  We obtain a formula for the number of genus one curves with a variable
complex structure of a given degree on a del-Pezzo surface that pass through an
appropriate number of generic points of the surface. This is done using
Getzler's relationship among cohomology classes of certain codimension 2 cycles
in $\overline{M}_{1,4}$ and recursively computing the genus-one Gromov-Witten
invariants of del Pezzo surfaces. Using completely different methods, this
problem has been solved earlier by Bertram and Abramovich, Ravi Vakil, Dubrovin
and Zhang and more recently using Tropical geometric methods by M. Shoval and
E. Shustin. We also subject our formula to several low degree checks and
compare them to the numbers obtained by the earlier authors.
