> What was coax cable invented for?
Underground power distribution.
To make money in the electric racket, you have to deliver large energy to customers (mostly industrial at first) without killing anybody along the way.
Overhead distribution is obvious, but ugly and still dangerous. The Great Blizzard of 1888(?) in NYC, some people were snowed-in for a week, and a large part of the problem was the maze of telegraph and telephone wires in the streets after the poles collapsed from ice. Edison's early attempts to distribute electric used underground vaults, like small subway tubes with wires strung on the walls, very expensive.
Siemens was watching insulation technology (before good insulation, there was little alternative to bare wires on glass knobs). He could make an insulated cable, but proximity to ground would suck a lot of power. And there is the backhoe effect (even before backhoes were common): if you lay a wire underground, a backhoe WILL find and cut it. So high-power electric underground was considered lossy and dangerous.
So he made a (single-phase) cable with the hot lead in the center and the grounded lead as a shield around the outside. Ground-loss was small. He rigged a run from his generator to a big factory, showed it powering huge motors. Then, in public demonstration, with part of the cable left above ground, he took his coat off, picked up an axe, and said he would cut the cable with full power flowing. After a moment of thought, some people fainted. Nevertheless, he swung and cut the cable. He was not harmed. The axe blade was wedged in firm contact with the grounded shield before it hit the center hot conductor. The generators shut-down as, of course, they have to do when bad things happen.
That may not be the first use of coax. But I like the drama even more than Westinghouse's streetcars.
If you want an exact citation I'll have to go into the bowels of the library.