A Magic Square (or is it?)

Sorry for taking so long but I've come up with a puzzle that I'm not sure is solvable but maybe you can help me with it?

Okay, at least a hundred of us know the traditional magic square right? You just arrange numbers in a square grid so that each cell contains one number and each row, column and diagonal of the grid has numbers that add up to the same total. But what we take a 5x5 grid and remove the middle cell? Now here's the question,

Is it possible to arrange all the numbers 1 to 24 (no decimals of course) so that:

1) Each cell has only one number inside it

2) Every straight line of 5 cells has numbers that add up to the same total

If it's possible, how is it done? And if no such grid exists, why?

(If you're ever feeling confused about rule 2, just refer to the image below this sentence.) Such a square is fairly easy to construct:

Start with any 5x5 magic square (actually only the rows/columns need to be equal, the diagonals can be ignored).
Permute the rows so that the cell containing 25 is in the middle row.
Permute the columns so that the cell containing 25 is in the middle column.
Remove the centre cell which now contains 25.

If we use the given magic square, we only need to swap row 3 and row 5 to put the 25 in the centre, resulting in this square:

17 24  1  8 15
23  5  7 14 16
11 18 xx  2  9
10 12 19 21  3
4  6 13 20 22