is O, they are always at the same distance from that point; consequently they will continually vary their distance from the other centre P. Thus, when a paddle board arrives at that point of its revolution at which the centre P lies precisely between it and the centre O, its distance from P is less than in any other position. As it departs from that point, its distance from the centre P gradually increases until it arrives at the opposite point of its revolution, where the centre O is exactly between it and the centre P; then the distance of the paddle board from the centre P is greatest. This constant change of distance between each paddle board and the centre P is accommodated by the variation of the angle between the radial arm L and the short paddle board arm K; as the paddle board approaches the centre P this gradually diminishes; and as the distance of the paddle board from P increases, the angle is likewise augmented. This change in the magnitude of the angle, which thus accommodates the varying position of the paddle board with respect to the centre P, will be observed in the figure. The paddle board D is nearest to P; and it will be observed that the angle contained between L and K is there very acute; at E the angle between L and K increases, but is still acute; at F it increases to a right angle; at G it becomes obtuse; and at K, where it is most distant from the centre P, it becomes most obtuse. It again diminishes at I, and becomes a right angle between A and B. Now this continual shifting of the direction of the short arm K is necessarily accompanied by an equivalent change of position in the paddle board to which it is attached; and the position of the second centre P is, or may be, so adjusted that this paddle board, as it enters the water and emerges from it, shall be such as shall be most advantageous for propelling the vessel, and therefore attended with less of that vibration which arises chiefly from the alternate depression and elevation of the water, owing to the oblique action of the paddle boards.[41] [Footnote 41: A paddle-wheel resembling this has lately been constructed by Messrs. Seawards. It has been charged by Mr. Morgan as being a colourable invasion of his patent, and the dispute has been brought into the courts of law.]