1905-6.] A New Form of Harmonic SyntKetiser. 231 
properties, but the mathematical investigation has shown that the 
principle employed in its construction gives an accuracy which is 
more than sufficient for all practical purposes. 
§ 11. Improvements suggested by the Mathematical Theory. 
The mathematical form of the expression for the deviation of 
the pen (both for the greatest and for the average deviation) indi- 
cates a method by which this may be reduced by the following 
modification of the instrument. It will be observed that r, the 
distance of the pulley pins from the centre of the harmonic wheels, 
appears in the formula for the error, raised to the second power. 
That is to say, the deviation increases as the square of the eccen- 
tricity of the pins. Suppose that this eccentricity were to be 
reduced so as to be but 1 /n of its former value, then the greatest 
deviation of each harmonic wheel would be ( r/n) 2 /(4l ), and the 
average deviation - (r/n) 2 /(6’3l). Of course the motion of the 
pen would be reduced to 1 /» of its former amplitude ; but this 
could be avoided by interposing some multiplying device between 
the pen and the end of the wire, so that the motion of the pen 
was multiplied n times over, and made as large as before. Magni- 
fying the motion of the pen magnifies the deviation in the same 
ratio, hence we should now have 
(the greatest deviation) = n • ^ = - . — , 
v & ’ U n U 
and 
(t* ! Tl) ^ 1 ^*2 
(the average deviation) = -wV^-==- . 
o ’61 n 6*3 1 
But these expressions can be reduced without limit by taking n 
sufficiently large. 
With the present value of r, were a number of additional harmonics 
to be added to the apparatus, it would be necessary, in order to keep 
the total deviation sufficiently small, to increase the vertical distance 
between the fixed and moving pulleys. This of course would be 
undesirable, because it would make the machine so much less 
compact, and would, if carried to any extent, destroy that porta- 
bility which is one of its valuable features. It is here then that 
the alternative method of reducing the deviation is likely to prove 
of value. Originally the throw of the eccentric pulleys in the 
