7. Conclusion

Jenkins ([35], Figure 8) and later Fellman and Eriksson [1] presented the association between the TWRs and the TRRs in the Prussian data [16]. They observed marked fluctuations for the different annual data but a good agreement between the TRR and the TWR for the total data

Our impression is that the finding already noted by Strassmann [20] was the birth of Hellin's law. Furthermore, Jenkins [35] stressed that Hellin's law is a first approximation. It is generally agreed that the main argument for Hellin's law is that the probabilities of additional ovulations and the fissions of fertilized eggs can be explained by stochastic models. Consequently, in large data sets, the averages could be stable and formulated by a mathematical relation (Hellin's law). A common argument for the discrepancies is that after the conceptions, there is a long process influenced by disturbing factors (intrauterine deaths, spontaneous abortions, etc., of one or more fetuses). Jenkins [35] and Komai and Fukuoka [39], for instance, assumed that differential mortality in utero of twins and triplets could be one such factor. Consequently, the final result often shows only a weak resemblance to the outcome of a simple stochastic process associated with the initial conceptions. Excesses of higher multiple maternities in old birth registers must be considered paradoxical. One explanation can be the results of the comparisons between the changes in the rates of singletons, twins and triplets during the time from conception to confinement discussed above. Another probable explanation is that systematic errors in the registers may cause biases in the data. This explanation is less plausible if the data are collected in

In his study of the rates of multiple maternities for total, "white" and "colored" in US populations (1922–1936), Strandskov [40] evaluated how well his data satisfy Hellin's law. Applying χ<sup>2</sup> tests, he found that in none of the populations tested did the observed plural

Based on hospital data, Sarkar [41] studied the TWR in India and on Ceylon (Sri Lanka). His paper is interesting because he defined the TWR as 1 : n and the triplet rate as 1 : m2, that is, he indirectly used a modified Hellin's law without any reference to Hellin. One finds a deficit of triplet maternities (m > n). In addition, one observes that on Ceylon the TWR was low (1 : 161:1), yielding a TWR of 6.21 per 1000. On Ceylon, the TRR followed Hellin's law more

Das [42] formulated Hellin's law such that "the frequency of twin confinements bears to that of total confinements a ratio which is equal to the ratio borne by the frequency of the triplet confinements to that of the twin confinements". This modified definition is in congruence with Strassmann's version of the law. He reviewed earlier studies concerning Hellin's law and stressed the discrepancies presented in them [3]. Das concluded that Hellin's law has no sound basis and that exceptions to the rule have been the rule. In a later paper, Das [43] also

zygosity of both twins and triplets. His mathematical analyses of these models did not support

Fellman and Eriksson [2] compared in Figure 3 the TWR and the transformed TRR and QUR for Sweden (1751–2000). For the period 1871–1960, there is a deficiency in the TRR. Fellman

. He constructed an advanced model based on the

different countries, as is the case in data in ([32, 35], Table 1 and Figure 1).

birth frequencies agree closely with Hellin's law [3].

.

exactly because it was 1 : 154:4<sup>2</sup>

considered the relation TRR <sup>¼</sup> ð Þ TWR <sup>2</sup>

Hellin's law [1].

set [32].

18 Multiple Pregnancy - New Challenges

It is generally agreed that the main argument for Hellin's law is that the probabilities of additional ovulations and the fissions of fertilized eggs can be explained by stochastic models. Consequently, in large data sets, the averages could be stable and formulated by a mathematical relation (Hellin's law). A common argument for the discrepancies is that after the conceptions, there is a long process influenced by disturbing factors (intrauterine deaths, spontaneous abortions, etc., of one or more fetuses) [32]. The discussion of Hellin's law is complicated by the fact that the law is a mathematical rule concerning theoretical rates, but all checks have to be based on empirically obtained rates. In fact, one can only check if the discrepancies are so large that they cannot be explained by random errors. If the discrepancies are small, an exact Hellin's law cannot be accepted. In this way, no exact proof to support the law can be obtained [32].
