1905 - 6 .] A New Form of Harmonic Synthetiser. 
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§ 9. Expression for the Average Deviation of the Pen. 
By the average deviation of the actual trace we mean the 
average value of the discrepancy between it and some S.H. curve 
chosen as a standard for comparison. In fairness this latter curve 
should, as was pointed out before, be that particular one which is 
most like the trace ; but the criterion of likeness is not to be the 
magnitude of the greatest difference between the two curves, as in 
the case discussed above, but the magnitude of the average 
difference. We have first of all then to ascertain the constants of 
the comparison curve, which — in general at least — will not 
be identical with the comparison curve determined before ; 
although the two comparison curves will have this in common, 
that both must be isoperiodic and cophasal with the actual trace. 
As regards isoperiodicity, what was said in the former case applies 
equally now ; as regards cophasality, the proof is simpler than 
before. It is as follow's. Consider any S.H. curve differing in 
phase from the trace by an amount a, and again by an amount 
— a, in both cases the average error of the trace, measured against 
this S.H. curve as a standard, would be the same. Here then is a 
function (the average deviation) which has equal values for values 
of its argument (the phase difference) that are equal in magnitude 
but opposite in sign. It follows therefore that when the phase 
difference is zero, the average deviation must be a maximum or a 
minimum. In this case obviously it will be a minimum — which 
proves our theorem. Accordingly, as in the former case, the com- 
parison harmonic must have the form (w + p cos t) ; and the 
difference between it and the trace at any time t will be 
e = \w +p cos t\-\m- J(l 2 + ?’ 2 - 2 Ir cos t)\ 
say, 
e=p COS t + V+ J(ol- 2/3 COS t), . . . -(lb) 
where 
v — w — m, a = l 2 + r 2 , and /3 = Ir. . . . -(16) 
The value of the average deviation, e' say, will be, 
* o 
and to complete the determination of the comparison S.H. we 
