Some years ago, before the enactment of the Ten Hours Factory Bill, I had the honour to be engaged by the Earl of Shaftesbury (then Lord Ashley) on an investigation of the average distances traversed daily by children and others employed as “piecers” in the cotton mills about Manchester and the surrounding districts. After taking the various dimensions of the mules or spinningjennies, and making the requisite observations as to the number of strokes of each machine per minute, and the average number of breakages of the threads at each stroke, I found that the necessary calculation involved a preliminary discussion of a somewhat curious mathematical problem.