Well, if I've understood your problem correctly, this is what you're asking:

You're standing at the North Pole. You travel out horizontally a distance of I miles, where "horizontally" is defined as "parallel to the equator", and drop down vertically until you reach the earth (where "vertically" is defined as "parallel to the earth's axis").

Max is correct - this only makes sense if I is less than or equal to the earth's radius r. If I > r, then you'll miss the earth entirely and shoot off into infinity.

It's a fairly straightforward problem, although I don't see any useful application for it. Using the pythagorean formula, the answer is

drop = r - sqrt(r^2 - I^2)