﻿an Harmonic Analyser. 123 



In the Fourier expansion 



y=f(x)=A + A l smx + A 2 sm 2x + . . +A n smnx+ "| ,^ 

 + B x cos x + B 2 cos 2x + . . + B n cos nx + . , J ' 



A.= -l ?/sin#d.r== - |cos.zdy= OR') 



B, = -i ycosxdx= - \sin xdy = RR' I 



Also/(#) may be expanded in the form 

 Ao+Cxsin (x- aO + Cjsin (2ar— « 2 ) + • • C n sin (n^-aj, (3) 

 A , B , C , and a being connected by the relations 



A„=C n cos a „l 

 -B B =C„ S in a „/' • • • • (4a > 

 or 



On = A n + JJ, 



— ! 



tan 



From (2) and (4b) it is evident that OR (fig. 2) is equal to 

 7t0,, and the angle Y'OR is equal to « L . 



If now the axis of the rolling wheel W makes n turns 

 while the tracer P moves over one complete period of the 

 curve (fig. 1) , the corresponding values of OR and the angle 

 Y'OR will be nirQ n and a n respectively. 



Various arrangements of mechanism are suggested for 

 connecting the rolling wheel with the tracer so as to satisfy 

 the above conditions ; the following seems the most suitable, 

 as it can be adapted for an instrument to give more than one 

 simple harmonic constituent term for one tracing of the 

 curve. 



The motion of the rolling wheel relative to the paper is 

 compounded of two simple movements : — (a) a pure rolling, 

 the distance rolled being equal to dy the element described 

 by the tracer P ; (b) a motion of rotation, the point of contact 

 of the wheel with the paper being the centre of rotation, and 

 the angle turned through from the initial line being propor- 

 tional to x the abscissa of P. The relative motion will, 

 therefore, be the same if the wheel be rolled along a straight 

 line fixed in the instrument, while the paper is made to turn, 

 the centre of rotation of the paper being the point of contact 

 of the wheel with it which is continually varying in position. 

 Fig. 1 represents diagram matically the mechanism. The curve 



