that the nature of my said invention, and manner of performing the same, are particularly described and ascertained as follows; that is to say: In the first place my said invention is established upon a mathematical theorem, which may be enumerated in the words here underlined, namely: If two equal rings or circular lines in the same plane, or in planes parallel to each other, be conceived to revolve each upon its respective centre in its own plane, with one and the same uniform velocity, and in the same direction with regard to parts of the rings or lines alike situated, and any point be taken in one of the rings or lines, and a right line be drawn from that point parallel to a line supposed to join the centres, until it meets the other ring or circle, then I say the right line so drawn will be equal to the line of distance between the centres, and will continue equal and parallel to that line of distance during the whole of every revolution so made. And the demonstrative proof of the said theorem being very easily deducible by any person acquainted with the elementary principles of geometry, and not being needful to enable a workman to perform my said invention, I have not for those reasons considered it to be requisite that I should enlarge upon the same in this place.

Secondly, the circle a, c, e, g, and the circle 1, 2, 3, 4, in Fig. 1, (Plate II.) denote the rings or circular lines before-mentioned, and Y and X denote the centres thereof; and the line 1 a parallel, and equal to X Y, the line of distance of the centres will continue equal and parallel to that line of distance in the positions 2 c, and 3 e, and 4 g, and in all other positions into which the point 1 can be brought during the uniform, equal, and similarly-directed revolutions of the two circles.

Thirdly, if a wheel which I shall here call the pitch wheel be constructed, and here denoted by the circle 1, 2, 3, 4, Fig. 1, having four or any other number of paddles a b,

a c,

ac, fe, hg, capable of being moved or shifted in position upon the respective axes or centres thereof, 1, 2, 3, 4, and the extremities of the said paddles, a, c, e, g, or any arms or connecting pieces equal in length to X Y, be attached (under the conditions before set forth) to another wheel or revolving piece, which I shall here call the connection wheel or piece, it will be a necessary consequence that if one of the said wheels be made to revolve, the other wheel will also revolve in the same direction and with the same velocity, and the paddles will continue parallel to X Y, and to each other (excepting only in the case wherein the said cases or centres may be governed by arms or connecting pieces, and the paddles themselves be also capable of having their positions on the said axis or centres varied, and then each paddle will continue parallel to itself as it may have been so first posited); and fourthly, Fig. 2, numbers 1, 2, and 3, denote by views of the nature of two vertical sections, and one ground plan or section, the said pitch wheel and connection wheel or piece fitted up and connected with the paddles; and in all the drawings, the same letters and figures indicate the same parts or things. B denotes the axis or arbor of the pitch wheel, which wheel is or may be made double in order to support the axles of the paddles; and C denotes a circular fixed piece which serves as an arbor for the connecting wheel or piece to revolve upon by its smooth and fitted socket; and the said circular piece is by the construction eccentric with regard to the pitch wheel, and sufficiently large to allow the arbor B of the pitch wheel to pass through it without interfering with the motion of the connecting wheel or piece. And further, at A 1, A 2, A 3, and A 4, are seen the arms or connecting pieces always parallel to the line between the centres XY; but in such constructions of machinery

machinery as do not require the arbor B, or any arbor or axis for the pitch wheel to pass through, as here delineated, then instead of the piece C, may be substituted a simple pivot or any other kind of centre to govern the circular motion of the connecting circle or piece. And moreover, in such cases as may require that the position of all the paddles should be changed at the same time, with regard to surrounding fixed objects, by a motion equivalent to that of weathering or changing their obliquity, I do produce such and the said change by any well-known means of construction, by which the centre X can be made adjustable in position around, and preserving constantly the same distance from the centre Y, and I do accordingly change the position of the said centre X, and thereby also the obliquity of the said paddles, by virtue or reason of the constant condition, that the arms A 1, A2, A3, and A 4, will continue parallel to the line X, Y, joining the centres of the pitch wheel and connection wheel, in what manner soever the said line X Y may at any time be placed. And I the said Robertson Buchanan do further declare, as to the practice of my said invention in the impelling of vessels, boats, barges, or rafts, I do fit up the same with one or more wheels, having paddles and other apparatus as before described, and do cause the pitch wheels to revolve by the action of steam or any other suitable first mover, and thereby to strike the paddles duly into and against the water, so as by the usual re-action to produce a progressive motion of and in the said vessels, boats, barges, or rafts; and I do fix or adjust the centre X in such manner, that the paddles shall have their surfaces of action perpendicular to that of the water, or nearly so, namely, with the upper borders or edges thereof inclined forward more or less by trial and observations, as the

velocity