The story
Bhagīratha wanted the ashes of his ancestors washed by the Gaṅgā, and she was in heaven. He stood on one foot for a thousand years to bring her down, and Śiva caught her in his hair on the way so the fall would not split the earth.
Daśaharā means 'that which takes away ten' — ten kinds of wrong, three of the body, four of speech, three of the mind. Ten is the day's number throughout: ten dips, ten lamps, ten of each offering.
It is kept as the river's own birthday, and the Gaṅgā is addressed as a mother rather than as a place.