Sheppard's correction formulae in several variables

1. A simplified method for the deduction of Sheppard's correction formulae for ordinary and factorial moments, and for semi-invariants, together with more general results than those previously known, especially with regard to the remainder term, has recently been given 1 . The method is based upon a very simple application of Euler—MacLaurin's formula, and the usual corrections are thereby shown to be valid for very general frequency curves, even for those without high contact in one or both directions. The note referred to deals with the case of one variable; the use of the method employed there can, however, easily be extended to an arbitrary number of variables, which extension it is the purpose of this paper to carry through. For the sake of formal simplicity the deductions are performed with two variables; the results can then immediately be generalized.