - usually with some mention of parentage or 
birth-place, epithets that define the god's domain or powers, and a 
celebration of those powers, along with a promise by the singer to 
glorify his subject.s All these elements are present in Hermes' en­
counter with the tortoise, whom he pictures immediately in her 
transformed state as a lyre. 

The subject is given first in line 25, where the poet notes the 
connection of the tortoise with music ('EpfJ.fj~ TO(' 1TpWTf,UTa XEAVV 
TEKrf,VaT' ao(,&)v). An invocation follows as Hermes begins his ad­
dress by identifying the animal at once as a uVfJ.J30Aov (30). The 
standard commentaries take this to mean an 'omen' (presumably of 
Hermes' imminent success),6 which of course it is. But the word here 
has broader implications. A uVfJ.{joAOll is also a token or symbol by 
which a god can be recognized, one which gives power over the deity 
and is therefore used when calling on him.7 Hermes acknowledges 
the important association of god and tortoise with his use of this 
term, which presages the role of the animal-turned-Iyre in the ex­
change that will ratify his reconciliation with Apollo.s The address of 

5 For discussion of this form see e.g. E. Norden, Agnostos Theos (Leipzig 1913), esp. 
143ff; R. Keyssner, Gottesvorstellung und Lebensauffassung in griechischen Hymnos (Stutt­
gart 1932). More recent studies pertinent here include R. Janko, "The Structure of the 
Homeric Hymns: A Study in Genre," Hermes 109 (981) 9-24, and W. Race, "As­
pects of Rhetoric and Form in Greek Hymns," GRBS 23 (I982) 5-14. The form is 
observable in most of the Homeric hymns. 

6 E.g. L. Radermacher, Der homerische Hermeshymnus (SitzWien 213.1 (19311); T. W. 
Allen, R. Halliday, E. E. Sikes, The Homeric Hymns2 (Oxford 1936); F. Cassola, Inni 
omerici (Milan 1975). 

7 This use can also be observed, for example, in the magical papyri: e.g. Pap. Graec. 
Mag. I 4.2304, 2311; II 7.786 (ELP71Ka O"ov Ttl U7JJ.LELa Kat TO: uVI-Lfjo'A.a TOV OVO/-LUTO~, 
iva /-LOL braK01)crn~, OTt O"OL E7TEV"X0/-LUL); see III index s. v. uVI-Lfjo'A.ov. 

8 The use of uVI-Lfjo'A.ov as part of an exchange, especially in rites of hospitality, is 
attested elsewhere. See P. Gauthier, Symbola: les etrangers et la justice dans les cites 
grecques (Nancy 1972) 62ff; W. Mtiri, LVI-L(30AOV: Wort- und Sachgeschichtliche Studie 
(Bern 1931). The same word recurs in line 527, where it may mean 'pact, agreement' 
(so Cassola), and in any case refers to the reconciliation being effected between these 
two gods. There, as in the earlier use, the connection between Hermes and the lyre 
(and hymnic form) that he invents remains a strong undercurrent in the scene. 



SHELMERDINE, SUSAN C., Hermes and the Tortoise: A Prelude to Cult , Greek, Roman and 
Byzantine Studies, 25:3 (1984) p.201 

204 HERMES AND THE TORTOISE 

the tortoise in her 'immortal' transformation as the lyre that Hermes 
will exchange for his acceptance as a god, and as a symbol of the 
hymn with which he soothes Apollo's anger, acts as an invocation by 
establishing her divine nature and power. 

The invocation is followed closely (31) by the common hymnic 
greeting, Xa'iPE, and the naming of the animal. Here Hermes focuses 
on the tortoise-shell (the only part of the animal in which he has any 
interest), asking where it came from (32-33),9 and using a string of 
epithets that foreshadow its transformation into the lyre (31-32): he 
calls the tortoise "lovely in shape" (q,vT,V EpoEuua), "played/beaten 
at the dance" (~OPO(:TV7rE), and "comrade of the feast" (Ba'To~ ETai­
PTI).10 Hermes draws his speech to a close with a promise to honor 
the tortoise (ovB' a1TOTtI-'T,UW, 35); this final element of the hymnic 
form leads directly into Hermes' next act, the killing of the tortoise. 

All the standard elements of hymnic language are thus present 
(and in the appropriate order) in this scene. The associations with 
sacrifice are less precise, but their presence too can be demonstrated. 
A sacrifice regularly includes the playing out of a kind of 'game' in 
which the victim becomes a willing participant in the rite. This was 
the case, for instance, at the Bouphonia, where the victim, by the 
sprinkling of barley on the ground, was induced to bow its head and 
'agree' to the sacrifice. Such ritual has been described by Karl Meuli 
as a 'comedy of innocence', enacted to defuse the guilt of the killer 
through the supposed acquiescence of the animal to its death.l1 In­
cluded also in the ritual is the actual killing, the cutting up of the 
animal, sometimes the stretching out of its skin, and a song of cele­
bration to inaugurate the event at which the rite occurs, often in the 
context of a feast.12 

9 This question, focusing on the 'birth-place' of the shell, takes the place of the usual 
mention of parentage or cult location, neither of which would be appropriate here. 

