122 Professors Perry and Ayrton on the Music 



whereas at any other place it is such that, acting like a cam 

 or tappet, it gives pure harmonic motion to a slider kept press- 

 ing on the roller, and only capable of radial motion. Such a 

 sliding piece, resting any where between K and C, receives one 

 complete harmonic motion during one revolution of the roller ; 

 if it rests anywhere between G and K, it completes two pure 

 harmonic motions during a revolution, three between J and 

 G, four between F and J, five between H and F, and six 

 between B and H. The amplitude gradually decreases as the 

 sliding piece is made to rest on places nearer the circular sec- 

 tion, where of course there is no up and down motion of the 

 sliding piece. 



Every section, therefore, of the roller has a curved outline, 

 of which the construction is easy. Thus suppose we want the 

 section which will give a slider five complete harmonic swings 

 in one revolution of the roller. Make the angle A B (fig. 2) 

 equal to one fifth of four right angles ; describe the circular 

 arc A B with as centre and radius A equal to R + r, 

 where R is the radius of each of the three circular parts of the 

 roller H, J, K, and r the radius of the small friction-wheel on 

 the end of the sliding piece ; make B D equal to half the 

 maximum swing we wish the slider to receive. Then with 

 centre B and radius B D describe the semicircle 159, which 

 divide into any number of equal parts (eight in the figure), and 

 let fall perpendiculars 22, 33, 44, &c. from the points of sub- 

 division on to the diameter D B 9, meeting it in the numbered 

 points. With as centre, describe a circular arc through 

 each of these numbered points in D B. Divide the angle 

 A B into twice the number of equal parts into which the 

 semicircle 159 was divided. Then draw a curve 1 C D through 

 the intersections of the first arc and first radius, the second 

 arc and second radius, &c. Finally, draw a great many equal 

 circles with radius r, the centres being in the curve ; then the 

 envelope DEF (fig. 3) is one fifth of the whole curved sec- 

 tion we wish the curved roller to possess ; and a template of 

 tin may be made to be used in the construction of the roller. 

 It will be found advisable to construct four templates for each 

 division of the roller, since, although a section of the har- 

 monic surface described by the centre of the little friction- 

 wheel formed by a plane passing through the axis of the large 

 roller in a straight line, such a section of the actual surface on 

 which the little wheel rests will be curved in consequence of 

 r, its radius, not being infinitely small. Our large roller was 

 made of hard wood ; but it would have been much better if it 

 had been made of cast iron or steel, since when of wood it, as 

 well as the little friction-wheel of the sliders, must be made 

 large to avoid abrasion ; but even when these are large it is 



