﻿Paper on a new Harmonic Analyser. 127 



troublesome, but the mathematics of link motions and radial 

 valve-gears become very simple when we consider, not merely 

 the fundamental simple harmonic motion, which is all that is 

 usually studied, but the octave, which is found to help or hurt 

 in the various forms. 



I was first attracted to this subject when studying the 

 beautiful but little-known valve-motion invented long ago by 

 Sir F. Bramwell, in which the only overtone is three times 

 the fundamental. 



Given any function completely, we can by a numerical 

 method, and with as much accuracy as we please, develop it 

 in Fourier's Series. In the < Electrician ' of Feb. 5th, 1892, I 

 published the numerical work of one example calculating from 

 23 ordinates. In the sheet which I here exhibit one of my 

 students, Mr. Fox, has done the same work by a graphical 

 method. Probably he is the very first to carry out the idea 

 of the late Prof. Clifford by descriptive geometry*. That is, 

 we have imagined the curve to be wrapped round the cylinder, 

 and it was surprising to find how rapidly its projections could be 

 drawn upon the two planes and their areas obtained by the 

 planimeter. We then imagined the curve to be wrapped 

 twice round and the projections drawn and their areas taken. 

 I wish I had time to dwell upon the interesting problems that 

 arose during the work, for example as to whether the area was 

 to be taken as positive or negative. However many loops 

 such a figure may possess, the well-known rule for autotomic 

 plane circuits (Thomson andTait's i Elements,' §445) is really 

 attended to by the planimeter. The direction of motion of 



* Note added May 29M. — The descriptive geometry method is fairly 

 quick, and may be made as accurate as one pleases, but of course it 

 cannot compare in quickness with the Henrici Analyser. 



It is obvious that by properly shaping one's cylinder, wrapping the 

 curve round it, and then finding the area of it, projected on a plane 

 parallel to the axis, one may develop an arbitrary function in a series of 

 any normal forms. Thus if Q(#) is any tabulated function of x, and y is 



the arbitrary function of x, and we wish to find the integral yy . Q(x) . dx 7 



the shape of the curve which must be used instead of a circle in the 

 Clifford construction is easy to find. It must be such that the cosine of 

 the angle which the short length bx of the curve makes with the trace of 

 the plane on which the projection is to take place shall be proportional 

 to Q(x), and several easy methods of drawing the curve or a series of 

 such curves may be found. Once found, there is no more difficulty in 

 developing any new arbitrary function in any series of normal forms 

 than Mr. Fox found with his Fourier Series. A series of curves will be 

 needed for a development in Zonal Harmonics, but only one curve will 

 be needed for the Zeroth Bessels. These curves, or shapes of sections of 

 cylinders, I am now proceeding to draw on a sufficiently large scale for 

 exact work. 



