235 
1918-19.] A “ Duplex” Form of Harmonic Synthetiser. 
In the author s paper of 1906, however, the result of the mathematical 
discussion is to show that the effect of the “ geometrical error ” on the 
accuracy of the instrument is much less than would be supposed. 
But in the mathematics a simplifying assumption had to be made 
(that the diameters of the pulleys employed were negligibly small), and 
its presence introduces some uncertainty in the conclusion arrived at. 
The present paper was commenced solely with the intention of getting 
rid of this assumption; but in the course of the work the fortunate 
discovery was made that a “ duplex ” disposition of the pulleys of the 
apparatus — entailing no addition of parts, but rearrangement only — 
would very nearly annul the error altogether. This result makes so great 
a change in the mathematical reasoning that it is better presented afresh, 
without reference to that given in the former paper. 
Definition of “Error.” 
Any harmonic synthetiser consists of a combination of mechanical units, 
each giving rise to a linear motion which is exactly, or very approxi- 
mately, simple harmonic. In the latter case, the measure of the error e is 
defined as “the displacement of the actual position of the moving part 
from its theoretical position, divided by the total range of movement.” 
We shall lead up to the Duplex Unit by considering as a simple 
example the “ Crank and Wire” Unit (fig. 1) employed D 
by Lord Kelvin. In all that follows, it is evident that 
without loss of generality the crank radius may be 
taken as unity. 
In fig. 3 let CA be a crank revolving about its 
centre C, and AB a cord passing round a pin at B, 
its end D being supposed to move in the straight line 
BD. The error of the movement of D, regarded as an approximation to 
S.H.M., is 
e = \{J(V + 1 ~ cos 0) + cos 6 - x) . . . (1) 
As an example the following table gives some corresponding values of 
6 and e. The calculation is best made by the more convenient formula 
(1. 
1 - cos 20\ 
-(-■ 
1 - cos 20V 
1 
\8 
x - cos 0 J 
\8 
cc — cos 0 ) 
x — cos 0 
the approximation of which is amply sufficient. The value 5 J 2 has been 
chosen for x to enable a comparison to be made later on with a duplex 
unit of the same dimensions (the case in which p — 5). 
