1905 - 6 .] A New Form of Harmonic Synthetiser. 219 
three S.H. wheels in all, the greatest deviation cannot be more 
than zT o an i n °h i but the important thing here is not the 
consideration of a temporary model, but is rather to ascertain 
whether the mechanical principle employed would be satisfactory 
in the case of a complete instrument possessing a large number of 
constituents. As that means a large value for u nf it becomes 
essential to ascertain the value of the deviation of each S.H. 
wheel with precision. 
How, in the mathematical reasoning employed above, to obtain 
an expression for the value of this deviation there may lurk a 
fundamental fallacy. In order to measure the deviation it is 
necessary to take some particular S.H.M. as a standard for 
comparison : and the S.H.M. which we choose for this purpose 
should be that to which the actual motion is most nearly 
akin. Unless, therefore, the S.H.M. that was employed in our 
former discussion can be shown to be the particular one which 
satisfies this condition, the result arrived at, being deduced 
from mistaken premises, must be misleading. 
In order to investigate this point, let ABODE in fig. 8 be the 
curve that would be drawn by the recording arrangement of Plate, 
fig. 2, were the latter actuated by the harmonic wheel of fig. 6 ; 
it being assumed that the wire has been sufficiently lengthened to 
allow its free end to come down over a pulley at Q, and be 
attached to the pen, which we arrange vertically underneath. 
Consideration of fig. 6 will show that AD must be symmetrical 
about the central line FC ; and that it will, except at the 
points A, C, and E. everywhere lie above the curve AGCHE, 
which represents the S.H.M. obtained by resolving in a vertical 
direction the circular motion of the pin S in fig. 6. e in equation 
(3) gives the difference between the ordinates to the two curves 
respectively ; and this difference we saw will be a maximum at 
the point £ = cos -1 (r/2Z) ; i.e. (as may easily be shown) at 
the point where ABODE cuts the time axis. How, were 
AGCHE to be bodily moved any short distance upwards, keeping 
it parallel to itself, then the length of its ordinates Avould all 
be altered by the same amount ; and therefore the maximum dis- 
crepancy between the two curves would still occur at t = cos -1 (r/2l) ; 
and therefore the value of the maximum discrepancy would be 
