302 SECTIONAL TRANSACTIONS.— A. 



Another interesting field for theoretical investigation is provided by the transition 

 from one type of binding to another shown by the oxides, sulphides, selenides, tellurides, 

 arsenides of certain divalent elements. Some have the sodium chloride structure, 

 others the opaque metallic nickel arsenide structure which has properties more like 

 those of an intermetallic compound. Again, there is the so-called ' adamantine ' 

 compounds of formula AX, with their characteristic tetrahedral arrangement in which 

 the binding appears to be homopolar throughout. 



The pressing need at the present juncture is for a theoretical technique which can 

 treat these complex interatomic bindings so as to obtain quantitative results for 

 comparison with experiment. 



Dr. J. D. Bernal. — 



Discussion on the previous papers. 



Prof. E. A. Milne, F.U.S.— Stellar Structure and the Origin of Stellar 

 Energy. 



Department op IVIathematics (A*). 



Dr. J. Henderson.— T^e Methods of Construction of the earliest Tables of 

 Logarithms. 



The paper gives a brief description of the methods adopted by Napier and 

 Briggs in the construction of their original logarithmic tables. Napier constructed 

 the first logarithmic table in the general sense, while Briggs followed with the 

 first logarithmic table with 10 as the base. In Napier's method the calculations 

 do not depend on the idea of a base; thej^ depend essentially on the association 

 of two progressions, one arithmetical and the other geometrical. Briggs, however, 

 used methods which depend on the fact that the base is 10; he formed a 

 skeleton table and then interpolated by the method of differences, which he 

 apparently understood completely. 



Dr. R. A. Fisher. — Inverse Probability. 



The controversy over ' inverse probability ' seems to be unique in the history of 

 mathematics. The reasons for the rejection of the classical theory are obvious and need 

 only be stated. Its retention in mathematical text-books is to be explained by the 

 fact that until recently no alternative method was available to give an account of 

 inductive reasoning. 



The method of maximum likelihood has no logical connection with inverse 

 probability, although it has been associated with it historically. Its derivation by this 

 path involves the introduction of arbitrary functions at two distinct stages, which 

 can be made to cancel each other. Likelihood is not a synonym for probability ; it 

 is a quantity, which, like probability, measures the degree of rational beUef , but it does 

 not obey the laws of probability. Statements about unknown samples of known 

 populations are made in terms of probability, statements about the unknown popula- 

 tions from which known samples are drawn are made in terms of likelihood. Likeli- 

 hood serves all the purposes necessary for the problem of statistical estimation. 



The invariant character of the percentile values does, however, make possible 

 certain statements in terms of probability respecting the values of the parameters of 

 populations. Statements of this type, which have very strangely been overlooked, are 

 available only when the observations are of quantitative variates, and not merely of 

 frequencies. They differ from the statements of inverse probability, both numerically 

 and logically ; the statements of inverse probability are absolute in the form, based on 

 a hypothetical super-population of an absolute character, but can never be verified, 

 for any further samples from the same population will alter the content of the state- 

 ments. The statements of the percentile method are relative in form and rigorously 

 demonstrable without any assumption as to the a priori distribution of the parameters. 



Dr. J. Wishart. — Combinatorial Methods in Problems of Sampling. 



In the statistical theory of sampling, we are concerned with finding the mean 

 values of powers and products of series of symmetric functions of the observations of 



