What is the origin of the terms 'jet' and 'prolongation' in differential geometry? I am just curious what is the reason for the terms 'jet' and 'prolongation' in differential geometry? Is there some mental imagery that these names are supposed to evoke? Or are they so-named because of some particular example that was later generalized? Or perhaps these names are completely arbitrary?

As I understand it, the word "jet" is meant to evoke the idea of a "spray" of curves through a point, or more accurately, their equivalence classes up to $k$th order contact.

Consider this section of the Wikipedia entry on jets#Jets_of_functions_from_the_real_line_to_a_manifold):

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(The jet of a map between manifolds is then defined in terms of the jets of curves.)

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