198 Mr. A. E. Donkin on an Instrument for the [Feb. 19, 



number of teeth, the relative angular velocities of the spindles can be regu- 

 lated at pleasure. The paper upon which the curve is to be drawn is carried 

 upon a rectangular frame, E F Gr H, capable of sliding horizontally up and 

 down in a direction parallel to that of the plane passing through the spin- 

 dles. This frame has a pair of rollers, E F and G- H, at each end connected 

 by tape-bands, between which the paper passes as the rollers turn. In order 

 to give a motion of revolution to the rollers, a wheel, L, is fixed upon the 

 axis of one of them whose teeth gear into those of a pinion, P Q, along- 

 side which the frame slides, and which is itself driven by one of the 

 vertical spindles. A connecting-rod, D M, is carried to the frame from the 

 crank of this spindle, so that upon turning the latter a vibratory motion 

 is given to the former ; and since the transverse motion of the paper also 

 depends upon the same spindle, a fixed pencil-point resting on it would 

 draw a simple harmonic curve whose amplitude would depend on the 

 radius of the crank, and wave-length on the transverse speed of the paper, 

 which can be regulated at pleasure by means contrived for the purpose*. 



A vibratory motion similar and parallel to that of the frame is given 

 to a small tubular glass pen, E-, so arranged as to move with its point 

 lightly resting upon the paper. This motion is communicated by a con- 

 necting-rod, C N, from the other crank, which is carried underneath the 

 sliding-frame and jointed to the lower end of a small vertical lever, S, to 

 whose upper end the arm carrying the pen is attached. 



The weight W serves to regulate the pressure of the pen on the paper, 

 as it can be screwed in or out. T is merely a pillar upon which the 

 change-wheels can be placed for convenience. 



If the pair of wheels on the spindles are now connected by the inter- 

 mediate one, it is plain that, upon turning either of the spindles by a 

 winch provided for the purpose, the two motions of the paper will be 

 combined with that of the pen, and the curve drawn will be that com- 

 posed of, the two simple harmonic ones, which would be the result of 

 separately combining the harmonic vibrations due to each crank with the 

 transverse motion of the paper. Thus if m and n are the numbers of teeth on 

 the pair of wheels respectively, the equation to the resultant curve will be 



2/=sin wo? + sin nx. 



This equation implies that not only are the radii of the cranks the same, 

 but also that they start parallel to each other and at right angles to the 

 vertical plane passing through their axes : both these conditions can, how- 

 ever, be altered ; and therefore the general form of equation to the curves 

 which the machine can draw will be 



y=a sn m#--a sn 



* It should be observed here that the vibratory motion thus given to the frame is not 

 truly harmonic. In order to make it so, a more complicated contrivance than the simple 

 crank and connecting-rod would have to be adopted ; but this would probably introduce, 

 through unavoidable play, an error greater than the present one, the length of the con- 

 necting-rods and the small size of the cranks rendering the latter nearly inappreciable. The 

 motion will, however, for the sake of convenience, be considered truly harmonic throughout 



