OK, I've been doing this problem for fun (it's a great problem, BTW!): http://www.physics.harvard.edu/academics/undergrad/probweek/prob76.pdf
Here is the solution: http://www.physics.harvard.edu/academics/undergrad/probweek/sol76.pdf
However, I'm having a problem understanding the solution. How does the first line lead to the second line in the simplification of $F(t + dt)$? It looks like the author has taken $Vdt$ to be zero to get $x/\ell$ but has left in $Vdt$ for the second term in line two. Is that legal?