226 Proceedings of Royal Society of Edinburgh. [sess. 
We have then in this way completely determined the S.H.M. 
to which the actual motion is most akin ; and we find it to he of 
the same period and phase (p. 221), and amplitude (equation 
14), hut situated with its time axis at a distance r 2 /(4Z) above 
the time axis of the actual motion. Now this means that if we are 
to regard the actual curve as an attempt on the part of one of the 
constituents of the instrument to draw this particular S.H. curve, 
and are to estimate the deviation of the actual curve accordingly, 
then when in using the instrument in practice we measure on the 
paper any ordinate of the trace produced by the pen, it must be done 
from a line drawn in in the position of the hypothetical time axis 
of the comparison S.H. curve, and not in the position of the time 
axis of the actual trace. In the above argument we have been 
considering only one of the constituent harmonics of the instru- 
ment ; but the quantity r 2 /(4Z) has the same value for all the 
harmonics; and hence when in practice a number are employed 
to draw a compound curve, the rule still applies. 
The true harmonic motion with which the actual motion is to 
be compared having been found, we now go on to ascertain the 
greatest discrepancy between the two, the “ deviation ” of the 
actual motion. 
This can easily be done by means of fig. 10, for the ordinates to 
the three curves from any point in that figure give the “ turning 
values” of the deviation function e for the values of p and v 
corresponding to the point. Now it will be remembered that the 
values of p and v we have been led to choose correspond to 
the point U on the y axis when the latter is moved up to 
bisect WK. Hence the deviation has the same numerical value, 
±|WK= ±r 2 /(4Z), at each of its three turning values. This 
then is the value of the maximum deviation ; and it will occur 
five times -in each complete revolution, namely, at the times 
t = 0, t — cos 1 \ 
l2 +r 2 H 
— ),* t ~ 7T, Z=COS _1 -l 
IV and 
rl p L 
rl p l 
t — 27t. The first three of these times are given by (9), and occur 
during the down stroke ; the existence of the last two, on the up 
stroke, is inferred at once from symmetry. 
* The angle is to be taken between 0 and - k . 
t The angle is to be taken between -k and 2 ir. 
