  We find convergent double series expansions for Legendre's third incomplete
elliptic integral valid in overlapping subdomains of the unit square. Truncated
expansions provide asymptotic approximations in the neighbourhood of the
logarithmic singularity $(1,1)$ if one of the variables approaches this point
faster than the other. Each approximation is accompanied by an error bound. For
a curve with an arbitrary slope at $(1,1)$ our expansions can be rearranged
into asymptotic expansions depending on a point on the curve. For reader's
convenience we give some numeric examples and explicit expressions for
low-order approximations.
