for the Composition of two Harmonic Curves. 225 



into those of a pinion, P Q, alongside 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 commu- 

 nicated by a connecting-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 

 intermediate one, it is plain that, upon turning either of the spin- 

 dles b} r 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 composed 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 



y = sin mx + sin nx. ■ ■ 



This equation implies not only that the radii of the cranks are 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, however, be altered ; and therefore the general form 

 of equation to the curves which the machine can draw will be 



y= a sin (mx -f a ) -f- b sin (nx + /3), 

 where a and b are the radii of the cranks, and a and (3 are depen- 

 dent on their relative inclinations to the above-mentioned vertical 

 plane at starting. 



As an example, suppose that a—b, while the ratio of m to n is as 

 2 to 1 ; then the above equation will represent the curve of pressure 

 for the octave. Similarly, if m is to n as 16 to 15, the resultant 



* 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 connecting-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. 



Phil Mag. S. 4. Vol. 48. No. 317. Sept. 1874. Q 



