1918-19.] A “Duplex” Form of Harmonic Synthetiser. 237 
Of course such a pair constitutes only one harmonic element of the 
complete apparatus, and in practice the wire, after passing over A, does 
not terminate, but is led down to the crank pin of the next pair, and so on. 
Fig. 6 (a). 
In each of the above figures the centres of the two top pulleys occupy the positions 
of the pins A and B in fig. 5 ; P, the centre of the other pulley, that of the 
crank pin P in fig. 5. 
Also the length of stroke yielded by the duplex unit is not materially 
inferior to that of a simple unit, as will be shown later. 
Device for Eliminating the Diameter of the Pulleys from the 
Mathematical Analysis. 
By merely arranging the wire round the pulleys in the manner shown 
in figs. 6 (a) or ( b ) their diameters do not 
need to be considered, and the mathematical 
theory which regards pulleys as points 
is no longer an approximation, but is 
exact. For it will be seen that in each 
case the total length of the wire from A 
to B is equal to its length in fig. 5 from 
A to B, plus a constant length equal to the 
circumference of a pulley. Probably the 
arrangement shown in (a) is not so good 
as that in ( b ) because it involves a crossed 
belt. 
Expression for the Error. 
Fig. 7 shows a duplex harmonic unit 
having a pen P attached to the free end of 
the wire ABCP, the pen being constrained 
to move in a vertical line. To investigate 
its error of motion it is convenient to take 
our fiducial point Q such that CQ = ABCP, for then QP = AB + BC. 
Fig. 7. — DA = D0 = DC=p, and 0B = 1. 
