Exact conditional distribution of a three-phase invariant in the space group P1. I. Derivation and simplification of the Fourier series

An exact expression is derived for the conditional probability density function of a three-phase invariant and the general result is applied to the space group P1. The expression for the conditional density is given in terms of a sixfold Fourier series. A straightforward numerical evaluation of this series, without further analysis, is extremely time consuming. Much of the present paper is therefore devoted to an exposition of symmetries hidden in the various summations. The computational effort required for the evaluation of the above expression is thereby reduced to manageable proportions in a number of interesting cases. Results of numerical computations of the exact conditional density are given in the second paper in this series. It is also shown that the exact expression for the conditional probability reduces to that given by Cochran [Acta Cryst. (1955), 8, 473-478].