  We present a fast method for evaluating expressions of the form $$ u_j =
\sum_{i = 1,i \not = j}^n \frac{\alpha_i}{x_i - x_j}, \quad \text{for} \quad j
= 1,\ldots,n, $$ where $\alpha_i$ are real numbers, and $x_i$ are points in a
compact interval of $\mathbb{R}$. This expression can be viewed as representing
the electrostatic potential generated by charges on a line in $\mathbb{R}^3$.
While fast algorithms for computing the electrostatic potential of general
distributions of charges in $\mathbb{R}^3$ exist, in a number of situations in
computational physics it is useful to have a simple and extremely fast method
for evaluating the potential of charges on a line; we present such a method in
this paper, and report numerical results for several examples.
