SUBSTITUTABILITY OF KIN AND THE CROW-OMAHA PROBLEM

The reasons for the existence of Crow-Omaha terminologies have long been debated because of difficulties in associating them with specific features of social organization or practice. However, by going back to the theories of earlier authorities like Josef Kohler, Radcliffe-Brown and Lévi-Strauss and using them to interpret some key ethnographies, it is possible to suggest why they exist. That is, from ego’s perspective, the vertical equations that define Crow-Omaha terminologies unite descent lines whose members in each generation can substitute for one another in relations with ego’s descent line, as marriage and other ties between those lines work themselves out over a time scale of, very often, several decades. Ultimately this can be linked to the long claimed links between Crow-Omaha terminologies and the prohibition of certain kin types in marriage, which typically act to delay the repetition of previous marriage alliances for one or more generations. It is suggested that Crow-Omaha terminologies have less to do with the prohibitions themselves than with these periods of delay.


Introduction
Crow-Omaha kinship terminologies have long posed a problem for those keen to associate terminological patterns with specific sociological features. These attempts continue, witness a recent collection discussing this issue edited by Thomas Trautmann and Peter Whiteley (2012). In fact the only terminological patterns that do show a logical fit with a definite sociological feature, namely cross-cousin marriage, are those described as prescriptive by Rodney Needham, whose language in these respects I general follow (e.g. Needham 1973) -and even then the expected correlation is by no means always found ethnographically.
No such logical fit is so dramatically evident in the case of Crow-Omaha terminologies, though explanations abound as to why they should exist at all. Most of the early theories were described by Robert McKinley in a paper published in 1971, though he was later attacked for an aspect of his own interpretation by Robert Barnes (1976;1984, Ch. 10). Nonetheless, both authors are useful to me in this article, though in their own ways: McKinley (q.v. 1971) by obviating the need to go through all the older theories, some of which are decidedly redundant; and Barnes because his book on the Omaha people themselves (1984) has been of immense value in guiding me through the maze of issues surrounding the actual Omaha ethnography and the reasons why the name "Omaha" became one of the most famous in the anthropology of kinship.
Perhaps the two most prominent explanations for Crow-Omaha terminologies, those that have survived the longest, are Radcliffe-Brown's theory that such terminologies are an expression of lineal unity (1952, Ch. 3) and Lévi-Strauss's observation (1966) that they identify lineages related to ego into which ego may not marry, at least in the present and immediately following generation(s). Both theories have been much derided, especially the former, which has perhaps worn less well over time; but the latter too is vitiated by the fact that the marriage avoidances that are the subject of that theory by no means need a Crow-Omaha terminology to accompany them, witness north Indian kinship systems (Tiemann 1970). However, while Radcliffe-Brown's theory is close to being tautological, it does at least focus on the terminologies themselves and their vertical, lineal equations. The fact that for Lévi-Strauss Crow-Omaha terminologies need marriage prohibitions to explain them, but not vice versa, means that his theory cannot be a universal one. It is, therefore, Radcliffe-Brown's position that constitutes the launch pad for the arguments presented in this article, it being understood that my modifications of his position still do not constitute the universal theory we have all been searching for -so far in vain, and perhaps in perpetuity.
Before proceeding further, I should make clear my own understanding of Crow-Omaha systems and equations, together with their implications for other aspects of kinship. Crow-Omaha equations appear first and foremost in the line of complementary filiation in Meyer Fortes's sense (e.g. the mother's patriline where there is patrilineal descent and patrilineal exogamy). Thus, in the case of Omaha-type terminologies, one frequently finds terminological equations uniting MF, MB, MBS, MBSS, MBSSS etc. (minimally, perhaps, MB and MBS), that is, equating the males of this line in successive generations. Omaha terminologies therefore express the operation of patrilineal descent, though not all societies with the terminology actually have the associated descent system.
What has tended to be puzzling since Josef Kohler (1975Kohler ( [1897) first tried to interpret these terminologies at the end of the nineteenth century are the similar vertical equations linking the corresponding female kin, with a focus on the equation M = MBD, possible including MZ as well through bifurcate merging. One explanation, suggested by Kohler himself, starts from the unexceptional fact that anywhere in the world, formally speaking, male ego's mother is his father's wife and his father's WBD is his own MBD. Among the Omaha, however, WBD may be one of the father's options for a second marriage, a rare event even among the Omaha, let alone elsewhere. For Kohler, this provides a sufficient explanation for the specific features of the Omaha terminology.
However, a principle of descent is also involved in these equations. Generally speaking, in societies of all types, it is perhaps logical at first sight to conceive of a woman's samesex successor as her own daughter. Certainly inheritance may sometimes go matrilineally from mother to daughter even where descent itself (in the sense of the transmission of social status) is patrilineal. For example, in India the stridhana or female "portion," usually consisting of female clothing and jewelry, goes matrilineally from mother to daughter in a society where patrilineal descent and inheritance predominate (see Tambiah 1973). Of course, this rule of matrilineal descent applies in any case in the typical matrilineal society, but Crow terminologies, which express matrilineal descent, do not do so through equations uniting mother and daughter (see next paragraph). However, in most patrilineal societies these two women are not lineally linked, at least not by birth: assuming exogamy of patrilines, the line, therefore, her same-sex successor must be traced through a male, most obviously her brother (ego's MB), and will therefore be ego's MBD-hence the reason for the puzzling equation. In short, once one has vertical equations for males, it is only logical to expect an equivalent series for females. Moreover, this set of vertical equations can often be extended further down successive generations, as happens with the male terms, and this can also proceed regularly through male links, thus including MBSD, MBSSD, etc. These latter specifications are not always included in the key textbook definitions of Omaha-type equations, yet they are implied by the logic of the classification and are found in many actual cases too, if not uniformly, then at least sufficiently frequently to invite an explanation such as Kohler's. Although, as just suggested, Radcliffe-Brown's theory of the significance of these terminologies has lasted less well than Lévi-Strauss's, it is still my starting point here. However, I am not just accepting Radcliffe-Brown's explanation hook, line, and sinker. Quite a lot of ethnography since his day, and some even before his time, shows that lineal unity is not just a thing in itself but may very well have a function (as it did for Lévi-Strauss, viz. the part it allegedly played in controlling marriage choices), though the function of lineal unity, if any, may vary from case to case.
In the present context, I argue that this function can be seen to have two main forms, which can also be connected: 1) Residual interests in, or claims to, property or prestations held, but not owned, by others are one possible reason for Crow-Omaha vertical equations. This will become clearer when I discuss examples below, but the scenario I am referring to runs roughly as follows. The members of descent group or descent line A are related (typically, it seems, by marriage, often in the deep past) to ego's descent group or descent line B, but are and remain separate from it. But further, A has a residual interest in property currently controlled by the members of B or has a claim to prestations that B owes to A, in such a manner that the lineally related members of A are substitutes for one another over time when it comes to these claims being recognized. Moreover, where the terminology is Crow-Omaha in type (and they need not be: see Parkin 2018), the vertical equations make this substitutability clear. This recognition of claims does not necessarily mean that they will be resolved at any particular point in time, and indeed in many cases it seems that such claims can never be resolved entirely if the relationship is to endure in the long term. It may therefore be the fact of substitutability that is the key here to interpreting the vertical terminological equations, not the interests or claims themselves. Among other things, this would avoid putting forward an argument that is simply a reprise of Radcliffe-Brown's doctrine of the unity of the lineage: while that unity may exist in and of itself, it may also express the substitutability of lineally related kin for particular, ethnographically varying reasons. Examples of this process are David Labby's (1976) work on Yapese property and Ross Bowden's (1983) work on the mortuary prestations of the Kwoma of Papua New Guinea, both discussed at length below, together with other similar cases.
