This paper is concerned with the application of symmetry principles to the reduction of terms in Fourier synthesis. For this purpose two theorems are first given, the reciprocal symmetry theorem and the additivity theorem. To illustrate the use of these in performing summations for symmetrical crystals, the form of the electron density function is derived for Fourier syntheses of projections corresponding to the 17 plane groups. The extension of the method to three-dimensional syntheses is also discussed in sufficient detail so that it can be applied to any space group.

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