Motivations for the term "jet" in the context of viscosity solutions for fully nonlinear PDE My question is very direct:

What are the motivations for the name "jet"(subjet, superjet) in the context of viscosity solutions for second order fully nonlinear elliptic PDE?

The definition of which can be seen in Crandall, Ishii, Lions:
Crandall, Michael G.; Ishii, Hitoshi; Lions, Pierre-Louis, User’s guide to viscosity solutions of second order partial differential equations, Bull. Am. Math. Soc., New Ser. 27, No. 1, 1-67 (1992). ZBL0755.35015.
see also https://arxiv.org/pdf/math/9207212.pdf.
 A: The machinery of Jets were introduced by Ehresmann as an geometrically invariant way of describing PDEs and their solutions in analogy to the way that vector fields and integral curves invariantly describe ODEs and their solutions.
For example, there is a way of geometrically describing symmetries here and so Noether's theorem.
It's worth noting that sprays describe 2nd order ODEs and are a sort of midway house. It comes from iterating tangent bundles - but this doesn't appear to generalise well - at least I haven't seen any generalisations of such, except in passing. Whereas the notion of a jet generalises the equivalence class of curves of a certain tangent order.
A: 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 kth order contact
A: Strictly speaking, he answer is that there is no motivation for the name "jet" in the context of viscosity solutions for second order fully nonlinear elliptic PDE, because it was initially introduced in the more basic framework of differential calculus/geometry.
Still one can wonder about when and where it was introduced. I suspected it was initially introduced in French and possibly by Bourbaki, and asked J-P. Serre in an email, and got the answer:

Il me semble que la notion de jet, et la terminologie
correspondante, sont dus à Ehresmann (mais pas
à Bourbaki), vers 1950. Je vous envoie ci-joint un article de 1961 qui
confirme ceci.

(It seems to be that the notion of jet, and the corresponding terminology, are due to Ehresmann (but not Bourbaki) around 1950. Here is an article from 1961 confirming this.)
The article is Une généralisation du calcul des jets et quelques prolongements généralisés de variétés différentiables (DOI link linking to ProjectEuclid, no paywall) by M. Kawaguchi. Its first two sentences are:

Le mot "calcul des jets" a fait son début dans l'article
[1] de Ch. Ehresmann. En 1951, Ch. Ehresmann s'est intéressé surtout aux
fondements de la geometrie differentielle. [1] Les prolongements d'une variété différentiable, Atti IV Congresso Unione mat.
Italiana, Taormina Ott, (1951), 1–9.

(The phrase "jet calculus" first appeared in the article [1] by Ch. Ehresmann. In 1951, Ch. Ehresmann was especially interested in foundations of differential geometry.)

I should add that in French "jet" has nothing to do with plane or engine.
"Jet" is the substantive of "jeter" = throw, launch. I can see the translations: spurt, jet, squirt, throw, spray. Also a derived phrase is "premier jet" = "first draft".