2) The other form, already discussed above, is Kohler's suggestion that Crow-Omaha terminologies are associated with secondary marriages in which a chain of patrilineally connected women are preferential second spouses for ego, and therefore substitutable for one another. This might, therefore, be considered a special case of 1), though it was dismissed by Radcliffe-Brown, Lévi-Strauss and others (see Barnes 1984: 156) precisely because of its reliance on second marriages that presumably few in the society concerned will ever contract.
The rest of this article explores these scenarios further by discussing several pertinent examples. I should emphasize here, however, that I am not claiming to have solved the Crow-Omaha problem, merely attempting to interpret one major theory of their occurrence in a slightly different light. My sources are exclusively published ethnographies and not primary fieldwork of my own.

The general substitutability argument (1 above)
One relevant case that has been discussed extensively, though for other reasons, is social organization among the inhabitants of the island of Yap, a US Trust Territory in Micronesia.
Much of this discussion relates to the interpretation of descent on the island, a topic wrapped up with David Schneider's attempts (1972 to deconstruct descent entirely for analytical and comparative purposes, even though he originally interpreted the island's kinship system and practices in conventional genealogical terms (Schneider 1953). This involved him going back on these earlier studies (ibid.), in which he talked freely about both patrilineal and matrilineal descent groups (respectively the tabineau and the genung) being present on the island. In his later book, A cri0que of the study of kinship (1984), he set out to repudiate his earlier position, now deciding that the tabineau was in reality just a land-holding corporation, and seeking similarly to dismiss the idea of the genung as a descent group, though it is evident throughout his discussion that he found it no easy task to make these revised arguments. A subsequent study by David Labby (1976) unequivocally supports Schneider's original view that both groups are descent groups. It is clear, however, when comparing these two works, that the genung is not stressed as a unit by the Yapese; indeed, although exogamous and therefore governing marriages, it is residentially dispersed (residence being patrilocal). The tabineau, by contrast, is clearly more prominent, being both a residential and a landholding unit, and discernibly patrilineal in that succession to its headship proceeds patrilineally; it is also exogamous (ibid.: 32). In addition, unlike the tabineau, the genung is quasi-secret, not something discussed openly or that one asks questions about. Schneider says (1953: 217) that a mother will inform her children of their membership in it and point out other children who belong to it in the interests of avoiding incest, but anything less like a corporate group would evidently be hard to find.
Schneider also says (ibid.: 218) that marriage has little formality, the ceremony often being omitted altogether, and that divorce is easy to enact (a situation often found in societies with matrilineal descent). This, however, is implicitly contradicted by Labby's detailed description of the very elaborate exchanges of material goods that take place in connection with marriages (1976: 38-44). Status differences play some part in choosing spouses, as women should not marry into families or kin groups of lower status than their own (Labby 1976: 26). Whether this indicates a formal superiority of wife-givers over wife-takers Labby does not say, but it seems unlikely, as neither he nor Schneider mention the existence of positive marriage rules, nor do they give any indication of other possible restrictions on affinal alliance, for example, whether there are any bans on two kin groups repeating alliances in the generations following the first marriage.
In addition, neither author draws attention to the fact that the Yap kinship terminology is a Crow-type one, despite both of them describing it in some detail; indeed, Labby does not mention the fact at all, and Schneider only does so in passing (1953: 215). In addition, the terminology is used only in reference, and even then there are restrictions on its use as such; it is not used in address, vocative use being by name alone (apparently regardless of age or gender). This may be the reason for both authors minimizing its importance. Nonetheless, taking the two main sources together (ibid.: 219; Labby 1976, Ch. 4 thaw for all +3 females, plus the line of FFFZD, FFFZDD, FFFZDDD. Similarly, tutu covers many males in +2, as well as the line of FFZS, FFZDS, FFZDDS, while 0taw refers to many +2 females plus the line of FFZD, FFZDD, FFZDDD. Then there is wa'ayengin, a reciprocal term for MB and ZC, but also extending the line of the latter to ZCC and ZCCC. Wolag, given as "sibling," also unites MZC and MZCC. Affines can also be involved in this process: thus, le'engin, W, MBW, also links MZSW and MZDSW. However, as can be seen, most of these equations involve patrilateral relatives, consonant, it seems, with the obvious importance of ego's FZ as mafaen, roughly owner, or joint owner with her brother, of the land on which ego is brought up and will continue to live if male (women marry virilocally). This right of (co-)ownership also extends to FZ's children and further descendants, and it exists regardless of the fact that neither she nor they live on the land they co-own. This is clearly the reason, Labby notes (ibid.: 54-5), for viewing her whole descent line as a terminological unit (citamangin, ci0ningin), as well as those of FFZ (tutu, 0taw) and FFFZ (thang e tu, thang e thaw) respectively, who are also implicated in co-ownership by virtue of the birth of their ascendants on this land and their departure from it at marriage. Indeed, the three lines headed, respectively, by FZ, FFZ and FFFZ are themselves called mafaen, in the order mafaen ni be'ec, or 'new mafaen'; mafaen ni le' or 'coconut mafaen,' a metaphor for this line's hardness and durability; and mafaen ni gapalou or 'black bird mafaen,' a reference to the likelihood of FFFZ no longer being alive and of her spirit having been eaten by a black bird (a starling important in divination). In addition, having enjoyed the fruits of the land for a long time, this line will have been paid off and therefore, of the three lines, will have the least claim on ego's tabineau. The line headed by FZ, by contrast, which is the closest to ego, has the most claim over the latter by virtue of its residual ownership of the land ego is occupying (ibid.: 54). This also seems to be an explanation for the equations under wa'ayengin, at least in so far as this term unites ZD, ZDD and ZDDD, as they are the future mafaen of this I use Labby's spellings where they differ from Schneider's. I do not provide a full analysis of this terminology, 2 nor indeed of any other, in this article.
The diagnostic terms in Crow-Omaha terminologies do not always restrict themselves to expressing lineal uni 3 ty and may well have referents in more than one terminological line. See, for a good example, Barnes on the Omaha themselves (1984: Tables 9, 10).
There was a vagueness about some of these terms among Labby's informants, as he indicates on his main chart tabineau, though they do not appear to be given a name as a line per se, as they are not mafaen to male ego himself (ibid.: 58).
Despite the importance of these relatives as actual or potential co-owners of the tabineau's land, I suggest that, as they are resident elsewhere, contact with them is infrequent, consonant with their terminological identity as units rather than as individuals. By contrast, matrilateral relatives are not defined vertically as units in the terminology since they potentially have different origins (there is no suggestion that, for example, male ego's mother and wife should come from the same genung), and they go to different tabineau on marriage; therefore, there is nothing to unite them.