10 This last epithet is the very one Apollo will use of Hermes himself, at 436, when 
the connection between god and lyre, suggested by his initial identification of her as his 
aVlJ-{30Aov (and, perhaps, by his substitution of the lyre in his own place in the cradle 
during his nocturnal adventures), is made complete. It is here that Hermes wins Apol­
lo's favor (436-38): 

BOVcfx}IIE /-LY';l(aIlWJTa 7TOVEV/-LEVE oom)~ ETatpE 

7TEII~KOVTa {3owv &IITa~ta Tawa J,LE/-LTJAa~. 
T,U'VXLw<; Kat E7TELTa BtaKPtIlEEU'(Jm QUo. 

11 K. Meuli, "Griechische Opferbrauche," in Phyllobolia fiir Peter von der Miihll (Ba­
sel 1946) 185-286 = Gesammelte Schri/ten II (Basel 1975) 907-1019. See also W. 
Burkert, Homo Necans, tr. P. Bing (Berkeley 1983) s. v. "Comedy of innocence," for a 
discussion of this idea. 

12 See Burkert (supra n.1 1) 3-12 for a discussion of the full rite of sacrifice and its 
variations (e.g. the selling of the victims' skins). Of particular importance as a model is 



SHELMERDINE, SUSAN C., Hermes and the Tortoise: A Prelude to Cult , Greek, Roman and 
Byzantine Studies, 25:3 (1984) p.201 

SUSAN C. SHELMERDINE 205 

Once again, the hymn's lyre episode contains each of these ele­
ments. As Hermes pledges to honor the tortoise, at the end of his 
'hymnic' address, he adds that she must aid him first (34-35). The 
only way she can do so (naturally) is by giving up her shell to be­
come the instrument on which he will sing her praises. Implicit in this 
'bargain', then, is the animal's assumed acceptance of her death, 
from which she will gain honor as a lyre sacred to the god. This is 
precisely the 'comedy of innocence' described by Meuli. The tortoise, 
of course, has no more agreed to her death than did any of the Bou­
phonia victims, but the inclusion of this 'bargain' between Hermes 
and the tortoise is important in marking the ensuing death as sugges­
tive of the regular 'game' of sacrificial ritual. 

The description of the killing itself is brief (41-42), but it uses 
language appropriate to sacrificial ritual, and remarkably similar to that 
which describes the cattle sacrifice in 118-19. Hermes is said to have 
"pierced out the life-force of the mountain tortoise" (alWv' egeroP'IJ­
O'ev opeO'KcpoLO xeAWV'IJ~), just as, in the later scene, he "pierced 
through the life [marrow] of the cows" (8L' alWva~ 'TE'Topr,O'a~). The 
repetition of aWJv and of 'TOPEW to describe both killings is note­
worthy, especially since we do not find this word combination in 
either the Iliad or the Odyssey.13 Conspicuous too is the image of the 
god in each case turning his victim (s) over before dealing the final 
blow (ava7T'lJAr,O'a~, 41; E:YKAivwv 8' e,roALv8e, 119). 

In the earlier passage, the unparalleled use of ava7TTjAr,<Ta~ (appar­
ently a form of *ava7TTjAEW from aVa7TliAAw), has baffled most com­
mentators, who have tried, unconvincingly, to emend the text. 14 

Even those who accept the form, and take it to mean 'swing/toss up', 
seem uncomfortable with the sense, which ought to refer to the 
motion of turning the animal onto its back, not tossing it in the air. 
Help may come, however, from archaeological evidence. The scene 
of a child swinging a tortoise (or a lizard), suspended by its tail on a 
short string is a motif found in vase painting by at least the second 

the sacrifice which Hermes in the hymn makes to the twelve gods 015-37), and which 
follows precisely this form (including the absence of the feasting). 

13 Scenes of sacrifice are common in both Homeric epics, and could have provided 
traditional language for these scenes (as they do for others in the hymn) had the hym­
nist wished to gloss over the two episodes. See e.g. 11. 1.315-18, 458-67; 7.520-23; 
Od. 3.432-63. 

14 For example, aVa1TtA:r1(m~ of 'pounding' the tortoise to death (Hermann; now 
accepted by Cassola); ava7TELp';/Va~ of impaling the creature (Stephanus). See Allen/ 
Halliday/Sikes (supra n.6) ad 41 for a discussion of the passage (in which they accept 
ava7T7JA:'J(]'a~), and a review of the conjectures. 



