1905-6.] A New Form of Harmonic Synthetiser. 
221 
Therefore the distance (or “ deviation ”) at any time t between 
the actual pen and the imaginary moving point of (5) is given by 
the equation, 
e =p cos ( qt + a) + w - [m — J(l 2 + r 2 - 2rl cos t) } . (6). 
But the motion of the pen given by (4) has a periodic time 27 r, 
and is, we know, very nearly S.H. ; and so the comparison S.H.M. 
must also have a periodic turn 27r, as otherwise the two would 
eventually get out of step. Therefore in (5) we may put q= 1. 
Further, it can be shown that the actual and comparison motions 
Fig. 9. — In the above figure the curve APB SC is the same as the curve 
ABODE in fig. 8, i.e. it has been drawn as if the length of 1 in 
the machine were but 2‘5, instead of 30. 
must be cophasal. In fig. 9 let the curve APBSC represent 
the actual trace of the pen ; and let DQETF be a S.H. curve, 
cophasal and isoperiodic with APBSC, but of any amplitude. 
There will be some point on the time axis for which the difference 
between the two curves is a maximum PQ. Because of the 
symmetry of both curves about HK, there must be an equal 
maximum difference ST between them at an equal distance from 
HK on the other side. Now suppose the curve DQETF to be 
shifted bodily a short distance to the right (say) ; and let it now 
cut the lines PQ, ST in the points Q' and T # ; then ST' will be 