North-west of Yap is the island of Truk (Chu(u)k in some later texts), associated especially with fieldwork done by the anthropologist Ward Goodenough. In fact, his first publication on the inhabitants of this island (1947) was a journal article written with, and clearly under the influence of, George Murdock, in which he unambiguously defines the Truk kinship terminology as a Crow one; a later, book-length publication by Goodenough alone (1951) is rather more cautious on this point. Nonetheless, as on Yap, descent per se (i.e., as a marker of identity or as a means to control marriage through exogamy) is fundamentally matrilineal, meaning, of course, that the children of a male ego are in a different matriline than his own. In addition, what the authors call a 'matrilineal descent line' is not only a lineage segment but also a property-holding corporation, the most significant property here being land (lineages per se are not property-holders).
However, as on Yap, under certain circumstances men can inherit patrilineally from their fathers as öfökür (a general term for -1 descendants). Most such inheritance is of the father's movables, but if land is involved, the inheritor must recognize the residual rights of those members of his father's matriline who are the actual owners. And, as these individuals apparently belong to a number of different generations, "The result is a kinship system of the Crow type, despite the characteristic Malayo-Polynesian paucity of distinct denotative terms" (Murdock and Goodenough 1947: 340). Thus, semei is F, FB, FZS, FZDS, or any male of father's lineage, while inei is FZ, FZD, or any female of father's lineage, both therefore covering other lineal kin in addition to the already specified genealogical relations. However, the MBC = C equation, which also defines Crow terminologies, does not appear here, except in so far as both of these kin types are subsumed under the general -1 category of öfökür 'children,' while MB is included under semei (mentioned above), or alternatively under the zerolevel term ääi mwään 'elder opposite-sex sibling' (Murdock and Goodenough 1947: 340-1).
The features described in the last sentence are seen as transitional by Goodenough in his later solo work, except on the island of Puluwat, where, at the time he was there, this transition had already taken place (1951: 96; see also below). He explains this transition as follows (ibid.): This redefinition is a logical outgrowth of the organization of lineages and descent lines as corporations whose members are regarded as siblings. It is possible that at one time a person stood in a dual relationship to his mother's brother's son just as he does now to his mother's brother, calling this cousin a sibling when they interacted simply as kinsmen but calling him a child when behaving as a member of a corporation toward one of their jëfëkyr. One might predict that if the same trend continues, reference to mother's brother as 'father' will give way to calling him 'older brother' in all contexts, as appears now to be the case on [the Truk island of] Puluwat.
In other words, Goodenough is speculating that Hawaiian equations linking referents in the same genera0on -namely, F = MB and MBS = B -is giving way to a Crow pattern that makes equations between genera0ons, namely MB = eB and B = S. Another passage, published as an addendum to the earlier joint publication but written by Goodenough alone (1947: 343), provides the following explanation, which, unlike the quotation above, seems to treat the situation purely synchronically rather than as a potential transition supported by precedent: The term ääi mwään … refers primarily to an elder sibling of the same sex, and is applied to maternal uncles only in connection with inheritance or other matters when it is important to distinguish between 'fathers' … of one's own and one's father's lineage.
Recall that MB ordinarily shares a term with 'father,' but on some occasions, says Goodenough, a distinction should be made between them. Goodenough also gives information on marriage rules and practices (1951: 120, 122-3). Thus, there is no marriage to consanguines, especially within the matrilineage, nor into one's father's lineage, and therefore there is no cross-cousin marriage, nor indeed any marriage to first cousins.
Another society with matrilineal descent are the Trobriand Islanders, whose terminology, generally reckoned to be of Crow type, has been much discussed by Malinowski, as well as Leach (1958) and Lounsbury (1965) in a famous debate. However, it has also been studied since Malinowski by Fathauer (1961) and Weiner (1979), and by Read and Behrens (1990) with regard to the underlying generative logic of the terminology. Fathauer gives tabu FMM, FZ, FZD, FZDD, plus a number of other specifications, some male, some in other lines; tama FMMB, FMB, F, FB, FZS, FZDS; and latu MBC, C, ZC ws; these were later confirmed by Weiner (ibid.).
Prominent in these discussions has been how to interpret the term tabu. Weiner states that tabu can denote both ancestral kin who are what she calls the "founders" of land (including the whole of the + 2 genealogical level), as well as those women in male ego's father's clan whom male ego can marry. The latter, inter alia, include the vertically linked specifications FZ, FZD and FZDD -a classic Crow series -as well as FFBDD, another member of the same matriclan (1979: 339-42) and a classificatory FZD. She further states (ibid.: 343) that her informants "emphasized the very great importance attached to a marriageable spouse for ego coming from the same clan as ego's father" (apparently here both genders of ego are meant). This is apparently connected with mortuary payments ensuring control over land, though Leach (1958) suggested that what accounted for the kin types under this term was first, the position of many of them as recipients of urigubu payments from WB to ZH and, more generally, wife-givers to wife-takers, and secondly, their status as relatives with whom 5 there are potentially hostile relations, often involving joking relations. Weiner tries to reconcile the two meanings of tabu given above, as well as revealing that the earlier assumption of a preference for FZD marriage was denied by her informants. Nonetheless it appears that the term tabu partially maps out a vertical series of, here, matrilineally related women in different generations whom male ego may marry. At all events, by marrying into their father's clan, A recent study by Mosko (2017: 366) also draws attention to the equivalences between kin involved in urigubu 5 payments, in which they may stand in for one another (especially junior kin for senior kin), though he does not relate that to the Crow shape of the terminology. In any case, the kin term tabu is evidently not relevant here: the kin types involved are linked by real or classificatory father-son ties, not at all like a Crow system, nor a matrilineal one, in so far as a man's successor within the matriline is his sister's son, not his own son. both male and female egos are repeating, though in reverse order, a previous alliance between their and their father's clan without the affinal alliance system or terminology being at all prescriptive. As is well known in the Trobriands, at least formally, the place of the father in procreation is denied, and he is often seen as an outsider, indicating a low degree of emphasis on marriage, coupled with relatively easy divorce.
