The disc engine
A steam engine could have been the answer. But the massive beam engines of the day were expensive to install and so inefficient that their running costs were enormous. Moreover they were so slow and prone to breaking the threads being processed, most mill owners contented themselves with using steam engines as pumps, lifting the water from the tail race of a wheel and returning it to the top.
There had been a partly completed steam engine at the mill and this was advertised in the sale advertisements. Prospective purchasers were assured that it could be completed for £150. James and Edward either rejected it at the outset or abandoned it later. No record of its ever being in service has been found. So steam was not regarded as a solution.
Ideally they wanted to get more power from the brook but this must be done without sacrificing the flow over the triple wheels. By the end of the 18th century there were fountains in some formal gardens including nearby Chatsworth, so the brothers would be familiar with the concept of water pressure. What nobody had done was to use its energy to drive machinery. But, they reasoned, if they could design a suitable engine and install it above the height of the Regulator, they could put the used water into the dam without loss to the wheels. Getting a head of water was simple. All they needed to do was go higher up the valley, build a dam and run a pipe down to their yet non-existent machine. All quite simple.
Diagrammatic section of disc engine
Essentially the engine had two components. There was a sphere (pale green) split across the centre line with a disc (dark green) fitted between the two halves. A drive shaft (grey) extended vertically from the top of the sphere. The sphere was mounted in a casing (red). It was free to tilt in any direction, but the tilt angle was limited by the design of the casing. The interior of the casing (not shown) fitted tightly around the sphere/disc envelope. Across the casing there was a radial web (blue) and the disc had a slot to accommodate it. The web therefore prevented the disc rotating.
(Click to enlarge)