SHELMERDINE, SUSAN C., Hermes and the Tortoise: A Prelude to Cult , Greek, Roman and 
Byzantine Studies, 25:3 (1984) p.201 

206 HERMES AND THE TORTOISE 

half of the fifth century D.C.15 If the practice was in fact common­
place, the poet may well have used this verb to suggest, at one 
stroke, both a child's game and the act of ritual killing.I6 For the play 
of the child-god exposes the underside of the tortoise, enabling him 
to kill the animal-and, as before, the animal becomes a 'willing 
victim' by its participation in the 'game'. 

The notion that sacrifice was a natural, if not necessary, accompani­
ment to the creation of a musical instrument is not new. Walter 
Burkert, for instance, asserts that "any new creation, even the birth 
of music requires ritual killing. Underlying the practical use of bone­
flutes, turtle-shell lyres, and the tympanon covered with cowhide is 
the idea that the overwhelming power of music comes from a trans­
formation and overcoming of death. "17 So the 'mock sacrifice' of 
the tortoise is the first step in this birth process. And the subsequent 
cutting up of the animal and treatment of her shell continues the 
rite. 

As he did with the description of the killing, the poet continues to 
link the lyre's creation with the later sacrifice, using language that 
bears a striking resemblance to his description of Hermes preparing 
the slaughtered cattle for the twelve gods. In each case, he stresses 
the deed (W~ aJL' e1To~ 'TE Kal. ep'YoJ) EJLT,8E'TO, 46; ep"M 8' ep'YoJ) 
o1Ta'E, 120), and makes explicit the act of cutting ('Ta~)J), 47, 120).18 
Since the tortoise's flesh is of no interest to Hermes-as the meat of 
the victim would be in a regular sacrificel9 -the poet describes here 
the cutting up, not of the animal herself, but of reeds to form a 

15 For an illustration see British Museum Catalogue IV (London 1896) F 101; a draw­
ing of the same vase in T. Reinach, DarSag I 695; a discussion of the practice, with a 
photograph of this oinochoe, in A. E. Klein, Child Life in Greek Art (New York 1932) 
13 and pI. XV.c. Cf also the interesting suggestion of J. Maxmin ("A Note on Prax­
iteles' Sauroktonos," G&R 20 [I973] 36-37) that the the famous statue of Apollo 
Sauroktonos may originally have portrayed that god in the act of catching a lizard at the 
end of a string held in his left hand. If so, we have an interesting parallel for the action 
of Hermes in the hymn. It may be objected that these representations cannot help us 
understand a hapax from a Homeric hymn. But if, as I believe, the Hermes hymn 
dates to the end of the sixth century, it seems entirely reasonable that the same child­
hood game could be reflected in all three examples. 

16 Part of the hymn's charm is exactly in this play between the images of Hermes the 
child and Hermes the god. Compare, among many examples, the description of Her­
mes snuggling back into his cradle after a full night of miraculous deeds (150-53), and 
his indignant response to Maia's scolding only a few lines later (163-65). 

17 Burkert (supra n.l1) 39. Cf the Pindaric account of the invention of the flute by 
Athene after the Gorgons' death, Pyth. 12.4-24. 

18 TaJ,LWV, while not itself formulaic, is placed in both lines at the trochaic caesura of 
the third metron. 

19 Note, however, that the later cattle rite also does not follow the regular pattern in 
this respect; although the meat is divided into portions, it is not eaten. 



SHELMERDINE, SUSAN C., Hermes and the Tortoise: A Prelude to Cult , Greek, Roman and 
Byzantine Studies, 25:3 (1984) p.201 

SUSAN C. SHELMERDINE 207 

bridge for the new instrument.2o He continues the image of sacrifice 
with vocabulary (repeated in the later scene) such as vWTa (48, 122) 
and pLVoLo/pt.vov~ (48, 126) to describe the victim's body, in spite of 
the awkward usage of PLVO~ for 'shell'. And he focuses finally on the 
stretching of hide (aJ.L~t ... TaVVCT(TE, 49~ ETavvuuuTO, 51~ EgETa­
vvmre, 126). As before, the regular process is perverted, as the verbs 
of 49 and 51 refer not to the discarded part of the victim but to the 
strings and hide (fjEpJ..UX f3oo~, 49) that Hermes stretches over the 
shell in constructing his lyre. 