Since much has already been written about the Trobriand case, thus giving it its classic status, I do not wish to add unduly to that corpus here. The upshot appears to be, however, that, roughly as on Yap and Truk, matrilineal descent is opposed by patrilineal descent associated with a rule of patrilocal residence, creating a structural tension with regard to the respective land rights of patrikin and matrikin and of associated gift exchanges. We may conclude that Crow-type equations in the Trobriands thus map out the line(s) of those who are due urigubu payments and are also the preferred source of spouses for male ego and his own line. 6 A particularly well-described example of a society with patrilineal descent and an Omaha terminology are the Kwoma, of the Sepik river area in Papua New Guinea, studied by both Williamson (1980) and Bowden (1983), though there are some differences of detail be- Fundamentally, Bowden's analysis of the Kwoma (1983) links their Omaha terminology to how the Kwoma maintain alliances set up by marriage in the following three generations, when repeat marriages between the two groups are banned. While patrilines of wifegivers and wife-takers can be identified here, the social system is predicated on the fact that they must ordinarily be kept distinct from one another. There are no positive marriage rules 8 associated with the practice of any form of prescriptive alliance. In Bowden's words, "Kwoma formally prohibit the repetition of marriages, symmetrically or asymmetrically, between affinally linked patrilines (but not clans as wholes) for several generations once a marriage has been contracted" (ibid.: 748). As a result, alliances are dispersed among patrilines and cannot be focused on specific patrilines, as they are in both the models and many actual instances of prescriptive alliance. Nonetheless, Bowden's argument is that alliances initially contracted through a marriage are continued into the following generations (up to four) in other ways than through marriage, mainly involving mortuary prestations, and that this is consistent, with some exceptions, with the characteristically Omaha equations in the termi-Malinowski himself says that strictly speaking urigubu payments are owed by a man to his married sister so as 6 to signal that the latter, like her children, retains rights in her matrilineage's land despite her moving away from it in a patrilocal marriage; also, when she dies, these payments cease, at least in principle. It is also clear, however, that in effect they also benefit her husband and possibly some of his own kin as well. These payments should also be seen in light of the strict rule of avoidance between brother and sister. These issues are discussed in Malinowski 1929Malinowski , 1935 In a private conversation to which I was a party back in the 1980s, Rodney Needham, who became notorious 7 for his dismissal of 'Crow-Omaha' as a 'type' of kinship system (e.g. 1971), suggested that Bowden's interpretation was as close as we were likely to get to an explanation for these terminological features. It was also crucial in my own epiphany in relation to the theme of this paper. The actual Omaha system also rules out direct exchange, which formally equates wife-givers and wife-takers. nology. Some of these prestations are transferred asymmetrically from wife-takers to wifegivers, mostly as bridewealth, which in the first generation goes from ZH (the groom) to WB and the latter's same-generation kin in his patriline. Similar asymmetric transfers are repeated in the second generation between the wife-taker's sons and the same wife-giver's patriline, represented by ego's MB. The transfers in these two generations are compulsory, but they may be succeeded by further discretionary prestations of this kind in the third and fourth generations, after which the initial marriage tie is no longer seen as relevant and the two lines may intermarry again. These later transfers therefore take place between the wife-taker's SS and SSS respectively and the corresponding survivors of the initial wife-giver's line. More formally, the relationships typically involved are ZSS to FMB in the third generation and ZSSS to FFMB in the fourth, though whether the senior, wife-giving parties to these relationships will still be alive at this stage is a moot point. Simultaneously with these transfers, food goes in the opposite direction, from wife-givers to wife-takers, and there is also a constant "symmetrical exchange of various domestic, social and political services" (ibid.: 750). However, further marriages in this four-generation period are ruled out -that is, a male ego cannot marry into the patrilines of BWB, MB, FMB or FFMB, nor of WB, nor may he marry wifetaking relatives such as ZHZ, ZD or FZD, as this would involve an illicit reversal of the direction of alliances.
In linking the relationships set up by these exchanges to the terminology, which has numerous Omaha-type equations, Bowden has to acknowledge some exceptions. Thus, WB and WF are distinguished here because ego is not obliged to give prestations to WF, even though he must do so to WB. Also, for reasons Bowden admits he cannot account for, WB has a separate term not only from WF but also from WBS and the latter's descendants, despite the fact that an equation between them would be expected in an Omaha-type terminology, though such an equation does occur in Kwoma Pidgin. However, the equation WBS = WBSS = (and implicitly = WBSSS etc.) does obtain here, as they are all WB's successors in the exchange relationship with the respective wife-taking line and inherit WB's claims to the wealth payments. Similarly, from the point of view of the second generation of wife-takers, while MF has a separate term from MB, as he (MF) is not involved in these transfers with ego, MB's lineal descendants are equated with MB for four generations in classic Omaha fashion, such that MB = MBS = MBSS = MBSSS. From the points of view of the third and fourth generations of wife-takers, there are also terms uniting the lineal descendants respectively of FMB and FFMB. However, once this series of generations dies out, mutual terminological recognition of these lines of wife-givers and wife-takers ceases, in line with their ability to renew the initial marriage alliance: at this point, indeed, they become formally unrelated.
Although at this point the Kwoma also say it is good to reverse the direction of alliance to even up the balance between the two groups, fundamentally the relationship between wife-takers and wife-givers within any period of up to four generations is asymmetric. This is confirmed by Williamson's slightly earlier work on the Kwoma (1980: 535-44). She starts by stating that the Kwoma prefer to think in terms of their marriage practices as symmetric, despite the fact that there should be no exchange of sisters or of actual FZDs. She also says that the value given to symmetry is expressed in a more specific preference for marriage to classificatory FZD, to the extent that wives are routinely placed in this category regardless of their real relationship to ego. Bowden explicitly states that there is no such marriage preference (1983: 763-4, n. 4), remarking that only four of the hundred marriages for which he collected genealogical data were between male ego and a woman who could be classed as FZD. However, for Williamson this liking for exchange symmetry is also expressed as a preference to take wives from the lineage of FMB, to which actual FZD would also belong in the theoretical model of patrilateral cross-cousin marriage (m.s.), and FMBSD is identified as a legitimate wife, who would be equated with actual FZD in the same model. Although 9 Williamson does not say as much, this would also represent male ego repeating the marriage of, not his father, but his FF. However, her further statement that this woman would be a wife for ego's son or BS (ibid.: 542) looks odd in these circumstances since it would involve a marriage between generations. The significance of FMB for Williamson is also that, once mortuary payments to him have been completed, the initial marriage alliance also comes to an end, meaning that this happens a generation earlier than in Bowden's account; more particularly, it also suggests that a re-alliance now becomes possible, hence the suitability of FMBSD as a spouse for male ego. Despite the stress on symmetry in affinal alliance, however, Williamson makes it clear that most exchanges are asymmetric, as are marriage prestations, and that the ideal of symmetry often breaks down in practice, even though one can speak of a rough balance in spouse exchange overall. The discrepancies between Bowden's and Williamson's accounts cannot be reconciled further here, but they are probably not enough to affect the overall picture of how Kwoma marriage practices work and the extent to which the terminology can be associated with them. 10 The practice of maintaining relationships between two patrilines in the generations after the initial marriage, when no further marriages are possible, is also noted, though in much less detail, by Forge (1971) for another Papua New Guinea people. He states: "The Eastern Abelam…consider that each marriage sets up a relationship that subsists for three generations between a man, his sons and son's sons, and the sub-clan that provided his wife, the wife-givers becoming mother's brother's sub-clan and father's mother's brother's subclan to the two succeeding generations" (1971: 137). This relationship takes the form of "very elaborate and numerous exchanges…between the two groups for about a hundred years," which are "unbalanced and dissimilar at all stages, and valuables flow only to, never from, the wife-givers' sub-clan" (ibid.). Presumably the Kwoma, like the Abelam, have sufficient genealogical memory to be able to tell who owes what to whom over such a long period and when the exchanges come to an end and marriages can start up again.
The Kwoma case also has some resonance for similar practices in parts of eastern Indonesia and Sumatra (discussed in Parkin 2018). Both regions have examples of matrilateral cross-cousin marriage or asymmetric prescriptive alliance (MBD/FZS marriage), there being no Crow-Omaha terminologies at all in these areas as far as I am aware. However, there are societies in these regions that practice a non-prescriptive form of matrilateral marriage, though it might be one or other development of a former prescriptive system and thus represent an evolution away from it. As with matrilateral prescription, strict distinctions are still This equivalence does not appear in the Kwoma terminology, in which FZD is ruwey and FMBSD is yey. Mar 9 riage to FMBSD appears as a possible marriage option in many societies with Crow-Omaha terminologies. I hope to return to the question of marriage choices in such societies in more detail in a future paper.
