230 
Proceedings of Royal Society of Edinburgh . [sess. 
_P 2 
9 „ 3/ 2 
cos 2 £c^ + I cos 2 tdt + cos 2 tdt 
Sir J 3rr 
£ 2 _ 
I in 
6-3.? 
very approximately. 
§ 10. Application of the Mathematical Results to the 
Present Instrument. 
In the case of the present instrument, where r is equal to 
\ inch and l to 30 inches, the greatest deviation is equal to r 2 /(4Z), 
equal to of an inch. This result shows the importance of 
using the true theory in the estimation of the deviation, for it 
will be recollected that the fallacious theory first advanced 
(p. 217 et seq.) made out the deviation to he Jq of an inch, that 
is, it just doubled it. 
As there are three harmonic wheels in the present instrument, 
the greatest possible deviation of the pen is q of an inch ; and, 
moreover, even this amount of deviation will only he present when 
the deviations of the three harmonic wheels attain their maximum 
values simultaneously, which in practice will be a rare event. For 
as a rule the harmonic constituent possessed of a variable ampli- 
tude will have its two harmonic wheels set in different phases, 
and in this case the full error will never be attained ; and, again, 
the two constituents may be set to incommensurable periods, a 
condition of affairs which has a like effect. 
The value of the greatest deviation given above is only true of 
course if all measurements of the curve produced by the pen be 
made from the depressed time axis spoken of before, but in 
practice the simplification occurs, that this axis is quite in- 
distinguishable from the other drawn at the height of the zero of 
the pen, for the amount of the depression is given by the same 
formula which gives the maximum deviation, and is therefore only 
of an inch. 
To sum up the points connected with the present instrument, 
not only does it appear to be satisfactory as regards its mechanical 
