The functions in question are those which involve only the differences of their arguments ; and the object of this note is to suggest that an elementary discussion of these functions would be a valuable addition to the usual school course in algebra.

In his memoir on the Schwarzian derivative, Cayley gives the name ‘diaphoric’ to these functions (Art. 13). Invariants, seminvariants, covariants of quantics are diaphoric functions of the roots. Norms of complex numbers and ‘circulants’ are diaphoric functions, a particular case of which already appears in books on algebra under the properties of cube roots of unity.