  In this work, we propose a double-series representation of the difference
between two dilogarithms with specific arguments. The summation is derived
combining a formula we recently found for the so-called Grotendieck-Krivine
constant, and an identity obtained by Lima for the difference of two
dilogarithms with arguments $\sqrt{2}-1$ and $1-\sqrt{2}$ respectively. We also
give an integral representation of the difference considered by Lima, on the
basis of a formula published by Brychkov.