The asymmetry of Kwoma practices is anyway a feature of all Omaha-type terminologies. All Crow-Omaha 10 terminologies characteristically distinguish patrilateral and matrilateral cross-cousins in making their vertical equations, and indeed they distinguish groups of wife-takers and wife-givers more generally. However, they do so without MBD marriage (usually banned in itself) or the repetitions of alliance between groups generation after generation that this asymmetric form of marriage prescription entails. maintained between superior wife-givers and inferior wife-takers. Again, there are delays in renewing marriages between any two groups for a number of generations, with relationships in this intervening period being pursued in other ways and with kin in succeeding generations tending to substitute for one another when exercising claims to property or prestations. What these kin in different generations mostly consist of, however, are chains of mother's brothers and their ascendants in different clans or lineages: they are not lineally connected within a clan or lineage. That is, it is the alliance aspect that is the focus rather than the lineal aspect. 11 Quite a lot of material relevant to the theme of this chapter comes from Africa. One example is John Beattie's studies of the Bunyoro of Uganda (1957,1958,1960). He clearly states that their terminology is of Omaha type (e.g. 1960: 53-5; 1957: 335), with many crossgenerational equations, though he does not offer a really satisfactory explanation for their occurrence. The Bunyoro have non-corporate exogamous patriclans (ibid.: 317) and ban marriage into the clans of mother and MM, as well as one's own clan (ibid.: 321). Residence is patrilocal (ibid.: 329). There is a slight preference for marriage into FM's clan, and such marriages are not unusual (ibid.: 339).
As just noted, there are many vertical lineal equations in the Bunyoro terminology, which Beattie sets out in a series of diagrams (1957: 331, 334, 337; 1958: 12, 18  Beattie's reasons for the existence of these vertical equations apply mainly to the lines of MB and FZH and do not go much beyond the obvious statement that they reflect ego's view of these lines being in some sense discrete units. "Thus a man thinks of the children of his father's sister, just as he thinks of the children of his own sister, primarily and collectively as children whom a woman of his own agnatic group has born to another clan" (1957: 333). And further, "Correspondingly, a Munyoro conceives the agnatic group to which he is linked by his mother, a group from membership of which he is excluded, as a unit" (ibid.: 335), with all Here too, it was Bowden's account of the Kwoma that was instrumental in my realizing what was involved in 11 these Indonesian cases.
the women and men respectively at or below ego's level having the same terms (as shown above). Another people with an Omaha terminology are the Haya, who live on the Tanzanian, southwestern shore of Lake Victoria (Reining 1972). The Haya have patriclans and patrilineages, but Reining focuses more on what she calls patrilines or enda in the Haya language. Most marriages are exogamous with respect to both clan and village; residence is usually patrilocal, but if the bride comes from her mother's clan, then residence may be in the latter's village of origin. The rules of marriage seem relatively simple at first sight. Thus, female ego may marry into her mother's clan, as we have just seen, but not her father's, while male ego can marry into neither clan of his parents. He can, however, marry into FM's clan, which normally means marriage to FMBSD (reciprocally FFZSS); that is, repeating a marriage that took place two generations earlier. There is also a sense that in the interim a reciprocal marriage takes place, that is, in the opposite direction: "The village patriline gives a daughter in one generation and takes a daughter back in the next" (ibid.: 103). This suggests that the marriage in the intervening generation is to FZD: although Reining does not say as much, some of the marriages in her diagrams on page 104 are clearly between FZD and MBS. This is reinforced by the additional information that "A model consisting of three intermarrying clans satisfies known genealogical data" (ibid.: 103), this being how the bare model of FZD/MBS marriage is generally diagrammed.
Regarding the terminology, Reining distinguishes the vocative or address terminology from the reference one, the former being largely generational in pattern, the latter with at any rate one Omaha term, kusikisa (literally, 'to install the heir'), linking MB, MBS and MBSS. 12 The data Reining provides for the terminology are not sufficiently detailed to tell whether the reciprocal equation, FZC = ZC, also occurs here. In explaining the one equation she does identify, Reining refers back to Lounsbury's suggestion (1964a) that Crow-Omaha terminologies should be linked to rules of succession and by implication to inheritance. Inheritance here is normally from father to son, occasionally to a SS, rarely to a brother. More particularly here, "When a man dies, the mother's brother of his primary heir oversees the installation of his successors" (Reining ibid.: 99). In other words, the heir's MB verifies the succession. However, if he too is deceased, his son and potentially his SS will act instead, an equivalence expressed by the term kusikisa.
Yet another case is the Baamba of western Uganda, studied by Edward Winter (1956). These have patrilineal descent, with exogamous lineages and a three-generation ban on repeating previous marriages between lineages. At the time Winter wrote, traditional exchange marriages were being replaced by bridewealth marriages because of missionary opposition to the former and colonial government prohibition of them (ibid.: 21-2, 73, 80ff., 172, 190). The Baamba also have several vertical equations in their kinship terminology, namely nukwayo, MFBS,MB,MBS,MBSS,MBSSS;ama,MFBD,M,MZ,MBD,MBSD,MBSSD,also HM;and mwaliana,FFZC,FZC,ZC,also FBDC,. Following Murdock, Winter identifies the terminology as Omaha in type (ibid.: 256), but does not explain why beyond stating that the patriline of MB and his lineal descendants will appear to ego as a unit since ego is related to them as a whole. However, MB will not view the line of ego (the ZS of MB) as a unit because MB is actually likely to have several sister's sons in different lineages, This may therefore be an instance of Kronenfeld's notion of layering (2009), another matter I hope to address who, by contrast, will all have him as their MB in common and thus gravitate towards him and his line in certain significant contexts of the MB-ZS relationship (ibid.: 186ff.).
The Bangwa of western Cameroon also have some indications of Omaha-type equations in their terminology, though their ethnographer, Brain, is not very clear on the matter and does not give a full or very useable terminology, which he finds "very sparse […] Bangwa categories are not clearly differentiated terminologically…" (1972: 47; see also diagram, ibid.: 48). Thus, atsen'ndia is MB, MBC, MZ, MZC, FB, FBC, ZC, possibly FBSC too, though betat is also indicated as linking FBC and FBSS. The first of these terms is also used for what Brain initially refers to as "a shallow clustering of kin, to whom ego is related through his mother" (ibid.: 53). Etymologically this term combines atsen, a word for a woman's lower body, with ndia, the term for house (obviously meant in a sociological sense, as well as referring to the physical structure). From the outset (ibid.: 53 ff.), Brain describes the atsen'ndia as an informal, affective "matrigroup," as opposed to the more formal and structured "patrigroup." Both are small kin groups rather than full lineages, the latter being much more concerned with what we normally understand by descent than the former (Brain sums up Bangwa kinship in general as bilateral, ibid.: 59-60). Brain even suggests that the atsen'ndia should be understood through "psychology rather than structural analysis" (ibid.: 53). It does have a range of descent and inheritance functions, apart from ego's relationship of mutual support and affection with his MB, in such a manner that suggests that these functions and feelings extend in part to other members of it -one can readily detect Radcliffe-Brown's influence here -but Brain does not make any explicit link with possible terminological equations of Omaha type. It also seems that the term atsen'ndia can be used for both a matrilineal grouping involving ties of kinship with and through one's mother back to an ancestress, and the patrilineage from which one's mother came upon marriage, making it a matrilateral group of relatives or even affinal in character, that is, alliance-oriented instead (ibid.: 65). It is in this latter sense that we would expect Omaha equations, if any, to map out a vertical line through the generations, though that is far from clear from the available information. Affinity is certainly important in extending useful ties: marrying an existing affine is disliked as a waste of an opportunity to create a new tie. Marriage to distant patrikin, such as a betat, is therefore preferred. ZHZ marriage and sororal polygyny are also ruled out here, as they are thought to be susceptible to future quarrels among those involved (ibid.: 165). In fact, the importance of affinity is such that Brain finds it easy to distinguish wife-takers from wife-givers in what he calls a relationship of "asymmetrical alliance" (ibid.: 166), the former evidently being patrikin and dependents of the latter, who are generally wealthier (ibid.: 114). He even goes so far as to suggest that the Bangwa are closer in this respect to the Lakher and similar groups in Asia practicing matrilateral cross-cousin marriage than to African near-neighbors like the Tiv and the Tallensi (ibid.: 187), though there is no suggestion of a prescriptive system here, nor even of any preference for marriage to cousins.
