  This paper explores some previously-unrecognized consequences of Lerch's 1905
formula for the Fermat quotient, with special attention to the sums which he
introduced in this context. A generalization of his result is proved, and a new
proof given of a sharpened result by Skula (2008). We also sharpen the criteria
given by Emma Lehmer in 1938 for a Wieferich prime to be simultaneously a
Mirimanoff prime.
