218 Proceedings of Pay al Society of Edinburgh. [sess. 
hence the turning values of e occur when t — o or 7 r, or when t = 
cos -1 (r/2Z). The two former values of t give the minimum values 
of the deviation ; the latter value of t gives the maximum value 
of the deviation, which on substitution of this value for t is found 
to be e = r 2 /(2Z). In the present machine r is equal to J inch, and 
l to about 30 inches, and so the value of e in the above formula 
becomes of an inch. 
One point remains. If fig. 7, which represents the actual 
arrangement of one of the harmonic wheels of the latter, he 
compared with fig. 6, it can he shown without difficulty that the 
motion of the free end of the wire Q in fig. 7 (P being supposed 
fixed) will b.e almost exactly double * the motion of the free end 
of the wire Q in fig. 6. 
On the other hand, however, there must he set against this 
that, in order to restrict the motion of the pen within the limits 
imposed by the width of the paper band, it is as a rule necessary 
to arrange a pulley between the end of the summation wire and 
the pen, on the principle employed in grandfathers’ clocks, so that 
the pen, like the clock-weight, has a motion only half as great as 
that of the wire. The net result of the atpove is that our formula 
for the deviation remains as before. 
§ 8. On the Necessity for finding the Deviation of the Pen with 
Exactness. 
At first sight it might seem that a deviation such as the above 
is wholly negligible, but in reality this is far from being the case. 
The above deviation is that of only one of the S.H. wheels of the 
instrument ; and it must be remembered that the greatest deviation 
of a trace compounded of n S.H. constituents will be n times the 
above quantity, for every now and then the deviations of all the 
different S.H. wheels will fall together with the effect of producing 
a total deviation in the position of the pen equal to their sum. 
Of course, as regards the present apparatus, which possesses but 
* The diameter of the small pulleys is 1 inch, which is also the distance 
between the upper pair ; and these are both distant about 30 inches 
from the lower pulley, whose circle of rotation is \ inch in diameter. Such 
being the dimensions of the machinery, we are probably led into no appreci- 
able error by having to substitute the above approximate statement for the 
exact one, which is too complicated to be used. 
