98 
Proceedings of Royal Society of Edinburgh. [sess. 
ordinates has been taken, an analysis carried to one-half that 
number of partials, or somewhat less, suffices to define the curve. 
A curve may he reconstituted graphically from its analysis, hut 
the process is laborious when the partials are many. Donkin 
invented an instrument called the harmonograph, which suffices to 
describe a curve compounded of any two harmonic curves. The 
principle of this machine could probably be extended so as to 
describe the curve represented by any given Bourierian analysis, 
each partial being set going in the given amplitude and at the 
given phase. This curve could be at once compared with the 
original, and their difference, if any, could be seen and analysed. 
Broadly speaking, the loudness of subjective impression varies with 
the amplitude of the vibrations causing it. But there are known 
exceptions, e.g ., tone is produced by beats, though, if the beat- 
curve were analysed, it is manifest that this tone would come out 
with an amplitude of zero. Hermann affirms* that not only beats, 
but every thing periodic in a phonogram, is heard as tone of 
the given period or pitch. It is certain, at any rate, that, in the 
case of a vocal phonogram, the tone on which it is sung may be 
heard from it distinctly, and yet the amplitude of that particular 
tone, the fundamental, may be almost zero in the analysis. But 
* See list of references at end of this paper. 
