216 
Proceedings of Boy al Society of Edinburgh. [sess. 
The whole machine runs without attention, and can safely he 
left to itself in cases where for any reason a prolonged tracing is 
required — as for example when the constituent S.H.Ms. have been 
set to incommensurable periodic times ; in which case of course 
the resulting trace never quite repeats itself, but is always taking 
new forms. [A machine was constructed by the author in which 
pendulums were employed to give the S.H.Ms. ; but although the 
machine was satisfactory in many respects, it was abandoned, 
chiefly because of the fact that the pendulums necessarily ran 
down, and so rendered the instrument unsuitable for any sort of 
prolonged work.] 
As a further example of work done by the instrument, and to 
show the interesting character of the composite curves, the traces 
shown in fig. 5 have been reproduced. 
§ 6. Mathematical Investigation of the Deviation from its 
True Position of the Pen of an Instrument on the above 
Principle. 
We now come to discuss the most important consideration in 
connection with such an instrument as the present — its accuracy. 
It hardly needs to be pointed out that a principle of construction 
possessed of great merits from the point of view of simplicity, 
smoothness of working, and so on, would nevertheless be of little 
value were the curves produced by the apparatus embodying it 
liable to appreciable errors of form. After being satisfied by 
actual trial that any suggested design of synthetiser is satisfactory 
on the mechanical side, there must always remain the essential 
question : “ In view of the principle adopted, what will be the 
accuracy of the trace produced ? ” 
Accordingly we must now go on to discuss the pretensions of the 
present apparatus from this point of view. 
It is a nice question whether in such instruments it is more 
important to minimise the greatest deviation or to minimise the 
average deviation. If the data of the resultant curve are to be 
deduced from measurements of it at a number of different places, 
then it would be desirable to have the average deviation as small 
as possible ; but if some single property of the resultant curve is 
to be ascertained, e.g. the point at which it cuts the time axis, 
