﻿new Harmonic Analyser. 113 



the curve is drawn rotate about an axis perpendicular to that 

 of the cylinder, and thus avoid the simple harmonic motion, 

 which is always a drawback, as it introduces a great deal of 

 friction. Both instruments are also large and heavy, practi- 

 cally fixtures in the room where they are used. 



§3. Clifford has given a beautiful graphical representation 

 of Fourier's Series, which I knew more fully from personal 

 communication than from the short paper published in 

 vol. v. of the Proceedings of the Lond. Math. Soc. 



His result may be stated thus : — " If the curve to be 

 analysed be stretched out in the direction of the x to n times 

 its base without altering the y, and then wrapped round a 

 cylinder with circumference c so that it goes n times round, 

 then the orthogonal projection of this curve on that meridian 

 plane which passes through the zero-point of the curve will 

 enclose an area which is proportional to the coefficient B a . 

 In the same way A w is got by aid of a plane perpendicular to 

 the first." 



It was this theorem which led me to the construction of an 

 Harmonic Analyser. It can easily be put in the following 

 form. Suppose the cylinder placed with its axis horizontal 

 and the tangent plane to its upper edge drawn. This edge 

 cuts the curve in n points. Let P be one of them. If now 

 the cylinder be turned, and if at the same time the tangent 

 plane be moved in its own plane in a direction perpendicular 

 to the edge of contact, the point P will trace a curve on it. 

 This plane will be the same as Clifford's curve in case the 

 motion of the tangent plane is simply harmonic, completing 

 one period for each rotation of the cylinder. The curve will 

 be completed, after n rotations of the cylinder. 



The same curve will be traced if the original, unstretched, 

 curve is wrapped (once) round the cylinder, whilst the tangent 

 plane completes n periods of its simple harmonic motions for 

 one revolution of the cylinder. 



We thus get in a fixed plane a curve whose area equals, in 

 some unit, the coefficients A„ or B n , and this area can be 

 determined by an ordinary planimeter. The curve, of course, 

 need not be drawn out, as long as the tracer of the plani- 

 meter is always at the point P it will describe the curve. 



This can easily be realized. A flat board, whose upper 

 surface forms a platform on which the planimeter can rest, 

 is placed by the side of the cylinder so that its upper sur- 

 face lies in the tangent plane. A straight-edge is fixed 

 above the upper edge of the cylinder. The tracer of the 

 planimeter is pressed against it and made to follow the point 

 P on the curve. After a complete revolution of the cylinder, 



Phil. Mag. S. 5. Vol. 38. No. 230. July 1894. I 