More obviously Omaha in type is the terminology of the Konkomba of northern Ghana (Tait 1961: 133-5). Thus, umwidza is MB, MBS, MBSS, ZS, and nawa is MZ, MBD, MBSD. Other terms with cross-generational equations are tshin, FZH, ZH, WB, WZ; tshuor (ms), WFF, WF, WM, WFFZ, WFZ, WMB; and tshapi (ws), HFZ, HZ. Some of these latter equations also occur in Omaha-type terminologies, such as FZH, ZH and WB, also WF and WFF (Barnes 1984: end chart for male ego). Tait does not explicitly acknowledge the presence of Omaha equations here, but his text does describe circumstances in which whole lineages may be treated as units, regardless of generation. Also, he states that "It is clear that kinship terms are extended through the generations only in the lineage of Ego's mother" (ibid.: 134). Some of the circumstances where this occurs relate to the role of matrikin in mortuary rituals (ibid.: 134-6), and, in general, matrilateral ties are clearly important (ibid.: 149, 150). The Omaha equations may also go along with marriage restrictions, as in Lévi-Straussian theory. Thus, FM's lineage or "extended house" is banned as a source of wives for male ego (ibid.: 78, 102), and MM's lineage might be banned as well (but cf. pp. 78 and 102), as it certainly is for female ego. The "extended house" of FZH is also banned for both male and female egos (ibid.: 103). More generally, repeating alliances between the same kin groups is only possible in the third generation (ibid.: 106). Exchange marriages are allowed (almost certainly within the same generation), but are rare, at a rate of less than five percent of all marriages (ibid.: 95).
Also of Ghana, on the central part of the coast, are the Fanti, studied over many years by David Kronenfeld, who has also commented on the skewing of the Crow-Omaha phenomenon more generally. The Fanti are one of the Akan peoples, closely related to the Ashanti, with a population of over a million. They have matrilineages, called ebusua, grouped into a small number of matriclans. Kronenfeld (2009: Ch. 1 [1973Ch. 2 [1980]) claims to have uncovered three different terminological structures or "subsystems" (2009: 16) among them. One of them need not detain us here, since really it amounts to no more than the "ex 13 tension," in his words, of terms for focal kin types outwards to remoter relatives of the same and even unrelated alters who can be assimilated to them in terms of age, gender, etc. -a familiar enough situation worldwide. More particularly, for our purposes, there is a terminology in day-to-day use that tends to the generational in pattern, plus a Crow terminology that is less used but is seen as more "correct" (ibid.:17, 49). The Fanti's Crow terminology has the following minimal diagnostic equations: na FZ, FZD, FZDD, FZDDD (also M, MZ -perhaps influenced by the generational ["unskewed" in Kronenfeld's words] terminology -and other specifications); egya F, FB, FZS, FZDS, FZDDS; and awofasi GC, MBC (both the latter with other specifications as well). (ibid.: 20, Figs. 1-3; also ibid.: 37).
Clearly the Crow ("skewed") terminology is linked to the assimilation of generations, but the Fanti themselves link one of its key features, the MBC = C equation, to potential matrilineal inheritance, something they talk about and reflect on a lot (ibid.: 33). However, neither the rules nor actual practice conform to a simple model of matrilineal inheritance from, e.g., MB to ZS. Although the deceased's matrilineage takes charge of that portion of the inheritance the deceased himself inherits (Kronenfeld adopts a solely male perspective in his 14 description), it generally observes a hierarchy of heirs. At the top of this hierarchy are the younger brother(s) of the deceased, as is common in West Africa, a practice that involves inheritance within the same generation, not between generations. Inheritance between generations only becomes an issue when the last of the brothers dies, whereupon the inheritance goes to the eldest son of the eldest sister, then to the eldest son of the next eldest sister, and so on down the age hierarchy within the same generation until, on the exhaustion of that genera-Kronenfeld also claims to have found a doubling of skewed and unskewed terminologies among the Ashanti, 13 another Akan people (ibid.: 49). He also mentions a fourth pattern among the Fanti, namely their use of some English terms to modify the existing patterns (ibid.: 49, 313 ff.). He links this to legal changes introduced by the modern Ghanaian state favoring some inheritance from father to son even among matrilineal peoples.
Some of the property the deceased acquired himself may go out of the lineage to his wives and children (ibid.: 14 33). Elsewhere Kronenfeld indicates that such transfers usually take place during the deceased's lifetime (ibid.: 311). tion, it goes one generation lower, to the eldest son of the eldest sister's eldest daughter, and so on. Despite the adelphic inheritance, generation is seen as fundamental in the sense that a nephew (presumably ZS is meant here) can inherit from an uncle (MB?), but not vice versa. In practice, however, because of the priority given to adelphic inheritance, the small size of movable legacies, which are soon dispersed or used up, and the unlikelihood that the potential residual heirs will live long enough to inherit anything, cross-generational inheritance is rare, only occurring, if at all, late in the inheritor's life (ibid.: 33-4). The link between matri 15 lineal inheritance and the Crow terminology, therefore, seems to reside more in people's minds and their collective thought than in reality. However, it is reinforced by the theoretical ability of a ZS to inherit the widow(s) and children of his MB, which Kronenfeld calls "an extended form of the levirate" designed to give the survivors "a kind of old-age insurance" (ibid.: 34). This too, though, is rare, with widows generally preferring to remarry or return to their natal homes as dependents of their male kin. There are also some ritual aspects to this relationship between ego and MBW/MBC, but in general the circumstances described above do not seem to be enough to account for the Fanti's Crow terminology in a sociological sense, being limited to giving the Fanti a cognitive understanding and set of symbolic meanings for the underlying principles of their society.

The secondary marriage hypothesis (2 above)
To understand fully the implications of WBD being a preferred secondary spouse in some societies with at least Omaha-type terminologies (Kohler's hypothesis, 1975 [1897[), we need to look at marriage practices in such societies in a general sense. A particularly good place to start, given the ample evidence on them (cf. Barnes 1984), are the Omaha themselves.
