The "London, Leeds, Wakefield, Barnsley, Sheffield and Mansfield MACHINE"

The "Machine" was a stagecoach "Intirely New, Hanging on Steel Springs". Considering how rough the roads were at this time, the addition of steel springs must have been a major attraction, and might have been the reason why the operators characterized their stagecoach as a "machine". 

At the foot of the poster, there is the following:

"N. B. The Proprietors will not be answwerable for Gold, Silver, Plate, Jewels, &c. except entered as such and paid for accordingly."

Widely used before passenger railroads were available, stagecoaches made long scheduled trips using stage stations or posts where the stagecoach's horses would be replaced by fresh horses. The business of running stagecoaches or the act of journeying in them was known as staging.