Finally, when the instrument has been completed, Hermes draws 
the event to a close with a song which not only tests the quality of 
the new lyre, but also celebrates the transformation of the tortoise 
after her 'sacrifice' (52-61) .21 The content of the song confirms that 
the power of music and of the hymnic form is the poet's real subject. 
The god sings first of the association of Zeus and Maia, his own 
conception and birth, and then the cave that was his and Maia's 
home. His subject, in short, is identical to that of the hymn up to this 
point. And the beginning of his song (aJ.L~t J1ia KpoviB'Y1V Kat Mat­
aBu KaUt.7TEfjt.AOV OV, 57-58) recalls in detail the introductory for­
mulae common to this and other hymns (e.g. 7.1-2, a!-'<Pi A."wvvuov 
I.EJ.LEA'Y1~ EPt.KVBEO~ viov ... w~). Here we can see in the myth a 
mirroring of the hymn's actual performance, as the singer glorifies 
the very instrument on which he plays, and the poet simultaneously 
celebrates his own art. 

This same celebration recurs in Hermes' second song (423-33), 
where the subject matter once again mimics (in part) that of the 
hymn. The god sings the origin of the world, the birth of the gods, in 
order of age, and how each one received his or her divine share.22 

20 47-48: m;~E S' ap' €V J.LETPOUrL TaJLWV SovaKa" KaAal-WW 1TEtp~Va" Sw VWTa Sw 
invoLo XEAdwry". The passage has been variously interpreted. Reinach (DarSag III 1438) 
explained these reeds as part of a frame, set within the hollowed shell, over which to 
stretch the ox-hide (49L but there is no need for such a device. The bridge was gen­
erally called the Sovag or KEpn<; (depending on the material from which it was made), 
but the plural here includes the reeds used to fasten the bridge to the shell. For a 
confirmation of this view, see now P. Phaklares,"XEA~," De/tion 32 (977) 218-33, 
who catalogues the archaeological remains of tortoise-shell lyres (all from the end of 
the Archaic period) and gives a detailed account of how the instrument was con­
structed. I am grateful to E. Vermeule for making me aware of this article, and to C. C. 
Vermeule for making it available to me. 

21 A similar song is also part of the cattle sacrifice, but it is postponed until the return 
to Apollo of his remaining animals (418), when it has the additional function of caus­
ing the brothers' immediate reconciliation. 

22 As the last born of the gods, Hermes will occupy a place of emphasis at the end of 
his song, particularly as he sings of the divine portion he wants to acquire (or, as the 
performance of this hymn attests, already has). 



SHELMERDINE, SUSAN C., Hermes and the Tortoise: A Prelude to Cult , Greek, Roman and 
Byzantine Studies, 25:3 (1984) p.201 

208 HERMES AND THE TORTOISE 

But he sings first of Mnemosyne, the mother of the Muses, whose 
servant he, like the singer of the hymn, has become (429-30).23 
Apollo's reaction is immediate: charmed by the music, and the hymn, 
he laughs with joy and forgets his anger, so that, by the end of 
Hermes' performance, he announces his willingness to accept a recon­
ciliation with his brother peacefully (";'(]"1JXUur; Kat E7TEtTa 8taKptVEE­
(TOat OLto, 438). It is in this reconciliation that Hermes exchanges the 
lyre (= crVJ.I,{3oAOV) for a share of divine power and achieves his 
original aim. 

The poet here completes the link between cattle theft/sacrifice and 
the lyre episode, as the healing power of the music and song resolves 
the conflict which Hermes initiated with his theft. This is the same 
ability of music to bring about peace and harmony-a force strong 
enough even to make gentle the harshness of Ares, god of war­
which Pindar praises in the opening lines of Pythian 1.24 The hymnist 
emphasizes this power inherent in the lyre (and the hymn it accom­
panies) by his treatment of their invention both as Hermes' first 
exploit and especially in terms of a (mock) sacrifice. If we understand 
this, we will not misunderstand the hymn. For, as it tells of Hermes' 
passage from new-born infant to god of craft, the hymn is at the 
same time celebrating its own origins, and those of the power of 
music, a power that comes from the conquering of death. The in­
vention of the lyre and the hymnic form is a prelude to Hermes' 
acquisition of cult, and to the performance of this hymn that honors 
him.25 

THE UNIVERSITY OF NORTH CAROLINA, GREENSBORO 

September, 1984 

23 This mention of the Muses recalls the regular invocation at the outset of all 
hymns, but the poet emphasizes their importance (and especially that of Mnemosyne, 
their creator) by beginning Hermes' song, and then reverting to speak of them. As so 
often in the hymn, the focus is on the music and the poetry. 

24 For other expressions of this power, compare e.g. the embassy to Achilles (It. 
9.186-91), which finds him sitting outside his tent, playing the lyre and, for a moment, 
forgetting his anger, or Hymn. Hom.Ap. 184ff, where the gods themselves are described 
as "having a care only for the lyre and song" when Apollo plays among them. 

25 I am grateful to J. Hamilton, G. Nagy, and the anonymous referees of this journal 
for their valuable suggestions on an earlier draft of this article, and to L. Koenen for 
his encouragement and advice at the outset. The flaws which remain in spite of their 
efforts are my own.