The Omaha are well known for marriage prohibitions that rule out the repetition of a marriage alliance between the same groups in the immediately following generations, viz. for a man, his mother's clan and the subclans of his FM, FMM, MM and MMM (Barnes 1984: 163). This has the effect of dispersing alliances among several alliance groups over time. Similar though not always identical prohibitions have become a defining feature of Crow-Omaha systems for many authorities, especially followers of Lévi-Strauss (cf. 1966), though they are not exclusive to them: that is, they are not necessarily associated with the vertical equations that characterize Crow-Omaha terminologies by definition. In the actual Omaha case, at least, Barnes (1984) says that these restrictions relate to close lineal kin, not their classificatory equivalents.
However, these restrictions do not prevent the repetition of marriages between descent groups in the same generation and may well go along with them. Indeed, in the Omaha case a group of lineage brothers may well marry a group of lineage sisters in another descent group within the same generation. One way of expressing this is to say that such marriages take place between BWZ and ZHB: in both cases, the assumption is that at least one marriage has taken place already in each direction. However, while Barnes (1984) considers marriage practices to be symmetric overall, sister exchange is not permitted and is evidently not practiced. Thus, while marriage between BWZ and ZHB (asymmetric) is allowed, marriage between ZHZ and BWB (symmetric) is not (NB: none of these specifications is given against any kin terms in the lists in Barnes, 1984: 132 ff., and Barnes himself does not mention them). How-One exception appears to be the 'stools' or chiefly positions (ibid.: 312). 15 ever, the actual Omaha terminology does have the equations F = FB = MZH; M = MZ = FBW; and FZ = MBW; though not MB = FZH. These specifications also form part of two classic Omaha-type vertical series respectively. It is tempting to see this as a relic of a prior period of prescription, though there is little evidence of this; but it is also consistent with the situation, just noted, in which descent lines in a single generation intermarry intensively but do not repeat the analogous marriages in later generations, as indicated by the lack of any affinal specifications for these terms. From ego's point of view, this could still be construed as representing the intermarriage of sibling sets. This would also be the case with cross-cousin marriage, but in that case cross cousins are clearly repeating the marriages of the previous generation and are definable because of this; thus, one would also expect cross-cousin specifications to be linked terminologically with affinal ones. This may not apply neatly to the actual Omaha case, given the ban on direct exchange mentioned above, but it could apply to other similar cases without this restriction.
But that is not all: among the actual Omaha, a man may marry several women from the same descent group but from different generations in what by definition will not be his first marriage. This is because they are all defined as wife's relatives, indicating that he has already married at least once. Thus, WZ, WFZ and WBD are terminologically equated as iho n 'ga (reciprocally ZH, BDH and FZH are equated under the term ishi'e). From the point of view of ego's children, the former group of specifications translate into MZ, MFZ and MBD, which are accordingly terminologically equated as iho n ,' though their reciprocals, ZS, BDS and FZS, are distributed between two different terms. Such marriages are far from universal, but they do appear to reflect known if subsidiary marriage preferences among the Omaha. A low take-up rate is not unusual for such preferences, and even first cross-cousin marriage has repeatedly been shown to be a minority practice, even where it is prescribed. One way of looking at this phenomenon is to return to the practice of BWZ-ZHB marriages noted above. Another way of conceiving these is to say that male ego finds a wife for his brother in the form of his own WZ. In marrying polygynously into his wife's descent group, however, he provides that wife not for his brother but for himself. 16 Kohler showed at the end of the nineteenth century (1975 [1897]) that many of the equations and distinctions in the actual Omaha terminology could be explained in terms of a preference for the polygynous marriages just mentioned, and Barnes subsequently identified yet more such equations (1984: 158-9). As already noted above, authorities coming after Kohler, especially Kroeber, Radcliffe-Brown, Murdock and Lévi-Strauss, rejected this explanation because it depended on preferences for marriages subsequent to ego's first marriage, marriages they considered less common than the first marriage (though in some cases they may still be practiced fairly often, as among the Omaha and Mapuche; see respectively, Barnes 1984: 156 and passim;Faron 1956Faron , 1961. Nonetheless, the structure of the actual Omaha terminology does correspond to the regular operation of these preferences without being defined by it in the same way as a prescriptive terminology would be, as Barnes makes clear. Barnes therefore appears to accept Kohler's basic premise, though does so rather cautiously (1984: 175, 217).
The fact that Barnes (1984 : Table 3) found many more equations to add to those Kohler uncovered that express the secondary marriage preferences discussed above for the This, incidentally, recalls a possible explanation for ZD marriage in parts of south India and the Amazon, ac 16 cording to which male ego takes his ZD as a wife not for his son, but for himself, 'anticipating,' as it were, ZD's marriage.
Omaha terminology means that the actual Omaha terminology expresses this sociological phenomenon as much as it does patrilineal unity, with which it is more usually associated. However, as Barnes points out (1984: 149, 151), the two phenomena are connected, given that some vertical equations in the actual Omaha terminology could be interpreted as uniting a line of patrilineally linked women with whom male ego might contract a secondary marriage. Similarly, from the point of view of ego's children, these women are also equated by the identical circumstance that they are women whom their father might marry secondarily.
There also appear to be other examples of terminologies with vertical equations uniting potential marriage partners. Gifford's early work on the Omaha-type terminology of the Miwok of California has long been seen as pointing in that direction in the context of the rather rare WBD marriage (1916; otherwise MBD marriage occurs here), as does Rivers' work (1914) on the Banks Islands in Melanesia, where the relevant category is MBW (reciprocally HZS) in a Crow terminology (both cases are summarized by McKinley 1971: 235-6). Not surprisingly, the Osage, closely related to the Omaha, also have evidence of preferential second marriages to WBD and WFZ, and M and MFZ, at least, have the same term (Nett 1952). Finally, the terminology of the Tlingit, a matrilineal people of north-west North America, has the Crow equations FZ = FZD and FB = FZS, which combine the marriage preferences for male nobles and commoner women respectively. In the former case, this is strengthened by the fact that another preference is for marriage to the wife of a deceased MB (which McClennan 1961: 114 calls 'nepotic widow inheritance'; cf. also the Banks Islands case, just mentioned), who would ideally also be FZ here. This is expressed by a symmetric equation in the +1 level of the terminology despite the distinction between patrilateral and matrilateral cross cousins in ego's genealogical level (cf. also FF = MMB in +2). One motive here is the desire, in this matrilineal society, for male ego to marry into his father's line (which would be different from his own, of course), which is achieved through FZD marriage. However, neither preference is stated to be a rule or prescription (de Laguna 1952;McClennan 1961). McClennan (ibid.) adds the information that FZ marriage is quite common, that FB/BD marriage is also known, and that fraternal polyandry also takes place here. Further, the neighboring Atna are said to have cross-cousin marriage with a clear preference for FZD over MBD, though whether this amounts to prescription, and whether the Atna are matrilineal too, is not made clear. The Haida of the NW coast, who are matrilineal, may have a preference for FZD marriage except among chiefs, who inherit their MB's wives as well as MB's positions; there is no evidence of FZ marriage (Moore 1963: 302-3). In the Solomon Islands too, the matrilineal Baniata have a Crow terminology and MBW as a possible marriage partner, alongside third cousins (Scheffler 1972: 350-2).
