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1. TheProblem .—When we have a set of quantitiesu 1 ,u 2 , ...um , which may be regarded as the observed values of certain unknown quantities U1 , U2 , ... Um , and we form the hypothesis that these U’s can be represented by a formula which either is completely determined bya priori considerations or involves constants which have to be determined from the data, how are we to decide whether the hypothesis is justified ? The classic method of testing a hypothesis of this kind is Prof. Karl Pearson’s "χ2 " method, published in 1900 (ref. 6). But in one important class of cases the accuracy of his formula is open to doubt. The problem has not yet been finally solved: the object of the present paper is to take a step towards its solution.

III. The fit of a formula for discrepant observations