In Crelle's Journal for 1844, at the end of a paper on Cubic Forms, Eisenstein gives the following results:—

where m is any odd number and ρ a primitive root of the equation zm = 1;

where

No demonstrations are given of these identities; but they are said by the author to be only particular cases of a very general equation, the complete theory connected with which he hoped to give on another occasion.

At p. 193 of the same volume of Crelle he returns to the subject, not however for the purpose of giving the promised theory, but to add several other results similar to those before given.