A rather similar situation is also found among the Pende of the Kasai province of the former Belgian Congo, for whom, at least at the time of de Sousberghe's study (1955), there was a strong preference for FZD marriage so as "to see our father's face again" (ibid.: 47ff.), i.e., to return to the father's clan through marriage. Indeed, the same term is used for both FZD and "wife." Formerly (perhaps as an alternative) there is also said to have been a preference for FZ in marriage for a similar reason. Thus, FZ and FZD can be linked as marriage preferences for a male ego, though they appear to have separate terms among the Pende, and no Crow-type equations are evident in de Sousberghe's description (he does not give a full terminology). Marriage is devalued in this strongly matrilineal society, as is the case in some other cases of societies with matrilineal descent. lies with WB, who is expected to provide ego with a replacement wife himself if asked to do so. In any case, we are also told that the BaVenda place a priority on WZ over WBD in these circumstances. Given the tendency to repeat marriage to MBD, she could presumably also be a sibling of WB if ego's patriline had taken more than one wife from WB's line.
In his work on the Mapuche of southern Chile, Faron (1956Faron ( , 1961 makes similar observations, as he sees WBD marriage in particular as important in certain, usually unsettled periods of Mapuche history as a way of cementing alliances between kin groups when the main alternative, MBD marriage, was ruled out for some reason or had become less popular. For Faron, it is part of a bundle of practices that includes polygyny more generally and the levirate and sororate as well, an explanation that is potentially relevant to other cases. But 17 he is also concerned to try and establish changes in terminological pattern over time and the reasons for them, since the Mapuche have only had an Omaha terminology in more recent periods of their traceable history. It is also clear that some of the differences in terminological patterns reflect regional variation, and some of Faron's reasoning for terminological change is somewhat speculative. Despite this, very real changes can be discerned in a chronology that, thanks to early Spanish dictionaries, can be traced back to about AD 1600. Thus, at that time, and indeed well into the eighteenth century, the terminology, such as we have a record of it, seems to have been mainly generational, especially in ego's generation, though with a partly bifurcate merging pattern in +1 and a few +2/-2 equations. By the early twentieth century the pattern had become Dakota-like (in Faron's terms) in ego's level, and indeed in the other two medial levels too. By the middle of the twentieth century, Faron himself was collecting terminologies with some Omaha equations, namely M = MBD and MBW = MM, MFZ, though not MB = MBS or FZC = ZC. Unfortunately what Faron means by 'Dakota' is not entirely 18 clear, but it appears to have been the same as Dravidian rather than Iroquois in Lounsbury's sense of the latter term (1964b). Also, given the almost complete lack of affinal terms in any of Faron's sources, including the terminologies he collected himself, we are prevented from diagnosing any of these terminologies properly.
Another problem is the matching of marriage practices with these various terminologies. Fundamentally, Faron appears to believe in a chronology that starts in recorded history with occasional MBD marriage, which is mentioned in eighteenth-century sources. Then we have a period of WBD marriage, as settled affinal alliance groups in the form of patriclans break down in the face of warfare with the Spanish and later the Chilean authorities. The argument here is that WBD marriage was better able to perpetuate alliances between families under these circumstances and that it was part of a package of alliance maintenance combining polygyny more generally with the levirate and sororate. Later, in the more settled period of the twentieth century, patrilines were re-established and WBD marriage gave way to MBD marriage, which is the preferred mode of affinal alliance today, though still not prescriptive (Faron reports no asymmetric prescriptive terminology). However, the Omaha equations in the later terminology reflect the earlier period of WBD marriage, while the earlier period of MBD marriage allegedly coincided with a longer period when the Mapuche terminology was The levirate involves a married woman taking her husband's brother as a spouse, normally after her husband's 17 death. The sororate involves a man taking his wife's sister (possible a "lineage" sister) as a second wife for himself, often polygynously. The second wife could, however, be another relative of the first, like WBD. This seems to be the burden of Faron's argument here.
There are some contradictions between Faron's different articles: I am relying on the full terminology set out 18 diagrammatically at 1956: 449. generational: i.e., the chronologies of terminological change and change in marriage patterns and preferences do not coincide. Moreover, only WBD is at issue as a marriage preference, 19 not also WZ and WFZ, as for the Omaha. At all events, for Faron the Mapuche are an example of a society where male ego can take WBD as his own wife, possibly at the expense of his own son (for whom she would be MBD), at least in certain periods. And despite the apparent lack of an asymmetric prescriptive terminology for any period, the standard diagram of such a terminology is only one of two in which male ego's WBD is his son's MBD, the other being the diagram of symmetric prescriptive alliance (assuming the male perspective is followed). This is not the case with the theoretical diagram of patrilateral cross-cousin marriage (FZD/ MBS), while among the Omaha, MBD, WBD and SW all have different terms and are in different lines in a manner that, expectedly, is not at all prescriptive (Barnes 1984). 20

Conclusion
The examples discussed in the first ethnographic section above show, with varying degrees of clarity, that Crow-Omaha equations express lineal unity not just because lineal unity itself exists, as Radcliffe-Brown tends to assume, but because lineages and clans have relationships of affinal alliance with one another. These relationships are often initiated through a marriage that cannot be repeated in the immediately following generations, but they also tend to be based on mutual ritual services or material claims, which may continue for several generations or even indefinitely, with successive kin in the descent line replacing one another over time. The whole lineage they belong to therefore assumes an identity in its own right, which may or may not also be expressed in the terminology. Therefore, this remains a partial explanation at best for the existence of Crow-Omaha terminologies, not a definitive one, and in all likelihood this is all we shall ever have.
Many of the examples in the second ethnographic section above, with their focus on secondary marriages to wife's kin, while relevant, can ultimately be seen as special instances of the substitutability hypothesis. At least that is the case with the Omaha, where a patrilineally linked group of wife's relatives are suitable partners. That is ruled out in the Mapuche case since only WBD is at issue. Again, this is only a partial explanation for the marriage patterns in societies with Crow-Omaha terminologies, a theme that deserves a paper in its own right. Terminologies like this express lineal unity by definition, but, as I hope to have shown here, it is not enough to leave it at that.
Ultimately, however, marriage preferences and lineal unity need to be combined within a single analysis if we are to arrive at a plausible hypothesis for the existence of Crow-Omaha terminologies. The possibility of renewing an alliance after the lapse of two or three generations (perhaps eighty to a hundred years) and the extent to which this happens are also important, as is the implied cyclicity involved in these practices. Indeed, another reason for the unity of some lineages that are marked out by Crow-Omaha equations is presumably the In 1967 Needham re-analysed the Mapuche case, concluding that the terminology was not prescriptive and 19 that MBD/FZS marriage was merely preferential. However, I prefer to go back to Faron's original accounts rather than rely on Needham here, as the latter had his own agenda and only partially succeeded in explaining the historical complexities of the Mapuche case.
Gabriela Piña, at the time of writing a PhD student at the London School of Economics who has done field 20 work with the Pehuenche subgroup of the Mapuche since 2011, has informed me that, while some MBD marriages take place, she has encountered no instances of WBD marriage. I am most grateful to Ms Piña for this information.
fact that marriages with them are impossible at present but will become possible at some point in the future, even if that future is several decades away. In this way, if in no other, Crow-Omaha terminologies express lineal unity over long time periods. In this respect they are unlike cross-cousin marriage and the prescriptive terminologies associated with them, where, at least formally, the repetition of an alliance is expected in the following generation, and the return to the starting point represented by an initial marriage accordingly takes place much sooner.