1918-19.] A “Duplex” Form of Harmonic Synthetiser. 241 
Equation (10) may be made to furnish a very simple approximation 
for p as a function of e by the use of (6), 
0 22 
p = 0*32 + 
V( - e )' 
and from this the following!: table of values was obtained : — ■ 
Table IY. 
— e 
•005 
*004 
•003 
■002 
•001 
*005 
•0001 
V 
3*43 
3*80 
4-34 
5-24 
7-28 
10T6 
22-32 
Extreme Compactness of Duplex Unit. 
We shall proceed to exhibit the use of the foregoing formulae by 
taking a numerical example. 
Let it be required to find the smallest duplex unit which will describe a 
simple harmonic motion, having an amplitude of 2^ inches and an obliquity 
error nowhere exceeding *01 of an inch. 
Here 
greatest er r ° r _ *01 _qq 9 
twice the amplitude 5 
hence, from Table IY, p = 5*24, the unit of measurement being the length of 
the crank arm. 
Now we find from Table III that when p = 5*24, an arm of 1 inch gives 
an amplitude of 1*408 inches. Hence, as we require an amplitude of 
2*5 inches, the crank must be 2 5/1 408, i . e . 1*776 inches long. 
Hence, finally, p is 5*24 x 1*776 = 9 306 inches in length. 
Comparison with Crank and Wire Unit. 
The significance of the figures just obtained will best be exhibited by 
showing how very large the ordinary crank and wire unit of fig. 1 would 
have to be to produce an equally good result. 
We shall first find the angle at which the greatest error of the unit 
occurs, and thence the value of the error. 
Suppose that the crank length in fig. 3 is r , and that the error e and 
its maximum value e are defined as before. 
Then 
€ = 2r {v^ 2 + 7,2 “ 2ajr cos 0 — x + r cos 0 j- > 
dO 2 r ly^r + r 2 - 2a?r cos 0 J 
VOL. XXXIX. 
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