Duhamel's principle is for finding the solution of PDEs. Different stew altogether.

en.wikipedia.org

I looked Duhamel theorem in Widder. Seems like the limit of a Stieltjes integral with special case Bliss. I couid find no other source. Is is not subsumed under Stieltjes integration?

Here is someone asking about Bliss's theorem. The person responding is clueless:

Here someone has excerpted the theorem and outline proof of Bliss's theorem from some old book:

I just saw this from a book. What is the real life application of it?

math.stackexchange.com

Bliss's theorem is a special case of Duhamel's principle. The most recent book that has a formulation and proof of it is Taylor and Mann's

*Advanced Calculus* (3rd edition, 1983).

Books on differential geometry (Schaum,..) motivate arc length.

But don't provide any kind of proof. The problem is that you don't quite have a Riemann sum. If you had one, then as the norm partition goes to zero, you get a Riemann integral, which is the definition of arc length. You don't quite have a Riemann sum because for a curve in the plane you have two functions x(t) and y(t), and you are using the mean value theorem on both. The points t' and t" inside an arc segment may not (and usually will not) coincide, so you can't get a Riemann sum for the formula for arc length.[/quote]